Zurich Lectures in Advanced Mathematics Edited by Erwin Bolthausen (Managing Editor), Freddy Delbaen, Thomas Kappeler (Managing Editor), Christoph Schwab, Michael Struwe, Gisbert Wüstholz Mathematics in Zurich has a long and distinguished tradition, in which the writing of lecture notes volumes and research monographs play a prominent part. The Zurich Lectures in Advanced Mathematics series aims to make some of these publications better known to a wider audience. The series has three main constituents: lecture notes on advanced topics given by internationally renowned experts, graduate text books designed for the joint graduate program in Mathematics of the ETH and the University of Zurich, as well as contributions from researchers in residence at the mathematics research institute, FIM-ETH. Moderately priced, concise and lively in style, the volumes of this series will appeal to researchers and students alike, who seek an informed introduction to important areas of current research. Previously published in this series: Yakov B. Pesin, Lectures on partial hyperbolicity and stable ergodicity Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry Sergei B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Pavel Etingof, Calogero-Moser systems and representation theory Guus Balkema and Paul Embrechts, High Risk Scenarios and Extremes – A geometric approach Demetrios Christodoulou, Mathematical Problems of General Relativity I Camillo De Lellis, Rectifiable Sets, Densities and Tangent Measures Michael Farber, Invitation to Topological Robotics Alexander Barvinok, Integer points in polyhedra Paul Seidel, Fukaya categories and Picard–Lefschetz theory Published with the support of the Huber-Kudlich-Stiftung, Zürich
Alexander H.W. Schmitt
Geometric Invariant Theory and Decorated Principal Bundles
Author: Prof. Alexander H. W. Schmitt Freie Universität Berlin Institut für Mathematik Arnimallee 3 D-14195 Berlin Germany E-mail:
[email protected] 2000 Mathematics Subject Classification (primary; secondary): 14L24, 14H60; 13A50, 14D20, 14F05, 14L30 Key words: Geometric invariant theory, reductive group, representation, group action, categorical, good, geometric quotient, principal bundle, decorated bundle, moduli space
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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1
Geometric Invariant Theory . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.1 Algebraic Groups and their Representations . . . . . . . . . . . . . . 17 1.1.1 Linear Algebraic Groups I — Definitions . . . . . . . . . . . 17 1.1.2 Representations . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1.3 Linear Algebraic Groups II — Linear Algebraic Groups as Subgroups of General Linear Groups . . . . . . . . . . . . . 21 1.1.4 Reductive Affine Algebraic Groups . . . . . . . . . . . . . . 23 1.1.5 Homogeneous Representations of the General Linear Group . 24 1.1.6 Faithful Representations and Extensions of Representations . 26 1.1.7 Appendix. The Reductivity of the Classical Groups via Weyl’s Unitarian Trick . . . . . . . . . . . . . . . . . . . . . 27 1.2 Geometric Invariant Theory for Affine Varieties — A First Encounter 30 1.2.1 The Categorical Quotient of a Vector Space by a Representation . . . . . . . . . . . . . . . . . . . . . . . 30 1.3 Examples from Classical Invariant Theory . . . . . . . . . . . . . . . 34 1.3.1 Algebraic Forms . . . . . . . . . . . . . . . . . . . . . . . . 34 1.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.3.3 The Invariant Theory of Matrices . . . . . . . . . . . . . . . 43 1.4 Mumford’s Geometric Invariant Theory . . . . . . . . . . . . . . . . 49 1.4.1 Good and Geometric Quotients . . . . . . . . . . . . . . . . 49 1.4.2 Quotients of Affine Varieties . . . . . . . . . . . . . . . . . . 51 1.4.3 Linearizations . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.5 Criteria for Stability and Semistability . . . . . . . . . . . . . . . . . 66 1.5.1 The Hilbert–Mumford Criterion . . . . . . . . . . . . . . . . 66 1.5.2 Semistability for Direct Sums of Representations . . . . . . . 86 1.5.3 Semistability for Actions of Direct Products of Groups . . . . 89 1.6 The Variation of GIT-Quotients . . . . . . . . . . . . . . . . . . . . . 92 1.6.1 The Finiteness of the Number of GIT-Quotients . . . . . . . . 92 1.6.2 Variation of the GIT-Quotients . . . . . . . . . . . . . . . . . 93 1.7 The Analysis of Unstable Points . . . . . . . . . . . . . . . . . . . . 95 1.7.1 A Few Words about GIT on Non-Algebraically Closed Fields 96 1.7.2 The Theory of the Instability Flag . . . . . . . . . . . . . . . 97 1.7.3 The Instability Flag in a Product . . . . . . . . . . . . . . . . 102
vi 2
Decorated Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Principal Bundles — Definitions and First Properties . . . . . 2.1.2 The Classification Problem for Decorated Principal Bundles . 2.2 Rudiments of the Theory of Vector Bundles . . . . . . . . . . . . . . 2.2.1 The Topology of Vector Bundles . . . . . . . . . . . . . . . . 2.2.2 The Riemann–Roch Theorem . . . . . . . . . . . . . . . . . 2.2.3 Bounded Families of Vector Bundles . . . . . . . . . . . . . 2.2.4 The Moduli Space of Semistable Vector Bundles . . . . . . . 2.3 Decorated Vector Bundles: Projective Fibers . . . . . . . . . . . . . . 2.3.1 Set-Up of the Moduli Problem . . . . . . . . . . . . . . . . . 2.3.2 Semistability of Swamps . . . . . . . . . . . . . . . . . . . . 2.3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 The Parameter Space . . . . . . . . . . . . . . . . . . . . . . 2.3.6 On the Geometry of the Moduli Spaces . . . . . . . . . . . . 2.3.7 The Chain of Moduli Spaces . . . . . . . . . . . . . . . . . . 2.4 Principal Bundles as Swamps . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Principal Bundles and Associated Vector Bundles . . . . . . . 2.4.2 Back to Some GIT . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Pseudo G-Bundles . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Semistable Reduction for Principal Bundles and the Proof of Theorem 2.4.1.8 . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 The Proof of Theorem 2.4.3.3 . . . . . . . . . . . . . . . . . 2.4.6 The Geometry of the Moduli Spaces — A Guide to the Literature . . . . . . . . . . . . . . . . . . . . 2.4.7 Appendix I: Some Remarks Concerning the Moduli Stack of Principal Bundles . . . . . . . . . . . . . . . . . . . . . . . . 2.4.8 Appendix II: Moduli Spaces for Principal Bundles with Reductive Structure Group via the Ramanathan–G´omez–Sols Method . . . . . . . . . . . 2.4.9 Appendix III: Anti-Dominant Characters . . . . . . . . . . . 2.5 Decorated Tuples of Vector Bundles: Projective Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Homogeneous Representations . . . . . . . . . . . . . . . . . 2.5.2 More on One Parameter Subgroups . . . . . . . . . . . . . . 2.5.3 The Moduli Problem of Tumps . . . . . . . . . . . . . . . . . 2.5.4 Proof of Theorem 2.5.3.7 . . . . . . . . . . . . . . . . . . . . 2.5.5 Properties of the Semistability Concept . . . . . . . . . . . . 2.5.6 Quiver Representations . . . . . . . . . . . . . . . . . . . . . 2.6 Principal Bundles as Tumps . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Principal Bundles and Associated Tuples of Vector Bundles . 2.6.2 The Relevant GIT-Quotients . . . . . . . . . . . . . . . . . . 2.6.3 Pseudo G-Bundles . . . . . . . . . . . . . . . . . . . . . . . 2.7 Decorated Principal Bundles: Projective Fibers . . . . . . . . . . . . 2.7.1 The Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Decorated Pseudo G-Bundles . . . . . . . . . . . . . . . . .
103 103 115 118 118 119 121 124 135 135 138 143 144 146 163 167 174 174 179 185 188 190 195 195 199 208 210 210 211 217 223 229 237 257 258 259 263 266 266 270
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Decorated Principal Bundles . . . . . . . . . . . . 2.7.3 Asymptotic Semistability . . . . . . . 2.7.4 Hitchin Pairs . . . . . . . . . . . . . 2.7.5 Fine Tuning of the Theory . . . . . . 2.8 Decorated Principal Bundles: Affine Fibers . 2.8.1 The Moduli Functors . . . . . . . . . 2.8.2 Comparison with Projective *-Bumps 2.8.3 Construction of the Moduli Spaces . . 2.8.4 Further Properties and Examples . . . 2.9 More Generalizations . . . . . . . . . . . . . 2.9.1 Positive Characteristic . . . . . . . . 2.9.2 Higher Dimensional Base Varieties . 2.9.3 Parabolic Structures . . . . . . . . .
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103 279 283 286 288 288 292 295 303 321 321 325 354
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
Introduction ccording to Hartshorne ([96], Section I.8), one of the guiding problems in Algebraic Geometry is the classification of algebraic varieties up to isomorphy. Let us briefly mention two (related) variants of the problem. The first one is the classification of complex projective varieties up to isomorphy, usually assumed to be smooth or mildly singular. The strategy is first to accomplish the birational classification of so-called minimal varieties and then relate any projective variety to a minimal model by some specific operations such as blow-down. The reader may consult [14] for the classical case of surfaces and [121], [47], and [145] for results in higher dimension. In the second one, we place ourselves in one particular projective space !n and wish to classify all closed subvarieties of this projective space up to projective equivalence. We remind the reader that two subvarieties of !n are called projectively equivalent, if there is an automorphism of the ambient space !n , i.e., an element of PGLn+1 ('), which carries the first variety onto the second one. This problem is just the abstract formulation of putting a system of homogeneous equations in the variables x0 , . . . , xn into a suitable normal form. In modern language, one states it as the slightly more general problem of classifying polarized varieties, i.e., pairs (X, L) which consist of a projective variety X and an ample line bundle L on it. Treatises of this theory are in [68], [17], and [215]. The two problems do overlap: A Fano manifold X, for example, yields the polarized variety (X, −KX ). Conversely, two B polarized varieties (X, L) and (X B , LB ) with dim(X) = dim(X B ), Ldim(X) = LB dim(X ) , and B Pic(X) ! Pic(X ) ! ( are projectively equivalent, if and only if they are isomorphic. As Hartshorne also describes, these classification problems usually fall into two parts. First, one has some discrete numerical invariants such as the Hilbert polynomial of a polarized variety (X, L) which yields a coarse subdivision of the class of all objects. Second, the objects with fixed numerical invariants usually come in positive dimensional families and one has to construct a moduli space for them, which is an algebraic variety whose points are in “natural” bijection to the set of isomorphy classes of the objects with fixed numerical data. Mumford has conceived his Geometric Invariant Theory (GIT) as a major tool for constructing such moduli spaces. To get a more concrete idea, let us look at a special case of the second classification problem, namely the classification of hypersurfaces in !n up to projective equivalence. The numerical invariant which we have to take into account is just the degree of a hypersurface, a positive integer. For fixed d ∈ (>0 , the hypersurfaces of degree d form the linear system |O(n (d)|. If '[x0 , . . . , xn ]d denotes the vector space of homogeneous
/
I 2
2
polynomials of degree d in the variables x0 , . . . , xn , then we have the identification 4 ^ 4 ^ |O(n (d)| = P '[x0 , . . . , xn ]d := '[x0 , . . . , xn ]d \ {0} /'. . The group PGLn+1 (') = SLn+1 (')/(µn+1 · $n+1 ) acts on !n as its automorphism group. It clearly induces an action of PGLn+1 (') on |O(n (d)| and P('[x0 , . . . , xn ]d ). This action may also be obtained in a different way: The group SLn+1 (') acts on the vector space '[x0 , . . . , xn ]d as a group of linear automorphisms by change of variables. This action descends to an action of SLn+1 (') on P('[x0 , . . . , xn ]d ). Since the center µn+1 · $n+1 acts trivially, it induces an action of PGLn+1 (') on P('[x0 , . . . , xn ]d ) which is—using the right conventions—the one that we have introduced before. Intuitively, the moduli space for hypersurfaces of degree d in !n will be the quotient P('[x0 , . . . , xn ]d )/ PGLn (').1 So far, we have no clue whether or in which sense we can construct the quotient P('[x0 , . . . , xn ]d )/ PGLn+1 (') as an algebraic variety. The same circle of ideas works in the more general context: Subvarieties of !n with fixed Hilbert polynomial are parameterized by a projective scheme, a so-called Hilbert scheme, which replaces the linear system in the above example, and, as before, the action of PGLn+1 (') on !n yields an action of PGLn+1 (') on this Hilbert scheme. Again, we are lead to the problem of forming quotients (see [215] for this general context). The guiding problem has thus evoked our interest in the following problem: Let G be an algebraic group, X a variety or, more generally, a scheme, and α: G x X −→ X an action of G on X. In which sense can we form the quotient of X by the action of G? One easily checks that, in general, the set of orbits does not carry a natural scheme structure. Thus, one first has to develop the appropriate notion of a quotient. The most general one is that of a categorical quotient which is denoted by X//G. Still one finds examples where even the categorical quotient does not exist as a variety, separated scheme, or just scheme. Thus, let us formulate the following more concrete problem: In the above situation, suppose that X is a variety or a scheme of finite type over '. Then, the task is to find a G-invariant open subset U ⊆ X, as large as possible, such that U//G exists as a variety or a scheme of finite type over '. Now, we restrict to the case where G is a reductive linear algebraic group (e.g., SLn (') or GLn (')). Then, Mumford’s GIT as developed in [155] is a formalism for finding such open subsets. (More recently, various generalizations have been discovered, e.g., [99].) To give the reader an impression, let us indicate the most basic case. For this, let *: G −→ GL(V) be a representation of G, i.e., * is a homomorphism of linear algebraic groups. Then, κ: G x V
−→ V
(g, v) 1−→ *(g)(v)
!
is an action of G on V by linear automorphisms. The '-algebra of regular functions on the affine variety V is '[V] = Sym. (V ∨ ) = d≥0 Symd (V ∨ ). We form the algebra
'[V]G =
% Sym (V ) d
∨ G
(1)
d≥0
1 We make the following abuse of notation: Although the actions will be usually left actions, we always divide from the right, for typographical reasons.
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3
of the functions that are constant on all G-orbits in V. A fundamental theorem of Hilbert asserts that '[V]G is again a finitely generated '-algebra, so that it belongs to an affine algebraic variety which we shall denote by V//*G. Moreover, the inclusion '[V]G ⊆ '[V] gives a G-invariant morphism π: V −→ V//*G. This maps exhibits V//*G as the categorical quotient of V by the action of G. It has several nice properties. One of them is that π maps the set of closed G-orbits in V bijectively onto V//*G. Thus, in this case, we may just take U = V as the whole variety. The representation * also supplies the action ^ 4 κ: G x P(V) −→ P(V) := V \ {0} /'. ^ 4 6 _ g, [v] 1−→ *(g)(v) . As we have implicitly observed in (1), the algebra '[V]G is graded. It therefore defines a projective variety P(V)//*G. This time, the inclusion '[V]G ⊆ '[V] of graded algebras gives rise to a G-invariant rational map π: P(V) $ P(V)//*G. The map π is defined in the point x = [v], if there is a non-zero homogeneous function f ∈ '[V]G of positive degree with f (v) % 0. Such a point is said to be *-semistable. The set P(V)*-ss of *-semistable points in P(V) is open and *-invariant, and the morphism π: P(V)*-ss −→ P(V)//*G exhibits P(V)//*G as the categorical quotient of P(V)*-ss by the induced G-action. Hence, we take U = P(V)*-ss in this case. In general, U will be a proper subset. However, its categorical quotient is a projective variety. It is clearly an important task to characterize the semistable points with a handy criterion. This is the so-called Hilbert–Mumford criterion. (This criterion is the main reason for the success of GIT in applications. It is still lacking in its strong form in the recent generalizations of GIT such as [99].) There are two things noteworthy here: The question of determining the invariant ring '[V]G with respect to the action of G := SLn+1 (') on V := '[x0 , . . . , xn ]d which we have introduced above was the topic of classical invariant theory. Hilbert managed to prove the finite generation of the invariant ring and the Hilbert–Mumford criterion in precisely that set-up. Along the way, he discovered his most famous results in commutative algebra, such as the Nullstellensatz. (The reader may have a look at the lecture notes [107].) The second point is that almost everything (especially in applications) is reduced in one way or another to the above results. As the main abstract (i.e., isolated from applications) results of GIT in which we are interested we note the following: Given a representation *: G −→ GL(V), leading to the actions κ: G x V −→ V and κ: G x P(V) −→ P(V), GIT provides us with: • The categorical quotient π: V −→ V//*G. • The G-invariant open subset U := P(V)*-ss and the categorical quotient π: U −→ P(V)//*G. • A characterization of U by means of the Hilbert–Mumford criterion. It is the aim of this book to provide a generalization of the above abstract results to a relative setting. To formulate it, let X be a connected smooth projective curve over the complex numbers. The input datum for our theory will be again a representation
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*: G −→ GL(V) of the reductive group G on the finite dimensional '-vector space V. If P −→ X is any principal fiber bundle with structure group G, then we may associate to it a vector bundle P* with fiber V, using the representation *. Thinking in terms of cocycles, P* is glued together with a cocycle in GL(V) which is the image under * of a cocycle in G which gives P. According to the actions κ: G x V −→ V and κ: G x P(V) −→ P(V), we want to classify affine and projective *-pairs. The former objects are pairs (P, σ) which consist of a principal G-bundle P on X and a section σ: X −→ P* and the latter objects are pairs (P, β) composed of a principal G-bundle P on X and a section β: X −→ P(P* ) := (P* \ { zero section })/'. . These objects may be viewed as families of points v ∈ V and x ∈ P(V) varying over X in the way that vector bundles on X may be considered as families of vector spaces varying over X: The bundle G := A ut(P) −→ X of (local) automorphisms of P −→ X is a group scheme over X, i.e., there are maps eX : X −→ G , the neutral section, iX : G −→ G , the inversion map, and mX : G xX G −→ G , the multiplication map, of varieties over X, such that the diagrams expressing the group axioms for these operations do commute. The fibers of G over X are affine algebraic groups which are isomorphic to G. Furthermore, there are the actions κX : G x P* −→ P* X
and
κX : G x P(P* ) −→ P(P* ). X
There are obvious isomorphy relations on the classes of affine and projective *-pairs. As usual, there are some natural discrete data to be considered. To this end, we look at X as a compact Riemann surface, i.e., as a complex manifold and eventually as an oriented topological manifold. Denote by Π(G) the set of isomorphy classes of topological principal G-bundles on X. (If G is connected, then Π(G) can be identified with the fundamental group π1 (G).) Then, to each (algebraic) principal G-bundle can be assigned a class in Π(G), its topological type. If we fix ϑ ∈ Π(G), then the topological vector bundle P* does not depend on the principal G-bundle P of topological type ϑ. Thus, it makes sense to speak about the cohomology class [β(X)] ∈ H . (P(P* ), () of the section β. This class naturally identifies with an integer l ∈ (. The program which will be carried out in the present book is the following: • We first formulate a notion of semistability for affine and projective *-pairs (which will depend on several parameters). This is, so to say, the Hilbert– Mumford criterion for *-pairs. • Having fixed the topological background data ϑ ∈ Π(G) and l ∈ ( as well as the stability parameters, we construct the moduli space M (*, ϑ, l) for semistable projective *-bumps2 of topological type (ϑ, l) as a projective scheme over '. • For fixed ϑ ∈ Π(G) and chosen stability parameters, we construct the moduli space M (*, ϑ) for semistable affine *-pairs (P, σ), where P has topological type ϑ, as a quasi-projective scheme of finite type over '. It comes with a projective map to an affine space. 2 Certain
generalizations of *-pairs needed for getting projective, i.e., compact moduli spaces.
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These results are clearly formal generalizations of the results of GIT which we have considered above: In fact, replacing the base manifold X by a point, we recover the results which we had declared before to be the most interesting ones. Note the important difference that we have to define a priori what the semistable objects are. The reason is that there is no scheme of finite type over ' which parameterizes the isomorphy classes of affine or projective *-pairs, even if we fix the topological background data. This “unboundedness” phenomenon is familiar from the theory of vector bundles. Thus, we first have to single out a bounded family of affine or projective *-pairs. This is what the notion of semistability first does for us. Having a bounded family, standard methods may be applied to construct a parameter space B which parameterizes isomorphy classes of affine or projective *-pairs in such a way that any isomorphy class from the bounded family under inspection does correspond to a point in B. The parameter space comes also with an action of a general linear group GL(Y). Thus, we are in the setting of GIT as we have described before. The final point to be checked is that the Hilbert–Mumford criterion that comes from GIT agrees with our “Hilbert–Mumford criterion”, i.e., the notion of semistability. The problem here is that our notion is intrinsic in terms of the *-pair whereas the notion coming from GIT depends on many unnatural choices. The hard part of the work really is to arrange everything in such a way that this last step works out nicely: Whereas it is comparably easy to establish for GLn (') and projective *-pairs (restricting to homogeneous representations), more and more elaborate tricks are necessary to pass via GLn1 (') x · · · x GLnt (') to arbitrary reductive groups and from projective to affine *-pairs. It is the main aim of these notes to present these methods in a fairly self-contained way. It should be noted that these results seem to be completely new for reductive groups other than products of general linear groups and, in the case of affine *-pairs in the above generality, also for GLn ('). In the end, we see that our results are certainly a formal generalization of GIT but also an application of it. After this outline of the main achievements of this monograph, we will look at potential applications. Let us have a brief glance at the case of principal bundles without extra structures. The best known reductive linear algebraic groups are automorphism groups of certain algebraic or geometric structures. The general linear group GLn (') is the group of linear automorphisms of 'n . This observation makes the notion of a principal GLn (')-bundle equivalent to the more familiar notion of a vector bundle of rank n. Likewise, the fact that PGLn+1 (') is the group of automorphisms of the projective space !n shows that the notion of a principal PGLn+1 (')-bundle is equivalent to the notion of a !n -bundle over X. Now, !n -bundles over X are examples of smooth projective manifolds, and their classification up to isomorphy over the base manifold X is equivalent to the classification of principal PGLn+1 (')-bundles over X. Thus, this special case relates to the above guiding problem in Algebraic Geometry. (Admittedly, since any projective bundle over the curve X is the projectivization of a vector bundle, the classification of !n -bundles over X can be expressed in terms of vector bundles as, e.g., in Section V.2 of [96]. The formalism of PGLn+1 (')-bundles is, however, the right framework when thinking about higher dimensional base varieties.) In a similar spirit, one can treat the classification of divisors in projective bundles, with respect to isomorphy which respects the embedding into the projective bundle and the map onto X, as the classification of certain projective *-pairs for the group GLn (').
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There is, however, a more exiting construction which gives a whole new horizon of applications: Treating X again as a topological manifold, we define its fundamental group π1 (X). We may equip π1 (X) with the discrete topology and view it as a complex * −→ X becomes a holomorphic princiLie group. In this way, the universal covering X pal π1 (X)-bundle. Choosing an appropriate open covering of X in the strong topology, * is thus determined by a cocycle with values in π1 (X). (Note that this cocycle is loX cally constant.) Next, we may give ourselves a representation ψ: π1 (X) −→ G. Then, we use ψ to transfer the cocycle from π1 (X) to G. This new cocycle is the gluing datum for a holomorphic principal G-bundle P ψ on X. By Serre’s GAGA theorems (see [194] and [195]), it is associated to an algebraic principal G-bundle which we also call P ψ . A classical theorem of Narasimhan and Seshadri [158] asserts that the assignment ψ 1−→ P ψ induces a bijection between the set of equivalence classes of irreducible representations ψ: π1 (X) −→ Un (') and isomorphy classes of stable vector bundles of degree 0. It even yields a homeomorphism between the corresponding moduli spaces. The representations are classified by a real analytic moduli space whereas the moduli space of stable bundles is a smooth quasi-projective variety. Thus, we have found an algebro geometric model for a topological space which has been defined in terms of the topology of X only. The tools of Algebraic Geometry become in this way available to study questions of purely topological nature. The generalization of this theorem to other reductive groups is due to Ramanathan [174]. Donaldson has interpreted these results in the framework of the Kobayashi–Hitchin correspondence [58]. Now, the Kobayashi–Hitchin correspondence has been widely extended (see, e.g., [13], [156], and [138]). It relates, among other things, semistable affine and projective *-pairs to solutions of certain vortex type equations. The moduli spaces which we have obtained will be again models for some gauge theoretically defined topological spaces. Thus, our results fill an apparent gap in the literature. The work of Hitchin, Donaldson, Simpson, and Corlette also brings us back to studying representations (the introduction to [38] gives a better account of this and precise references): Representations of the fundamental group π1 (X) in (non-compact) real forms of a reductive linear algebraic group G, such as U(p, q) for GL p+q ('), lead to interesting algebro geometric objects, and our results provide moduli spaces. In various situations, where moduli spaces were known before, these constructions were exploited to gather information on the moduli spaces of representations (topological spaces defined in terms of the topology of X) by studying their algebro geometric models (see, e.g., [38] and [39]). Looking at Table 3 in [40], we see that some other moduli spaces among those which we construct here for the first time will become important, too. Detailed Content of the Book The first chapter is an introduction to Geometric Invariant Theory (GIT) as developed by Mumford in his famous book [155]. As mentioned earlier in the introduction, we will mainly deal with actions of a reductive linear algebraic group G on a vector space V or a projective space P(V) by means of a representation *: G −→ GL(V). In order to be reasonably self-contained, we will first review the theory of linear algebraic groups and their representations on finite dimensional vector spaces. The crucial notions are irreducible and completely reducible representations. Since, in this book, we will al-
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ways be working over the field of complex numbers, we may introduce reductive linear algebraic groups as those groups for which all finite dimensional representations are completely reducible. As a first illustration, we verify that finite groups and tori, e.g., '. , are reductive linear algebraic groups. In the appendix to Section 1.1, we present a method due to Hermann Weyl which allows to check that the classical groups such as GLn (') and SLn (') are reductive. Then, it also follows that GLn1 (') x · · · x GLnt (') is reductive. This is a very important fact, because almost all the actions at which we will look in the applications are induced by representations of products of general linear groups. Subsections 1.1.5 and 1.1.6 discuss some more specific topics on representations which are mainly technical tools for the moduli space constructions in the second chapter. The main references for this section are the books by Borel [30] and Kraft [123]. In Section 1.2, we start to look at the problem of forming the quotient of a vector space by the action of an algebraic group G via a representation *: G −→ GL(V). First, we observe that we cannot parameterize the G-orbits in V in any useful way by an algebraic variety, if there are non-closed orbits. Then, we derive how the globally defined regular functions on any quotient should look like. A classical theorem by Hilbert states that the ring of these functions is finitely generated, if the group G is reductive. Thus, we get an affine algebraic variety as potential quotient. We formulate the basic properties of this quotient: Basically it is as good as it can be (keep in mind that G must be reductive). Finally, we present the important notions of stable and semistable points and of nullforms. The nullforms are precisely those points in V which map to the same point in the quotient as 0 ∈ V. (The nullforms have to be “thrown away”, if one wants to form the quotient of P(V)!) The Hilbert–Mumford criterion tells us how to detect the stable and semistable points (whence also the nullforms). Before proceeding to the proofs of the fundamental theorems stated in Section 1.2, we would like to see them in action. This happens in Section 1.3. We will speak there about some specific representations which were studied in the classical literature on invariant theory. The most prominent one is the representation of SLn (') on algebraic forms of degree d in n variables. In a more geometric language, one studies here the classification of projective hypersurfaces of degree d in the projective space !n−1 up to projective equivalence. We will evaluate the Hilbert–Mumford criterion in several examples and also compute the invariant rings in some situations, or, equivalently describe the resulting quotient. Another instructive example is the action of GLn (') on (n x n)-matrices by conjugation. Here, one can explicitly determine the quotient and compare it with the Jordan normal form. The reader should study this example very carefully and reflect what it tells us about properties of the quotient. Interesting generalizations arise when studying the action of GLn (') on tuples of matrices or, more generally, quiver representations. Here, it is in general impossible to obtain a complete list of normal forms. Consequently, it is more interesting to find out as much about the quotient as possible. Section 1.4 is devoted to the fundamental concepts of GIT. In order to correctly appreciate GIT quotients, we first define good and geometric quotients according to Mumford. The defining properties are certain natural requirements on a quotient which have two important consequences: First, a good quotient is also a quotient in the categorical sense. Second, a good quotient can be patched up from affine quotients. Hence,
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we first study the theory of affine quotients. This means we will first prove Hilbert’s theorem on the finite generation of the ring of invariants. Then, we check that the quotient we thus obtain is really a good one in the sense of Mumford. Here, we have followed the exposition in the book by Dieudonn´e and Carrell [53]. If *: G −→ GL(V) is a representation of the reductive group G and if V ss is the set of semistable points in V, then we get the G-invariant (open) subset P(V)ss := V ss /'. ⊆ P(V). The result of Hilbert grants that we may form the quotient π: P(V)ss −→ P(V)//*G. The properties of good quotients imply that this is also a good quotient. The salient feature here is that P(V)//*G is again a projective variety. In order to apply these findings to a wider range of examples, one has to use linearizations. So, assume that G is a reductive linear algebraic group, X is a projective variety, and σ: G x X −→ X is an action of G on X. A linearization of σ consists of a representation *: G −→ GL(V) and a G-equivariant closed embedding ι: X '→ P(V). We may define X ss := P(V)ss ∩ X. Then, we obtain the following commutative diagram !" 7 P(V)ss X ss πX
3
X//(*,ι)G
π
!"
3 7 P(V)//*G,
and X//(*,ι)G is a good quotient of X ss and a projective variety. The choice of a linearization is a parameter in the theory. Note that any given linearization (*, ι) of σ may be multiplied by a character χ of G: The linearization χ · (*, ι) := (*χ , ι) features the representation *χ : G −→ GL(V), g 1−→ χ(g) · *(g). We use this to study (all possible) linearizations of a '. -action on a projective space: Let λ: '. −→ GL(V) be a representation. It leads to an action λ: '. x P(V) −→ P(V). Of course, (λ, idP(V) ) is a linearization of λ. Next, we can form (λk , vk ) where λk is the k-th symmetric power of λ and vk is the k-th Veronese embedding. If χd : '. −→ '. denotes the character z 1−→ z−d , we thus get the family χd · (λk , vk ), k ∈ (>0 , d ∈ (, of linearizations. It is easy to verify that the quotient depends only on the ratio d/k ∈ 3. A priori, we get an infinite family of possible quotients. However, we can easily determine the semistable points for each linearization and check that we get only a finite number of open subsets which arise as subsets of semistable points with respect to different linearizations and consequently also only finitely many possible quotients. Moreover, it is possible to understand the relationship between different quotients. Although this seems to be only a peculiar example, it is a very important one. Indeed, we will see in Section 1.6 that it has far reaching consequences. For the basic formalism of GIT, we have used the sources [155] and [160]. Before we proceed to Section 1.6, we will prove and study in Section 1.5 the Hilbert–Mumford criterion. It is the main reason for the success of GIT in applications. Note that, so far, we have only an abstract formalism which attaches to a group action on a projective variety and a linearization of that action an open subset of semistable points for which the good quotient exists as a projective variety. With the definition of semistability, it is almost impossible to find the semistable points. On the other hand, using the Hilbert–Mumford criterion, one often gets nice intrinsic characterizations of semistable points. (Recall that we have already studied meaningful examples in Sec-
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tion 1.3.) After presenting Richardson’s proof of the Hilbert–Mumford criterion from [24], we will evaluate it in several attractive examples. Notably, we investigate King’s example of χ-semistable quiver representations [120]. The last two subsections contain some refined semistability criteria which are useful in some special problems. In Section 1.6, we address the issue of the linearization as a parameter more seriously. First, we show that, given G, X, and σ: G x X −→ X as before, there are only finitely many G-invariant open subsets which occur as open subsets associated to a linearization of σ. This interesting and important fact was independently obtained by Dolgachev and Hu [57] and by Białynicki-Birula [21]. We give a transcription of Białynicki-Birula’s approach to the GIT setting, originally published in [185]. Next, we would like to understand how the quotients to two different linearizations of a given action are related. To this end, we discuss the master space construction of Thaddeus [214] and use the semistability criteria from Section 1.5.3 to reduce to the case of a '. action with which we are already familiar. This is a simplified version of the results in [57] and [214]. The final section of the first chapter is devoted to a certain refinement of the Hilbert– Mumford criterion: Look again at a representation *: G −→ GL(V) and at a nullform v ∈ V. By the Hilbert–Mumford criterion, there is a one parameter subgroup λ: '. −→ G, such that limz→∞ λ(z) · v = 0. The question is whether one can find a one parameter subgroup with this property, such that the convergence to zero is the “fastest possible” and “how unique” this one parameter subgroup is. The solution is due to Bogomolov [29], Hesselink [105], [106], Kempf [118], and Rousseau [181]. Their uniqueness result is that the parabolic subgroup Q associated to λ is unique. An essential consequence of this result is that the Hilbert–Mumford criterion remains true over non-algebraically closed fields of characteristic zero. Another useful application is due to Ramanan and Ramanathan [173] who associate to the nullform v a point [v∞ ] ∈ P(V) which is semistable for the action of a Levi subgroup L of the instability parabolic subgroup Q for the canonical linearization modified by a certain character χ. All these results are exposed with almost no proof, following the paper [173]. Finally, we mention a result by the author [189] on the instability one parameter subgroup for an unstable point in a product P(V1 ) x P(V2 ). The content of Section 1.8 is crucial for many of the constructions of Chapter 2. We have written Chapter 1 entirely in the language of complex algebraic varieties in order to make it accessible to a large audience. As prerequisite, a good acquaintance with Chapter I of Hartshorne’s “Algebraic Geometry” [96] should suffice. The reader who is familiar with the theory of schemes will have no trouble in extending all the results to the setting of schemes of finite type over '. Indeed, the results will be used in that framework in the second chapter. The foundations of GIT were, of course, put down in Mumford’s book. His book is, however, considered to be rather technical. More user friendly accounts have been given since, including [160], [170], [56], and [153]. The reader may replace or complement some sections with these references. The main novelty of our exposition is the elementary discussion of the finiteness of the number of different quotients for the same action and the variation of the quotients. Moreover, the results of Section 1.8 do not seem to have been included in text books either. Finally, we have tried to highlight some phenomena or facts which have counterparts in the theory of moduli spaces which will be developed in the second chapter.
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The second chapter is more in the style of a research monograph. The reader will need here some familiarity with the theory of schemes. Again, Hartshorne’s book will amply suffice. However, it would be very useful, if the reader had some ideas about the concept of moduli spaces or spaces representing certain functors. A basic example of such a moduli space is projective n-space or more generally the projectivization of a vector bundle. Its universal property is given in [96], Chapter II, Proposition 7.12. An important generalization of this example are Graßmannians (see Lecture 6 in [95]). The reader who has mastered the example of Graßmannians is well-prepared for all kinds of parameter and moduli spaces which he or she will encounter in our book. Additional introductions to the concept of a moduli space are contained in [160] and Lecture 4 and 21 of [95]. Together with Chapter 1, these prerequisites should be sufficient for attacking Chapter 2. Section 2.1 introduces the classification problem whose solution will occupy the rest of Chapter 2. In order to properly state it, we need the basic notions of the theory of principal bundles. Since there is no standard textbook which covers this theory, we will give a brief account of this theory, following an exposition of Serre [195]. In Section 2.2, we discuss or review the theory of vector bundles on complex algebraic curves. The reader should be aware that there are several excellent introductions to this topic, including [160], [135], [116], and, for bundles of rank two, [153]. To begin with, we will present the classification of topological vector bundles on a smooth projective curve. Then, we state the Riemann–Roch theorem for coherent sheaves and reduce it to the familiar case of line bundles. Section 2.2.3 discusses the crucial notion of a bounded family of vector bundles. Boundedness is a necessary condition for constructing moduli spaces. Unfortunately, the family of vector bundles of fixed topological type is not bounded. We will give numerical extra conditions which ensure boundedness. One of the possible conditions is the famous semistability. In that section, we will also present Grothendieck’s quot scheme. Using it, we reduce the classification problem for semistable vector bundles of given topological type to the problem of forming the quotient of a certain quasi-projective variety by the action of a reductive linear algebraic group. In Section 2.2.4, we will sketch the exact procedure how the techniques from Chapter 1 are applied. The hard work will begin in Section 2.3. We address the solution of the classification problem in case the structure group is a general linear group and the representation is a homogeneous one. This is the core also for the subsequent constructions: We will devise various tricks and methods in order to reduce everything to it. After reviewing the classification problem, we give an example how, for a concrete choice of the representation, the general problem specializes to the classification of interesting algebraic varieties. Then, we give the notion of semistability. This time, it will depend on a parameter, namely, a positive rational number. We have included two examples where the semistability concept comes in an easier form. Afterwards, the construction of the moduli space begins. We first check the boundedness of semistable objects, using the criterion from Section 2.2.3. Then, we construct the parameter space with its group action and evaluate the Hilbert–Mumford criterion. The latter is the hardest part. It origins from our paper [187] and is a refinement of techniques developed by Simpson [204] and Huybrechts and Lehn [114], [115]. After having constructed the moduli spaces, we describe (without proof) two basic geometric properties they enjoy
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under certain conditions. Section 2.3 concludes with the chain of moduli spaces. This parallels the finiteness of GIT quotients from Chapter 1. We briefly mention the work of Thaddeus [213] (without explicitly addressing the Verlinde formula). The following section on moduli spaces of principal G-bundles should be regarded as a first highlight of this monograph. There, we give a full construction of the moduli space of semistable principal G-bundles with connected reductive structure group. Although these moduli spaces were constructed over 30 years ago by Ramanathan [175], [176], they haven’t been treated in textbooks, so far. The approach we will present origins from the papers [186] and [188]. The basic idea is to use the results on decorated vector bundles. In order to do so, one has to describe principal G-bundles as vector bundles with additional structures. In Section 2.4.1, we will take the first steps in that direction. Moreover, we will discuss the notion of semistability, the concept of S-equivalence and polystability, and state the main theorem on the existence of moduli spaces. In the subsequent section, we study some GIT problems which naturally appear in our approach. Afterwards, we introduce pseudo G-bundles. These are certain generalizations of principal G-bundles. The advantage is that one can easily associate to a pseudo G-bundle a decorated vector bundle. This gives a notion of semistability which depends on a parameter δ and makes the construction of the moduli space for δsemistable pseudo G-bundles as a projective variety fairly easy. The “miracle” is that the moduli space thus obtained is the moduli space of semistable principal G-bundles. After presenting the proofs of these assertions, we give a brief survey on the literature on moduli spaces of principal G-bundles. In three appendices, we have collected some remarks on the moduli stack, a sketch of the construction of the moduli space for nonsemisimple reductive structure groups, and a verification of the fact that our notion of semistability indeed coincides with Ramanathan’s. Section 2.5 is dedicated to the structure group G := GLr1 (') x · · · x GLr1 ('). There are two novel and important aspects here: a) The group G has lots of characters which, of course, enter the definition of semistability and b) we have to choose a faithful representation κ: G −→ GL(W). This introduces even more parameters into the theory. In the first two sections, we conceive some tools in the representation and invariant theory of G which help us develop an efficient formalism for the complicated objects we are dealing with and prepares us for the construction of the moduli spaces. After presenting the moduli problem and the main result on the existence of moduli spaces in Section 2.5.3, we proceed to the construction of the moduli spaces. The faithful representation κ allows us to reduce to the case of decorated vector bundles. The details of the construction are rather tricky and technical. After having constructed the moduli spaces, we study the asymptotic behavior of the semistability concept. Again, the fact that G has many characters gives us various directions in which we can look at the asymptotics. To get from the very general and abstract results to more concrete situations, we discuss our results in the special case of quiver representations. We will see that we obtain a generalization of King’s result on moduli spaces of quiver representations to the setting of vector bundles on curves. Specializing even further, we give elements of the theory of holomorphic chains from the paper [2]. In that setting, we can easily describe and study several important phenomena which conjecturally extend to more general quiver problems. In view of the importance of quiver representations in representation theory, we hope that this special case of our construction will have
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interesting applications in the future. An immediate one concerns the determination of the Betti numbers of some representation spaces of the fundamental group π1 (X) of the Riemann surface X. If we want to treat classification problems involving principal G-bundles with reductive but non-semisimple structure group, then the techniques of Section 2.4 are not perfectly suited. The approach of that section is based on embeddings of the structure group G into a special linear group SL(W). Now, G has non-trivial characters but SL(W) does not. Hence, we cannot extend the characters from G to SL(W). On the other hand, the characters of G are important parameters for the semistability concept. The way out is to embed G into a group of the shape H := GLr1 (') x · · · x GLr1 ('). Indeed, one can arrange that any character of G extends to a character of H (at least up to positive multiples). In this way, we may use the results of Section 2.5 in order to treat classification problems for principal G-bundles with arbitrary reductive structure group G. The set-up will be explained in Section 2.6. Most of the arguments are straightforward generalizations of their counterparts in Section 2.4. One can use these methods to construct moduli spaces for principal G-bundles without additional structures (see [80]). Since the details are rather awkward and we already have constructed the moduli spaces in Section 2.4 with different methods, this application is omitted here. In the following section, we come to the real novelties of this volume. We are now able to cover the classification of principal G-bundles, G a (not necessarily connected nor semisimple) reductive linear algebraic group, together with sections in the projective bundle that is associated via a previously fixed representation *: G −→ GL(V). Here, there are still some technical assumptions on *. In Section 2.7.1, we will define the notion of semistability for the objects under consideration and formulate the main result on the existence of moduli spaces. Afterwards, we will introduce some more general objects, called decorated pseudo G-bundles, and define a crude notion of semistability for them. The benefit is that we can harvest the projective moduli spaces for semistable decorated pseudo G-bundles from Section 2.5 and Section 2.6. In order to derive the main result of Section 2.7, we have to carefully analyze the behavior of semistability for decorated pseudo G-bundles when certain parameters become large. This is done in Section 2.7.2. The conclusion of this analysis is that the moduli spaces of semistable decorated principal G-bundles are special examples of the moduli spaces of decorated pseudo G-bundles for certain stability parameters. Next, we will also study the asymptotic behavior of the semistability concept for decorated principal Gbundles. This is crucial for the results in Section 2.8. As a first application, we will explain in Section 2.7.4 how we obtain the moduli space of semistable Higgs bundles together with a natural compactification as an example of the general result. In Section 2.7.5, we address the subtle point of representations *: G −→ GL(V) whose kernel contains a positive dimensional central torus. Here, the notion of semistability may be relaxed a bit. Section 2.8 finally presents the main result of this monograph: The semistability concept and the moduli spaces for principal G-bundles which are decorated by a section in the vector bundle that is associated via a previously fixed representation *: G −→ GL(V). Here, there is absolutely no restriction on *. Again, we will first introduce the notion of semistability and state the result on the existence of moduli spaces. This result will be reduced to the main technical result from Section 2.7. In order to do so, we
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* from *. Then, we can have to cook up a homogeneous representation * *: G −→ GL(V) associate, for any principal G-bundle P, to a section σ: X −→ P* a section β: X −→ P(P** ). We check that this assignment is finite-to-one on the isomorphy classes and compatible with semistability. Therefore, we can use this assignment to construct the moduli spaces for semistable affine *-pairs (P, σ) from those for semistable projective * *-pairs (P, β). An important point here is that we do not expect our moduli spaces to be projective in general. Instead, they should be projective over an affine variety which depends on the GIT-quotient V//*G via a generalized Hitchin map. After the construction of the moduli spaces, we will discuss in Section 2.8.4 some extensions and examples of our general results. In particular, we will show how we can remove the technical assumption in Section 2.7, how we recover the moduli space of Bradlow pairs, and how we get, as a new example, moduli spaces of Higgs bundles for real reductive groups. At the end, we will again discuss representations *: G −→ GL(V) whose kernel contains a positive dimensional central torus. The proofs leading to the main results of Chapter 2 are already very technical and lengthy. Still, one might ask for even more general results. The following two directions of generalization seem very natural: a) Extend the results to base fields of positive characteristic; b) Extend the results to base varieties of higher dimensions. A more specialized extension c) asks for equipping the vector and principal bundles with parabolic structures. Objects of this kind arise in connection with the investigation of representations of the fundamental group of an open Riemann surface in a connected real or complex reductive group. In Section 2.9, we will explain what we know about these potential extensions. In positive characteristic, the business becomes very complicated, if principal G-bundles with non-classical structure groups are involved and there the theory is still in its beginnings. Over smooth higher dimensional base varieties over a field of characteristic zero, the results are as general as over curves. There are only some fine points to be observed. Finally, on a curve over the field ', the introduction of parabolic structures poses no problem at all. Acknowledgments I owe my interest and foundations in Geometric Invariant Theroy as well as the incentive to this work to my teacher Christian Okonek. I learned about the general moduli problem treated in this book from him and Andrei Teleman during the work on [164]. The results obtained in Section 2.8 were originally planned as part of a joint project with Okonek and Teleman (which has been abandoned for the time being). Chapter 1 is based on my lectures at the University of Zurich in 1996/1997 and 2006 and at the University of Barcelona in 1997. I would like to thank those institutions for the opportunity to lecture on the material and the audience for their interest in the respective courses. The first portions of Chapter 2 were written during my visit to the Universidad Complutense de Madrid in 2003. The work seriously started while I was staying at the IHES in 2005. I thank those institutions for their hospitality. ´ Discussions with Ch. Okonek, A. Teleman, L. Alvarez-C´ onsul, O. Garc´ıa-Prada, T. G´omez, I. Sols, J. Heinloth, and many others provided important input to this work. J. Heinloth carefully read most of the manuscript and discovered many mistakes
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and unclear passages. Additional proof-reading was done by G. Hein. Their comments were very valuable for improving the exposition. P. Gothen kindly forwarded me some extracts of [72]. Last but not least, the referee pointed out several inaccuracies, suggested improvements, and contributed a wealth of interesting and important historical remarks as well as additional references. I am very grateful for this support. While preparing the current volume, I benefitted from financial support from the following sources: The DFG via a Heisenberg fellowship, via the Schwerpunkt program “Globale Methoden in der Komplexen Geometrie—Global Methods in Complex Geometry”, and via the SFB/TR 45 “Perioden, Modulr¨aume und Arithmetik algebraischer Variet¨aten—Periods, Moduli Spaces and Arithmetic of Algebraic Varieties ”, the DAAD via the “Acciones Integradas Hispano-Alemanas” program, contract number D/04/42257, the European Commission through its sixth framework program “Structuring the European Research Area” and the contract Nr. RITA-CT-2004-505493 for the provision of Transnational Access implemented as Specific Support Action. Notation and Conventions General. — We have tried to follow the standard terminology of Algebraic Geometry such as in Hartshorne’s book [96]. The ground field will be ' unless otherwise specified. In the first chapter, the reader may think of varieties in the classical sense, i.e., in the one defined in Chapter I of [96]. We write $n for the unit (n x n)-matrix. Sets. — If S is a set and n a positive integer, we write S x n for the n-fold cartesian product S x · · · x S . We write {pt} for a set containing exactly one element. Categories. — A groupoid is a category in which all morphisms are isomorphisms. Schemes and varieties. — A scheme will be a scheme of finite type over the complex numbers. A variety is a scheme which is reduced and irreducible. For a cartesian product X x Y = X xSpec(.) Y of schemes, we let πX : X x Y −→ X and πY : X x Y −→ Y be the projections onto the first and the second factor, respectively. If X is a scheme or a variety and W is any subset, then W stands for its closure in the Zariski topology (with its induced reduced scheme structure). If X is a projective scheme and F is a coherent OX -module, the cohomology groups of F are finite dimensional '-vector spaces, and we set hi (F ) := hi (X, F ) := dim. (H i (X, F )), i ≥ 0. An open subset U ⊂ X of the variety X is said to be big, if the complement X \ U has codimension at least two in X. Algebraic groups. — In the standard reference [30], the theory of reductive groups is developed only for connected groups. We will slightly deviate from this: A reductive group need not be connected. (This allows to include the orthogonal groups.) However, we require a semisimple group to be connected, so that a semisimple group does not have any non-trivial character. Vector bundles . — By a standard abuse of language, we do not distinguish between vector bundles (geometric objects) and locally free sheaves (see [96], Exercise II.V.18): A geometric vector bundle is identified with its sheaf of sections. Recall that the projective bundle !(E) associated to a vector bundle E on the variety X is Pro j(S ym. (E)), i.e., it is the bundle of hyperplanes in the fibers of E or, equivalently, lines in the fibers of the dual vector bundle E ∨ . This applies, in particular, to vector spaces, so that !(V) stands for (V ∨ \ {0})/'. , V being a finite dimensional '-vector space. Occasionally,
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we will also use P(E) := (E \ zero section)/'. ! !(E ∨ ) for the projective bundle of lines in the fibers of E. If E is a vector bundle on a curve and F ⊂ E is a subsheaf, then ^ 4 F := ker E −→ (E/F )/Tors(E/F ) is a subbundle of E which coincides with F in all but finitely many points. We will refer to F as the subbundle generated by F . (Note that we have deg(F) ≥ deg(F ).) Semistability conditions . — If a certain object such as a one parameter subgroup or a subbundle occurs in a definition of semistability, we will always assume that it is non-trivial. (This prevents us from defining stable objects in a way that they will never exist.)
Chapter 1
Geometric Invariant Theory 1.1 Algebraic Groups and their Representations n this section, we will introduce the basic notions and results from the theory of linear algebraic groups. Standard references are the books [44], [30], [113], and [207]. We will also discuss some useful results concerning representations of linear algebraic groups. For this, we use the references [53], [124], and [123].
"
1.1.1 Linear Algebraic Groups I — Definitions A linear (or affine) algebraic group is a group object G in the category of affine algebraic varieties. In more down to earth terms, this means that G is an affine algebraic variety, equipped with a neutral element e ∈ G, a regular map µ: G x G −→ G, the multiplication, and a regular map inv: G −→ G, the inversion, such that the tuple (G, e, µ, inv) fulfills the axioms of a group. Remark 1.1.1.1. A linear algebraic group G is non-singular as an algebraic variety. First, we may find a non-singular point g ∈ G. For any other point h ∈ G, the left multiplication by hg−1 is an automorphism of the variety G which maps g to h, so that h is a non-singular point of G, too. Exercise 1.1.1.2. Let G be a linear algebraic group. Describe the extra structures on its coordinate algebra '[G] that correspond to the data e, µ, and inv, respectively. A homomorphism between linear algebraic groups is a regular map which is at the same time a homomorphism of groups. A (closed) subgroup H of an algebraic group G is a closed subvariety of G which is also a subgroup. One checks that the kernel of a homomorphism χ: G −→ GB between linear algebraic groups is an example for a subgroup of G. Example 1.1.1.3. i) For the time being, the main example of a linear algebraic group will be the general linear group GLn (') (compare Remark 1.1.1.4): The neutral element is the identity matrix, multiplication is matrix multiplication, which is obviously
18
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regular, and the inversion is the formation of the inverse matrix. The regularity of that operation results from Cramer’s rule. In particular, we see that the multiplicative group '. := GL1 (') of the field ' is an algebraic group. Moreover, you already know some closed subgroups of GLn ('). One of them is the special linear group SLn ('). It is described by the polynomial equation det = 1; SLn (') is the kernel of the homomorphism det: GL n (') −→ '. between linear algebraic groups. ii) The special orthogonal group is the closed subgroup of SLn (') defined through F L SOn (') := g ∈ SL n (') | gt · g = $n . iii) Define the matrix
X J :=
0 −$n
$n 0
J .
This matrix equips '2n with the non-degenerate and antisymmetric bilinear form β(v, w) := vt · J · w,
∀v, w ∈ '2n .
The isometry group of the pair ('2n , β) is the symplectic group F L Sp2n (') := g ∈ GL 2n (') | gt · J · g = J . Observe that Sp2n (') is actually a subgroup of SL2n ('). iv) A homomorphism χ: G −→ '. is called a character of G. The characters of G form an abelian group which is denoted by X(G). For G = GLn (') and any r ∈ (, the map g 1−→ det(g)r is a character of G. Conversely, one shows that any character of GLn (') is of that shape. This may be deduced from the fact that the coordinate algebra of GLn (') is isomorphic to the ring '[xi j , i, j = 1, . . . , n; det−1 ]. v) If G1 ,. . . ,Gn are algebraic groups, then their cartesian product G1 x · · · x Gn with the neutral element e = (e1 , . . . , en ) and componentwise multiplication and inversion is also an algebraic group. vi) A linear algebraic group T which is isomorphic to '. n is called a(n) (algebraic) torus. (Note that '. n is an algebraic group by virtue of v).) For its character group, we find X(T ) ! X('. n ) ! (n . In the latter identification, a vector α = (α1 , . . . , αn ) ∈ (n yields the character (z1 , . . . , zn ) 1−→ zα1 1 · · · · · zαn n of '. n . vii) A one parameter subgroup of G is a homomorphism λ: '. −→ G. The one parameter subgroups of a torus T also form a free abelian group X. (T ) of finite rank. Given a character χ and a one parameter subgroup λ of T , the composed homomorphism χ ◦ λ: '. −→ '. is given as z 1−→ zγ for a uniquely determined integer γ. We set Fλ, χ4 := γ. In this way, we obtain the perfect pairing ([30], 8.6) F., .4: X. (T ) x X(T ) −→ (, i.e., the induced homomorphism X. (T ) −→ X(T )∨ is an isomorphism.
Remark 1.1.1.4. Viewing GLn (') and SLn (') as subsets of affine variety Mn ('), one sees that GLn (') is the open subvariety { det % 0 } while SLn (') is the closed subvariety { det = 1 }. We have the morphism α:
GLn (') −→ Mn (') x ' . g 1−→ (g, det(g)−1 )
S 1.1: A G
19
This yields the description GLn (') = { (g, t) ∈ Mn (') x ' | det(g) · t = 1 }, so that we have realized GLn (') as a closed subvariety of the affine variety Mn (') x '.
1.1.2 Representations Suppose V is a finite dimensional complex vector space and G is a linear algebraic group. We consider V as an affine algebraic variety. A (left) action of G on V is a regular map σ: G x V −→ V, satisfying the axioms: 1. For any g ∈ G, the map σg : V −→ V, v 1−→ σ(g, v), is a linear isomorphism; σe = idV . 2. For any two elements g1 , g2 ∈ G, one has σg1 ·g2 = σg1 ◦ σg2 . Giving the action σ is the same as giving the homomorphism *: G g
−→ GL(V) 1−→ σg .
In this correspondence, one associates to a homomorphism * the action σ: G x V (g, v)
−→ V 1−→ *(g)(v).
For an action σ: G x V −→ V, we will abbreviate σ(g, v) to g · v, g ∈ G, v ∈ V. In the above situation, V is also said to be a (left) G-module and the homomorphism * to be a (rational) representation of G on V. Let V and W be two G-modules. A linear map l: V −→ W is G-equivariant or a homomorphism of G-modules, if l(g · v) = g · l(v), for all g ∈ G and all v ∈ V. Two representations *i : G −→ GL(Vi ), i = 1, 2, are called equivalent or isomorphic, if there is an isomorphism of G-modules between V1 and V2 . To a given family *i : G −→ GL(Vi ), i = 1, 2, . . . , n, of representations, we may associate new representations, using constructions from Linear Algebra, e.g., *1 * · · · * *n : G −→ GL(V1 * · · · * Vn ), by setting (*1 * · · · * *n )(g)(v1 * · · · * vn ) := *1 (g)(v1 ) * · · · * *n (g)(vn), for g ∈ G and vi ∈ Vi , i = 1, . . . , n. Exercise 1.1.2.1. Construct further representations such as direct sums *1 5 symmetric powers Symr *, or exterior powers r *.
-
· · · - *n ,
For any representation *: G −→ GL(V), its dual or contragredient representation *∨ : G −→ GL(V ∨ ) on the dual space V ∨ is defined by *∨ (g)(l): v 1−→ l(*(g)−1 (v)), for g ∈ G, l ∈ V ∨ , and v ∈ V.
20 S 1.1: A G
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Exercise 1.1.2.2. i) What would happen, if we defined *∨ (g)(l): v 1−→ l(*(g)(v))? ii) Let *: G −→ GL(V) be a representation. Show that V and V ∨ are isomorphic G-modules, if and only if there is a non-degenerate bilinear form F ., . 4: V x V −→ ' which is G-invariant, i.e., F g · u, g · v 4 = F u, v 4 for all g ∈ G and all u, v ∈ V. We derive the representations
^ 4 *∨d := Symd (*∨ ): G −→ GL Symd (V ∨ ) .
The meaning of these representations will be discussed in Section 1.2.1.
Example 1.1.2.3 (Representations of '. ). Let *: '. −→ GL(V) be a representation of '. . We claim that there are a basis v1 , . . . , vn of V and integers γ1 ≤ · · · ≤ γn , with J T XT n n zγi · αi · vi , for all g ∈ G. *(z) αi · vi = i=1
i=1
We begin with the following result: Lemma 1.1.2.4. Assume H is a subset of elements of GLn (') which commute with each other. i) The elements of H may be simultaneously trigonalized, i.e., there exists an element g ∈ GLn ('), such that g · H · g−1 is contained in B, the subgroup of upper triangular matrices in GLn ('). ii) If all the elements of H are diagonalizable, then they may be brought simultaneously into diagonal form, that is, there is a matrix g ∈ GLn ('), such that g · H · g−1 is a subset of the subgroup D of diagonal matrices in GLn ('). Proof. We prove the statements by induction on n. For n = 1, there is nothing to show. Now, let n be arbitrary and assume that the lemma is known for m < n. Ad i). If all elements of H are scalar matrices, there is nothing to prove. Otherwise, pick a “non-scalar” element h0 ∈ H, an eigenvalue λ0 ∈ ' of h, and let E(λ0 ) be the eigenspace of h0 with respect to the eigenvalue λ0 . For any other element h ∈ H and v ∈ E(λ0 ), we compute ^ 4 ^ 4 (1.1) h0 h(v) = h h0 (v) = h(λ0 · v) = λ0 · h(v), so that h(v) ∈ E(λ0 ). Therefore, the subspace E(λ0 ) ! 'n is invariant under multiplication with any element h ∈ H. Applying the induction hypothesis to E(λ0 ) and 'n /E(λ0 ) does the trick. Ad ii). Again, we may assume that there is an element h0 ∈ H which is not a scalar matrix. Let λ1 , . . . , λ s , s ≥ 2, be the distinct eigenvalues of h0 . Since h0 is assumed to be diagonalizable, 'n decomposes as the direct sum E(λ1 ) - · · · - E(λ s ). The computation in (1.1) shows that any element of H preserves this decomposition. Any element h ∈ H thus yields an automorphism of E(λi ), i = 1, . . . , s, which is again diagonalizable. Applying the induction hypothesis to E(λ1 ),. . . ,E(λ s ) concludes the argument. "
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21
Now, let W := { z ∈ '. | ∃n > 0 : zn = 1 } be the subgroup of roots of unity. It is dense in '. and so is its image H := *(W) in *('. ). We may apply the lemma in order to exhibit a basis v1 , . . . , vn , providing the isomorphism β: GL(V) −→ GLn ('), such that β * *B : '. −→ GL(V) ! GLn (') maps W to the group D of diagonal matrices. Since D is evidently closed in GLn ('), we find *B ('. ) ⊆ *B (W) ⊂ D.1 Our initial assertion follows by suitably reordering the basis v1 ,. . . ,vn . " .n Example 1.1.2.5 (Representations of tori). Let T = ' be a torus and *: T −→ GL(V) a representation of T on the vector space V. The same argument as above shows that * is diagonalizable. More precisely, the T -module V is isomorphic to χ∈X(T ) Vχ with F L Vχ := v ∈ V | *(t)(v) = χ(t) · v ∀t ∈ T , χ ∈ X(T ).
!
1.1.3 Linear Algebraic Groups II — Linear Algebraic Groups as Subgroups of General Linear Groups The aim of this section is to demonstrate that any linear algebraic group is isomorphic to a closed subgroup of a general linear group. This is one of the first results one encounters in the theory of linear algebraic groups. The techniques used in its proof fit neatly in the framework of these notes and will be taken up several times. Let G be a linear algebraic group with multiplication µ: G x G −→ G. The morphism µ corresponds to the comultiplication µ. : '[G] −→ '[G] * '[G] on the coordinate algebra. Define left actions *l : G x '[G] −→ '[G] and *r : G x '[G] −→ '[G] of the group G on its coordinate algebra via *l (g, f ) := *r (g, f ) :=
*lg ( f ): h 1−→ f (g−1 h), *rg ( f ): h 1−→ f (hg),
∀g, h ∈ G, f ∈ '[G].
These two actions commute with each other, i.e., *lg1 *rg2 = *rg2 *lg1 , for all g1 , g2 ∈ G.
Proposition 1.1.3.1. Let V ⊂ '[G] be any finite dimensional subspace. Then, there exists a finite dimensional subspace W ⊂ '[G] which contains V and which is invariant under both *l and *r , that is, *l (G, W) ⊆ W and *r (G, W) ⊆ W. Proof. It is certainly sufficient to look at the case when V is generated by a single element f ∈ '[G]. Denote by U the subspace which is generated by all elements of the form *lg ( f ), *rg ( f ), g ∈ G. This subspace is apparently invariant under the actions *l and *r . It remains to show that U is finite dimensional, i.e., that all the *lg ( f ) and *rg ( f ) are contained in a finite dimensional subspace of '[G]. To this end, we consider the action Γ: (G x G) x G −→ G (g1 , g2 ; h) 1−→ g1 · h · g−1 2 . 1 Since the image of a homomorphism between algebraic groups is closed, we have equality at the first stage.
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S 1.1: A G 22
' fi with suitable elements fi ∈ '[G x G] and * We may write Γ . ( f ) = ni=1 fi * * fi ∈ '[G], i = 1, . . . , n. For g1 , g2 , h ∈ G, one has . −1 (*lg1 ◦ *rg2 )( f )(h) = f (g−1 1 · h · g2 ) = Γ ( f )(g1 , g2 ; h) =
so that (*lg1 ◦ *rg2 )( f ) =
'n
−1 i=1 fi (g1 , g2 )
n T i=1
·* fi ∈ F * f1 , . . . , * fn 4.
* fi (g−1 1 , g2 ) · fi (h), "
We will also need the following observation: Lemma 1.1.3.2. A finite dimensional subspace V ⊂ if µ. (V) ⊂ V * '[G].
'[G] is *r -invariant, if and only
Proof. Pick a basis f1 , . . . , fn for V and complete it by * f1 , . . . , * fm (# V) to a basis for a finite dimensional subspace H ⊂ '[G] with µ. (V) ⊂ H * '[G]. For f ∈ V, we may ' ' * write µ. ( f ) = ni=1 fi * ri + m i=1 fi * si , so that *rg ( f ) =
n T
ri (g) · fi +
i=1
m T
si (g) · * fi .
i=1
Thus, *rg ( f ) is contained in V, if and only if s1 (g) = · · · = sm (g) = 0. This immediately yields the claim. " We are now in the position to establish the main result: Theorem 1.1.3.3. Let G be a linear algebraic group. Then, there exists a positive integer n, such that G is isomorphic to a closed subgroup of GLn ('). Proof. Write '[G] = '[ f1 , . . . , fn ]. By Proposition 1.1.3.1, we may assume that the generators f1 , . . . , fn are linearly independent and span a *r -invariant subspace V. Then, Lemma 1.1.3.2 grants the existence of elements mi j ∈ '[G], i, j = 1, . . . , n, such that µ. ( f j ) =
n T
fi * mi j ,
j = 1, . . . , n.
i=1
Consequently, we define α: G g
−→ GL n (') ^ 4 1−→ mi j (g) . i, j
This is obviously a morphism between affine algebraic varieties. Exercise 1.1.3.4 ([30], 1.10). 1) Show that α is a homomorphism, i.e., (mi j (g1 g2 ))i, j = (mi j (g1 ))i, j · (mi j (g2 ))i, j , for all g1 , g2 ∈ G. 2) Prove that α is a closed embedding, by verifying that α. : '[GLn (')] −→ '[G] is surjective. The properties established in the preceding exercise prove that α maps G isomorphically onto a closed subgroup of GLn (') and conclude the proof. "
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23
1.1.4 Reductive Affine Algebraic Groups This section contains the discussion of the central notion of a reductive affine algebraic group. In a certain sense, the reductive groups are exactly those for which Geometric Invariant Theory works (see Remark 1.4.2.3, ii). Let G be an affine algebraic group, V a finite dimensional complex vector space, and *: G −→ GL(V) a representation of G on V. We say that the subspace W ⊆ V is *-invariant, if *(g)(W) ⊆ W, for all g ∈ G. In that case, the group G acts on W, and we may restrict * to the representation *|W : G −→ GL(W). We call * irreducible, if V does not contain *-invariant subspaces apart from {0} and V. A representation is called completely reducible, if it decomposes as a direct sum of irreducible representations. In other words, there are *-invariant subspaces V1 , . . . , Vn , such that 1) V = V1 - · · · - Vn , 2) *|Vi is irreducible for i = 1, . . . , n, and 3) * = *|V1 - · · · - *|Vn . The group G is termed reductive, if any finite dimensional representation of G is completely reducible. Remark 1.1.4.1. i) The modules corresponding to irreducible and completely reducible representations are called simple and semisimple modules, respectively. ii) For a reductive group, the representation theory is determined by its “simple” building blocks, i.e., the irreducible representations. Exercise 1.1.4.2 (Compare [123], Satz, p. 69). Verify the following statement: Lemma 1.1.4.3. Let G be an affine algebraic group, V a finite dimensional vector space, and *: G −→ GL(V) a representation. Then, the following assertions are equivalent: 1. The representation * is completely reducible. 2. Every *-invariant submodule U of V possesses a direct complement, i.e., there is a *-invariant submodule W with V = U - W as G-module. Every finite group is in a natural way an affine algebraic group. Theorem 1.1.4.4 (Maschke). Every finite group is reductive. Proof. Let *: G −→ GL(V) be a representation of G and W ⊂ V a proper, non-trivial G-invariant subspace. According to Lemma 1.1.4.3, we have to find a G-invariant subspace U, such that V = U - W as G-module. First, we find a complementary * i.e., V = U * - W as '-vector space. Associated to that decomposition, subspace U, π: V −→ W. We now define π: V −→ W there is the '-linear projection operator * ' via π(v) := g∈G g · (* π(g−1 · v)). It is rather obvious that π is linear, G-equivariant, and surjective (in fact, π(w) = |G| · w, for w ∈ W). The space U := ker(π) is a complementary submodule to W. " Example 1.1.4.5. i) In Example 1.1.2.3 and 1.1.2.5, we have seen that tori are reductive. ii) The additive group 4a of ' is not reductive. We have 7X J 0 1 λ !!! 4a ! 0 1 ! λ ∈ ' . 9X J( 1 Then, is a 4a -invariant subspace of '2 without direct complement. 0
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Exercise 1.1.4.6. Suppose that G and H are reductive affine algebraic groups. Show that G x H is reductive, too. Remark 1.1.4.7. To be very precise, one should use the term “linearly reductive” for the property that we have called “reductive”. This notion makes sense also over fields of positive characteristic. Unfortunately, in positive characteristic, the only linearly reductive algebraic groups are finite groups whose order is coprime to the characteristic and tori, or products of such groups. There is a notion of “reductivity” which is defined intrinsically (see [30], [113], and [207]). In characteristic zero, this notion is equivalent to “linearly reductive” (see [113], [123]), whence our abuse of language is justified. In positive characteristic, it is equivalent to “geometrically reductive” (see Remark 1.2.1.8 and [200]) which is weaker than “linearly reductive”, but suffices to develop Geometric Invariant Theory. In that weaker sense, special and general linear groups are examples for reductive groups also in positive characteristic.
1.1.5 Homogeneous Representations of the General Linear Group We start with a representation *: GLn (') −→ GL(V) of the general linear group. Fix a basis v1 , . . . , vm for V. In this way, we obtain an isomorphism GL(V) ! GLm ('). Using this isomorphism, * is given by an (m x m)-matrix of regular functions fi j ∈ '[GLn (')] = '[Mn (')][det−1 ], i, j = 1, . . . , m. If the fi j lie in the subring '[Mn (')], we say that * is a polynomial representation. Remark 1.1.5.1. For any representation *, there is a non-negative integer c, such that * * det#c is a polynomial representation. This shows that the polynomial representations play a peculiar rˆole in the representation theory of GLn (').
Theorem 1.1.5.2. Suppose that *: GLn (') −→ GL(V) is a polynomial representation. Then, there are positive integers a1 , . . . , ab , such that V is a submodule of
% W# , b
ai
i=1
W := 'n .
Proof ([124], proof of 5.3, Proposition). Since the representation * is polynomial, it extends to the map * *: Mn (') h
−→ Mm (') (! End(V)) ^ 4 1−→ fi j (h) . i, j=1,...,m
It is multiplicative in the sense that * *(h · hB ) = * *(h) · * *(hB ),
∀ h, hB ∈ Mn (').
We look at the GLn (')-action on Mn (') given as g . h := h · g−1 ,
g ∈ GL n ('), h ∈ Mn (').
This provides '[Mn (')] with the GLn (')-module structure (g • f )(h) := f (g−1 . h),
g ∈ GL n ('), f ∈ '[Mn (')], h ∈ Mn (').
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25
(Observe g • f = *rg ( f ) in the notation of Section 1.1.3.) Given λ ∈ V ∨ , we define the linear map ϕλ : V
−→
v
1−→
'& [Mn (')]
^ 4Z h 1−→ λ * *(h)(v) .
Obviously, ϕλ (v)(id) = λ(v) for all v ∈ V, and we compute ^ 4 ^ 4 ϕλ (gv)(h) = λ * *(h)(gv) = λ * *(hg)(v) = ϕλ (v)(g−1 . h) = ^ 4 = (g • ϕλ )(v) (h), ∀g ∈ GL n ('), v ∈ V, h ∈ Mn ('), i.e., ϕλ is a map of GLn (')-modules. Next, we let λ1 , . . . , λm be a basis for V ∨ . Our discussion above shows that ϕ: V
−→
v
1−→
'^ [Mn (')]$m
ϕλ1 (v), . . . , ϕλm (v)
4
is an injective map of GLn (')-modules. For any vector space U, the symmetric power Symd (U) is, as a representation of GL(U), the quotient of the tensor power U #d . For a decomposable vector v1 * · · ·* vd ∈ U #d , we write v1 · · · · · vd for its image in Symd (U). Then, for any d ∈ (≥0 , the map Symd (U) −→ U #d T vσ(1) * · · · * vσ(m) v1 · · · · · vd 1−→ σ∈Σm
is an injective map of GL(U)-modules. Note that Mn (')∨ ! W $n as GLr (')-module (with respect to the module structure “.”) and that ^ 4 '[Mn (')] = Symd Mn (')∨ .
% d≥0
The assertion of the theorem follows easily from the information gathered so far.
"
Remark 1.1.5.3. The representations of GLn (') on W #m can be explicitly decomposed into irreducible representations and, thus, are completely reducible (see [124], §3). The above theorem, therefore, implies that any polynomial representation of GLn (') is completely reducible. With Remark 1.1.5.1, one then concludes that any representation of GLn (') is completely reducible, so that GLn (') is an example of a reductive group. In Section 1.1.7, we will present a different proof for this fact. A representation *: GLn (') −→ GL(V) is homogeneous of degree deg(*) ∈ (, if ∀z ∈ '. .
*(z · id .n ) = zdeg(*) · id V , In the following, we set, for c > 0, n &8
'
n
Z#−c
:=
X&8 n
'
n
Z#c J∨
.
26 S 1.1: A G
26 S 1.1: A G
Corollary 1.1.5.4. Let *: GLn (') −→ GL(V) be a homogeneous representation. Then, there exist non-negative integers a, b, and c, such that * is a direct summand of the representation *a,b,c of GLr (') on n &^ 4#a Z$b &8 Z#−c Wa,b,c := 'n * 'n . Proof. According to Remark 1.1.5.1, we may find c ≥ 0, such that the representation *B := * * det#c on n Z#−c &8 V* 'n is polynomial. It suffices to prove the assertion for the homogeneous and polynomial representation *B . By Theorem 1.1.5.2, V is a submodule—whence a direct summand—of
% W# . b
ai
i=1
We must obviously have
a1 = · · · = ab = deg(*B ),
and this settles the claim.
"
1.1.6 Faithful Representations and Extensions of Representations Let G be a reductive affine algebraic group. By Theorem 1.1.3.3, there is a faithful representation ι: G −→ GL(V), i.e., ι is also an embedding (in fact, a closed embedding by [30], 1.4 Corollary). Theorem 1.1.6.1. Let *: G −→ GL(U) be a representation of the group G on the finite dimensional complex vector space U. Then, there exists a representation * *: GL(V) −→ GL(W), such that * is a direct summand of the representation * * ◦ ι. Roughly speaking, this means that any representation of G may be extended to a representation of GL(V), using the faithful representation ι. Proof ([124], 5.4, Proposition 1). Let m := dim(U). The construction used in the proof of Theorem 1.1.5.2 may be used to construct an injective map of G-modules ϕ: U −→ '[G]$m .
Here, the G-module structure on '[G] comes from the map g 1−→ *rg (see Section 1.1.3). On the other hand, the closed embedding ι corresponds to the surjective homomorphism ι# : '[GL(V)] −→ '[G] of '-algebras. Note that '[GL(V)] is a GL(V)module by means of the assignment g 1−→ *rg . Via g − 1 → *rι(g) , it becomes a G# module, too. It is not hard to see that ι is a map of G-modules. It follows readily from Proposition 1.1.3.1 that there is a finite dimensional GL(V)-invariant subspace W ⊂ '[GL(V)]$m , such that (ι# )$m (W) contains ϕ(U). This gives * *. Indeed, W ! (ι# )$m (W) - W B as G-module, because G is reductive, and (ι# )$m (W) ! U - W BB for the same reason. "
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27
1.1.7 Appendix. The Reductivity of the Classical Groups via Weyl’s Unitarian Trick Although we have declared the notion of a “reductive linear algebraic group” to be a central ingredient of Geometric Invariant Theory, we have verified reductivity only in a few cases. There are several ways to establish that certain groups are reductive. The methods of classical algebra allow to show that GLn (') and SLn (') are reductive. This is done, e.g., in the books [53] and [124]. In order to get a rounded theory of reductive groups, one defines the notion of a reductive linear algebraic group intrinsically within the theory of linear algebraic groups, i.e., without reference to representations. The reader may consult any book on linear algebraic groups for that. Then, one may see that this intrinsic notion of reductivity agrees—over '(!)—with the notion we use, either by linking the representation theory of the group G to the representation theory of its Lie algebra g or by linking the representation theory of G to that of a compact real form K of G. The first method is outlined in [113], Chapter V, §§13-14, whereas the second one is exposed in [123], Anhang II. Here, we will present some parts of the second approach, following [123], so that the reader might get an idea why, e.g., GLn (') is reductive. It is originally due to Weyl ([219], Chapter VIII, B.11). Lie Groups As is apparent from the definition of an algebraic group, there are analogous concepts in other categories. Here, we will need the notions of complex and real Lie groups which are the group objects in the category of complex and differentiable manifolds, respectively. The groups we are looking at are thus endowed with the additional structure of a complex or real manifold, such that the group law and the inversion are given as holomorphic or differentiable maps. It should be clear that any linear algebraic group gives in a natural way rise to a complex and a real Lie group. For instance, the general linear group GLn (') can obviously be interpreted in either way. Most of the notions we have introduced, so far, readily generalize to Lie groups. We may speak of representations of Lie groups2 , of the irreducibility and complete reducibility of a representation, so that we may also talk about reductive Lie groups. Representations of Compact Real Lie Groups Assume that K is a compact real Lie group. It is intuitively clear that one might generalize the proof of Maschke’s theorem (Theorem 1.1.4.4) in order to show that K is reductive, if there were an appropriate procedure for integration over the group. This is indeed possible. Theorem 1.1.7.1 (Haar). There is a unique linear functional C (K, ')
−→
f
1−→
'C K
f (g)dg
2 Also for real groups, we will look at representations *: G −→ GL(V) where V is a finite dimensional complex vector space.
28 S 1.1: A G
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on the space of continuous complex valued functions on K which satisfies the following properties: K 1. K dg = 1. 2. For all h ∈ K, f ∈ C (K, '): C C C f (h · g)dg. f (g · h)dg = f (g)dg = K
K
K
Proof. [217].
"
Example 1.1.7.2. i) For a finite group, one has C 1 T f (g)dg = f (g). · |K| g∈K K ii) For K = U1 ('), one finds C 2π ^ C 4 1 f exp(iϕ) dϕ. · f (g)dg = 2π 0 K Corollary 1.1.7.3. Suppose that *: K −→ GL(V) is a representation of the compact real Lie group K on the finite dimensional complex vector space V. Then, there exists a K-invariant Hermitian product F., .40 on V, i.e., F g · v, g · w 40 = F v, w 40 ,
∀v, w ∈ V, g ∈ K.
Proof. If F., .4 is any Hermitian product on V, one defines C F v, w 40 := F g · v, g · w 4dg, ∀v, w ∈ V. K
This is then a K-invariant Hermitian product.
"
Corollary 1.1.7.4 (Hurwitz/Schur). Any compact real Lie group K is reductive. Proof. Let *: K −→ GL(V) be a finite dimensional representation of K. Fix a Kinvariant Hermitian product on V. If U is any K-invariant closed subspace, then the direct complement W with respect to the fixed K-invariant Hermitian product is a Ksubmodule, such that V = U - W as K-module. By Lemma 1.1.4.3, V is completely reducible. " Given a matrix h ∈ Mn ('), we let h. := h be the transpose of the complex conjugate of h. Recall that the unitary group is defined as F L Un (') := g ∈ GL n (') | g. · g = $n . t
In other words, Un (') is the isometry group of 'n equipped with the standard Hermitian form. Another consequence of Corollary 1.1.7.3 is the following:
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29
Corollary 1.1.7.5. Let K ⊂ GLn (') be a compact real subgroup. Then, K is conjugate to a subgroup of Un ('). This implies, in particular, that Un (') is a compact real subgroup of GLn (') which is maximal with respect to inclusion among all compact real subgroups of GLn ('). Moreover, we see that any two maximal compact subgroups of GLn (') are conjugate. The reader may consult [31], Chapter VII, for a general statement of this kind. Real Forms of Reductive Linear Algebraic Groups Note that the map .
: GLn (') −→ GLn (') g 1−→ g.
is a real structure on the complex Lie group GLn ('), i.e., an anti-holomorphic involution. The set of real points of this real structure, that is, the set of fixed points of the involution, is exactly the unitary group. Thus, Un (') is a compact real form of GLn ('), and GLn (') is the complexification of Un ('). Remark 1.1.7.6. i) One verifies 4 4 ^ ^ dim/ Un (') = dim. GLn (') = n2 . ii) The involution . induces the '-anti-linear map Mn (') −→ Mn ('), m 1−→ −m. , on the Lie algebra Mn (') of GLn ('). The fixed points of this involution form the Lie algebra F L un (') := m ∈ Mn (') | m + m. = 0 of Un (').
Proposition 1.1.7.7. The set Un (') is Zariski-dense in GLn ('). Proof. Let H be the Zariski-closure of Un ('). This is a closed subgroup of the algebraic group GLn (') (cf. [30], 1.3, Proposition). Since GLn (') is connected, it suffices to prove ^ 4 dim(H) = dim GLn (')
in order to conclude that H = GLn ('). The Lie algebra h of H clearly contains un ('). Since h is a complex vector space, it contains also ' ·un ('). Now, i·un (')∩un (') = {0} and dim/ (un (')) = n2 , so that—as 1-vector spaces— h ⊇ i · un (') - un (') = Mn ('). Thus, dim(H) = dim. (h) = n2 = dim. (GLn (')), and we are done. Theorem 1.1.7.8. The group GLn (') is reductive.
"
30
S 1.2: GIT A V 30
Proof. Let *: GLn (') −→ GL(V) be a representation. It restricts to a representation * *: Un (') −→ GL(V). Any GLn (')-invariant subspace U possesses a Un (')-invariant direct complement W. It suffices to show that W is GLn (')-invariant. For this, we define F L NGL n (.) (W) := g ∈ GLn (') | g · W ⊆ W . By [30], 1.7, Proposition, NGL n (.) (W) is a closed subgroup of GLn ('). It contains Un ('), so that NGLn (.) (W) = GLn ('), by Proposition 1.1.7.7. This means that W is " GLn (')-invariant, and we are done.
Example 1.1.7.9. i) Show that the special linear group SLn ('), the special orthogonal group SOn ('), and the symplectic group Sp2n (') are invariant under involution . of GLn (') and GL2n ('), respectively. ii) Use i) to show that SLn ('), SOn ('), and Sp2n (') are reductive linear algebraic groups, too.
1.2 Geometric Invariant Theory for Affine Varieties — A First Encounter ere, we will formulate the central problem of taking the quotient of a vector space by the action of an algebraic group coming from a representation. After discussing the fundamental problems, we will state the most basic results of Geometric Invariant Theory without proof and illustrate them by several examples. (The proofs will follow in Section 1.4.)
#
1.2.1 The Categorical Quotient of a Vector Space by a Representation Let G be an affine algebraic group, V a complex vector space, and *: G −→ GL(V) a representation of G on V. The G-orbit of v ∈ V is the (locally closed) subset F L G · v := orbG (v) := *(g)(v) | g ∈ G . We define an equivalence relation on V, by saying that v1 and v2 ∈ V are equivalent, if they have the same orbit. Denote by V/*G the set of equivalence classes. Since V carries the structure of an affine algebraic variety, the following question seems natural: Problem 1.2.1.1. Does V/*G carry the structure of an affine algebraic variety, such that the projection map π* : V −→ V/*G is regular? The answer to the above question is “no” in general. Assume that V/*G is an affine algebraic variety with coordinate algebra A. The elements of A are the regular functions on V/*G. As regular functions on V, they are constant on any G-orbit in V. Conversely, a regular function on V which is constant on all orbits defines a (settheoretic) function V/*G −→ '. We may postulate that this function is regular, i.e., an
S 1.2: GIT A V
31
element of A. Define '[V]* as the subalgebra of '[V] consisting of those functions which are constant on all G-orbits. We would expect 4 ^ (1.2) V/*G = Specmax '[V]* . Example 1.2.1.2. Look at the action σ:
'. x 'n (z, v)
−→ 1−→
'n
z·v
that belongs to the representation *: '. −→ GLn ('), z 1−→ z · $n . As point set, V/*G = !n−1 ∪ {0}. A function on 'n which is constant on any line is a constant function. Thus, '['n ]* = ' and Specmax(') = {pt}. The essential lesson of this example is the following: A function f ∈ '[V]* is continuous in the Zariski topology and therefore not only constant on any orbit O, but also on the closure O of any orbit. In the above example, the orbit '. · v is closed if and only if v = 0. After all, we may only hope that we may equip the set of closed orbits in V with a “natural” structure of an affine algebraic variety. In the above example, this worked out. We stick to the approach suggested by (1.2) and make the following definition ^ 4 V//*G := Specmax '[V]* . We have overlooked one crucial point:
Problem 1.2.1.3. Is '[V]* a finitely generated '-algebra? In general, this will not be the case. However, for reductive groups, there is the following basic result: Theorem 1.2.1.4 (Hilbert). If G is reductive, then algebra.
'[V]* is a finitely generated '-
Let us have a closer look at the algebra '[V]* . We have ^ 4 ^ 4 V = Specmax Sym. (V ∨ ) = Specmax Symd (V ∨ ) ,
%
! i.e., '[V] =
d∈#≥0 d
∨
Sym (V ). As in Section 1.1.3, one checks that the action of G on V induces a left action *l : G x '[V] −→ '[V] with *l (G, Symd (V ∨ )) ⊆ Symd (V ∨ ). In particular, *l yields a family ^ 4 *ld : G −→ GL Symd (V ∨ ) , d ∈ (≥0 , d∈#≥0
of representations. It is easy to verify that *ld agrees with the representation *∨d that occurred in Section 1.1.2. Setting F L Symd (V ∨ )* := f ∈ Symd (V ∨ ) | *∨d (g)( f ) = f ∀g ∈ G , we find
'[V]* =
% Sym (V ) . d
∨ *
d∈#≥0
Assuming that G is reductive, we have the affine algebraic variety V//*G, and the inclusion '[V]* ⊂ '[V] yields the projection map π* : V −→ V//*G.
32 S 1.2: GIT A V
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Lemma 1.2.1.5. The regular map π* induces a bijection between the set of points of V//*G and the closed G-orbits in V. More precisely, the following holds true: If W1 and W2 are two disjoint, closed, G-invariant subsets of V, then there exists a function f ∈ '[V]* with f|W1 ≡ 0 and f|W2 ≡ 1. We distinguish the points in V according to the structure of their orbits: An element v ∈ V is said to be *-stable, if the orbit of v is closed and has the same dimension as G; *-semistable, if the origin 0 is not contained in the closure of the orbit of v; not *-semistable or *-nullform, if 0 lies in the closure of the orbit of v. If it is clear of which representation * we are talking, we will simply speak of stable points and so on. Remark 1.2.1.6. By definition, stable points are semistable. A point v ∈ V is semistable, if there is a non-constant homogeneous element in '[V]* that does not vanish in v. In other words, v is a nullform, if and only if all non-constant homogeneous *-invariant functions do vanish in v. Exercise 1.2.1.7. Let G be a linearly reductive affine algebraic group, *: G −→ GL(V) a representation, and v ∈ V \ {0} a fixed point for the induced G-action on V. Show that there is a homogeneous function f ∈ '[V]* of degree one which does not vanish in v. Use only the definition of linear reductivity and elementary results on representations. Remark 1.2.1.8. In general, an affine algebraic group G is said to be geometrically reductive, if, for any representation *: G −→ GL(V) and any G-invariant vector v % 0 in V, there is a homogeneous function f ∈ '[V]* of positive degree with f (v) % 0. By the exercise, linearly reductive groups are easily seen to be geometrically reductive. (The converse is true only in characteristic zero!) The most important tool for finding the stable and semistable points is provided by the following theorem: Theorem 1.2.1.9 (Hilbert–Mumford criterion). The point v ∈ V fails to be stable (semistable), if and only if there is a non-constant one parameter subgroup λ: '. −→ G with lim λ(z) · v = w (= 0) z→∞
for an appropriate point w ∈ V. Remark 1.2.1.10. Note that limz→∞ λ(z) · v lies, if it exists, in the closure of the orbit of v. Thus, the necessity of the stated condition is evident. We conclude by a theorem of Hilbert’s which is useful for determining generators for '[V]* . Theorem 1.2.1.11. Let I1 , . . . , I s ∈ '[V]* be invariant homogeneous functions whose common vanishing locus is exactly the set of nullforms. Then, '[V]* is the integral closure of '[I1 , . . . , I s ] in the field '(V). Proof. [210], Theorem 4.6.1 & Corollary 4.6.2.
"
Together with the following result, this theorem gives a handy criterion for a variety to be a quotient. We will apply it several times in the examples.
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33
Proposition 1.2.1.12. Let f : X −→ Y be a dominant and finite morphism between irreducible algebraic varieties. If Y is normal and there is a non-empty open subset U ⊂ Y, such that f| f −1 (U) : f −1 (U) −→ U is bijective, then f is an isomorphism. The proof rests on the following fact: Proposition 1.2.1.13. Let f : X −→ Y be a finite morphism between irreducible algebraic varieties, and let n be the degree of the field extension '(Y) ⊆ '(X). Then, there is a non-empty open subset U ⊂ Y, such that f −1 (y) consists of n distinct points for all y ∈ U. Proof. Since the extension of the function fields is finite and separable, there exists a primitive element ζ ∈ '[X], such that '(X) = '(Y)[ζ]. Let f (x) = xn +a1 xn−1 +· · ·+an be the minimal polynomial of ζ. One checks that one may find a principal open affine * = Specmax(A) ⊆ Y, such that a1 ,. . . ,an are in A and f −1 (U) * = Specmax(B) subset U n−1 with B = A · 1 - A · ζ - · · · - A · ζ as A-module. Now, let D ∈ A be the discriminant * as the subset where D does not of the polynomial f (see (1.4)). We define U ⊆ U vanish. This is non-empty, because f has no multiple zero. Let y ∈ U be a point with maximal ideal m x ⊂ A and write a for the class of an element a ∈ A in A/m x ! '. Then, B/m x · B ! '[x]/ f (x), f = xn + a1 xn−1 + · · · + an . The discriminant of f is D % 0, by construction. Thus, f has precisely n distinct zeroes and f −1 (y) = Specmax(B/m x · B) = { x1 , . . . , xn }. This establishes our contention.
"
Proof of Proposition 1.2.1.12. By Proposition 1.2.1.13, the extension '(Y) ⊆ '(X) of the function fields has degree one, whence '(Y) = '(X). This means that f is a birational map. Since a finite morphism is affine, we may assume without loss of generality that X = Specmax(B) and Y = Specmax(A). The map f # : A −→ B is an integral extension, because f is finite. Since A and B have the same quotient field and A is assumed to be normal, we must have that A = B. " Remark 1.2.1.14. Proposition 1.2.1.12 also holds under the weaker assumption that f is affine, dominant, and the codimension of X \ f (X) is at least two. Indeed, since f is affine, we are immediately reduced to the case that X and Y are affine varieties. We first check that f is birational. Our assumption grants that f is generically finite in the sense of [96], Exercise II.3.7. Therefore, we may find a non-empty open affine subset U, such that f|U : f −1 (U) −→ U is finite. Our above argument therefore shows that '(Y) = '(X), so that f is birational. Now, we have the diagram
'(Y) = '(X) ∪
∪
'[Y] → '[X].
It remains to show that '[Y] −→ '[X] is surjective. Suppose h ∈ '[X] \ '[Y]. Since h fails to be a regular function on Y, its set Z of poles is a divisor in Y, by the normality of Y and Krull’s principal ideal theorem. By our assumption, f (X) ∩ Z % ∅. From this, one easily sees that h cannot be regular as a function on X either (see, e.g., [140], Lemme, p. 233), a contradiction.
34
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We will establish Theorem 1.2.1.4, 1.2.1.9, and Lemma 1.2.1.5 in a more general context in Section 1.4 on Mumford’s Geometric Invariant Theory. In order to gain some familiarity with these results, we will discuss several examples within the next section.
1.3 Examples from Classical Invariant Theory n classical invariant theory, the objects of study were the actions of the special linear groups on spaces of algebraic forms. This problem is equivalent to the problem of classifying hypersurfaces in projective space up to projective equivalence. Thus, it gives a good illustration how important classification problems in Algebraic Geometry may be formalized in terms of group actions on algebraic varieties. For this reason, we present a wealth of examples regarding this problem. Finally, we state without proof the fundamental results concerning the invariant theory of matrices and a more recent generalization of them to representations of quivers. We conclude the section by an example for a quotient by a special orthogonal group in order to have some example where the group acting is not a special or general linear one.
"
1.3.1 Algebraic Forms An algebraic form of degree d on 'n is the restriction to the diagonal of a symmetric d-multilinear form n 'D!!!!!!!!!WB!!!!!!!!!\ x · · · x 'n −→ '. d -times By polarization, the restriction to the diagonal allows to recover the whole multilinear form, so that, by definition of the symmetric powers, an algebraic form of degree d is the same as a linear map ϕ: Symd ('n ) −→ '.
Therefore, the algebraic forms of degree d on 'n form the vector space Symd ('n ) . For the low degrees d = 2, 3, and 4, we speak of quadratic, cubic, and quartic forms, respectively. For the low dimensions n = 2, 3, and 4, we talk about binary, ternary, and quaternary forms, respectively. Let (e1 , . . . , en ) be the standard basis of 'n and (x1 , . . . , xn ) the dual basis of 'n ∨ . There is the isomorphism ∨
Symd ('n ) ϕ/
7 Symd ('n ∨ )
∨
7
'
ν=(ν1 ,...,νd ) ∈{ 1,...,n }x d
ϕ(eν1 · · · · · eνd ) · xν1 · · · · · xνd .
(1.3)
We identify the vector space Symd ('n ∨ ) with the vector space '[x1 , . . . , xn ]d of homogeneous polynomials of degree d in the variables x1 , . . . , xn . For an algebraic form ϕ of degree d on 'n and its associated polynomial f , we find the relationship f (α1 , . . . , αn ) = ϕ(α · · · · · α),
∀α := (α1 , . . . , αn )t ∈ 'n .
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35
The Action of SLn (') by Substitution of Variables
The group SLn (') acts on '[x1 , . . . , xn ]d via
7 '[x1 , . . . , xn ]d vd : SLn (') x '[x1 , . . . , xn ]d / 7 g · f : α 1−→ f (gt · α). (g, f ) In the notation of Section 1.1.2, we have vd = (ι∨d )−1 , ι: SLn (') ⊂ GLn (') being the inclusion. We formulate the following list of natural tasks: t
1. Describe the set '[x1 , . . . , xn ]d /vd SLn ('), i.e., find for every orbit a particularly “nice” representative, a so-called normal form. 2. Compute the invariant ring, i.e., the coordinate algebra of the quotient variety '[x1 , . . . , xn ]d //vd SLn ('). This involves the determination of a set of generators for the invariant ring. 3. Find the (semi)stable forms and the nullforms. By Theorem 1.2.1.11, this might help to solve the second problem, too. Remark 1.3.1.1. i) Of course, there is the analogous action of GLn (') on the vector space '[x1 , . . . , xn ]d . However, all forms are nullforms for this action, because the center of GLn (') acts via non-trivial homotheties. ii) It is important to note that we have just formulated a classification problem in Algebraic Geometry. Let H ⊂ !n−1 be a hypersurface of degree d. Then, we find a polynomial f ∈ '[x1 , . . . , xn ]d , such that H = { f = 0 }. Two hypersurfaces H1 = { f1 = 0 } and H2 = { f2 = 0 } are projectively equivalent, if there is a matrix g ∈ GLn (') with f1 = g · f2 . This is equivalent to the fact that there is an automorphism of the ambient space !n−1 which carries H2 into H1 . Therefore, the set ('[x1 , . . . , xn ]d \{0})/vd GLn (') parameterizes the projective equivalence classes of hypersurfaces of degree d. (Observe that this set equals !('[x1 , . . . , xn ]d ∨ )/ SLn ('). Refer also to Example 1.4.3.15.) Apparently, two projectively equivalent hypersurfaces are isomorphic. Conversely, one may show that two isomorphic smooth hypersurfaces of dimension at least 3 are projectively equivalent. In !3 , the same assertion holds for hypersurfaces of degree d % 4. A proof may be found in the book [87]. (The authors of [87] make the assumption d % n for n ≥ 3. This may be avoided by the use of the cohomology ring.) Finally, the case of surfaces of degree 4 in !3 belongs to the realm of K3-surfaces. Here, the notions of “isomorphy” and “projective equivalence” are still equivalent on the complement of countably many Zariski closed subsets. For this, we refer the reader to the paper [147].
1.3.2 Examples In this section, we give a number of examples which illustrate how the solutions to the above questions may look like in concrete examples.
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36 S 1.3: C I T
Quadratic Forms The theory of quadratic forms is familiar from Linear Algebra. Indeed, for a homogeneous polynomial q of degree 2, there is a uniquely determined symmetric (n x n)matrix S q , such that q(α1 , . . . , αn ) = αt · S q · α,
∀α = (α1 , . . . , αn )t ∈ 'n .
In addition, one verifies S g·q = g · S q · gt ,
∀g ∈ GL n (').
Linear Algebra teaches us that, given a symmetric matrix S ∈ Mn ('), there is a matrix g ∈ GLn ('), such that g·S ·gt is a diagonal matrix with ones and zeroes on the diagonal. This implies the following result: Lemma 1.3.2.1. Suppose q ∈ '[x1 , . . . , xn ]2 . Then, there are a matrix g ∈ GLn (') and a natural number m ∈ { 0, . . . , n } with g · q = x21 + · · · + x2m . (For m = 0, we read this as g · q = 0.) Next, we look at the action of SLn (') on '[x1 , . . . , xn ]2 . The discriminant of a quadratic form q is given as Δ(q) := det(S q ). The discriminant is certainly invariant under the SLn (')-action. Using Lemma 1.3.2.1, one verifies: Corollary 1.3.2.2. Assume q ∈ '[x1 , . . . , xn ]2 , and set δ := Δ(q). For δ % 0, the form q is equivalent to the form qδ := δx21 + x22 + · · · + x2n . Otherwise, there is a number m ∈ { 0, . . . , n − 1 }, such that q is equivalent to x21 + · · · + x2m . We even have the following strengthening of the corollary: Theorem 1.3.2.3. W := '[x1 , . . . , xn ]2 //v2 SL n (') = Specmax('[Δ]). Proof. Let I ∈ '[W] be an invariant polynomial. The “general” quadratic polyno' mial may be written as 1≤i≤ j≤n κi j xi x j . The coordinate algebra of the affine variety '[x1 , . . . , xn ]2 is, thus, given as '[κi j ; 1 ≤ i ≤ j ≤ n], and I is a polynomial in the κi j . We define IΔ ∈ '[Δ], by replacing κ11 by Δ, κii , i = 2, . . . , n, by 1, and the remaining indeterminates by 0. The polynomial I − IΔ takes the value zero on each qδ . Since " I − IΔ ∈ '[W], Corollary 1.3.2.2 implies I − IΔ ≡ 0. Exercise 1.3.2.4. Prove that q is a nullform, if and only if Δ(q) = 0. Determine the dimension of the stabilizer group of a form q. (The description as a matrix problem may help.) Which are the (semi)stable points in '[x1 , . . . , xn ]2 ?
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Binary Forms We write a binary form f of degree d as + ad+1 xd2 . f = a1 xd1 + a2 x1d−1 x2 + · · · + ad x1 xd−1 2 Recall that SL2 (') acts triply transitively on the projective line !1 ([30], 10.8). Hence, f is equivalent to a form of the shape λ · x1d−i xi2 , 0 ≤ 2i ≤ d, λ ∈ ', or I λ · xµ11 xµ22 (x1 − x2 )µ3 di=µ1 +µ2 +µ3 +1 (x1 − βi x2 ), λ ∈ '. , βi ∈ ' \ { 0, 1 }. Let us determine the stable and semistable forms, as well as the nullforms. The property of being stable, semistable, or a nullform is invariant under the action of SL2 ('). Moreover, any one parameter subgroup may be diagonalized, by 1.1.2.3. Using the Hilbert–Mumford criterion (Theorem 1.2.1.9), we have to find out for which forms f = a1 xd1 + · · · the limit X J z lim ·f z−1 z→∞ does exist or equals 0. We compute: Lemma 1.3.2.5. i) The limit exists, if and only if a1 = · · · = a; d / = 0. 2 = 0. ii) The limit equals zero, if and only if a1 = · · · = a; d+1 2 / Rephrasing this observation in intrinsic terms, we see: Lemma 1.3.2.6. i) A binary form of degree d is stable, if and only if it does not have a zero of multiplicity ≥ d2 . ii) A binary form of degree d is semistable, if and only if it does not have a zero of multiplicity > d2 . For odd d, the notions “stable” and “semistable” do agree. We conclude this general discussion by introducing a classical invariant of binary forms which is available in any degree, the so-called discriminant. For two binary forms f = a1 xd1 + · · · + ad+1 xd2 and h = b1 xe1 + · · · + be+1 xe2 , the resultant of f and h is defined as !! !! !! a1 a2 · · · ad+1 !! !! !! a1 a2 · · · ad+1 !! !! . . .. .. · · · !! !! · · · !! ! a1 a2 · · · ad+1 !! !! , R( f, h) := !!! !! b1 b2 · · · be+1 !! !! !! b1 b2 · · · be+1 !! !! . . .. · · · .. !! !! · · · !! ! b1 b2 · · · be+1 ! the displayed matrix having e + d rows. As is well-known, the condition R( f, h) = 0 holds, if and only if f and h share a common factor. Furthermore, R(g· f, g·h) = R( f, h) for all g ∈ SL2 ('). The discriminant of the form f is then declared to be a1 · Δ( f ) := R( f, f B ).
(1.4)
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The polynomial Δ is, by construction, an invariant polynomial which vanishes in the form f exactly when f has a multiple zero. Details on the resultant and the discriminant may be found in [216], §§33-35. Binary Cubic Forms According to Lemma 1.3.2.6, a binary cubic form k is either a nullform or a stable form. The first case is equivalent to k having a double or triple zero, i.e., to Δ(k) = 0. The discriminant of k = a1 x31 + a2 x21 x2 + a3 x1 x22 + a4 x32 is calculated by the formula Δ(k) = 27a21 a24 − a22 a23 − 18a1 a2 a3 a4 + 4a1 a33 + 4a32 a4 . If k is a nullform, it is equivalent to 0, x31 , or x21 x2 . For Δ(k) % 0, k is equivalent to a form kδ := δ · x1 x2 (x1 − x2 ), δ4 = Δ(k). Note that Δ(kδ1 ) = Δ(kδ2 ), if and only if there exists a fourth root of unity ζ with ζ · δ1 = δ2 .
Exercise 1.3.2.7. Show that, for any fourth root of unity ζ, there is a matrix g ∈ SL2 (') which carries the form kδ into the form kζ·δ . Theorem 1.3.2.8. Any invariant of binary cubic forms is a polynomial in the discriminant. Proof. Let I ∈ '[a1 , a2 , a3 , a4 ] be an invariant polynomial, and define * I ∈ '[a], by setting a2 = −a3 = a and a1 = 0 = a4 . By Exercise 1.3.2.7, * I must be a polynomial in a4 , that is, there exists a polynomial % I with * I(a) = % I(a4 ). We declare IΔ := % I(Δ) ∈ '[Δ]. Again, the identity I ≡ IΔ follows from the fact that I and IΔ agree on the forms kδ . " Binary Quartic Forms Here, we confine ourselves to a brief summary of the known results. They will be used in the next example. We write a binary quartic form as q := a1 x41 + 4a2 x31 x2 + 6a3 x21 x22 + 4a4 x1 x32 + a5 x32 . One may then construct the following invariants: I
:=
J
:=
a1 a5 − 4a2 a4 + 3a23
a1 a3 a5 + 2a2 a3 a4 − a1 a24 − a22 a5 − a33 .
The curious reader may consult the references [53], [107], [153], [170], [191], or [210] for information how these invariants are obtained. With the above invariants, the discriminant for binary quartic forms becomes D = I 3 − 27J 2 . Claim 1.3.2.9. A binary quartic q is a nullform, if and only if I(q) = J(q) = 0.
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Proof. If q is a nullform, then it is equivalent to x31 x2 , x41 , or 0. One easily checks I(q) = J(q) = 0. If q is semistable, then it has either two zeroes of multiplicity 2 or there is a simple root. In the first case, q is equivalent to λ · x21 x22 , λ ∈ '. , and one sees I(q) = 3λ2 % 0. In the second case, q may be transformed into a form of the shape λx41 + µx31 x2 + x1 x32 , and one finds I(q) = −µ/4 and J(q) = −λ/16. One of these invariants does not vanish, because q has no triple zero. In addition, we immediately recognize that I and J are algebraically independent. " Theorem 1.3.2.10. The ring of invariants of binary quartic forms is generated by the two—algebraically independent—invariants I and J. Proof. As we have observed before, I and J are indeed algebraically independent, so that the ring '[I, J] is normal and Specmax('[I, J]) ! +2 . The inclusion '[I, J] ⊆ '[V]* gives rise to a finite and surjective morphism 4 ^ 4 ^ ϕ: Specmax '[V]* −→ Specmax '[I, J] ! +2 between normal affine varieties. We would like to see that ϕ is an isomorphism. By Proposition 1.2.1.12, it suffices to check that there is a non-empty open subset U ⊆ Specmax('[I, J]), such that ϕ−1 (U) −→ U is a bijection. We choose U := { D = I 3 − 27J 2 % 0 }. Let f be a quartic form with D( f ) % 0. Then, f decomposes as a product of distinct linear forms. In suitable coordinates, we may write f = λx41 + µx31 x2 + x1 x32 . We now find I( f ) = −µ/4 and J( f ) = −λ/16, and the assertion becomes evident. " Another proof which refers to the geometric meaning of the above invariants and their relationship to the cross ratio of four points in !1 is contained in [160], p. 96ff. Pairs, Consisting of a Cubic Form and a Linear Form The aim of this section is the determination of the ring of invariants A for the action of ∨ ∨ SL2 (') on Sym3 ('2 ) - '2 . Exercise 1.3.2.11. Show that β: '2 x '2 B
(h, h )
−→
'
1−→ det(h|hB )
is a non-degenerate SL2 (')-invariant bilinear form. Conclude that '2 and '2 are isomorphic as SL2 (')-modules. Remark 1.3.2.12. According to the preceding exercise, we may as well study the action ∨ of SL2 (') on Sym3 ('2 ) - '2 . The invariant ring B for this action is the so-called ring of covariants for binary cubic forms. The generators and relations for this ring are computed in [210], Proposition 3.7.7, with the symbolic method. The reader is advised to check that the result is—under the above SL2 (')-equivariant identification ∨ ∨ ∨ of Sym3 ('2 ) - '2 with Sym3 ('2 ) - '2 —the same which we will state below. We begin by constructing several invariants in A. The bilinear map ∨
µ: Sym3 ('2 ) - '2 (k, l) / ∨
∨
7 Sym4 ('2 ∨ ) 7 k·l
40 S 1.3: C I T
S 1.3: C I T 40
is clearly equivariant and surjective. For this reason, it yields an inclusion of the ring of invariants of binary quartics into A. By Theorem 1.3.2.10, all invariants of binary quartics are polynomials in the invariants I and J. Furthermore, the invariants of binary cubics give elements in A. Recall that all such invariants are polynomials in the discriminant Δ (Theorem 1.3.2.8). As the fourth invariant, we introduce the resultant R. On a pair (k, l) with k = a1 x31 + a2 x21 x2 + a3 x1 x22 + a4 x32 and l = b1 x1 + b2 x2 , R takes the value R(k, l) = a1 b32 − a2 b1 b22 + a3 b21 b2 − a4 b31 . Theorem 1.3.2.13. One has A = '[I, J, Δ, R]. Here, the elements I, Δ, and R are algebraically independent, and there is the single relation 27 · J 2 =
1 · ΔR2 + I 3 . 256
(1.5)
Proof. For the moment, let us assume that I, Δ, and R are algebraically independent and that Relation 1.5 holds. Since (1/256)ΔR2 + I 3 is obviously an irreducible polynomial, we may apply [96], II, Exercise 6.4, and infer that '[I, J, Δ, R] is integrally closed in '(I, J, Δ, R). As in the proof of Theorem 1.3.2.10, we will demonstrate that 4 4 ^ ^ ∨ ∨ ϕ: Specmax '[Sym3 ('2 ) - '2 ]SL2 (.) −→ Specmax '[I, J, Δ, R] is a bijective map over a non-empty open subset. To this end, we define U as the open subset containing those pairs (k, l) for which none of the invariants I, J, Δ, and R vanishes. This means that we may assume k = λx31 + µx21 x2 + (1/ν)x32 and l = νx1 . We calculate I(k, l) = −µν/4, J(k, l) = −λν/16, and R(k, l) = −ν2 . This determines the numbers λ, µ, and ν up to a common sign. The substitution x1 − 1 → −x1 and x2 1−→ −x2 carries (k, l) into (− f, −p), and establishes our contention. Claim 1.3.2.14. An element (k, l) fails to be semistable, if and only if all the invariants I, J, Δ, and R vanish in (k, l). Proof. To see this, we first locate the nullforms with the Hilbert–Mumford criterion. Exercise 1.3.2.15. Demonstrate that a pair (k, l) is a nullform, if and only if l2 divides the cubic form k or if l = 0 and k has a quadratic factor. The conditions stated in the exercise are equivalent to the following three conditions: 1. k has a double factor. 2. l divides k. 3. k · l has a triple factor. The first condition corresponds to Δ(k) = 0; Condition 2. to R(k, l) = 0; and the third condition to I(k · l) = 0 = J(k · l). The latter results from the fact that a binary quartic form is a nullform, if and only if it has a triple zero. This verifies the claim. "
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41
Now, we establish the independence of I, Δ, and R. By definition, this means that the map
'[x1 , x2, x3 ]
−→
'[I, Δ, R]
x1 , x2 , x3
1−→
I, Δ, R
from the polynomial algebra to the subring generated by I, Δ, and R is an isomorphism. Thus, we have to verify that it is injective. Suppose there were polynomials h1 , . . . , hn with hn % 0 and Rn h1 (I, Δ) + · · · + hn (I, Δ) = 0. For pairs of the form Pab := (x31 + ax1 x22 , bx1 ), we compute R (Pab ) = 0, Δ (Pab ) = 4a3 and I (Pab ) = a2 b2 /12. The above dependence relation implies hn (4a3 , a2 b2 /12) = 0 for all a and b. Therefore, hn = 0, contradicting the assumption hn % 0. Thus, we have shown that I, Δ, and R are algebraically independent. The discriminant of a binary quartic form is given by the formula D = I 3 − 27J 2 . Since D(k · l) vanishes, if and only if l|k or Δ(k) = 0, the invariant D lies in the ideal generated by R · Δ. In other words, there is a relation of the form I 3 − 27J 2 = (c1 I + c2 Δ + c3 R)RΔ, c1 , c2 , c3 ∈ '. We determine the coefficients with the pairs Qab := (x31 +ax1 x22 , bx2 ) and find Δ (Qab ) = 4a3 , R (Qab ) = b3 , I (Qab ) = −(ab2/4), and J (Qab ) = 0. Hence, c1 = c2 = 0 and c3 = −(1/256). This checks Relation 1.5. " Ternary Cubic Forms In this section, we will have a look at the possible degenerations among orbits of ternary cubic forms. Normal forms for ternary cubics are derived, e.g., in [123], p. 42ff. Theorem 1.3.2.16. Any ternary cubic form k % 0 may be transformed by an appropriate matrix in SL3 (') into one belonging to the following list. Shape of the curve { k = 0 } Non-singular cubic Cubic with ordinary double point Cubic with cusp Smooth conic with transverse line Smooth conic with tangent line Three independent lines Three lines from a pencil Double line and simple line Triple line
Normal form λ · (x31 + x32 + x33 ) − 3t · x1 x2 x3 , λ ∈ '. , t ∈ ', (t/λ)3 % 1 λ · (x31 + x21 x3 − x22 x3 ), λ ∈ '. x31 − x22 x3 λ · x1 (x21 − x2 x3 ), λ ∈ '. x2 (x21 − x2 x3 ) λ · x 1 x 2 x 3 , λ ∈ '. x1 x2 (x1 + x2 ) x21 x2 x31
Here, the stable forms are those defining a smooth cubic, the semistable forms are those defining a smooth cubic, a nodal cubic, a conic with a transverse line, or three independent lines. The remaining forms are nullforms. (The reader may verify this with the Hilbert–Mumford criterion as an exercise.)
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We refer the reader to the book [210] and [170], Section 0.14, for the determination of the ring of invariants of ternary cubic forms. Let us turn to degenerations among the orbits, i.e., we would like to determine which forms may lie in the closure of the orbit of a given form. Let k be a ternary cubic form. If the curve { k = 0 } is smooth, then k is stable, and its orbit is closed and eight dimensional. The following claim describes the possible degenerations among orbits of semistable forms. Claim 1.3.2.17. i) Let k be a ternary cubic form which defines a nodal cubic. Then, the closure of the orbit of k contains a form which describes the union of a conic and a transverse line. ii) Let k be a ternary cubic form which describes the union of a conic and a transverse line. Then, the orbit closure SL3 (') · k contains a form defining the union of three independent lines. Proof. i) Without loss of generality, we may write k = λ · (x31 + x21 x3 − x22 x3 ), λ ∈ '. . √3 √3 √3 We carry out the substitution x1 − 1 → 2x1 , x2 − 1 → −zx 2 − 2x1 , x3 1−→ −( 2/2z)x3 . It √ 3 transforms k into the form kz = 2λ· x1 (x21 − x2 x3 )−z 22 x21 x2 . We see 2λ· x1 (x21 − x2 x3 ) ∈ SL3 (') · k. ii) Suppose kB = 2λ · x1 (x21 − x2 x3 ). Replace x1 by −zx1 and x2 by −(1/z)x2. Then, B k goes to kzB := −2λ · z3 x31 + 2λ · x1 x2 x3 . " We write O(k) for the orbit of k. We depict the possible degenerations as follows: ^ 4 O λ · (x31 + x21 x3 − x22 x3 )
4 ^ 7 O 2λ · x1 x2 x3 .
^ 4 7 O 2λ · x1 (x2 − x2 x3 ) 1
We infer that the orbit of a form λ · x1 x2 x3 , λ ∈ '. , is closed: The orbit O(λ · x1 x2 x3 ) is always six dimensional and the above degenerations show that six is the minimal dimension the orbit of a semistable ternary cubic form can have. A form in O(λ · x1 x2 x3 )\O(λ· x1 x2 x3 ) would be semistable and have an orbit of dimension strictly less than six, impossible by the above. Claim 1.3.2.18. Among orbits of ternary cubic nullforms, there are the following possible degenerations: ^ 4 O x31 − x22 x3
^ 4 7 O x2 (x2 − x2 x3 ) 1 7 O(x2 x2 ) 1
^ 4 7 O x1 x2 (x1 + x2 ) 7 O(x3 ) 1
7
7 0.
Proof. Look at kz := x2 (x21 − x2 x3 )+zx31 , z ∈ '. . One immediately checks that { kz = 0 } is an irreducible cubic curve with a cusp in [0 : 0 : 1]. By Theorem 1.3.2.16, the forms kz are all contained in O(x31 − x22 x3 ), so that x2 (x21 − x2 x3 ) ∈ O(x31 − x22 x3 ). Next, we define kzB := x1 (x1 (x2 + zx3 ) + x22 ), z ∈ '. . The zero set of kzB is the union of a smooth conic with a tangent line. The same argument as above proves that x1 x2 (x1 + x2 ) ∈ O(x2 (x21 − x2 x3 )). The remaining degenerations are evident. "
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43
Further Results As soon as the number of variables or the degree of the forms increases, the explicit computations become much harder. Let us indicate two more cases where the above considerations have been carried out. These are the cases of ternary quartic forms [35] and quaternary cubic forms. A summary of classical results in the latter case is contained in [182]. In the paper [42], Brundu and Logar present a new approach to the classification of quaternary cubic forms based on computer algebra systems and find new results. In general, the use of modern computers allows to implement efficient algorithms for determining rings of invariants and obtaining more examples. We refer to the book [50] for the state of the art account.
1.3.3 The Invariant Theory of Matrices
In this section, we consider the action of GLn (') on the vector space Mn (')$ s by simultaneous conjugation. We will summarize some important results without proof. At the end, we present a more recent generalization, namely representations of quivers. These will show up again in Section 1.5.1 and Chapter 2, Section 2.5.6, p. 75ff. The Jordan Normal Form As a warm up, we discuss the familiar action of GLn (') on Mn (') by conjugation, i.e., g · m := g · m · g−1 , g ∈ GLn ('), m ∈ Mn ('). The equivalence relation induced on Mn (') by this action is similarity of matrices. By Linear Algebra, we know that any matrix m is similar to a matrix of the shape λ1
1 .. .
.. ..
. .
1 λ1
..
.
λk
1 .. .
.. ..
. .
. 1 λk
(1.6)
Exercise 1.3.3.1. Write the matrix from (1.6) as mB := m s + mn where m s is the matrix with the same diagonal entries as mB and zeroes elsewhere. Show that mB , and therefore m, is similar to the matrix m s + t · mn for every t ∈ '. . Conclude that m s lies in the closure of the GLn (')-orbit of m (which is the same as the GLn (')-orbit of mB ).
Let Dn ⊂ Mn (') be the subspace of diagonal matrices. The above exercise implies that the morphism ϕ: D '→ Mn (') −→ Mn (')//GLn (')
S 1.3: C I T 44
S 1.3: C I T 44
is surjective. The symmetric group in S n acts on Dn by permutation of the diagonal entries and that two matrices in Dn are similar if and only if they belong to the same S n -orbit. Thus, the morphism ϕ factorizes over a surjective morphism ϕ: Dn //S n −→ Mn (')//GLn ('). The coordinate ring of Dn //S n is, of course, well-known. In fact, using the coordinate functions xi : Dn −→ ', diag(λ1 , . . . , λn ) 1−→ λi , i = 1, . . . , n, and the elementary symmetric functions σ1 (λ1 , . . . , λn ) = λ1 + · · · + λn ,. . . , σn (λ1 , . . . , λn ) = λ1 · . . . · λn , we have
'[Dn ] = '[x1 , . . . , xn]
and
'[Dn //S n ] = '[Dn ]S
n
= '[σ1 , . . . , σn ].
This theorem may be found in most Algebra textbooks, e.g., [216], §33. Instead of using the elementary symmetric functions σ1 , . . . , σn , one may equally well work with the symmetric Newton functions s1 , . . . , sn , defined as si (λ1 , . . . , λn ) := λi1 + · · · + λin , i = 1, . . . , n. One has the following result: Theorem 1.3.3.2.
'[s1 , . . . , sn ] = '[σ1 , . . . , σn ] = '[x1 , . . . , xn ]S
n
.
Proof. [210], Proposition 1.1.2.
"
The functions σ1 ,. . . ,σn on Dn //S n extend to functions on Mn (')// GLn ('): Up to a sign, they are the coefficients of the characteristic polynomial of an (n x n)-matrix. In particular, ϕ is a closed embedding. Since ϕ is also surjective, it is an isomorphism. Using Theorem 1.3.3.2, we may use the symmetric Newton functions as coordinate functions on Dn //S n . Writing them as functions on Mn // GLn ('), we conclude: Corollary 1.3.3.3. Let xi j , i, j = 1, . . . , n, be the coordinate functions on Mn (') and 2 := (xi j )i, j . Then,
'[Mn (')]GL (
n .)
= '[xi j , i, j = 1, . . . , n]GLn (.) = '[Trace(2), . . . , Trace(2n )].
Remark 1.3.3.4. The above theorem appears already—with a different proof—in the book [93] (the Russian original dates from 1948). The isomorphism Dn //S n ! Mn (')//GLn (') is a special case of a result of Chevalley on reductive groups. We will encounter it in Exercise 1.5.1.39. Tuples of Matrices In the next step, we study the action of GLn (') on Mn (')$ s : g · (m1 , . . . , m s ) := (g · m1 · g−1 , . . . , g · m s · g−1 ),
g ∈ GLn ('), (m1 , . . . , m s ) ∈ Mn (')$ s .
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45
Theorem 1.3.3.5. Let xijk , i = 1, . . . , s, j, k = 1, . . . , n, be the coordinate functions on the affine space Mn (')$ s , and set 2i := (xijk ) j,k , i = 1, . . . , s. Then, the invariant ring '[Mn (')$s ]GLn (.) is generated by the invariants Trace(2i1 · · · · · 2il ), such that l ≤ n2 + 1. Remark 1.3.3.6. The study of the above action, its invariant ring, and related rings has a long history. We refer the reader to the article [65] for a discussion. A common reference for this result is Procesi’s article [171]. The theorem is, however, slightly older: 1) The fact that the invariants Trace(2i1 · · · · · 2il ), l ∈ %, generate the invariant ring was known much earlier (see Sibirski˘ı’s work [201] and [202]; Formanek refers to a paper by Kirillov from 1967) and 2) the bound on the number l was obtained independently by Razmyslov and published in [177] two years before Procesi’s article. In order to obtain the bound on l, Procesi and Razmyslov study the relations among the above invariants and prove that they follow from a “multilinear variant” of the theorem of Cayley–Hamilton. For s = 1, this variant of the Cayley–Hamilton theorem appears in the book of Gurevich ([93], §30) who attributes it to Lopshitz. Marinˇcuk and Sibirski˘ı computed the invariant rings for some special values of n and s (see [201], [202], [141], [142]). Exercise 1.3.3.7. i) Investigate the case n = s = 2. Prove that
'[M2 (')$2 ]GL (
2 .)
= '[T*1 , . . . , T*5 ].
For this, we declare the following invariants for a pair (m1 , m2 ) ∈ M2 (') - M2 ('): T*1 (m1 , m2 ) := Trace(m1 ), T*3 (m1 , m2 ) := Trace(m2 ),
and T*5 (m1 , m2 ) :=
T*2 (m1 , m2 ) := det(m1 ), T*4 (m1 , m2 ) := det(m2 ), Trace(m1 · m2 ).
Verify also that T*1 ,. . . ,T*5 are algebraically independent. (Hence, M2 (')$2 // GL2 (') ! +5 .) ii) Show that a tuple (m1 , . . . , m s ) ∈ Mn (')$ s fails to be semistable, if and only if m1 , . . . , m s may simultaneously be brought into upper trigonal form with zeroes on the diagonal. (Use Theorem 1.2.1.9.) Note: In general, one may study the action of a reductive linear algebraic group G on Lie(G)$ s that is induced by the adjoint representation. Richardson [178] determines, among other things, the unstable tuples (x1 , . . . , x s ) ∈ Lie(G)$ s . Plugging in G = GLn ('), this implies the above result. Quiver Representations — A not so Classical Example A quiver is a quadruple Q = (V, A, t, h) with the finite sets V, consisting of the vertices, and A, consisting of the arrows, and the maps t, h: A −→ V which associate to an arrow its tail and head. Such a combinatorial object can be easily depicted (see Figure 1.1).
46 S 1.3: C I T a
t(a)
4
•5 #
46 S 1.3: C I T
h(a)
7
•" 0 7 • 3! 3 33 33 34 •
•6
Figure 1.1: A quiver. A representation of Q (in the category of vector spaces) is a tuple (U; f ) = (Uv , v ∈ V; fa , a ∈ A) which consists of finite dimensional '-vector spaces Uv , v ∈ V, and linear maps fa : Ut(a) −→ Uh(a) . In other words, in a representation, we assign to each vertex a finite dimensional '-vector space and to each arrow a linear map from the vector space at the tail to the vector space at the head of the arrow. The tuple (dim(Uv ), v ∈ V) is called the dimension vector of the representation (U; f ). We call two representations (Uv1 , v ∈ V; fa1 , a ∈ A) and (Uv2 , v ∈ V; fa2 , a ∈ A) isomorphic, if there are isomorphisms Ψv : Uv1 −→ Uv2 with −1 , ∀a ∈ A. fa2 = Ψh(a) ◦ fa1 ◦ Ψt(a) Now, we fix a dimension vector n = (nv ∈ (≥0 , v ∈ V), and define Rep(Q, n)
:=
% Hom('
nt(a)
a∈A
GL(Q, n)
:=
2 v∈V
, 'nh(a) )
GLnv (').
In addition, there is the group action α: GL(Q, n) x Rep(Q, n) −→ Rep(Q, n) ^ 4 ^ 4 (gv , v ∈ V), ( fa , a ∈ A) 1−→ gh(a) ◦ fa ◦ g−1 t(a) , a ∈ A . It is clear that the set of orbits Rep(Q, n)/ GL(Q, n) equals the set of equivalence classes of representations of Q with dimension vector n. Example 1.3.3.8. The formalism we have just introduced may be viewed as a way to describe complicated classification problems in Linear Algebra. Some of them are familiar from the introductory courses. 7 • , the Gauß algorithm shows that Rep(Q, n)/ GL(Q, n) a) For the quiver • is parameterized by the rank of the linear map f : 'n1 −→ 'n2 . b) For the quiver • 0 , the Jordan normal form provides nice representatives for the GL(Q, n)-orbits in Rep(Q, n). 7 c) The normal form problem for the quiver • 7 • was solved by Kronecker (see [70]). d) In general, it is not possible to write down normal forms for all dimension vectors. A well-known “wild” problem is the normal form problem for the quiver %•0
,
S 1.3: C I T
47
i.e., the problem classifying pairs of non-commuting matrices up to simultaneous conjugation by invertible matrices. For this, one would need normal forms for the so-called indecomposable objects. More generally, we can look at the same problem for tuples of s matrices. For s = 1, the indecomposable objects are Jordan blocks, and one has a one-parameter family of Jordan blocks in any dimension, the parameter being the eigenvalue. For s > 1, one has families of non-equivalent indecomposable objects whose dimension grows so rapidly with the dimension of the underlying vector space that it is impossible to obtain lists of normal forms in all dimensions. Note that we have in all cases the affine variety Rep(Q, n)// GL(Q, n). In view of Example d), this seems quite important. The reader may consult the book [70] for these questions and their link to the representation theory of algebras. A path in the quiver Q is a sequence (a1 , . . . , a s ) of arrows, such that h(ai−1 ) = t(ai ), i = 1, . . . , s − 1. An oriented cycle is a path o = (a1 , . . . , a s ), such that h(a s) = t(a1 ). Exercise 1.3.3.9. Find the oriented cycles in the quiver displayed in Figure 1.1. Fix a dimension vector n. For any oriented cycle o = (a1 , . . . , a s ), we define to : Rep(Q, n) ( fa , a ∈ A)
−→ ' 1−→ Trace( fas ◦ · · · ◦ fa1 ).
The elements to are certainly invariant under the action of GL(Q, n). In fact, the following holds true: Theorem 1.3.3.10 (LeBruyn/Procesi). The invariant ring '[Rep(Q, n)// GL(Q, n)] is generated by the invariants to , o an oriented cycle in Q. Furthermore, it suffices to take into account oriented cycles o = (a1 , . . . , a s ) with &T Z2 s≤1+ nv . v∈V
Proof. [134].
"
Remark 1.3.3.11. i) Note that Theorem 1.3.3.5 is a special case of Theorem 1.3.3.10. ii) For quivers without oriented cycles, the GIT quotient Rep(Q, n)// GL(Q, n) is trivial, by the above theorem. We will explain below (p. 75ff) an application of the general Geometric Invariant Theory to the above quiver problems, due to King, which leads to highly non-trivial moduli spaces even when the quiver has no oriented cycles. A Quotient by a Special Orthogonal Group Let W be an r-dimensional complex vector space which is equipped with a non-degenerate symmetric bilinear form β: W x W −→ '. The group A > ! G := SO(W, β) := g ∈ SL(W) !! β(g · w, g · wB ) = β(w, wB ) ∀w, wB ∈ W (1.7) is isomorphic to SOr ('). The standard action of SL(W) on W induces an action of SL(W) on Hom(W, 'r ).
48 S 1.3: C I T
S 1.3: C I T 48
Remark 1.3.3.12. i) Note that Hom(W, 'r ) ! (W ∨ )$r as an SL(W)-module. ii) Since β is a non-degenerate and G-invariant bilinear form, the isomorphism W ! W ∨ induced by it is an isomorphism of G-modules.
We may restrict the SL(W)-action on Hom(W, 'r ) to G. The next result describes the quotient of Hom(W, 'r ) by this G-action. Lemma 1.3.3.13.
L F Hom(W, 'r )//G ! /r := (h, z) ∈ Symr x ' | z2 − det(h) = 0 .
Here, Symr is the vector space of symmetric (r x r)-matrices. Proof. We identify the space Symr with the subspace of Hom('r ∨ , 'r ), consisting of symmetric homomorphisms, that is, homomorphisms f : 'r ∨ −→ 'r with f ∨ = f , and ' with Hom(5r ('r ∨ ), 5r 'r ). Define the morphism * π: Hom(W, 'r ) ϕ
−→ Symr ^ 4 ϕ ϕ∨ 1−→ 'r∨ −→ W ∨ ! W −→ 'r .
Here, the isomorphism W ∨ −→ W comes from the non-degenerate symmetric bilinear π(ϕ), det(* π(ϕ))), is form β on W. Then, the morphism π: Hom(W, 'r ) −→ /r , ϕ 1−→ (* clearly G-invariant and, thus, descends to a morphism π: Hom(W, 'r )//G −→ /r . Note that π is birational as it is an isomorphism over the open subset Isom(W, 'r )/G of isomorphisms. We point out that /r is a normal variety (see [96], Exercise II.6.4). According to Proposition 1.2.1.12, it remains to show that π is also finite. For this, we use Hilbert’s results 1.2.1.11 which states that the coordinate ring of Hom(W, 'r )//G is integral over any subring generated by invariants which cut out the nullforms. This means that we have to check that ϕ is a nullform if and only if * π(ϕ) ≡ 0. The latter condition means that Im(ϕ∨ ) is an isotropic subspace for the induced non-degenerate symmetric bilinear form on W ∨ , i.e., Im(ϕ∨ ) ⊂ Im(ϕ∨ )⊥ . (Here, “⊥ ” refers to the orthogonal complement with respect to β.) It remains to verify that such a ϕ is indeed a nullform. To this end, we have to define an appropriate one parameter subgroup of G. First, we introduce a general construction. Let β0 be the standard bilinear form on 'r . We may fix an isomorphism (W, β) ! ('r , β0 ) in order to identify G with SOr ('). Let - ⊂ 'r be a non-trivial isotropic subspace of dimension, say, s and -⊥ ⊂ 'r its orthogonal complement with respect to β0 . This subspace has dimension r − s. Let w1 , . . . , w s be a basis for - and complete it to a basis w1 , . . . , wr−s of -⊥ . (Recall that - ⊆ -⊥ .) ⊥ Set - := F w s+1 , . . . , wr−s 4. Its orthogonal complement - intersects -⊥ precisely in -. ⊥ ⊥ Choose vectors wr−s+1 , . . . , wr ∈ - which complete w1 , . . . , w s to a basis of - . Then, w1 , . . . , wr is, indeed, a basis for 'r . Define γ := (γ1 , . . . , γr ) := ( −1, . . . , −1, 0, . . . , 0, 1, . . . , 1 ). D!!!!!!!WB!!!!!!!\ D!!WB!!\ D!!WB!!\ sx
(r−2s) x
sx
S 1.4: M’ GIT
49
Using our explicit description of the basis w1 , . . . , wr for 'r , we see that λ! : ' . z
−→ SLr (') r r Z &T T 1−→ ci · wi 1−→ zγi · ci · wi i=1
i=1
is a one parameter subgroup of SOr ('). Returning to our original setting, we associate to the isotropic subspace - of W that corresponds to Im(ϕ∨ ) under the isomorphism W ! W ∨ a one parameter subgroup λ! of G with the above construction. It is now easy to check that µ(ϕ, λ! ) = −1. This shows that ϕ is a nullform, by the easy direction of the Hilbert–Mumford criterion (compare Remark 1.2.1.10). " Exercise 1.3.3.14. i) In the above setting, we define O(W, β) by using the elements of GL(W) with the property stated in (1.7). This group is isomorphic to Or (') and acts on Hom(W, 'r ), too. Show that Hom(W, 'r )//O(W, β) ! Symr . ii) Formulate and prove the analogous results for an even dimensional vector space W which is equipped with a symplectic form.
1.4 Mumford’s Geometric Invariant Theory n this section, we present the central formalism of Geometric Invariant Theory. It is the heart of the first chapter. We first discuss various notions of quotients and relate them to each other. In the subsequent sections, we proceed to the construction of quotients of affine varieties and explain how a gluing construction yields more general results.
"
1.4.1 Good and Geometric Quotients Suppose V is an algebraic variety and G an affine algebraic group. Let σ: G x V −→ V be an action of G on V. The notion of an action is completely analogous to the one of Section 1.1.2 (except for the obsolete notion of “linearity”). A good quotient of Y with respect to the action σ is a pair (Y, π) which is composed of an algebraic variety Y and an affine, surjective morphism π: V −→ Y, such that the following requirements are met: 1. π is σ-invariant, i.e., π ◦ πV = π ◦ σ. Here, πV : G x V −→ V is the projection onto the second factor. 2. For every open subset U ⊆ Y, the homomorphism π# : OY (U) −→ OV (π−1 (U)) maps the algebra OY (U) isomorphically onto the algebra OV (π−1 (U))σ of σ-invariant functions on π−1 (U). 3. For every σ-invariant, closed subset Z ⊆ V, the set π(Z) is closed.
50
S 1.4: M’ GIT 4. If Z1 and Z2 are disjoint, σ-invariant, closed subsets of V, then their images π(Z1 ) and π(Z2 ) are disjoint as well.
A geometric quotient is a good quotient, such that every fiber π−1 (y), y ∈ Y, consists of a single orbit. Lemma 1.4.1.1. Let σ: G x V −→ V be an action of the affine algebraic group G on the variety V, and suppose (Y, π) is a good quotient. Then, the pair (Y, π) enjoys the following universal property: For any variety X and any σ-invariant morphism ϕ: V −→ X, there is a uniquely determined morphism ϕ: Y −→ X with ϕ = ϕ ◦ π. In particular, (Y, π) is unique up to canonical isomorphy. Proof. Note that we are dealing with varieties only, so that a morphism is determined by its underlying map of sets. Since π is surjective, ϕ: Y −→ X, π(v) 1−→ ϕ(v), v ∈ V, is the unique possibility of a set theoretical map with ϕ = ϕ ◦ π. This proves uniqueness. In order to show that ϕ is a well-defined map of sets, we must verify that, for two points v1 , v2 ∈ V, the condition ϕ(v1 ) % ϕ(v2 ) implies π(v1 ) % π(v2 ). Since the fibers ϕ−1 (ϕ(v1 )) and ϕ−1 (ϕ(v2 )) are disjoint, σ-invariant, closed subsets of V, Property 4. of a good quotient implies π(v1 ) % π(v2 ). Next, let U ⊆ X be an open subset. Then, ϕ−1 (U) = π−1 (ϕ−1 (U)) is an open subset in V. Property 3. of a good quotient grants that ϕ−1 (U) is open in Y. Therefore, the map ϕ is continuous in the Zariski-topology. By assumption, the morphism ϕ is σ-invariant. For any open subset U ⊆ X, the image of the homomorphism OX (U) −→ OV (ϕ−1 (U)) lies in OV (ϕ−1 (U))σ . Hence, we obtain the algebra homomorphism 4 ^ 2. OX (U) −→ ϕ. (π. OV )σ (U) = ϕ. (OY )(U) = OY ϕ−1 (U) . If we choose U to be affine, then the above homomorphism of algebras describes a morphism ψU : ϕ−1 (U) −→ U which satisfies ϕ|U = ψU ◦ π, by construction. The uniqueness statement of the beginning proves ψU = ϕ|U , so that ϕ is a regular map. " A pair (Y, π) with the universal property stated in Lemma 1.4.1.1 is called a categorical quotient. As in the case of a vector space, we will restrict to actions by reductive affine algebraic groups. First, we will verify the existence of the good quotient for an affine variety V. In the section on linearizations, we will discuss the options in the general situation. In that case, we may only expose σ-invariant, open subsets of V for which a good quotient exists. Exercise 1.4.1.2 (See [160]). Let V be an algebraic variety, G an algebraic group, and σ: G x V −→ V an action of G on V. i) Let (Y, π) be a good or geometric quotient for V by G. Show that, for every open subset U ⊆ Y, the pair (U, π|π−1 (U) ) is a good or geometric quotient, respectively, for π−1 (U) by G. ii) Let π: V −→ Y be a G-invariant morphism, such that there is an open covering (Ui )i∈I with the property that, for any i ∈ I, the pair (Ui , π|π−1 (Ui ) ) is a good or geometric quotient for π−1 (Ui ) by G. Prove that (Y, π) is a good or geometric quotient, respectively, for V by G.
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Exercise 1.4.1.3 (See [155], Chapter 0; [215], Chapter 3). Translate the results of this section into the language of schemes. In particular, prove Lemma 1.4.1.1 in that context. This exercise may be continued during the following sections.
1.4.2 Quotients of Affine Varieties The quotients of affine varieties are the building blocks for general GIT quotients. Existence of Good Quotients for Affine Varieties Let σ: G x V −→ V be an action of the reductive affine algebraic group G on the affine variety V. As in Section 1.1.3, we define a left action * of G on the coordinate algebra '[V] of V via *(g, f ): v 1−→ f (g−1 · v), ∀g ∈ G, f ∈ '[V], v ∈ V. As in the proof of Proposition 1.1.3.1, one checks that any element f ∈ '[V] is contained in a finite dimensional, G-invariant subvectorspace W ⊂ '[V] on which G acts by a rational representation. In general, we define a rational representation of the reductive group G on the finitely generated '-algebra A to be a map τ: G x A −→ A, satisfying the following axioms: 1. For any g ∈ G, the map τg : f 1−→ g · f := τ(g, f ) is a '-algebra automorphism of A; τe = idA . 2. For all g1 , g2 ∈ G, one has τg1 ◦ τg2 = τg1 g2 . 3. Every element f ∈ A is contained in a finite dimensional subvectorspace W ⊂ A on which G acts by a rational representation. Remark 1.4.2.1 (The Reynolds operator). If A is an infinite dimensional '-vector space and τ: G x A −→ A satisfies the above properties with “'-algebra automorphism” replaced by “'-linear automorphism”, we say that A is a locally finite G-module. Locally finite modules basically behave like finite dimensional modules. For example, it is rather obvious that any locally finite G-module is a direct sum of finite dimensional simple modules. In particular, the submodule AG of G-invariant elements in A is a direct summand of A. The resulting linear projection A −→ AG is called the Reynolds operator. It behaves functorial with respect to G-module homomorphisms. The Reynolds operator is only available in characteristic zero. Theorem 1.4.2.2 (Hilbert). Let G be a reductive algebraic group, A a finitely generated '-algebra, and τ: G x A −→ A a rational representation of G on A. Then, the algebra F L Aτ := f ∈ A | τg ( f ) = f ∀g ∈ G is finitely generated.
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The proof of this result will be given in the following section. Remark 1.4.2.3. i) This theorem goes back to Hilbert. It was formulated and proved in the above set-up by Nagata who also brought it to positive characteristic. For this reason, it is sometimes called the theorem of Hilbert–Nagata. ii) The converse to Hilbert’s theorem also holds true: First, Nagata ([157], see also [53] or [89], §8, p. 47) gave an example of a representation of a non-reductive affine algebraic group G on a finitely generated '-algebra A, such that the corresponding ring of invariants is not finitely generated. Later, Popov ([169], see also [89], Theorem 8.2) proved that any non-reductive affine algebraic group possesses a representation on a finitely generated '-algebra with non-finitely generated ring of invariants. iii) It is interesting to note that there is a classical finiteness result, due to Weitzenb¨ock ([218], see also [89], Theorem 10.1), for the non-reductive group 4a (see Example 1.1.4.5). It states that, for a representation *: 4a −→ GL(V) on a finite dimensional '-vector space V, the ring of invariants Sym. (V ∨ )&a is finitely generated. We may apply Theorem 1.4.2.2 to the initial situation of an action σ: G x V −→ V of the reductive group G on the affine variety V. Then, *: G x '[V] −→ '[V] is a rational representation of G on '[V]. The algebra '[V]* is finitely generated, and we set V//G := Specmax('[V]* ). The inclusion '[V]* ⊂ '[V] yields the morphism π: V −→ V//G. Theorem 1.4.2.4. The pair (V//G, π) is a good quotient for V with respect to the action σ. Exercise 1.4.2.5. Suppose that, in the situation of the theorem, W '→ V is a G-invariant closed subvariety. Use the Reynolds operator to conclude that the natural morphism W//G −→ V//G is a closed embedding. (This fails, in general, over fields of positive characteristic.) Remark 1.4.2.6. In general, Winkelmann [220] has shown that, given an action * of a not necessarily reductive affine algebraic group G on an affine variety V, the ring of invariants '[V]* , though not necessarily finitely generated, is the algebra of regular functions of a quasi-affine variety Y. However, the inclusion '[V]* ⊂ '[V] leads only to a G-invariant rational map π: V $ Y, so that Y need not be the categorical quotient. Proofs for Theorem 1.4.2.2 and 1.4.2.4 We start with the proof of Hilbert’s finiteness theorem, following [53], Chapter 3. Proof of Theorem 1.4.2.2 For f ∈ A, we let M f be the (finite dimensional) subvectorspace of A that is generated by the elements of the form g · f , g ∈ G, and N f ⊂ M f the subvectorspace that is generated by the elements of the form g1 · f − g2 · f , g1 , g2 ∈ G. Lemma 1.4.2.7. Given f ∈ A, there exists an element f B ∈ N f with f . := f + f B ∈ M f ∩ Aτ .
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Proof. If f ∈ N f , we may simply choose f B = − f . Otherwise, N f ∩ ' · f = {0}. For every g ∈ G, we have g· f = f +(g· f − f ). The first summand lies in ' · f and the second one in N f , i.e., M f = '· f -N f . In other words, N f is a one codimensional G-submodule of M f . Since we assume G to be reductive, there is a τ-invariant complement to N f . f - N f as G-module. Note that We, thus, find an element * f ∈ Aτ , such that M f = ' · * * the fact that f generates a submodule merely implies the existence of a character χ of G with g · * f = χ(g) · * f , for all g ∈ G. But, if χ were non-trivial, then we could find g0 ∈ G with * f = (1/(χ(g0) − 1)) · (g0 · * f−* f ), whence * f ∈ Nf . " ' ' Lemma 1.4.2.8. For elements f1 , . . . , fr ∈ Aτ , we find Aτ ∩ ri=1 A · fi = ri=1 Aτ · fi . Proof. We proceed by induction on r. For r = 0, there isn’t anything to prove. Suppose ' ' f = ri=1 hi fi ∈ Aτ ∩ ri=1 A · fi . According to Lemma 1.4.2.7, we choose hBr ∈ Nhr with ' hr +hBr ∈ Aτ . The fact f ∈ Aτ implies 0 = g1 f −g2 f = ri=1 (g1 hi −g2 hi ) fi , for all g1 , g2 ∈ 'r−1 G, so that (g1 hr − g2 hr ) fr = − i=1 (g1 hi − g2 hi ) fi . Since hBr ∈ Nhr , this observation ' * grants the existence of elements * hi ∈ A, i = 1, . . . , r − 1, with hBr fr = r−1 i=1 hi fi . By the ' * induction hypothesis and (hr + hBr ) ∈ Aτ , we infer that f − (hr + hBr ) fr = r−1 i=1 (hi − hi ) fi 'r−1 τ is contained in i=1 A · fi . This verifies the assertion. " Remark 1.4.2.9. In the proof of the lemma, we have only used the fact that in a finite dimensional G-module V a submodule W of codimension one possesses a direct complement. If one uses the rather elementary fact that A is as a G-module isomorphic to an infinite direct sum of finite dimensional irreducible G-modules, one can slightly simplify the argument (see [113], Section 14.3). Lemma 1.4.2.7 will be used again, anyway.
Now, we look at the case of the polynomial algebra A = '[x1 , . . . , xn ] and assume that the action of G preserves degrees, that is, for g ∈ G and a homogeneous polynomial f of degree d, the polynomial g· f is again homogeneous of the same degree d. Suppose f ∈ Aτ is an invariant polynomial. In the ring A, we decompose f = f1 + · · · + fk where fi is homogeneous of degree di , i = 1, . . . , k, and di % d j for i % j. For any g ∈ G, we have g · f − f = 0, and g · fi is homogeneous of degree di . Therefore, we must have g · fi − fi = 0, i = 1, . . . , k. For this reason, the algebra Aτ inherits the grading from A. Let Aτ+ be the irrelevant ideal. In A, the formula Aτ+ = (A · Aτ+ ) ∩ Aτ holds true. Write ' A · Aτ+ = F f1 , . . . , fr 4, fi ∈ Aτ+ . By Lemma 1.4.2.8, we have Aτ+ = ri=1 Aτ · fi . The next contention proves that Aτ is finitely generated.
!
Claim 1.4.2.10. Let B = i≥0 Bi be a graded '-algebra with B0 = '. If the irrelevant ideal B+ is generated by the elements b1 , . . . , br , then B = '[b1 , . . . , br ]. In particular, B is a finitely generated '-algebra. Proof. We may choose the bi to be homogeneous of degree at least 1. Let b ∈ B be homogeneous. We carry out an induction on deg(b). If deg(b) = 0, so that b ∈ ', ' the assertion is evident. Now, let deg(b) = d > 0 and write b = ri=1 hi bi . Here, deg(hi ) = deg(b) − deg(bi ) < d, so that hi ∈ '[b1 , . . . , br ], i = 1, . . . , r, by the induction hypothesis. "
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Finally, we treat the general case. Write A = '[a1 , . . . , ar ], such that the generators ai form the basis of a G-invariant subvectorspace (compare the proof of Theo' rem 1.1.3.3). For g ∈ G, write g·ai = rj=1 τi j (g)·a j , i = 1, . . . , r. Next, let '[x1 , . . . , xr ] be a polynomial ring and set, for g ∈ G, B
τ (g, xi ) =
r T
τi j (g) · x j .
j=1
This defines a degree preserving action τB of G on the ring '[x1 , . . . , xr ], and the projection p: '[x1 , . . . , xr ] −→ A, xi 1−→ ai , i = 1, . . . , r, is G-equivariant. Hilbert’s theorem is now an immediate consequence of ^ 4 B p '[x1 , . . . , xr ]τ = Aτ . Here, the inclusion ⊆ is obvious. For the converse inclusion, suppose f = p( * f) ∈ B f − f. ∈ Aτ . By Lemma 1.4.2.7, there exists an element f . ∈ '[x1 , . . . , xr ]τ with * ' * * g1 ,g2 ∈G ' · (g1 f − g2 f ). Since p is G-equivariant, we derive T ^ 4 ' · g1 p( *f ) − g2 p( *f ) = {0}. p( * f − f .) ∈ g1 ,g2 ∈G
Therefore, f = p( f . ) ∈ p('[x1 , . . . , xr ]τ ). B
"
Proof of Theorem 1.4.2.4 The morphism π is affine and G-invariant. In order to see that it is surjective, let w ∈ V//G and mw = F f1 , . . . , fr 4, f1 , . . . , fr ∈ '[V]* , be its maximal ideal. By ' Lemma 1.4.2.8, the ideal I := ri=1 '[V] · fi ⊂ '[V] is a proper ideal. If n ⊂ '[V] is a maximal ideal that contains I and if v ∈ V is the point defined by this maximal ideal, then n ∩ '[V]* = mw and π(v) = w. Next, we establish Property 2. It suffices to check the property for open subsets of the form (V//G) f , f ∈ '[V]* . In this case, the assertion amounts to the equality ('[V]* ) f = ('[V] f )* . Since f is G-invariant, this equality is clear. For Property 3. and 4., we will make use of the following result: Claim 1.4.2.11. Let Z1 and Z2 be disjoint, closed, G-invariant subsets of V. Then, there exists a function f ∈ '[V]* , such that f|Z1 ≡ 0 and f|Z2 ≡ 1. Let us, for the moment, believe in the truth of this claim. For Z1 , Z2 , and f as above, we view f as a function on V//G. Then, the claim tells us f|π(Z1 ) ≡ 0 and f|π(Z2 ) ≡ 1. In particular, π(Z1 ) ∩ π(Z2 ) = ∅. Finally, let Z be a G-invariant, closed subset of V. For any point w ∈ V//G with w # π(Z), the subsets Z and π−1 (w) are disjoint, closed, and G-invariant. Consequently, w # π(Z). This means that π(Z) is closed. Property 4. results from Property 3. and the above observation. " Proof of Claim 1.4.2.11. Let I1 and I2 be the ideals of Z1 and Z2 , respectively. Since Z1 and Z2 are disjoint, we have I1 + I2 = '[V]. Pick f1 ∈ I1 and f2 ∈ I2 with
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f1 + f2 = 1. For all g1 , g2 ∈ G, one has g1 · f1 − g2 · f1 = −(g1 · f2 − g2 · f2 ). We deduce Z &T T ' · (g1 · f1 − g2 · f1 ) ⊂ I2 ∩ ' · g · f1 ⊂ I2 ∩ I1 . g1 ,g2 ∈G
g∈G
' According to Lemma 1.4.2.7, we may choose a function f1. ∈ '[V]* ∩ ( g∈G ' · g · f1 ) ' with f1 − f1. ∈ g1 ,g2 ∈G ' · (g1 · f1 − g2 · f1 ) ⊂ I1 ∩ I2 (s.a.). The condition f1. ∈ I1 . implies f1|Z1 ≡ 0. Since f1 − f1. ∈ I2 , we see ( f1 − f1. )|Z2 ≡ 0. Finally, the property . f1|Z2 ≡ 1 results from f1 + f2 = 1, and we derive f1|Z ≡ 1. " 2 Exercise 1.4.2.12. i) Let V be an algebraic variety, G a reductive algebraic group, and σ: G x V −→ V an action of G on V. Suppose (Y, π) is a good or geometric quotient for V by G. Show that, for any variety Y, (Y x Z, (π x idZ )) is a good or geometric quotient, respectively, for V x Z by G. ii) (From [175] and [75].) Let V1 and V2 be algebraic varieties, G a reductive algebraic group, σ1,2 : G x V1,2 −→ V1,2 G-actions, and ϕ: V1 −→ V2 a G-equivariant and proper morphism. Assume that the good quotients (Y1,2 , π1,2 ) do exist. Use i) to conclude that the induced morphism π: Y1 −→ Y2 is again proper. (Recall from [96] that a morphism between varieties is proper, if it is universally closed.)
1.4.3 Linearizations Let V be an algebraic variety which is equipped with an action σ: G x V −→ V by the reductive affine algebraic group G. Our goal is to find σ-invariant, open subsets V0 and V1 , such that the good quotient V0 //G and the geometric quotient V1 //G do exist. These subsets should be as large as possible. To this end, one first looks for σ-invariant, open, and affine subsets Ui ⊆ V and forms the quotients Ui //G. In Section 1.4.2, we have demonstrated that these quotients exist. In the next step, we may try to glue the parts Ui //G to a “big” quotient. Most naively, one could define V1 as the union of all σ-invariant, open, affine subsets which possess a geometric quotient. Unfortunately, the resulting quotient will not be separated, in general, i.e., it will not be a variety (see Example 1.4.3.1). Thus, we will have to be more careful in our choice of the open, affine, invariant subsets. For this purpose, we will use the concept of a linearization. Example 1.4.3.1. The automorphism group of the projective line !1 is the affine algebraic group PGL2 ('). Choose N ∈ %, N ≥ 3, and look at the action of PGL2 (') on !1 N that is defined by g · (x1, . . . , xN ) := (g · x1 , . . . , g · xN ). Set F L Ui jk := (x1 , . . . , xN ) ∈ !1 N | xi , x j , and xk are distinct .
Here, i, j, and k are natural numbers with 1 ≤ i < j < k ≤ N. The map PGL2 (') −→ U123 ⊂ !1 3 , g − 1 → (g · 0, g · 1, g · ∞), is an isomorphism of varieties ([30], Section 10.8). This enables us to construct the isomorphisms Φi jk : PGL 2 (') x !1 N−3 −→ Ui jk . For example, we use Φ123 (g, (x4, . . . , xN )) := g · (0, 1, ∞, x4, . . . , xN ). Let PGL2 (') act on PGL2 (') x !1 N−3 by left multiplication on the first factor. Then, the isomorphisms
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Φi jk are equivariant, and we infer that the geometric quotient Pi jk := Ui jk // PGL2 (') does exist and is isomorphic to !1 N−3 . Given two triples i, j, k and l, m, n, the subset Ui jk ∩ Ulmn is invariant under the PGL2 (')-action. The images of this set in Pi jk and Plmn are canonically isomorphic. Hence, we may paste the Pi jk together to form the pre-variety P. The pre-variety P enjoys all the properties of a geometric quotient, except for being a variety. To see that it is not separated, we consider the curves C1 := { [0, 1, t, ∞, 0, . . . , 0] ∈ P124 | t ∈ !1 } and C2 := { [0, 1/t, 1, ∞, 0, . . . , 0] ∈ P134 | t ∈ !1 }. For t ∈ '. , both curves yield the same curve C ⊂ P. The point [0, 1, ∞, ∞, 0, . . . , 0] ∈ C1 does not lie in C2 , so that the curve C admits more than one extension in P. The General Theory Let p: L −→ V be a line bundle (in the geometric sense). A linearization of the action σ in L is an action σ of G on the line bundle L, satisfying the following two axioms: 1. The projection p is G-equivariant, i.e., p ◦ σ = σ ◦ p. 2. For g ∈ G and v ∈ V, the induced map Lv −→ Lg·v , w 1−→ g · w, is linear. A given linearization σ in L induces further linearizations of the action σ, for example, σ#r in the tensor powers L#r , r ∈ %, and σ∨ in the dual line bundle L∨ . Suppose that s: V −→ L is a G-equivariant section of the line bundle L. Then, the set V s := { v ∈ V | s(v) % 0 } is a σ-invariant, open subset of V. We call a point v ∈ V σ-semistable, if there are a positive number r and a G-equivariant section s of L#r , such that V s is affine and contains v; we say that v is σ-polystable, if, in addition, the orbit G · v is closed in V s . Finally, a point v ∈ V is σ-stable, if there exist r ∈ (>0 and a G-equivariant section s ∈ H 0 (V, L#r ), such that V s is affine, v ∈ V s , dim(G · v) = dim(G), and all orbits of the action of G on V s are closed (in V s ). We say that the action of G on V s is closed. As usual, we will simply speak about stable points and so on. The set of σ-semistable points is denoted by Vσss and the set of σ-stable points by Vσs . These sets are open in V. For Vσss , this is clear, and for Vσs , it will result from Lemma 1.4.3.4. Remark 1.4.3.2. i) In Section 1.2.1, we have already defined stable and semistable points for the action of a reductive affine algebraic group G on a vector space V. If we look at the resulting action σ on !(V ∨ ) together with its induced linearization σ in O((V ∨ ) (1) (see below), then [v] ∈ !(V ∨ ) is σ-(semi)stable, if and only if v is (semi)stable in the sense defined before (see also Remark 1.2.1.6). ii) The variety Vσss is always quasi-projective. Indeed, the ampleness of L|Vσss is an immediate consequence of the following theorem: Theorem 1.4.3.3 ([92], II, 4.5.2 & 4.5.10). Let V be a pre-variety and L a line bundle on V. Then, L is ample, if and only if, for every point v ∈ V, there are r ∈ % and a section s ∈ H 0 (V, L#r ), such that s(v) % 0 and V s is affine. Here is another characterization of stable points: Lemma 1.4.3.4. Let v ∈ V be a point and s ∈ H 0 (V, L#r ) a G-invariant section with s(v) % 0, such that V s is affine, dim(G · v) = dim(G), and G · v is closed in V s . Then, v is stable.
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Remark 1.4.3.5. Stable points may now be described as polystable points with finite stabilizer. Proof. By general properties of the fiber dimension, the set F L Z := w ∈ V s | dim G · w < dim G is closed in V s . Since V s is affine and Z and G·v are disjoint, G-invariant, closed subsets, there exists a G-invariant function f ∈ '[V s ] with f (v) = 1 and f|Z ≡ 0. We may choose µ so large that sµ · f ∈ H 0 (V, L#µ·r ). Then, V sµ · f is an affine, G-invariant subset that contains v, and the induced G-action on V sµ · f is closed, because Z ∩ V sµ · f = ∅. " Let us discuss a more geometric approach to linearizations. Further details may be found in [155] and [215]. We first look at a projective space !(W), W being a finite dimensional '-vector space. Let G be an affine algebraic group and σ: G x !(W) −→ !(W) an action of G on !(W). Suppose σ is a linearization of this action in O((W) (1). This yields the linear action of G on the vector space W = H 0 (!(W), O((W) (1)) with g · s: !(W) −→ O((W) (1), v 1−→ g · s(g−1 · v). Finally, let τ: G x W ∨ −→ W ∨ be the dual action. One easily verifies that the action τ lifts the action σ, that is, for the canonical projection π: W ∨ \ {0} −→ !(W), the property π(τ(g, w)) = σ(g, π(w)) holds true for all g ∈ G and w ∈ W ∨ \ {0}. Conversely, such a lifting τ yields a linearization of the given action in the tautological line bundle O((W) (−1) and, therefore, also a linearization in O((W) (1). The two processes are inverse to each other. Exercise 1.4.3.6. Show that (semi)stable and polystable points may be characterized in the following way: A point v ∈ !(W) is semistable, if and only if 0 # G · w (closure in W ∨ ), for any lift w ∈ W ∨ \ {0} of v. A point v ∈ !(W) is polystable (stable), if and only if, for any lift w ∈ W ∨ \ {0} of v, the orbit G · w is closed in W ∨ (and its dimension equals dim(G)). (Thus, we recover the notions defined after Remark 1.2.1.8.) Now, let V be a quasi-projective variety and σ: G x V −→ V an action of the affine algebraic group G on V. Let L be an ample line bundle on V and σ: G x L −→ L a linearization of the action σ. For r ∈ % sufficiently large, there exists a finite dimensional subspace W ⊆ H 0 (V, L#r ), such that the rational map α: V $ !(W) is a locally closed embedding. As usual, W may be chosen as a G-invariant subspace. Again, we also define a linear action τ of G on W ∨ which yields the action τ of G on !(W). The morphism α is equivariant with respect to these actions, and the restriction G x L#r −→ L#r of the linearization in O((W) (r), coming from τ, gives the linearization σ#r . Theorem 1.4.3.7. If r is sufficiently large, there exists a finite dimensional subspace W ⊂ H 0 (X, L#r ), such that the induced embedding α: V −→ !(W) has the following features: 1. A point v ∈ V is σ-(poly)stable, if and only if α(v) is (poly)stable. 2. A point v ∈ V is σ-semistable, if and only if α(v) is semistable and G · α(v) ∩ !(W)ss ⊂ V.
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Proof (Cf. [155], Amplification 1.8, p. 36; [215], Proposition 3.37). Ad 1. We begin = by fixing sections si ∈ H 0 (V, L#r ), i = 1, . . . , t, such that Vσss = ti=1 V si . Let OV (V si ) = '[ f1i , . . . , fmi i ], i = 1, . . . , t. We may choose r and W in such a way that si ∈ W and si · fµi ∈ W, for i = 1, . . . , t and µ = 1, . . . , mi . To do so, we might have to replace the si by suitable powers. If v ∈ V si , then α(v) ∈ U si , so that α(v) is semistable. By assumption, the orbit of v is closed in V si . Observe that V si is mapped onto a closed subvariety of U si . This results from the assumption on the si · fµi which tells us that the homomorphism '[U si ] −→ '[V si ] is surjective. The orbit of α(v), thus, is closed in U si , i.e., α(v) is polystable. If v ∈ Vσs , that is, if v is stable, then we obviously have dim(G · α(v)) = dim(G). Since α(v) is polystable, it is also stable, by Lemma 1.4.3.4. Let us now establish the converse. Let v ∈ V be a point, such that α(v) is polystable. * := V ∩ U s , We select a G-invariant section s ∈ H 0 (!(W), O((W) (*)) with v ∈ U s . Set U * * and let U be the closure of U in U s . Then, the sets G · v and U \ U are G-invariant, disjoint closed subsets of U s , and there is an invariant function f ∈ '[U s ] with f (v) = 1 and f|U\U* ≡ 0. By the choice of f , * ∩ (U s ) f = U ∩ (U s ) f . V ∩ (U s ) f = U If µ ∈ % is large enough, then sµ · f =: sB ∈ H 0 (!(W), O((W) (µ · *)). Furthermore, U sB = U s ∩ U f and V sB = V ∩ U sB = U ∩ U sB . Hence, V sB is affine. By construction, v ∈ V sB , so that v ∈ Vσss . If α(v) is even stable, we may start with a section s, such that the orbits of the G-action on U s are closed and then proceed as before. For trivial reasons, we have dim(G · v) = dim(G), and the action on V sB is closed, because it is closed on U s (⊇ V sB ). Ad 2. The stated condition grants that G · α(v) ∩ U s and U \ U are disjoint, so that we may argue as for 1. Conversely, if v is σ-semistable, then α(v) is semistable as well. By 1., the unique closed orbit in G · v is also closed in !(W)ss . This immediately implies the claim. " We now present the main theorem on the existence of quotients. Theorem 1.4.3.8. Let G be a reductive affine algebraic group, and V, σ, and L as above. Suppose σ: G x L −→ L is a linearization of the action σ. i) The good quotient of Vσss with respect to the induced G-action does exist and is a quasi-projective variety. We denote it by (V//σG, π) . If V is projective and L is ample, then the quotient V//σG is projective, too. ii) In V//σG, there is an open subset U s with π−1 (U s ) = Vσs , such that (U s , π|Vσs ) is the geometric quotient of Vσs for the induced action by G. Proof. If r ∈ % is sufficiently large, then there are invariant global sections s1 , . . . , sn ∈ H 0 (V, L#r ), such that the Ui := V si , i = 1, . . . , n, are affine subsets which cover the variety Vσss . The action of G on V induces actions of G on the Ui . Since the Ui are affine varieties, the good quotients (Wi , πi ) of the Ui by these actions exist, according to Theorem 1.4.2.4. The functions si j := s j /si are G-invariant and regular on Ui , so that we may interpret them as elements of '[Wi ]. Set Wi j := { w ∈ Wi | si j (w) % 0 }. Then, −1 we obviously have π−1 i (Wi j ) = U i ∩U j = π j (W ji ). By Exercise 1.4.1.2, (Wi j , πi|Wi j ) and (W ji , π j|W ji ) are both good quotients for Ui ∩ U j with respect to the induced G-action.
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Thus, there is a uniquely determined isomorphism ψi j : Wi j −→ W ji with ψi j ◦ πi = π j . Since the ψi j are uniquely determined, they obey the cocycle rule ψi j ◦ ψ jk = ψik . Therefore, we may glue the Wi to a pre-variety W. By the compatibility conditions on the πi , there is also a morphism π: Vσss −→ W with π|Ui = πi . It remains to be shown that W is quasi-projective (and, therefore, in particular a variety). The Wi form an affine covering of W, and the si j ∈ OW (Wi ∩ W j ) satisfy the cocycle condition. Hence, they define a line bundle M over W, such that π. (M) = L#r . Moreover, it is clear that the si ∈ H 0 (W, M), i.e., there are sections sections si ∈ H 0 (V, L#r ) descend to sections * 0 . * si ∈ H (W, M) with π (* si ) = si , i = 1, . . . , n. Note that W*si = Wi , i = 1, . . . , n. By Theorem 1.4.3.3, the line bundle M is ample. Finally, (W, π) is a good quotient, by Exercise 1.4.1.2. If V is a projective variety and L is an ample line bundle, then we may equivariantly embed V into !(W), W := H 0 (V, L#r ). Let R be the homogeneous coordinate ring of V in !(W). The linearization induces a degree preserving representation * of G on R. The quotient V//σG is then the projective variety Proj(R* ). This verifies the first part of the theorem. The second part is proved along similar lines. Here, we choose the sections si ∈ H 0 (V, L#r ), i = 1, .., n, in such a manner that the sets Ui := V si are affine and such that all orbits of the induced G-action on Ui are closed. " Remark 1.4.3.9. The above construction provides the following objects: A variety V//σG, a morphism π: V −→ V//σG, and an ample line bundle M on V//σG. We first note that the line bundle depends only on the number r. In fact, let s1 , . . . , sn and sB1 , . . . , sBm be two sets of sections as in the above proof. We define W := F s1 , . . . , sn 4 and W B := F sB1 , . . . , sBm 4. It is clearly enough to look at the case W B ⊆ W. The evaluation map W * OV −→ L defines the rational map V $ V//σG −→ !(W), ι
and we find M = ι. (O((W) (1)). Now, we compose this morphism with the projection !(W) $ !(W B ). The resulting rational map is the one associated to the evaluation map W B * OV −→ L. Thus, it yields the same line bundle M on V//σG. Next, we claim that, for the line bundles M and M B that are constructed with respect to the numbers r and rB , respectively, there are positive numbers µ and µB , such that M #µ ! M B#µ . B
We may assume that rB = t · r, for some t > 0. Let M be constructed from the sections s1 , . . . , sn ∈ H 0 (V, L#r ). Then, the construction with respect to the sections st1 , . . . , stn ∈ B H 0 (V, L#r ) supplies the line bundle M #t . By what we have observed before, this line bundle does agree with M B . In general, let us call two line bundles L and LB on the variety V equivalent, if there B exist positive integers µ and µB with L#µ ! LB#µ . A polarization is then the equivalence class [L] of an ample line bundle L on V. To summarize, we stress that the constructions performed in the proof of Theorem 1.4.3.8 give a quasi-projective variety V//σG together with a polarization [M].
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Exercise 1.4.3.10. Let G be a reductive group acting on the varieties V and W, and suppose we are given a G-equivariant and affine morphism ϕ: V −→ W. If σ: G x L −→ L is a linearization of the G-action on W, then we may pull it back to get a linearization ϕ. (σ): G x ϕ. (L) −→ ϕ. (L). Show that ϕ−1 (Wσss ) ⊆ Vϕss. (σ) . In particular, the categorical quotient of V by G exists, if ϕ(V) ⊆ Wσss . Exercise 1.4.3.11. Let G be a reductive linear algebraic group which acts on the varieties X and Y, and let ϕ: X −→ Y be a G-equivariant affine morphism. Assume that the good quotient Y −→ Y//G exists. i) Show that the good quotient X −→ X//G also exists and comes with an affine morphism ϕ: X//G −→ Y//G. ii) If ϕ is finite, prove that ϕ is finite, too. (Recall Exercise 1.4.2.12, ii).) Example 1.4.3.12 (!n as GIT-quotient). In topology, one introduces !n as the quotient ('n+1 \ {0})/'. . In Algebraic Geometry, one may do so as well. For this, we look at the following linearizations of the given '. -action on 'n+1 \ {0} in the trivial line bundle: σd :
'. x('n+1 \ {0}) x ' (z, v, t) /
7 ('n+1 \ {0}) x ' 7 (z · v, zd · t), d ∈ (.
We have H 0 ('n+1 \ {0}, O.n+1 \{0} ) = '[x0 , . . . , xn ], and we immediately discover that the σd -invariant sections are precisely the homogeneous polynomials of degree d. For d < 0, there are no invariant sections, and, for d = 0, only the constant ones. In these cases, there are no semistable points. If d > 0, we may work with the sections si := xdi , i = 0, . . . , n. Going through the construction in the proof of Theorem 1.4.3.8 with these sections evidently produces !n as the GIT quotient, and the induced ample line bundle M agrees with O(n (d). Remark 1.4.3.13. More generally, one can construct the Graßmannian Gr(s, r) of sdimensional quotients of 'r as the GIT-quotient for the action of GLs (') on the vector space Hom('r , ' s ), 0 < s < r (see [153], Section 8.1). Example 1.4.3.14 (Weighted Projective Spaces). Let V = V1 - · · · - V s be the direct sum of finite dimensional complex vector spaces and (d1 , . . . , d s ) a tuple of positive integers. We look at the '. -action 4 ^ α: '. x V1 - · · · - V s −→ V1 - · · · - V s 4 ^ 4 ^ z, (v1 , . . . , v s ) 1−→ zd1 · v1 , . . . , zds · v s . Then, the quotient !(d) := (V \ {0})//'. exists as a normal projective variety. It is the weighted projective space for the weights d = (d1 , . . . , d1 , . . . , d s , . . . , d s ). (The number di is repeated ri := dim(Vi ) times.) The theory of weighted projective spaces is studied in detail in the papers [16] and [55]. We will give a brief sketch and present a construction which will be useful to us in Chapter 2 on various occasions. To this end, we look at the linearization 4 ^ 4 ^ σ: '. x (V \ {0}) x ' −→ (V \ {0}) x ' ^ 4 ^ 4 z, (v, t) 1−→ z · v, z · t .
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If vi = (vi1 , . . . , viri ) is a basis for Vi , i = 1, . . . , s, and if 4 ^ x11 , . . . , x1r1 , . . . , x1s , . . . , xrss is the basis which is dual to the resulting basis of V, then one checks 4 ^ !(d) ! Proj '[x11 , . . . , x1r1 , . . . , x1s , . . . , xrss ] . D!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!WB!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\ =:R
!
In this picture, the ring R is not graded in the usual way, but by assigning to the variable xiν the weight di , ν = 1, . . . , ri , i = 1, . . . , s. We write R = i≥0 Ri . For every positive integer d, we define the subring R(d) :=
%R
(id)
.
i≥0
By the Veronese embedding,
^ 4 Proj R(d) ! Proj(R).
For a suitable common multiple d of the integers di (one may, in fact, take the least common multiple in our setting), the ring R(d) is generated in degree one, i.e., by the elements in Rd . (The reader may refer to [154], III.8, for these observations.) Therefore, we have the surjective homomorphism Sym. (Rd ) −→ R(d) of graded algebras which gives rise to the closed embedding
!(d) '→ !(Rd ).
%
Let us consider the vector space
.d :=
t=(t1 ,...,t s ):t1 d1 +···+t s d s =d
Symt1 (V1∨ ) * · · · * Symts (V s∨ ).
Note that there is the obvious surjection .d −→ Rd , so that we altogether find the surjective homomorphism Sym. (.d ) −→ R(d) of graded algebras and with it the closed embedding ιd : !(d) '→ !(.d ). This embedding might also be described in a slightly more functorial manner. Indeed, over V, there are the tautological sections si : OV −→ Vi * OX which give rise to the homomorphisms τi : Vi∨ * OV −→ OV . Thus, we may form the homomorphism % τd :=
%
t=(t1 ,...,t s ):t1 d1 +···+t s d s =d
Symt1 (τ1 ) * · · · * Symts (τ s ): .d * OV −→ OV .
The restriction of % τd to V \ {0} provides a morphism V \ {0} −→ !(.d ) which factorizes over the closed embedding ιd . This construction will be used several times in the second chapter.
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Example 1.4.3.15 (Projective Hypersurfaces). In Section 1.3.1, we studied the representation vd : SLn (') −→ GL(Vd ), Vd := '[x1 , . . . , xn ]d . As we have emphasized before, this representation provides an action of SLn (') on !(Vd ∨ ) together with a natural linearization. Hence, GIT supplies the quotient !(Vd ∨ )// SLn ('). One can show that smooth hypersurfaces are stable. Furthermore, there is a homogeneous polynomial Δ in the coefficients of the polynomials in Vd , such that the hypersurface V( f ) is smooth, if and only if Δ( f ) % 0. Thus, the set { Δ % 0 }/ SLn (') carries the structure of a quasi-projective variety. It may be viewed as the moduli space of smooth hypersurfaces of degree d in !n−1 . GIT also provides the natural compactification !(Vd ∨ )// SLn ('). Be careful that the latter space parameterizes semistable hypersurfaces with respect to a more delicate equivalence relation which identifies hypersurfaces, if and only if their orbit closures intersect within the semistable locus. In Section 1.3.1, this equivalence relation and the geometry of semistable hypersurfaces were highlighted for plane cubic curves, i.e., “hypersurfaces” defined by ternary cubic forms. More information on the classification of hypersurfaces is contained in [155]. Since the GIT process we have described so far depends on the choice of a linearization, the following question naturally emerges: Given a variety V, an affine algebraic group G, an action σ: G x V −→ V of G on V, and a line bundle L on V, how many linearizations of σ in L are there? One possibility to modify a given linearization σ: G x L −→ L is to multiply it by a character of G: If χ: G −→ '. is a character of G, we declare the linearization χ · σ, by defining G x Lv −→ Lg·v , (g, w) 1−→ χ(g) · σ(g, w), for every point v ∈ V. If V is a projective variety, this is indeed the only freedom we have: Lemma 1.4.3.16. Assume, in the above setting, that V is a projective variety. Then, two linearizations σ1,2 : G x L −→ L of the G-action in V differ only by a character of G. Proof. From the two linearizations σ1 and σ2 , we construct the linearization σ1 * σ∨2 : G x V x ' −→ V x ' of the G-action in the trivial line bundle. This linearization is of the form (g, v, z) 1−→ (g · v, * σ(g, v) · z), σ *: G x V −→ '. being a nowhere vanishing function. Since V is projective, we have H 0 (G x V, OG x V ) = H 0 (G, OG ). Hence, * σ ∈ H 0 (G, OG. ). Since ∨ σ1 * σ2 is an action, * σ must be a character. " Remark 1.4.3.17. If V is not projective, then the above result remains true for connected reductive groups with trivial character group. The techniques are beyond the scope of our notes (see [155], Proposition 1.4). The proof of existence of linearizations requires deeper results on reductive groups and algebraic varieties which we do not wish to discuss. In the applications which we have in mind, there will always be some apparent linearizations. For the sake of completeness, we quote the following statement ([155], Corollary 1.6): Theorem 1.4.3.18. Let G be a reductive affine algebraic group, X a normal variety, σ: G x X −→ X an action of G on X, and L a line bundle on X. Then, there exist a number n ∈ (>0 and a linearization σ: G x L#n −→ L#n of the action σ in L#n .
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Exercise 1.4.3.19. Give an example of G, X, σ, and L as in the theorem, such that there is no linearization of σ in L. Polarized '. -Quotients
In this section, we will study an action λ: '. x !(W) −→ !(W) in more detail. In particular, we will clarify how the sets of (semi)stable and polystable points look like for different linearizations. The discussion is taken from the paper [164]. Suppose λ: '. x W ∨ −→ W ∨ is a lifting of the action λ. As explained after Lemma 1.4.3.4, λ yields a linearization of λ in O((W) (1), denoted again by λ. According to Example 1.1.2.3, the module W ∨ decomposes as W∨ =
%W . m
i=1
∨ i
In this decomposition, Wi∨ is the non-trivial eigenspace to the character χdi : '. −→ '. , z 1−→ zdi , and we assume d1 < · · · < dm . Let v ∈ !(W) and w ∈ W ∨ \ {0} a lift of v. We set L F λ (v) := min di | w has a non-trivial component in Wi∨ dmin L F λ (v) := max di | w has a non-trivial component in Wi∨ . dmax λ Theorem 1.4.3.20. i) The point v ∈ !(W) is λ-semistable, if and only if dmin (v) ≤ 0 ≤ λ dmax (v). λ λ (v) = 0 = dmax (v) ii) The point v ∈ !(W) is λ-polystable, if and only if either dmin λ λ or dmin (v) < 0 < dmax (v).
Proof. With respect to a suitable basis for W ∨ , consisting of eigenvectors, the lift w of v will have the coordinates (w1 , . . . , wn ), such that, for z ∈ '. , we find λ
λ
z · (w1 , . . . , wn ) = (0, . . . , 0, zdmin (v) · wi0 , . . . , zdmax (v) · wir , 0, . . . , 0). Using Exercise 1.4.3.6, the assertion is obvious.
"
Exercise 1.4.3.21. Which points are stable? Let us describe all possible linearizations of λ in ample line bundles. The given representation λ corresponds to a linearization of the '. -action in O((W) (1). The kth powers, k ≥ 1, of this linearization may also be described in terms of representations. We have the kth symmetric power λk : '. x Symk (W ∨ ) −→ Symk (W ∨ ) of λ. Using the identification of Symk (W ∨ ) with Symk (W)∨ (compare Section 1.3.1), λk provides an action of '. on !(Symk (W)). The Veronese embedding vk : !(W) −→ !(Symk (W)) is then '. -equivariant, so that λk leads to a linearization of λ in O((W) (k). So far, we have found a linearization of λ in each line bundle O((W) (k), k ∈ (>0 . According to Lemma 1.4.3.16, we obtain any other such linearization by multiplying one of the above list by a character of '. . Define the representation λkd : '. x Symk (W ∨ ) −→ Symk (W ∨ ), (z, w) 1−→ z−d · λk (z, w), k ∈ (>0 , d ∈ (. This supplies a linearization of λ
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in O((W) (k) which we also denote by λkd . All possible linearizations of λ in ample line bundles are, therefore, given by the members of the family λkd , k ∈ (>0 , d ∈ (. Any of these linearizations yields a polarized variety (Qkd , [Mdk ]) as the quotient. B B If d/k = d B /kB , then (Qkd , [Mdk ]) and (QkdB , [MdkB ]) are isomorphic as polarized varieties. t·k This results from the fact that λt·d —as a representation—is just the t-th symmetric power of λkd , t ≥ 1. Observe that, for any point v ∈ !(W), 4 λkd ^ λ dmin vk (v) = k · dmin (v) − d,
4 λkd ^ λ (v) − d. dmax vk (v) = k · dmax
This leads to the following result: λ Theorem 1.4.3.22. i) A point v ∈ !(W) is λkd -semistable, if and only if dmin (v) ≤ λ (d/k) ≤ dmax (v). λ λ (v) = (d/k) = dmax (v) ii) A point v ∈ !(W) is λkd -polystable, if and only if either dmin λ λ or dmin (v) < (d/k) < dmax (v). In particular, any point v ∈ !(W) is polystable with respect to the linearization λkd , for an appropriate choice of d ∈ ( and k ∈ (>0 .
To an integer i with 1 ≤ i ≤ 2m, we associate a subset Ii of following list: 3 \ [d1, dm], if i = 2m i+1 }, {d if i is odd Ii := 2 (d i , d i +1 ), if i % 2m is even. 2 2 Corollary 1.4.3.23. The equality !(W)ss = λk d
!(W)ssλ
3, according to the
holds, if and only if there is an
kB dB
index i ∈ { 1, . . . , 2m }, such that Ii contains both d/k and d B /kB . Moreover, if i is even and d, k with d/k ∈ Ii and di± , ki± with di− /ki− = di/2 and + + di /ki = di/2+1 are given, then
!(W)ssλ ⊆ !(W)ssλ k d
k− d−
and
!(W)ssλ ⊆ !(W)ssλ k d
k+ d+
.
All in all, there are 2m notions of semistability for the given '. -action λ. Denote the corresponding unpolarized quotients by Qi , i = 1, . . . , 2m. Note that Q2m = ∅. The final observation in Corollary 1.4.3.23 shows that there are maps
Q1
%% %% % % %$ %
Q2 1 11 11 11 12
Q3
Q2m−2' '' .. '' . '' .. . '' . *.. + Q2m−3 Q2m−1
(1.8)
between those quotients. Remark 1.4.3.24. One easily sees that, for i = 1, . . . , 2m, there is a k > 0, such that Qi = Q2j .
For an odd index i ∈ { 3, . . . , 2m − 3 }, we set Q±i := Qi±1 . The resulting morphisms π±i : Q±i −→ Qi are isomorphisms away from the closed subset !(Wi ) ⊂ Qi . The
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65
exceptional locus of π+i and π−i is given by F L !+i := v ∈ !(W) | lim z · v ∈ !(Wi ) //'. z→0 F L = v = [wi , . . . , wm ] ∈ !(Wi - · · · - Wm ) | wi % 0 //'. and
!−i
:= =
F F
L v ∈ !(W) | lim z · v ∈ !(Wi ) //'. z→∞
L v = [w1 , . . . , wi ] ∈ !(W1 - · · · - Wi ) | wi % 0 //'. ,
respectively. Note that !+i and !−i are weighted projective bundles over !(Wi ). We see that the quotients Q−i and Q+i are birationally equivalent and that the birational transformation Q−i $ Q+i is a weighted blow down followed by a weighted blow up. Exercise 1.4.3.25. Formulate the corresponding results for arbitrary projective varieties with a '. -action. Remark 1.4.3.26. The papers [22] and [88] supply an intrinsic description of the sets of semistable points of '. -actions in terms of the fix point locus. This description will be used in Section 1.6.1. In the above setting, the fix point locus of the '. -action on !(W) has the connected components Fi := !(Wi ), i = 1, . . . , m. We define, for i = 1, . . . , m, F L Xi+ := v ∈ !(W) | lim z · v ∈ Fi z→0
= Xi−
:= =
!F (Wi · · · Wm ) \ !(Wi+1L v ∈ !(W) | lim z · v ∈ Fi z→∞ !(W1 · · · Wi ) \ !(W1 -
-
-
-
-
-
· · · - Wm ),
· · · - Wi−1 ),
and, for i < j, Ci j
:= (Xi+ \ Fi ) ∩ (X −j \ F j ) ^ 4 = !(Wi - · · · - W j ) \ !(Wi+1 - · · · - W j ) ∪ !(Wi - · · · - W j−1 ) .
Corollary 1.4.3.27. i) If k · di − d % 0, for i = 1, . . . , m, we set i0 := max{ k · di − d < 0 }. It follows that ) Ci j . !(W)ssλk = d
1≤i≤i0 i0 +1≤ j≤m
ii) If k · di0 − d = 0, then
!(W)ssλ
k d
= Xi−0 ∪ Xi+0 ∪
)
Ci j .
1≤i≤i0 −1 i0 +1≤ j≤m
Example 1.4.3.28 (Induced polarizations). Let W be a finite dimensional complex vector space. We look at a representation λ of '. on the dual space, such that there
66
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is a decomposition W ∨ = W1∨ - W2∨ , such that Wi∨ is the eigenspace to the weight z 1−→ zdi , i = 1, 2, and d1 < d2 . We want to determine the polarized quotients (!(W)//λkd '. , [Mdk ]). For d/k = di , i = 1 or i = 2, this quotient equals (!(Wi ), [O((Wi ) (1)]). If d1 < d/k < d2 , then the unpolarized quotient is simply !(W1 ) x !(W2 ), and the projection map ^ 4 πkd : !(W) \ !(W1 ) ∪ !(W2 ) −→ !(W1 ) x !(W2 ) is the natural one. Claim 1.4.3.29. One has [Mdk ] = [O((W1 ) x ((W2 ) (kd2 −d, −kd1 +d)]. For any two positive integers m, n, there are numbers d ∈ ( and k ∈ (>0 , such that the induced polarization is given by [Mdk ] = [O((W1 ) x ((W2 ) (m, n)]. Proof. Let M = O((W1 ) x ((W2 ) (m, n) be a representative of the induced polarization. In . view of the description of πkd , we see that πkd (H 0 (M)) = Symm (W1 ) * Symn (W2 ). If m n Sym (W1 ) * Sym (W2 ) shows up as an eigenspace, it is with respect to the character χ−(md1 +nd2 )+((m+n)/k)d . Finally, we must have −(md1 + nd2 ) + ((m + n)/k)d = 0, i.e., m(kd1 − d) + n(kd2 − d) = 0. This gives the first assertion. In order to check the second contention, we will have to find, for given m and n, positive integers k and r as well as an integer d, such that the two equations kd2 − d −kd1 + d
= =
rm rn
are satisfied. This is an easy exercise.
"
1.5 Criteria for Stability and Semistability n concrete applications, it will be important to explicitly determine the open subsets of stable and semistable points. The definition of stability and semistability is not very helpful for this task. If V is a projective variety, then the so-called Hilbert–Mumford criterion works. It is indeed the central tool in GIT and one of the main reasons for its success in applications. Therefore, the first aim of this section is to outline the proof of this important result. Later, we will present some more specific methods for detecting the stable and semistable points. These are tailored to more special situations, such as actions coming from direct sums of representations or by products of reductive groups.
"
1.5.1 The Hilbert–Mumford Criterion We have already encountered the Hilbert–Mumford criterion as Theorem 1.2.1.9. Here, we will state an apparently more general version. For this, we let X be a projective variety, G a reductive affine algebraic group, σ: G x X −→ X an action of G on X, and σ: G x L −→ L a linearization of this action in the line bundle L. Let λ: '. −→ G be a one parameter subgroup of G. Since X is projective, we may form, for any point
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67
x ∈ X, the limit point x∞ := limz→∞ λ(z) · x. The point x∞ is clearly a fix point for the
'. -action coming from λ. The linearization σ provides a linear action of the group '.
on the one dimensional vector space Lv∞ . If l is a generator for L x∞ , then this action is of the form z · l = zγ · l, for a certain integer γ which does not depend on the choice of l. We now set µ(x, λ) := µσ (x, λ) := −γ. (1.9) Exercise 1.5.1.1. Let *: G −→ GL(V) be a representation of G. We study the resulting action on !(V) with its linearization in O((V) (1). Suppose λ: '. −→ G is a one param' ' eter subgroup. Choose a basis v1 , . . . , vn of V, such that *(λ(z))( ci ·vi ) = ni=1 zγi ·ci ·vi , for all z ∈ '. and all tuples (c1 , . . . , cn ) of complex numbers. For a non-trivial linear ' form l = ni=1 ci · v∨i , verify that ^ 4 ^ 4 F L µ* [l], λ := µ [l], λ = − min γi | l(vi ) % 0 . Derive the following rules for the µ-function: i) For any point x ∈ !(V), any g ∈ G, and any one parameter subgroup λ of G, one has µ(x, λ) = µ(g · x, g · λ · g−1 ). Here, (g · λ · g−1 )(z) := g · λ(z) · g−1 , for all z ∈ '. . ii) For any point x ∈ !(V) and any one parameter subgroup λ of G, we have µ(x, λ) = µ( lim λ(z) · x, λ). z→∞
Theorem 1.5.1.2. If, in addition to the above assumptions, L is an ample line bundle, then a point x ∈ X is (semi)stable, if and only if, for any non-trivial one parameter subgroup λ of G, the inequality µ(x, λ)(≥)0 is verified. Here, the symbol “(≥)” means that the weak inequality “≥” is requested for “semistable” and the strict inequality “>” for “stable”. Exercise 1.5.1.3. Reduce Theorem 1.5.1.2 to Theorem 1.2.1.9, using Exercise 1.4.3.6 and 1.5.1.1 and the fact that, for a representation *: G −→ GL(V), the stabilizer of a point v ∈ V with closed orbit in V is reductive.3 Theorem 1.5.1.2 follows from the result stated below. Theorem 1.5.1.4. Let G be a reductive affine algebraic group and *: G −→ GL(V) a representation of G on the finite dimensional vector space V. Assume v ∈ V and that W is a non-empty *-invariant closed subset of G · v \ G · v. Then, there exist a point w ∈ W and a one parameter subgroup λ: '. −→ G, such that w = limz→∞ λ(z) · v. Before we proceed to the proof of that result, let us give two applications. 3 This can be seen as follows: By [30], §6, the categorical quotient G/G always exists and is isomorphic v to the orbit G · v. By our assumptions, it is therefore affine. By a theorem of Matsushima, this implies that G v is reductive (see [146], Th´eor`eme 3; [19], Theorem 1; [139], Section 1.2).
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Exercise 1.5.1.5. Show that a point x is polystable, if and only if, for every non-trivial one parameter subgroup λ of G, the inequality µ(x, λ) ≥ 0 holds true and, additionally, for every one parameter subgroup λ with µ(x, λ) = 0, there is an element g ∈ G with lim λ(z) · x = g · x. z→∞
Exercise 1.5.1.6. Let G be a reductive group and *: G −→ GL(V) a representation. We now present a way to compactify the quotient V//*G. To this end, note that * - #: G −→ GL(V - ') gives an action on ! := !(V ∨ - ') together with a linearization σ in O( (1) (see the discussion following Lemma 1.4.3.4). i) Use the Hilbert–Mumford criterion to show that a point [v, ε] ∈ ! is semistable, if and only if either ε % 0 or ε = 0 and v ∈ V is semistable (as defined after Lemma 1.2.1.5; compare Exercise 1.4.3.6). ii) Let π: ! $ !//σ G be the quotient morphism. Define L F U := [v, 1] ∈ ! | v ∈ V ! V. Show that U is a saturated G-invariant open subset of the set of σ-semistable points, i.e., of the form π−1 (U) for some open subset U of !//σG. iii) Conclude U ! V//*G. (Therefore, the projective variety !//σG may be seen as a natural compactification of the affine variety V//*G.) iv) Can you prove (some of) these properties without reference to the Hilbert– Mumford criterion? We will explain the proof of Theorem 1.5.1.4 after Richardson which is given in [24], §4. This proof makes use of some facts on complex Lie groups. Recall that any non singular variety X may be equipped in natural way with the structure of a complex manifold. The corresponding topology on X is called the strong topology [154]. Now, there are two topologies on X at our disposal. For a subset Z ⊆ X, we denote by Z the s closure in the Zariski topology, and by Z the closure in the strong topology. If Z is a s constructible subset, then Z = Z ([154], p. 84). We won’t define what a constructible subset is, but we remind the reader that images of morphisms are constructible subsets, according to a result of Chevalley’s ([30], AG 1.3). If X is, in fact, an algebraic group, then X carries also the structure of a complex Lie group. The group law remains, of course, the same. In the following, we will use the notion “torus” only in the algebraic sense defined in 1.1.1.3, vi). Let G be an affine algebraic group. A maximal torus of G is a subgroup T which is a torus and is maximal with respect to inclusion among all subgroups of G which are tori. It is known ([30], Corollary 11.3) that any two maximal tori T and T B of G are conjugate, that is, there is an element g ∈ G with T B = g · T · g−1 . In particular, T and T B have the same dimension. Theorem 1.5.1.7 (Cartan Decomposition). Let G be a reductive affine algebraic group and T a maximal torus of G. Then, there is a compact real Lie subgroup H ⊂ G, such that G = H · T · H.
S 1.5: C (S)S Proof. [104], Theorem 1.1.
69 "
Example 1.5.1.8. For GLn ('), the group D, consisting of all diagonal matrices, is a maximal torus, and one has GLn (') = Un (') · D · Un (') ([123], Satz 4, p. 289). Proof of Theorem 1.5.1.4 In the first step, we verify the statement in the case that G is an algebraic torus. In Exercise 1.1.2.5, we have seen that the representation * can be diagonalized. For the following considerations, we choose a basis with respect to which the action is given by diagonal matrices. In the sequel, we will write D for the space of diagonal matrices within Mn (') and D for the group of diagonal matrices within GLn ('). Step 1. — To begin with, we point out that W must contain the closed orbit in G · v \ G · v. Without loss of generality, we may, therefore, assume that W agrees with the closed orbit. Let (v1 , . . . , vn ) be the coordinate presentation of v with respect to the chosen basis. After possibly renumbering, we may assume that vi % 0, i = 1, . . . , r, and vi = 0, i = r + 1, . . . , n. Then, we form the submodule Y := { (y1 , . . . , yn ) ∈ V | yr+1 = · · · = yn = 0 }. After replacing V by Y, we will have r = n, i.e., all coordinates of v are non-zero. The orbit map orb(D, v): D −→ V, diag(α1 , . . . , αn ) 1−→ (α1 v1 , . . . , αn vn ), is thus an isomorphism. If the orbit G · v fails to be closed, as we assume, then *(G) ⊂ D isn’t closed either. The character group X(D) is isomorphic to (n and is generated by the characters χi with χi (diag(α1 , . . . , αn )) := αi , i = 1, . . . , n. Let X. (D) be the group of one parameter subgroups of D (see Example 1.1.1.3) and F . , . 4: X. (D) x X(D) −→ ( the perfect pairing as defined in Example 1.1.1.3. We define the subgroup F L X B := χ ∈ X(D) | χ|*(G) ≡ 1 F L = χ | F * ◦ λ, χ 4 = 0 for all one parameter subgroups λ of G . Note that the homomorphism G −→ D gives rise to (and is in fact determined by) the homomorphism X(D) −→ X(G). The group X B is just the kernel of this homomorphism. It contains so to speak the equations for the subtorus *(G) ⊂ D (see [30], §8, for more details on tori and diagonalizable groups). The set of characters that extend to regular functions on D is given by the semigroup n AT
> ! γi · χi !! γi ≥ 0, i = 1, . . . , n .
i=1
We intersect this semigroup with X B and obtain the semigroup P :=
n AT
> ! γi · χi ∈ X B !! γi ≥ 0, i = 1, . . . , n .
i=1
Let R(G) be the closure of *(G) in D. The semigroup P provides us with regular functions on D which are constant on R(G) and do not vanish on R(G). In addition,
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we set J := { j | ∃ p = γ1 χ1 + · · · + γn χn ∈ P with γ j > 0 }. In a suitable numbering, we will have J = { k, . . . , n }, for some k ∈ { 1, . . . , n + 1 }. For every element g ∈ R(G) ' and every index i ∈ J, we then have χi (g) % 0. (Recall that ni=1 γi · χi stands for the function D −→ ', (d1 , . . . , dn ) 1−→ χ(d1 )γ1 · · · · · χ(dn )γn .) The fact that orb(D, v) is an isomorphism implies the following: Lemma 1.5.1.9. For a point vB = (vB1 , . . . , vBn ) in the closure of G · v, we have vBi % 0 for i ∈ J. This lemma gives us some control over G · v. In particular, if J = { 1, . . . , n }, then R(G) would be contained in D. Since the image of a group homomorphism is always closed ([30], 1.4.(a)), it follows that R(G) = *(G), contradicting our assumptions. Consequently, k > 1. Let X be the group of characters of G and X. the group of one parameter subgroups. As before, we have the bilinear map F . , . 4: X. x X −→ (. We also declare X% := X *# 3, X.,% := X. *# 3, and xi := F* ◦ . , χi 4: X.,% −→ 3. We may view xi as an ' element of X% , and we find X B *# 3 = { ni=1 γi · χi | γ1 · x1 + · · · + γn · xn = 0 }. Claim 1.5.1.10. Let U be a finite dimensional vector space over 3 and u1 , . . . , uh , uh+1 , . . . , um elements from U, such that the following conditions hold true: ' 1. If, for ri ∈ 3≥0 , i = 1, . . . , m, one has the equation m i=1 ri ui = 0, then ri = 0, for i = 1, . . . , h. ' 2. There exist elements si ∈ 3>0 , i = h + 1, . . . , m, such that m i=h+1 si ui = 0. Then, there exists an element l ∈ U ∨ with l(ui ) < 0, for i = 1, . . . , h, and l(ui ) = 0, for i = h + 1, . . . , m. By this claim, we find an element l ∈ X.,% , such that l(xi ) < 0, for i = 1, . . . , k − 1, and l(xi ) = 0, for i = k, . . . , n. If κ ∈ (>0 is sufficiently large, then λ := κ · l will be a one parameter subgroup with limz→∞ λ(z) · v = (0, . . . , 0, vk , . . . , vn ) =: v∞ . To conclude, we have to prove v∞ ∈ W, i.e., we must verify that the orbit G · v∞ is closed, If it were not closed, we could apply our considerations to v∞ instead of v and obtain a one parameter subgroup λB with limz→∞ λB (z) · v∞ = vB = (0, . . . , 0, vBk+1 , . . . , vBn ). For i = k, . . . , n, we then have either vBi = 0 or vBi = vi . Now, Lemma 1.5.1.9 implies vBi % 0, i = k, . . . , n. This means vB = v∞ , a contradiction. (The conclusion k > 1 implies that v∞ does not lie in the orbit of v, whence vB does not lie in the orbit of v∞ either.) It remains to prove the above claim. Set U/ := U *% 1, Z = U/F uh+1 , . . . , um 4, and Z/ := Z *% 1. Then, the convex hull C of the classes u1 , . . . , uh in Z/ does not contain the origin. In fact, if there are ri ∈ 1≥0 with r1 u1 + · · · + rh uh = 0, then there are also ri ∈ 3≥0 with this property. This signifies that there are th+1 , . . . , tm ∈ 3, such that r1 u1 + · · · + rh uh + th+1 uh+1 + · · · + tm um = 0. By Condition 2., we may choose the ti to be positive. Then, 1. implies that r1 = · · · = rh = 0. Since 0 # C, there is a hyperplane which is defined over 3 and passes through the origin and which does not intersect C. In other words, there is a linear map l: Z −→ 3, such that C ⊂ { u ∈ Z/ | (l *% id/ )(u) > 0 }. We finally define l as the composition of −l and the projection map U −→ Z. "
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Step 2. — By Step 1, it suffices to exhibit a torus T ⊂ G with W ∩ T · v % ∅. Let us assume that there were no such torus and select a maximal torus T ⊂ G. Let z be an arbitrary point, belonging to the G-orbit of v. By our assumption, W and T · z are disjoint, closed, T -invariant subsets. Indeed, remembering z = g · v, for some g ∈ G, T · z = T · g · v = g · (g−1 · T · g) · v, and W ∩ g · (g−1 · T · g) · v = g · (W ∩ (g−1 · T · g) · v) = ∅, because g−1 · T · g is a maximal torus, too. By Lemma 1.2.1.5, there exists a T -invariant function fz which vanishes on W and takes the constant value 1 on T · z. We set Uz := { y ∈ V | fz (y) % 0 }. In view of the Cartan decomposition, we may fix a compact Lie subgroup H ⊂ G with G = H · T · H. The orbit H · v is compact as well, so that we may find points z1 , . . . , zn with H · v ⊂ Uz1 ∪ · · · ∪ Uzn . The function f : V −→ 1, y 1−→ | fz1 (y)| + · · · + | fzn (y)|, is continuous in the strong topology of V. It, therefore, admits a positive minimum on the compact set H · v and, since it s s is T -invariant, also on (T · H) · v and (T · H) · v . The subsets W and (T · H) · v are disjoint, because f vanishes on W. Since W is invariant under the action of G, the s s subsets W and H · (T · H) · v are also disjoint. The set H · (T · H) · v is closed in the s strong topology, because H is compact. It contains G · v and is contained in G · v , i.e., we have s s (1.10) H · (T · H) · v = G · v = G · v. The last equality results from the fact that G · v, as the image of a morphism, is cons structible. Equation 1.10 leads to a contradiction as W ⊂ G · v and W ∩ H ·(T · H) · v = ∅. " Remark 1.5.1.11. The above proof is exposed in a slightly different fashion in [123], Sections 2.2 & 2.4. A different elementary proof, working only in the case that 0 is contained in G · v, is given in the paper [211]. Example 1.5.1.12 (One Parameter Subgroups of SLr (')). By Example 1.1.2.3, given a one parameter subgroup λ: '. −→ SLr (') of the special linear group, there are a basis w = (w1 , . . . , wr ) of 'r and a weight vector γ = (γ1 , . . . , γr ), such that ' • γ1 ≤ · · · ≤ γr and ri=1 γi = 0, ' ' • λ(z) · ci · wi = zγi · ci · wi , for all (c1 , . . . , cr ) ∈ 'r and all z ∈ '. . Conversely, the data w and γ with the above properties define, via the formula in the second item, a one parameter subgroup λ(w, γ) of SLr ('). ' Any weight vector γ = (γ1 , . . . , γr ) with γ1 ≤ · · · ≤ γr and ri=1 γi = 0 may be decomposed as r−1 T γi+1 − γi (i) γ= · γr (1.11) r i=1 with
4 ^ γr(i) := i − r, . . . , i − r , i, . . . , i , D!!!!!!!!!!!WB!!!!!!!!!!!\ D!WB!\ ix
i = 1, . . . , r − 1.
(r−i) x
Exercise 1.5.1.13. Expanding the construction which was used in the proof of Lemma 1.3.3.13, you may try to describe the one parameter subgroups of Or (') (which are the same as the one parameter subgroups of SOr ('), because the latter is the connected component of the identity of the former). See [82] for more details.
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Example 1.5.1.14 (Weights in Graßmannians). Suppose 0 < s < r, and let 4 := Gr(s, r) be the Graßmann variety of s-dimensional quotients of 'r [87]. The group SLr (') acts on 4 via _ 6 _ 6 g−1 k g · k: 'r −→ ' s := 'r −→ 'r −→ ' s . This action is transitive, i.e., there is a single orbit. Nevertheless, we will carry out the computations, because they are very useful for many applications, e.g., when looking at the SLr (')-action on products of Graßmannians [160]. In Chapter 2, similar computations are important for constructing moduli spaces (p. 133f). On 4, there exists the universal quotient k& : 'r * O& −→ Q& , i.e., for x ∈ 4, the quotient k&|{x} agrees with the quotient represented by x. There is the surjection s
∧ k& :
s 8
'r
*
O& −→ L& := det(Q& ).
5 This surjection yields the Pl¨ucker embedding ι: 4 '→ ! := !( s 'r ) with ι. (O( (1)) = L& and, thus, a linearization of the SLr (')-action. Let λ(w, γ) be a one parameter subgroup of SLr (') (see Example 1.5.1.12). For [k: 'r −→ ' s ], we obtain the filtration ^ 4 {0} ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vr = 'r with Vi := k Fw1 , . . . , wi 4 , i = 1, . . . , r. Exercise 1.5.1.15. i) Set i j := min{ i = 1, . . . , r | dim(Vi ) = j }, j = 1, . . . , s, and show s T ^ 4 µ [k], λ(w, γ) = − γi j . j=1
ii) Conclude r−1 4 T 4 ^ γi+1 − γi ^ · µ [k], λ(w, γr(i) ) . µ [k], λ(w, γ) = r i=1
iii) Prove that 4 ^ 4 ^ 4 ^ µ [k], λ(w, γr(i) ) = i · dim ker(k) − r · dim Fw1 , . . . , wi 4 ∩ ker(k) . This exercise shows that [k] is (semi)stable, if and only if dim(L ∩ ker(k)) dim(L) (≤) , dim(ker(k)) r
for every subspace {0} ! L ! 'r .
Of course, this condition is always violated for L = ker(k), so that there are no semistable points. Remark 1.5.1.16. In the last example, Property ii) in 1.5.1.15 plays a decisive rˆole. If this property fails, computations with the Hilbert–Mumford criterion may get very nasty. For this reason, we will present below some more specialized criteria for stability and semistability which may help in such difficult situations.
S 1.5: C (S)S Example 1.5.1.17. Consider the action of SL3 (') on [ f ] be the class of the matrix 0 1 0 f := 0 0 1 , 0 0 0
73
!(M3 (')) by conjugation.
Let
and λ and λB the one parameter subgroups that are given with respect to the standard basis by the weight vectors (−2, 1, 1) and (−1, −1, 2), respectively. Then, we compute µ([ f ], λ) = 0 = µ([ f ], λB ), but µ([ f ], λ((e1 , e2 , e3 ), (−3, 0, 3))) = −3. Example 1.5.1.18. In this important example, we will provide a uniform and general framework in which both Example 1.5.1.14 and Example 1.5.1.17 may be understood. We assume, for simplicity, G = SLr ('). The arguments may be extended without much ado to arbitrary reductive groups. Let T ⊂ SLr (') be the maximal torus, consisting of the diagonal matrices, such that the product of the diagonal entries equals 1. (Of course, any other maximal torus would do as well.) Let w = (e1 , . . . , er ) be the standard basis of 'r and set K := 1≥0 · γr(1) + · · · + 1≥0 · γr(r−1) .
For every one parameter subgroup λ of SLr ('), there exists an element g ∈ SLr ('), such that g · λ · g−1 = λ(w, γ) with γ ∈ K ∩ (n . Suppose we are given a representation *: SLr (') −→ GL(V). By Exercise 1.5.1.1, we see ^ 4 x ∈ !(V)ss ⇐⇒ µ g · x, λ(w, γ) ≥ 0 for all γ ∈ K ∩ (n and all g ∈ G. *
!
The restriction of the representation * to the torus T provides us with the decomposition V ∨ = χ∈X(T ) Vχ , and we define F L ST(*) := χ ∈ X(T ) | Vχ % {0} as the set of states of *. For χ ∈ ST(*), let πχ : V ∨ −→ Vχ be the projection map. Assume x ∈ !(V) and that v ∈ V ∨ is a lift of x. The set F L ST(*, x) := χ ∈ ST(*) | πχ (v) % 0 is called the set of states of the point x. For a subset I of ST(*) and χ ∈ I, we set F L K(I, χ) := K(*, I, χ) := γ ∈ K | Fλ(w, γ), χ4 ≤ Fλ(w, γ), χB 4 ∀χB ∈ I . Here, the product Fλ(w, γ), χ4 is, for γ ∈ K ∩ (n , defined as above and, for γ ∈ K, by linear extension. We find ) K= K(I, χ). (1.12) χ∈I
The cone K(I, χ) is the intersection of finitely many rational half spaces, i.e., it is a so-called rational polyhedral cone (see [62]), and (1.12) is a fan decomposition. A
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face of K(I, χ) is the intersection of K(I, χ) with a supporting hyperplane, that is, a hyperplane, such that the cone is contained in one of the closed half spaces associated to this hyperplane. One dimensional faces are referred to as edges. For any edge k, we may choose a primitive integral generator γ , i.e., γ ∈ k ∩ (n and (1/m) · γ # k k k K(I, χ) ∩ (n , for all m ∈ (>0 . Then, one has [62] T 1≥0 · γ . K(I, χ) = k
k
= We define Γ(I, χ) = { γ | k is an edge of K(I, χ) } and Γ(I) = χ∈I Γ(I, χ). k This decomposition into cones has the property, that, for x ∈ !(V), χ ∈ ST(*, x), ' and γ = lk · γk ∈ K(ST(*, x), χ) with lk ≥ 0, for all k, the formula T ^ 4 T ^ 4 µ x, λ(w, γ) = lk · µ x, λ(w, γk ) = − lk · Fλ(w, γk ), χ4 holds true. Therefore, we find the following version of the Hilbert–Mumford criterion: ^ 4 ^ 4 ⇐⇒ µ g · x, λ(w, γ) ≥ 0, for all γ ∈ Γ ST(*, g · x) , g ∈ G. x ∈ !(V)ss * The advantage in this formulation is that γ belongs to a finite set of weight vectors. = Set Γ(*) := I⊆ST(*) Γ(I). This is still a finite set of weight vectors. We conclude ^ 4 x ∈ !(V)ss ⇐⇒ µ g · x, λ(w, γ) ≥ 0, for all γ ∈ Γ(*) and all g ∈ G. * Here, the advantage is that Γ(*) depends only on * and not on the point x. For an arbitrary reductive group G, one would work with the closure of a Weyl chamber W(B, T ) of G instead of K [30]. Example 1.5.1.19. Let us look at the representation *: SL2 (') −→ GL(Sym2 ('3 )), and let w = (e1 , e2 , e3 ) be the standard basis of '3 . Then, (e1 · e1 , e1 · e2 , e1 · e3 , e2 · e2 , e2 · e3 , e3 · e3 ) is a basis for Sym2 ('3 ). Set z1 0 0 ! ! ! . 0 z 0 z , z ∈ ' T := . 2 ! 1 2 0 0 z−1 z−1 1
2
One checks ST(*) = { χ−2,−2 , χ−1,0 , χ0,−1 , χ0,2 , χ1,1 , χ2,0 } with χi1 ,i2 :
and
1
z1 0 0
T 0 z2 0
0 0 −1 z−1 1 z2
/
7
'.
7 zi1 · zi2 2 1
P λ(w, (γ1 , γ2 , −γ1 − γ2 )), χi1 ,i2 = i1 · γ1 + i2 · γ2 .
Either K(I, χ) is empty or the whole of K, except for the case χ2,0 # I, χ1,1 # I, and { χ0,−1 , χ0,2 } ⊂ I. In that case K(I, χ) = ∅, for χ # { χ0,−1 , χ0,2 }, and F L K(I, χ0,−1 ) = γ ∈ K | γ2 ≥ 0 = 1≥0 · (−1, 0, 1) + 1≥0 · (−2, 1, 1) L F K(I, χ0,2 ) = γ ∈ K | γ2 ≤ 0 = 1≥0 · (−1, 0, 1) + 1≥0 · (−1, −1, 2).
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Let [l] ∈ !(Sym2 ('3 )) be the class of a linear form l: Sym2 ('3 ) −→ '. The cone decomposition associated to ST(*, [l]) contains non-empty, proper subcones of K, if and only if l(e1 · e1 ) = 0 = l(e1 · e2 ) and l(e1 · e3 ) % 0 % l(e2 · e2 ). We will generalize this example in Chapter 2, Section 2.8.4. Quiver Moduli after King As another illustration how the Hilbert–Mumford criterion works and how different linearizations may influence the result, we will present here the important results from King’s paper [120]. We will, however, use a slightly different approach. First, we look at the action of GLn (') on Mn ('). In order to arrive at compact moduli spaces in the end, we look at the action of GLn (') on !(Mn (')∨ - ') (compare Exercise 1.5.1.6). There is the surjective homomorphism '. x SLn (') −→ GLn (') with finite kernel. For the task of constructing the good quotient, it does not matter whether we work with the group GLn (') or with the group '. x SLn ('). Speaking about the action of '. x SLn ('), we notice that '. acts trivially. Therefore, it suffices to study the SLn (')-action on !(Mn (')∨ - '). An easy application of the Hilbert– Mumford criterion, observing Exercise 1.3.3.7, yields the following result: Proposition 1.5.1.20. Let v := [m, ε] be a point in !(Mn (')∨ - '), λ: '. −→ SLn (') a one parameter subgroup, and W• (λ) : W0 := {0} ! W1 ! · · · ! W s ! W s+1 := 'n the associated flag. i) We have µ(v, λ) > 0, if and only if there is an index i ∈ { 1, . . . , s }, such that fm (Wi ) $ Wi , fm : 'n −→ 'n , x 1−→ m · x. ii) The inequality µ(v, λ) < 0 holds, if and only if ε = 0 and fm (Wi ) ⊆ Wi−1 , i = 1, . . . , s + 1. Now, we come to the setting of quivers. Let Q = (V, A, t, a) be a quiver, n = (nv , v ∈ V) a dimension vector, and Rep(Q, n) the associated representation space of Q. For any ordered pair (v, vB ) of vertices in Q, we let b(v, vB) be the number of arrows a ∈ A with t(a) = v and h(a) = vB and F L b := max b(v, vB) | v, vB ∈ V . Note that we may write
% %
Rep(Q, n) =
(v,vB )∈V x V
=
(v,vB )∈V x V
Hom('nv , 'nvB )$b(v,v ) B
^ 4 B Hom 'nv , ('nvB )$b(v,v ) .
For each pair (v, vB ) ∈ V x V, we have the embedding ('nvB )$b(v,v ) (x1 , . . . , xb(v,vB ) ) B
'→ ('nvB )$b 1−→ (x1 , . . . , xb(v,vB ) , 0, . . . , 0).
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All in all, we obtain the GL(Q, n) equivariant embedding ^ ^ 4 4 B Hom 'nv , ('nvB )$b(v,v ) '→ Hom 'nv , ('nvB )$b .
%
%
(v,vB )∈V x V
(v,vB )∈V x V
There is also the GL(Q, n)-equivariant embedding ^ 4 Hom 'nv , ('nvB )$b '→
%
%
(v,vB )∈V x V
^ 4 Hom ('nv )$b , ('nvB )$b
(v,vB )∈V x V ^ 'b fv,vB , (v, vB )
4 ∈ V xV . fv,vB , (v, v ) ∈ V x V 1−→ ' Setting n := b · v∈V nv , we may view GL(Q, n) as a subgroup of GLn (') and define SL(Q, n) := GL(Q, n) ∩ SLn ('). Remark 1.5.1.21. Clearly, ^ 4 Hom ('nv )$b , ('nvB )$b ^
4
B
%
(v,vB )∈V x V
is just Hom('n , 'n ). Thus, using the above constructions, we can associate to any representation ( fa , a ∈ A) ∈ Rep(Q, n) an endomorphism f : 'n −→ 'n . As above, we only have to look at the action of SL(Q, n). Observe that any one parameter subgroup λ: '. −→ SL(Q, n) may also be viewed as a one parameter subgroup of GL(Q, n) or SLn ('). As a one parameter subgroup of GL(Q, n), we write it as a tuple λ = (λv , v ∈ V), λv : '. −→ GLnv ('). As a one parameter subgroup of SLn ('), it yields a weighted flag (W• (λ), α• (λ)) inside 'n . Let γ1 < · · · < γ s+1 be the distinct 's weights in the weight vector i=1 αi · γn(di ) , di := dim(Wi ), i = 1, . . . , s. Then, using the formalism of Example 1.5.1.36, & Z$b L F Wi = Wvi , Wvi = x ∈ 'nv | λv (z)(x) = zγi · x ∀z ∈ '. , v ∈ V, i = 1, . . . , s,
% v∈V
and Wi =
% W = &% W i
j=1
j
v∈V
v,i
Z$b
,
Wv,i :=
% W , v ∈ V. i
j=1
j v
With this notation, we obtain from Proposition 1.5.1.20: Proposition 1.5.1.22. Let r := [ fa , a ∈ A, ε] be a point in !(Rep(Q, n)∨ - ') and λ: '. −→ SL(Q, n) a one parameter subgroup. i) We have µ(r, λ) > 0, if and only if there are an index i ∈ { 1, . . . , s } and an arrow a ∈ A, such that fa (Wt(a),i ) $ Wh(a),i . ii) The inequality µ(r, λ) < 0 holds, if and only if ε = 0 and fa (Wt(a),i ) ⊆ Wh(a),i−1 , i = 1, . . . , s + 1. Let f : 'n −→ 1.5.1.21) and
'n be the endomorphism associated to ( fa , a
∈ A) (see Remark
W• (λ) : W0 := {0} ! W1 ! · · · ! W s ! W s+1 := 'n the associated flag in 'n . The second condition in the above proposition is then equivalent to f (Wi ) ⊆ Wi−1 , i = 1, . . . , s + 1.
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Corollary 1.5.1.23. A point r := [ fa , a ∈ A, ε] ∈ !(Rep(Q, n)∨ - ') is an SL(Q, n)nullform, if and only if ε = 0 and the map f : 'n −→ 'n is nilpotent. Proof. The only thing which is not totally obvious from Proposition 1.5.1.22 is the assertion that r is unstable, if ε = 0 and f is nilpotent. Let m be the last natural number with f m & 0. Let v be a fixed vertex. For every path p of length m, starting in, say, v p and ending in v, there is a morphism f p : 'nv p −→ 'nv , and we define Wv as the subspace generated by the subspaces Im( f p ), p a path as above. We leave it as an exercise to the reader to construct a one parameter subgroup λ: '. −→ SL(Q, n), using data as below, such that the weighted flag in 'n is & Z 4$b ^ Wv {0} ! ! 'n , (1) .
% v∈V
By Proposition 1.5.1.22, ii), µ(r, λ) < 0, so that r is unstable.
"
A one parameter subgroup λ = (λv , v ∈ V): '. −→ SL(Q, n) may be specified by the following data: • a tuple (wv , v ∈ V) where wv = (wv,1 , . . . , wv,nv ) is a basis for 'nv , v ∈ V; • a tuple (γ , v ∈ V) where γ = (γv,1 , . . . , γv,nv ) is a vector of integers. v
v
These data will be subject to the conditions below: • for v ∈ V, γv,1 ≤ · · · ≤ γv,nv ; ' 'v • v∈V ni=1 γv,i = 0. We denote such a one parameter subgroup by λ(wv , γ , v ∈ V). Next, we want to modify v the given linearization of the SL(Q, n)-action on !(Rep(Q, n)∨ - ') in O(N), N = 0, by a character χ of SL(Q, n). Such a character is of the form χ(av , v ∈ V): SL(Q, n)
−→
(mv , v ∈ V)
1−→
'2.
det(mv )av ,
v∈V
for a suitable tuple (av , v ∈ V) of integers. If r ∈ !(Rep(Q, n)∨ - ') and λ: '. −→ SL(Q, n) is a one parameter subgroup, we write µ(r, λ) for the quantity defined with respect to the original linearization in O(1) and µχN (r, λ) for the quantity which is computed with respect to the linearization in O(N) which has been modified by the character χ. Then, for any r ∈ !(Rep(Q, n)∨ - ') and any one parameter subgroup λ: '. −→ SL(Q, n), the following formula holds true µχN (r, λ) = N · µ(r, λ) + F λ, χ 4.
(1.13)
Now, we may easily generalize the construction in Example 1.5.1.18. This gives us a certain finite set Γ(Q, n) of tuples (γ , v ∈ V) of integral weight vectors subject to the v above conditions, such that the function µ(., .) has to be evaluated only at one parameter subgroups of the form λ = λ(wv , γ , v ∈ V) with (γ , v ∈ V) ∈ Γ(Q, n). By (1.13), the v
v
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same is true for the function µχN (., .). Suppose that (γ , v ∈ V) is a tuple of integral v weight vectors as above, that (wv , v ∈ V) and (wBv , v ∈ V) are two tuples of bases, and that λ = λ(wv , γ , v ∈ V) and λB = λ(wBv , γ , v ∈ V) are the resulting one parameter v v subgroups. Then, there is an element g ∈ SL(Q, n) with λB = g · λ · g−1 , whence F λ, χ 4 = F λB , χ 4. We may, therefore, set 1 P 1 P (γ , v ∈ V), χ := λ(wv , γ , v ∈ V), χ , for any tuple (wv , v ∈ V). v
v
Now, assume > A! ! !! N > max !!F (γ , v ∈ V), χ 4!! !! (γ , v ∈ V) ∈ Γ(Q, n) . v
v
(1.14)
For a character χ = χ(av , v ∈ V), a representation ( fa , a ∈ A) ∈ Rep(Q, n) is called χ-(semi)stable or (av , v ∈ V)-(semi)stable, if for any collection (Wv , v ∈ V) consisting of subspaces Wv ⊆ 'nv , v ∈ V, such that {0} ! ( v∈V Wv )$b ! 'n and f (Wt(a) ) ⊆ Wh(a) , a ∈ A, the inequality ' ' v∈V av · nv v∈V av · dim(Wv ) (≤) ' (1.15) ' v∈V dim(Wv ) v∈V nv
!
is verified. We call ( fa , a ∈ A) χ-polystable or (av , v ∈ V)-polystable, if there are j j collections of subspaces (Wvj , v ∈ V) and of maps ( faj : Wt(a) −→ Wh(a) ), j = 1, . . . , t, such that
'n
v
=
% W , v ∈ V, j=1 j
j v
and
fa =
% f , a ∈ A, t
j=1
j a
j
j
and the representation (Wv , v ∈ V, fa , a ∈ A)“∈” Rep(Q, (dim(Wv ), v ∈ V)) is χ-stable.4 In other words, a polystable representation is the direct sum of stable representations of lower dimension.
Exercise 1.5.1.24. i) Given χ = χ(av , v ∈ V), m ∈ (>0 , and χB = χ(m · av , v ∈ V), check that ( fa , a ∈ A) is χ-(semi/poly)stable, if and only if it is χB -(semi/poly)stable. ii) Assume we are given χ = χ(av , v ∈ V), m ∈ (, and χB = χ(m + av , v ∈ V). Show that ( fa , a ∈ A) is χ-(semi/poly)stable, if and only if it is χB -(semi/poly)stable.
Theorem 1.5.1.25. Fix a character χ = χ(av , v ∈ V) and suppose N satisfies (1.14). Then, for a point r = [ fa , a ∈ A, ε] ∈ !(Rep(Q, n)∨ -'), the Hilbert–Mumford criterion µχN (r, λ)(≥)0 is satisfied for all non-trivial one parameter subgroups λ: '. −→ SL(Q, n), if and only if the following conditions are verified: 1. ε % 0 or f is not nilpotent (see above); 2. ( fa , a ∈ A) ∈ Rep(Q, n) is χ-(semi)stable. 4 To
j
make sense out of this, one needs to choose isomorphisms Wv ! being χ-stable does not depend on the choice of these isomorphisms.
"dim(Wvj ) , v ∈ V. The property of
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Proof. We first prove the direction “=⇒”. If ε = 0 and f were nilpotent, then, by Proposition 1.5.1.22, we would find a one parameter subgroup λ: '. −→ SL(Q, n) with µ(r, λ) < 0. By definition of Γ(Q, n), this one parameter subgroup may be chosen of the form λ(wv , γ , v ∈ V) with (γ , v ∈ V) ∈ Γ(Q, n). But then, (1.13) and (1.14) v v imply µχN (r, λ) < 0, a contradiction. This establishes 1. For 2., let (Wv , v ∈ V) be a collection of subspaces Wv ⊆ 'nv , v ∈ V, such that {0} ! ( v∈V Wv )$b ! 'n and ' f (Wt(a) ) ⊆ Wh(a) , a ∈ A. Set iv := dim(Wv ), v ∈ V, and i := v∈V iv , and choose the tuple (wv , v ∈ V) of bases in such a way that
!
Wv = F wv,1 , . . . , wv,iv 4,
v ∈ V.
Next, the tuple (γ , v ∈ V) of weight vectors is defined by v ^ 4 γ := i − n/b, . . . , i − n/b, i, . . . , i , v ∈ V. v D!!!!!!!!!!!!!!!!!!WB!!!!!!!!!!!!!!!!!!\ D!WB!\ iv x
(nv −iv ) x
Then, λ := λ(wv , γ , v ∈ V) is a one parameter subgroup of SL(Q, n) with µ(r, λ) = 0, v thanks to 1. Therefore, by (1.13), 4 ^ P T 1 av · nv · i − (n/b) · iv (≥)0. µχN (r, λ) = (γ , v ∈ V), χ = v
v∈V
The latter inequality is just ' ' v∈V av · dim(Wv ) v∈V av · nv (≤) ' . ' v∈V dim(Wv ) v∈V nv Finally, we prove the converse direction “⇐=”. Since 1. holds true, by assumption, Proposition 1.5.1.22 grants µ(r, λ) ≥ 0, for every one parameter subgroup λ of SL(Q, n). Next, we look at a one parameter subgroup λ = λ(wv , γ , v ∈ V) with (γ , v ∈ V) ∈ v v Γ(Q, n). If µ(r, λ) > 0, then also µχN (r, λ) > 0, by the assumption on N and (1.14). Hence, we may assume µ(r, λ) = 0. Let (W• (λ), α• (λ)) be the associated weighted flag, W• (λ) : {0} ! W1 ! · · · ! W s ! 'n . Then, Wi =
&
%W v∈V
v,i
Z$b
,
i = 1, . . . , s.
!
Since µ(r, λ) = 0, Proposition 1.5.1.22 shows that (Wv,i , v ∈ V) is, for every index i ∈ { 1, . . . , s }, a collection of subspaces, such that {0} ! Wi = ( v∈V Wv,i )$b ! 'n and fa (Wt(a),i ) ⊆ Wh(a),i , a ∈ A. Furthermore, for a given vertex v ∈ V, we find P 1 Wv,i = wv,1 , . . . , wv,dim(Wv,i ) . By the construction which we have used in the first part of the proof, we may, thus, associate to any i ∈ { 1, . . . , s } a one parameter subgroup λi = λ(wv , γi , v ∈ V). It is not v hard to check that s T λ= αi · λi , for some αi ∈ 3≥0 , i = 1, . . . , s. i=1
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Thus, µχN (r, λ) = F λ, χ 4 =
s T
αi · F λi , χ 4.
i=1
By the same argument that has already been used in the first part of the proof, the condition F λi , χ 4(≥)0 is equivalent to ' ' v∈V av · dim(Wv,i ) v∈V av · nv ' (≤) ' . v∈V dim(Wv,i ) v∈V nv Since the latter inequality is satisfied by assumption, we are done.
"
Exercise 1.5.1.26. Set ! := !(Rep(Q, n)∨ - '), and let πχ : ! $ !//σχN SL(Q, n) be the quotient that we have constructed above, based on the linearization in O(N) associated to χ. Let Rep(Q, n)χ-ss ⊂ Rep(Q, n) be the subset of χ-semistable representations. Show that there is an open subset U ⊂ !//σχN SL(Q, n) with χ-ss . π−1 χ (U) = Rep(Q, n)
The quasi-projective variety Mχ (Q, n) := U = Rep(Q, n)χ-ss // SL(Q, n) may thus be viewed as the moduli space of χ-semistable representations of Q with dimension vector n. Remark 1.5.1.27. The quasi-projective variety Mχ (Q, n) is actually the moduli space that was constructed by King in [120]. Thus, our approach gives a little more, namely the natural compactification !//σχN SL(Q, n) of Mχ (Q, n). This viewpoint will be very important for our moduli space constructions in the second chapter. On the other hand, King does not need the artificial compactification of Rep(Q, n) and gives a construction which formally parallels the construction of !n from the '. -action on 'n+1 . Exercise 1.5.1.28. The action of SL(Q, n) on ! is associated to the representation *: SL(Q, n) −→ GL(Rep(Q, n)). If we perform the above constructions with the trivial character 0, then we do the same thing as in Exercise 1.5.1.6. i) Prove that there is a morphism fχ : !//σχN SL(Q, n) −→ !//σ0N SL(Q, n) which induces the projective morphism fχ0 : Mχ (Q, n) −→ M0 (Q, n) ! Rep(Q, n)//* SL(Q, n). ii) Show that M0 (Q, n) = {pt}, if Q does not contain any oriented cycle. Conclude that Mχ (Q, n) is projective in this case. Remark 1.5.1.29. This observation illustrates the meaning of King’s construction. If we are interested in the classification problem associated to quivers, then the natural approach in Section 1.3.3 via categorical quotients of vector spaces may yield unsatisfactory results. King’s definition of χ-semistability restricts the class of admissible
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objects, but provides us with better moduli spaces. An interesting illustration is the case of a quiver with vertices 1 and 2 and h arrows from 1 to 2. This quiver contains no oriented cycles, so that M0 (Q, n) is just a point. Using Exercise 1.5.1.24, one checks that there is exactly one additional non-trivial notion of semistability. (Exercise: Formulate it in a parameter independent way.) Thus, we get one new projective moduli space M. (Q, n). This will, in general, be a non-trivial variety (see Chapter 2, Remark 2.2.4.14). Exercise 1.5.1.30. We look at the quiver Q = 2 −→ 1 −→ 0. Determine all stability parameters χ for which there exist χ-semistable representations. For which parameters χ and dimension vectors n do there exist stable representations? (Hint: You may distinguish the cases a) r0 > r1 % r2 , b) r0 < r1 > r2 , c) r0 % r1 = r2 , and d) r0 = r1 = r2 .5 In each of these cases, you will find some non-trivial subchains, such that the semistability condition on these subchains gives conditions on the stability parameter. The solution to this exercise is Theorem 5.1 in [2].) Parabolic Subgroups and One Parameter Subgroups A proper subgroup P of an affine algebraic group G is said to be a parabolic subgroup, if the quotient G/P is a projective variety. A Borel subgroup of G is a connected solvable subgroup B of G which is maximal with respect to inclusion among the connected solvable subgroups of G. Example 1.5.1.31. i) By [30], 11.1, Borel subgroups are parabolic subgroups. They are minimal with respect to inclusion among the parabolic subgroups. ii) Assume G = GLr (') or G = SLr ('). Let {0} ! W1 ! · · · ! W s ! 'r be a flag. The stabilizer P of this flag is a parabolic subgroup of G, and every parabolic subgroup of G arises in this way. The homogeneous space G/P is the flag variety that parameterizes flags {0} ! U1 ! · · · ! U s ! 'r with dim(Ui ) = dim(Wi ), i = 1, . . . , s. For s = 1, we recover the Graßmannians. In this picture, the Borel subgroups come from complete flags, i.e., flags with s = r − 1 and dim(Wi ) = i, i = 1, . . . , r − 1. Let us recall some facts from the Section “The flag complex” in Mumford’s book [155]. Some information is also contained in Springer’s book [207]. Proposition 1.5.1.32. Let G be a reductive affine algebraic group and λ: '. −→ G a one parameter subgroup. Then, F L QG (λ) := g ∈ G | gB := lim λ(z) · g · λ(z)−1 exists in G z→∞
is a parabolic subgroup of G. Furthermore, gB · λ(z) · gB −1 = λ(z), for every z ∈ '. and every g ∈ QG (λ). Proof. Let us sketch the proof from [155]. We fix a faithful representation ι: G '→ GLr ('). Then, we obviously have QG (λ) = ι−1 (QGLr (.) (λ)). Here, we view λ as a one parameter subgroup of GLr ('), using the embedding ι. We may find a basis ' ' w = (w1 , . . . , wr ) of 'r and integers γ1 ≤ · · · ≤ γr with λ(z)·( ci ·wi ) = zγi ·ci ·wi , for 5 The
other possible cases are obtained by dualizing from a) and c).
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all z ∈ '. and all tuples (c1 , . . . , cr ) of complex numbers. There are indices i1 , . . . , i s , i0 := 0, i s+1 := r, with γ j = γik , for j = ik−1 + 1, . . . , ik , k = 1, . . . , s + 1, and γi1 < · · · < γis+1 . These data define the flag W• : {0} ! W1 ! · · · ! W s ! W s+1 = 'r , Wk := F w1 , . . . , wik 4, k = 1, . . . , s + 1. One readily verifies that QGLr (.) (λ) is the stabilizer of this flag. The element gB = limz→∞ λ(z) · g · λ(z)−1 is a block diagonal matrix. More precisely, gB lies in GL(F w1 , . . . , wi1 4) x · · · x GL(F wis +1 , . . . , wr 4), whence it commutes with λ. " Remark 1.5.1.33. i) In the books quoted above, one takes the limit z → 0 in order to define the parabolic subgroup PG (λ). Thus, we have QG (λ) = PG (−λ).
(1.16)
ii) Any parabolic subgroup of a connected reductive affine algebraic group G is of the form QG (λ), for an appropriate one parameter subgroup λ of G [207]. iii) If we are given an injective homomorphism ι: G '→ H between reductive affine algebraic groups, then we obviously find QH (λ) ∩ G = QG (λ). For later reference, we also mention the following constructions (see [207] for more details): ^ 4 F L Ru QG (λ) := g ∈ G | lim λ(z) · g · λ(z)−1 = e ⊂ QG (λ) z→∞
is the unipotent radical of QG (λ), and F L LG (λ) := g ∈ G | λ(z) · g = g · λ(z) ∀z ∈ 4m (k) ⊂ QG (λ)
(1.17)
is a(!) Levi subgroup of QG (λ). Then,
^ 4 QG (λ) ! Ru QG (λ) " LG (λ).
iv) Let G '→ H be an inclusion among reductive linear algebraic groups. In view of Part ii) of this remark, it is evident that any parabolic subgroup of G extends to a parabolic subgroup of H. However, if Q is a parabolic subgroup of H, then Q ∩ G need not be a parabolic subgroup of G. As an example, look at the action of H := SL2 (') on the projective line !1 . Let x := [1 : 1] and Q = H x the H-stabilizer of x. Since the action of SL2 (') on !1 is transitive, we have H/Q ! !1 , so that Q is a parabolic subgroup of H. Let 7X J! 0 !! t 0 . G := ! '. . t∈' 0 t−1 ! The G-orbit of x is { [t : t−1 ] | t ∈ '. } ! '. , G ∩ Q = {±$2 } is the G-stabilizer of x, and G/(G ∩ Q) = G · x ! '. . Hence, G ∩ Q is not a parabolic subgroup of G. (Indeed, the closed subgroups of G are finite cyclic groups and G itself, and only G is a parabolic subgroup.) Exercise 1.5.1.34. Describe the above subgroups explicitly for G = SLr (') and G = GLr (').
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Proposition 1.5.1.35. Let *: G −→ GL(V) be a representation of the reductive group G. For every point x ∈ !(V), every one parameter subgroup λ: '. −→ G, and every element g ∈ QG (λ), one has µ(x, λ) = µ(x, g−1 · λ · g). Proof. We set y := limz→∞ λ(z) · x and gB := limz→∞ λ(z) · g · λ(z)−1 . Then, we compute ^ 4 4 ^ 4 ^ lim g−1 · λ(z) · g · x = lim g−1 · λ(z) · g · λ(z)−1 · λ(z) · x = (g−1 · gB ) · y. z→∞
z→∞
Using the rules for the function “µ” stated in Exercise 1.5.1.1, we infer ^ 4 µ(x, g−1 · λ · g) = µ (g−1 · gB ) · y, g−1 · λ · g = µ(gB · y, λ) = =
µ(y, gB −1 · λ · gB ) = µ(y, λ) = µ(x, λ).
Here, we have also used the fact that gB commutes with λ which was established in Proposition 1.5.1.32. " Example 1.5.1.36. A weighted flag in 'r is a pair (W• , α• ) which consists of a flag W• : {0} ! W1 ! · · · ! W s ! 'r and a tuple α• = (α1 , . . . , α s ) of positive numbers in ([1/r]. Let *: SLr (') −→ GL(V) be a finite dimensional representation of the special linear group. Every one parameter subgroup λ: '. −→ SLr (') yields the weighted flag (W• (λ), α• (λ)) in the following manner: The one parameter subgroup λ defines the integers γ1 < · · · < γ s+1 and the decomposition into non-trivial eigenspaces
'r :=
%W , s+1 i=1
i
L F W i := w ∈ 'r | λ(z)(w) = zγi · w ∀z ∈ '. .
Then, W• (λ) : {0} ! W1 := W 1 ! W2 := W 1 - W 2 ! · · · ! W s := W 1 - · · · - W s ! 'r and
γi+1 − γi , i = 1, . . . , s. r It follows easily from Proposition 1.5.1.35 and (1.11) that µ* (., λ) depends only on the weighted flag of λ. Since, moreover, any weighted flag arises from a one parameter subgroup of SLr ('), one can say that the real test objects for the Hilbert–Mumford criterion for SLr (') are weighted flags in 'r . The reader should keep this in mind for the definition of semistable objects in the relative setting. α• (λ) = (α1 , . . . , α s )
with αi :=
Remark 1.5.1.37. If G = GL(V), then the group QG (λ) has been constructed as the stabilizer of the flag V• (λ) : {0} ! V1 ! V2 ! · · · ! V s ! V from the weighted flag (V• (λ), α• (λ)) of λ. Note, that if λB is conjugate to λ, then dim(ViB ) = dim(Vi ) and αBi = αi , i = 1, . . . , s.
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The Adjoint Action Let G be a connected reductive linear algebraic group and σ: G x G −→ G, (g, h) 1−→ g · h · g−1 the adjoint action of G on itself. We would like to study the affine variety G//G for this action. To do this, fix a maximal torus T ⊂ G. Let F L N (T ) := g ∈ G | g · T · g−1 = T be the normalizer of T . Then, T is a normal subgroup of N (T ), and W := W(G, T ) := N (T )/T is a finite group, known as the Weyl group of G (see [30], 11.19). (Since any two maximal tori inside G are conjugate, the (isomorphy class of the) Weyl group does not depend on T .) Note that W acts in a canonical way on T . It is our aim to verify: Claim 1.5.1.38. The inclusion map T '→ G induces an isomorphism T/W ! G//G. For an element g ∈ G, we let CG (g) be its conjugacy class in G, i.e., its orbit under the adjoint action. An element t ∈ G is said to be semisimple, if it is contained in a maximal torus of G. By [30], 9.2, Theorem, the conjugacy class CG (t) of a semisimple element t ∈ G is a closed subset in G. Conversely, let g ∈ G be an element, such that CG (g) is closed. We claim that g is semisimple. First, we note that g lies in a Borel subgroup B ⊂ G ([30], 11.10, Theorem). Since a Borel subgroup is a parabolic subgroup ([30], 11.2, Corollary), we find a one parameter subgroup λ: '. −→ G with B = QG (λ). It is clear that
g∞ := lim λ(z) · g · λ(z)−1 ∈ LG (λ). z→∞
Moreover, LG (λ) is a maximal torus in B (by [30], 11.23, Proposition, (i)) and hence in G ([30], 11.3, Corollary). So, the element g∞ is semisimple. We assume that the orbit of g under the adjoint action is closed. Therefore, g and g∞ are conjugate, whence g is semisimple, too. So far, we have seen that the canonical morphism T −→ G//G is surjective. It is clearly invariant under the action of W on T , so that we also have a surjective morphism q: T/W −→ G//G. Since T and G are smooth, whence normal, the quotient varieties T/W and G//G are also normal (this is implied by Theorem 1.2.1.11; see also [123], II.3.3, Satz 4). By Zariski’s Main Theorem (Proposition 1.2.1.12, [30], 18.2, Theorem), we have to verify that q is bijective. This means that, given t, tB ∈ T and g ∈ G with g · t · g−1 = tB , there exists also an element h ∈ N (T ) with h · t · h−1 = tB . To see this, we let L F ZG (tB ) := γ ∈ G | γ · tB · γ−1 = tB
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be the centralizer of tB and Z := ZG (tB )0 the connected component of the neutral element. This is a closed subgroup of G ([30], 1.7, Proposition (c)), and T and g · T · g−1 are two maximal tori of Z. By [30], 11.2, Corollary, they are conjugate in Z. Hence, we find an element n ∈ Z with n · (g · T · g−1 ) · n−1 = T . Note that n · g ∈ N (T ), and we have (n · g) · t · (n · g)−1 = n · (g · t · g−1 ) · n−1 = n · tB · n−1 = tB . This settles our claim.6 " Exercise 1.5.1.39. Let Ad: G −→ GL(g) be the adjoint representation of G on g := Lie(G) ([123], II.2.3). Let T ⊂ G be a maximal torus and t ⊂ g its Lie algebra. Generalize the above arguments to show that there is an isomorphism t/W −→ g//G. This is a theorem of Chevalley (not published by himself, see [94]). Recall that you know all the material of this section for G = GLn (') from Section 1.3.3. The reader may look at Section 7.5 in [170] for more information on such “Chevalley sections” Remark 1.5.1.40. The invariant ring '[t]W is a polynomial ring. If G is simple, the degrees of the generators are computed in [33], Chapter VIII, §8, no 3, Corollaire 3, using [32], Planche I-VI. Estimates for the Weights of Some Special Representations In the following, *a,b,c will stand for the induced representation of GLr (') on the vector space r &^ 4#a Z$b &8 Z#−c Wa,b,c := 'r * 'r , a, b, c ∈ (≥0 . Then, !(Wa,b,c ) = !(Wa,b,0 ) and Wa,b,c ! Wa,b,0 as SLr (')-modules. ' (i) Let w = (w1 , . . . , wr ) be a basis for 'r and γ = r−1 i=1 αi · γ , αi ∈ 3≥0 , an integral a weight vector. Let I be the set of all a-tuples ι = (ι1 , . . . , ιa ) with ι j ∈ { 1, . . . , r }, j = 1, . . . , a. For ι ∈ I a and k ∈ { 1, . . . , b }, we define wι := wι1 * · · · * wιa , and wkι := (0, . . . , 0, wι , 0, . . . , 0) ∈ Wa,b,0 , wι occupying the k-th entry. The elements wkι ∨
with ι ∈ I a and k ∈ { 1, . . . , b } form a basis for Wa,b,0 . We let wkι , ι ∈ I a , k ∈ { 1, . . . , b } ' ∨ ∨ be the dual basis of Wa,b,0 . Now, let [l] ∈ !(Wa,b,0 ) where l = akι · wkι . Then, there exist k0 and ι0 with akι00 % 0, such that ^ ^ 4 4 4 ^ ∨ µ*a,b,c [l], λ(w, γ) = µ*a,b,0 [l], λ(w, γ) = µ*a,b,0 [wkι00 ], λ(w, γ) , 7 and, for any other k and ι with akι % 0, ^ ∨ 4 4 ^ µ*a,b,c [l], λ(w, γ) ≥ µ*a,b,0 [wkι ], λ(w, γ) . We also find that, for i ∈ { 1, . . . , r − 1 }, L ^ 4 F ∨ µ*a,b,0 [wkι00 ], λ(w, γ(i) ) = ν · r − a · i, ν = # ι j ≤ i | ι0 = (ι1 , . . . , ιa ), j = 1, . . . , a . 6 The
7 The
last argument was taken from [155], Chapter 2, Lemma 2.8. notation “µ* ” was introduced in Example 1.5.1.1.
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One concludes: Lemma 1.5.1.41. i) For every basis w = (w1 , . . . , wr ) of 'r , every integral weight 'r−1 vector γ = i=1 αi · γ(i) , αi ∈ 3≥0 , and every point [l] ∈ !(Wa,b,c ) r−1 &T
Z
r−1 Z &T ^ 4 αi · a · (r − 1) ≥ µ*a,b,c [l], λ(w, γ) ≥ − αi · a · (r − 1).
i=1
i=1
ii) For every basis w = (w1 , . . . , wr ) of ' , every two integral weight vectors γ = 1 'r−1 'r−1 (i) (i) i=1 αi · γ , αi ∈ 3≥0 , γ = i=1 βi · γ , βi ∈ 3≥0 , and every point [l] ∈ !(Wa,b,c ) r
2
r−1 Z ^ 4 4 &T ^ βi · a · (r − 1). µ*a,b,c [l], λ(w, γ + γ ) ≥ µ*a,b,c [l], λ(w, γ ) − 1
2
1
i=1
1.5.2 Semistability for Direct Sums of Representations In this section, we follow our article [184].8 Let G be a reductive affine algebraic group and V1 ,. . . ,V s finite dimensional complex vector spaces. We give ourselves the representations *i : G −→ GL(Vi ), i = 1, . . . , s. The direct sum * := *1 - · · · - * s of these representations provides us with an action of G on !(V), V := V1 - · · · - V s , together with a linearization in O((V) (1). The *i also yield, for every tuple ι = (ι1 , . . . , ιt ) with 0 < t ≤ s, ιi ∈ { 1, . . . , s }, i = 1, . . . , t, and ι1 < · · · < ιt , an action of G on !ι := !(Vι1 ) x · · · x !(Vιt ), and, for any tuple (k1 , . . . , kt ) of positive integers, a linearization of this action in the very ample line bundle O(ι (k1 , . . . , kt ). The following theorem reduces the determination of the semi- and polystable points in the space !(V) to the determination of the semi- and polystable points in the spaces !ι . Theorem 1.5.2.1. Let v = [v1 , . . . , v s ] be a point in !(V), vi being the component in Vi∨ , i = 1, . . . , s. Then, the following assertions hold true: i) The point v is *-semistable, if and only if there are an index tuple ι as above as well as positive integers k1 , . . . , kt , such that vιi % 0, for i = 1, . . . , t, and the point ([vι1 ], . . . , [vιt ]) ∈ !ι is semistable with respect to the linearization in O(ι (k1 , . . . , kt ). ii) The point v is *-polystable, if and only if, for the index tuple ι with vi % 0
⇐⇒
i ∈ { ι1 , . . . , ιt },
there are positive integers k1 , . . . , kt , such that ([vι1 ], . . . , [vιt ]) ∈ respect to the linearization in O(ι (k1 , . . . , kt ).
!ι is polystable with
Remark 1.5.2.2. For “stable”, only the converse direction in ii) holds true. In order to find the stable points, one has to use the characterization “stable = polystable with finite stabilizer”. 8 Thereby,
we also correct a mistake in that reference.
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Proof of Theorem 1.5.2.1. Ad i). By definition, the point v is semistable, if and only if there exist an integer k > 0 and a G-invariant section s ∈ H 0 (!(V), O((V) (k)) with s(v) % 0. Now, we have 4 ^ Symk1 (V1 ) * · · · * Symks (V s ). H 0 !(V), O((V) (k) = Symk (V) =
%
k1 +···+k s =k
The representation Symk (*) respects this decomposition, so that we also have 4G ^ 4G ^ Symk1 (V1 ) * · · · * Symks (V s ) . H 0 !(V), O((V) (k) = Symk (V)G =
%
k1 +···+k s =k
An invariant section s ∈ H 0 (!(V), O((V) (k)) with s(v) % 0, therefore, exists, if and only if there are an index ι as above, positive integers k1 , . . . , kt , and a section sB ∈ (Symk1 (V1 ) * · · · * Symks (V s ))G with sB (v) % 0. This is just the assertion we have made. Ad ii). The second contention follows by an easy induction from Part ii) in the result stated below. Theorem 1.5.2.3. Let vB = ([vι1 , vι2 ], [vι3 ], . . . , [vιt ]) ∈ !(Vι1 - Vι2 ) x !(ι3 ,...,ιt ) . Then, the following statements are true: i) The point vB is semistable with respect to the linearization in O(k, k3 , . . . , kt ), if and only if 1. either ([vιi ], [vι3 ], . . . , [vιt ]) ∈ !(ιi ,ι3 ,...,ιt ) is, for i = 1 or i = 2, semistable with respect to the linearization in O(k, k3 , . . . , kt ) 2. or there are positive integers n, k1 , and k2 with nk = k1 + k2 , such that ([vι1 ], [vι2 ], [vι3 ], . . . , [vιt ]) ∈ !(ι1 ,ι2 ,ι3 ,...,ιt ) is semistable with respect to the linearization in O(k1 , k2 , nk3 , . . . , nkt ). ii) The point vB is polystable with respect to the linearization in O(k, k3 , . . . , kt ), if and only if 1. either ([vιi ], [vι3 ], . . . , [vιt ]) ∈ !(ιi ,ι3 ,...,ιt ) is polystable with respect to the linearization in O(k, k3 , . . . , kt ), for i = 1 and vι2 = 0 or for i = 2 and vι1 = 0 2. or there are positive integers n, k1 , and k2 with nk = k1 + k2 , such that ([vι1 ], [vι2 ], [vι3 ], . . . , [vιt ]) ∈ !(ι1 ,ι2 ,ι3,...,ιt ) is polystable with respect to the linearization in O(k1 , k2 , nk3 , . . . , nkt ). Assertion i) is proved as before. For the second claim, we use Exercise 1.5.1.5. First, assume that vB is polystable. By i), we know that there are non-negative rational numbers κ1 and κ2 with κ1 + κ2 = k and t ^ 4 ^ 4 T ^ 4 κ1 · µ [vι1 ], λ + κ2 · µ [vι2 ], λ + ki · µ [vιi ], λ ≥ 0,
(1.18)
i=3
for every one parameter subgroup λ of G. We have to investigate what happens if equality holds in (1.18). There are two possibilities: The first one is that there are nonnegative rational numbers κBi % κi , i = 1, 2, with κB1 + κB2 = k, such that (1.18) holds
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for κB1 , κB2 , and all λ. In this case, the same is true for κBBi := (κi + κBi )/2, i = 1, 2. If equality occurs in (1.18) for κBB1 , κBB2 , and λ, then we must necessarily have µ([vι1 ], λ) = µ([vι2 ], λ). Otherwise the left hand expression in (1.18) would become negative for λ and (κ1 , κ2 ) or (κB1 , κB2 ). Define 4 ^ vBB := [vBBι1 , vBBι2 ], [vBBι3 , . . . , vBBιt ] := lim λ(z) · vB . z→∞
The equality µ([vι1 ], λ) = µ([vι2 ], λ) grants that vBBι1 % 0 % vBBι2 . It follows that 4 ^ 4 ^ lim λ(z) · [vι1 ], [vι2 ], [vι3 , . . . , vιt ] = [vBBι1 ], [vBBι2 ], [vBBι3 , . . . , vBBιt ] .
z→∞
This point is still semistable, so that vBB is also *-semistable, by i). There exists an element g ∈ G with vBB = g · vB , because vB is polystable. The second possibility is that there is only one pair (κ1 , κ2 ) with the respective properties. This means that, for all ν > 0, all G-invariant sections of O(νk, νk3 , . . . , νkt ) which do not vanish in vB are elements of Symνκ1 (Vι1 ) * Symνκ2 (Vι2 ) * Symνk3 (Vι3 ) * · · · * Symνkt (Vιt ). This easily yields the claim. Next, we come to the converse assertion. To this end, we first remark that, for any pair (κ1 , κ2 ) with κ1 + κ2 = k and any one parameter subgroup λ, the inequality t 4 T 4 4 ^ ^ ^ ki · µ [vιi ], λ µ(vB , λ) ≥ κ1 · µ [vι1 ], λ + κ2 · µ [vι2 ], λ +
(1.19)
i=3
is verified. By assumption, we may choose (κ1 , κ2 ) in such a way that the right hand side is non-negative for all one parameter subgroups λ. We will assume κ1 · κ2 % 0, because the other cases are easier to handle. For a one parameter subgroup λ with µ(vB , λ) = 0, we consequently have t ^ 4 ^ 4 T ^ 4 κ1 · µ [vι1 ], λ + κ2 · µ [vι2 ], λ + ki · µ [vιi ], λ = 0. i=3
There is an element g ∈ G, such that ^ 4 ^ 4 lim λ(z) · [vι1 ], [vι2 ], [vι3 ], . . . , [vιt ] = g · [vι1 ], [vι2 ], [vι3 ], . . . , [vιt ] , z→∞
because ([vι1 ], [vι2 ], [vι3 ], . . . , [vιt ]) is a polystable point. We may clearly replace vB by g · vB . Then, ([vι1 ], [vι2 ], [vι3 ], . . . , [vιt ]) is fixed by λ. Therefore, there exist integers γ1 and γ2 with λ(z)·vιi = zγi ·vιi , i = 1, 2. Note that γi = µ([vιi ], λ), i = 1, 2. If γ1 = γ2 , then vB is also fixed by the '. -action belonging to λ, whereas, for γ1 < γ2 , the inequality 0 = κ1 · γ1 + κ2 · γ2 +
t T
t T ^ 4 4 (1.19) ^ ki · µ [vιi ], λ < k · γ2 + ki · µ [vιi ], λ ≤ µ(vB , λ)
i=3
holds true. But, we have assumed µ(vB , λ) = 0.
i=3
"
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1.5.3 Semistability for Actions of Direct Products of Groups Here, we will study the following situation:9 We are given two reductive groups G and H and a representation *: G x H −→ GL(W) on the finite dimensional '-vector space W. By means of restriction, we find the representations γ and λ of G and H, respectively. These representations commute with each other, i.e., for any g ∈ G and any h ∈ H, one has γ(g) · λ(h) = λ(h) · γ(g). If, conversely, γ and λ are commuting representations of G and H, respectively, then they also define a representation * of G x H. The goal is to express *-semistability in terms of γ- and λ-semistability. This works only in a refined sense as we shall now explain. Set Qγ := !(W)//γG, and let πγ : !(W)ss γ −→ Qγ be the quotient map. If n is sufficiently large, then there is the embedding jn : Qγ '→ !(Symn (W)G ). Since γ and λ commute with each other, the group H acts on Qγ , Symn (W)G , and !(Symn (W)G ), and jn is H-equivariant with respect to these actions. Therefore, the H-action on !(Symn (W)G ) comes with a linearization which, by restriction, also yields a linearization of the Haction on Qγ in the line bundle O((Symn (W)G ) (1)|Qγ . As we have seen in Remark 1.4.3.9, the resulting notion of semistability does not depend on n. Thus, we may speak of the H-semistable and H-polystable points, and we denote the corresponding subsets by ps Qss γ and Qγ , respectively. ss Theorem 1.5.3.1. i) The set of *-semistable points in !(W) is the set π−1 γ (Qγ ) (⊆ ss !(W)γ ). In addition, we have !(W)//*(G x H) ! Qγ //H. The right hand quotient is formed with respect to the H-action which has been linearized in the way outlined above. ps ps ps ii) For the polystable points, the equality !(W)* = !(W)γ ∩ π−1 γ (Qγ ) holds true.
Remark 1.5.3.2. The analogous statement for stable points is false. Proof. i) We first assume that x ∈ !(W) is γ-semistable and that πγ (x) lies in Qss γ . For n G large n, the point jn (πγ (x)) ∈ !(Sym (W) ) is H-semistable. Hence, there exist k ≥ 1 and an H-invariant section 4H ^ 4H ^ s ∈ H 0 !(Symn (W)G ), O((Symn (W)G ) (k) = Symk Symn (W)G which does not vanish in jn (πγ (x)). The section s may also be viewed as an element of Symkn (W)G x H . As such, it corresponds to a (G x H)-invariant section s ∈ O((W) (kn) which does not vanish in x, and we see that x is also *-semistable. For the converse, suppose x ∈ !(W)ss * . For large m, there exists a (G x H)-invariant 0 section s ∈ H (O((W) (m)) which is non-zero at x. This section is an H-invariant element in Symm (W)G . This interpretation tells us that x belongs to !(W)ss γ. The claim on the equality of the categorical quotients is a direct consequence of the results we have proved, so far, and the universal property 1.4.1.1 of the quotient. ps ps ii) First, we deal with the case x ∈ !(W)γ ∩ π−1 γ (Qγ ). As we have shown in i), x is a *-semistable point. Thus, we have a *-polystable point y ∈ (G x H) · x ∩ !(W)ss *. ps = H · π (x), because π (x) ∈ Q . Therefore, It follows πγ (y) ∈ H · πγ (x) ∩ Qss γ γ γ γ 9 The
discussion is again taken from [164].
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there is an element h ∈ H with πγ (x) = h · πγ (y) = πγ (h · y). The closures of the G-orbits of x and h · y in !(W)ss γ intersect. Since x is a γ-polystable point, we deduce G·x ⊂ G · (h · y)∩ !(W)ss . Consequently, we have x ∈ (G x H) · y∩ !(W)ss γ * = (G x H)·y. Since y is *-polystable, the same is true of x. ps Now, assume x ∈ !(W)* . We fix a γ-polystable point y ∈ G · x ∩ !(W)ss γ . It is evident that πγ (y) = πγ (x). By i), we have πγ (x) = πγ (y) ∈ Qss γ . Applying i) the other way round, we recognize that y is *-semistable. By assumption, the orbit (G x H) · x is closed in !(W)ss * . Thus, there are g ∈ G and h ∈ H with y = g · h · x. The point g−1 · y is γ-polystable, and, since γ and λ commute with each other, the point x = h−1 · (g−1 · y) is γ-polystable, as well. We only have to check that πγ (x) lies in ps ps ps Qγ . To this end, let z ∈ !(W)γ be such that πγ (z) ∈ H · πγ (x) ∩ Qγ . (Recall that this implies that z is *-polystable.) Then, the points πγ (z) and πγ (x) are mapped to the same point of Qγ //H = !(W)//* (G x H). We know that the quotient map π* : !(W)ss * −→ !(W)//*(G x H) identifies the points in !(W)//*(G x H) with the closed orbits in !(W)ss* . Since x and z are both *-polystable, (G x H)·x = (G x H)·z. We infer H·πγ (x) = H·πγ (z). Thus, we have finally seen that the orbit of πγ (x) is closed in Qss " γ. Exercise 1.5.3.3. i) Let G and H be reductive linear algebraic groups and consider an action α: (G x H) x X −→ X. Assume that the good quotient π: X −→ X//(G x H) exists. Show that the good quotients πG : X −→ X//G and πH : X//G −→ (X//G)//H also exist and that (X//G)//H ! X//(G x H). (1.20) (Recall Exercise 1.4.3.11.) Conversely, if the good quotients πG : X −→ X//G and πH : X//G −→ (X//G)//H both exist, then π: X −→ X//(G x H) also exists (and one has (1.20)). ii) Verify that, under the assumptions in i), the proof of Theorem 1.5.3.1, ii), works and shows that the (G x H)-orbit of a point x ∈ X is closed, if and only if its G-orbit is closed in X and the H-orbit of πG (x) is closed in X//G. Remark 1.5.3.4. Given a (not necessarily reductive or affine) algebraic group G, an algebraic variety V, and an action σ: G x V −→ V, one defines a quotient variety to be a pair (Y := V/G, π) which consists of an algebraic variety Y and a surjective and open morphism π: V −→ Y, such that a) two points have the same image under π, if and only if they lie in the same orbit, and b) π. (OV )G = OY . The statement in Exercise 1.5.3.3, i), remains true in that setting (see [30], 6.10 Corollary, [180], Proposition 2). (The proof is similar. One needs the following universality property of a quotient variety: If Z is any other variety, and G acts on Y x Z by g · (y, z) = (g · y, z), then the quotient variety is (Y x Z, (v, z) 1→ (π(v), z)). In other words, the quotient is universal for base πY changes of the form Y x Z −→ Y. Recall that we have verified this for GIT quotients in Exercise 1.4.2.12.) Applications to (G x '. )-Actions
Let *: G x '. −→ GL(W) be a representation on the finite dimensional '-vector space W. We will be mostly interested in the resulting representation γ: G −→ GL(W). The criterion which we shall prove in this section describes how the γ-semistable points in
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!(W) may be determined from the G-semistable points in the '. -quotients of !(W)
(see Section 1.4.3). As an exercise, the reader may check that this criterion follows also from Theorem 1.5.2.1 and a suitable generalization of Example 1.4.3.28. Here, we will follow the direct, more natural path. We need more notation. Let λ: '. −→ GL(W) be the induced representation of . ' , and λ the induced action of '. on !(W). This situation was studied in detail in Section 1.4.3. In particular, we have defined the sets Ii , i = 1, . . . , 2m. For any ps index i, there are the corresponding subsets !(W)i ⊂ !(W)ss i of poly- and semistable ss points and the quotient Qi . Let πi : !(W)i −→ Qi be the respective projection map. Moreover, for any d ∈ ( and k ∈ (>0 , there are the quotient Qkd = Qi and the line bundle Lkd . Recall that i is determined by the condition d/k ∈ Ii . The G-action on !(W) induces G-actions on the varieties Qkd together with natural linearizations in the line bundles Lkd . The variety Qkd and the associated notion of semistability on Qkd merely depend on the ratio η := d/k. For i ∈ { 1, . . . , 2m } and η ∈ Ii , we therefore simply speak of the η-polystable and η-semistable points in Qi . Theorem 1.5.3.5. For x ∈ !(W), the following conditions are equivalent: 1. The point x is γ-semistable (γ-polystable). 2. There are an index i ∈ { 1, . . . , 2m } and a parameter η ∈ Ii , such that x ∈ !(W)ss i ps (x ∈ !(W)i ) and such that πi (x) is an η-semistable (η-polystable) point. Proof. We give the arguments for “semistable”. In the other case, the same reasoning applies. Let x ∈ !(W) be a γ-semistable point. Since the representations λ and γ commute with each other, the representation λn : '. −→ GL(Symn (W)) yields the representation λB : '. −→ GL(Symn (W)G ). As we have seen in Theorem 1.4.3.22, there exist k ∈ (>0 and d ∈ (, such that πγ (x) is semistable with respect to the concept of semistability on Qγ , coming from the representation (λB )kd . Replacing n by kn, we may assume that πγ (x) is semistable with respect to the notion induced by (λB )d , for a suitable integer d. Now, we look at the representation ^ 4 (γn x λnd ): G x '. −→ GL Symn (W) . The point x is γ-semistable, by assumption, whence it is also γn -semistable. Its image in the quotient is, by our choice of d and n, also semistable with respect to the induced notion of semistability. According to Theorem 1.5.3.1, the point x is (γn x λnd )semistable. If we apply that theorem in the other order, i.e., we divide by the '. -action before we divide by the G-action, then we see x ∈ !(W)ss i and that πi (x) is η-semistable in the quotient. Here, η := d/k, and i ∈ { 1, . . . , 2m } is the index with η ∈ Ii . Assume 2. This means that there are integers d and k > 0, such that η = d/k and the point x is (γk x λkd )-semistable. In particular, it is γk - and, thus, γ-semistable. " Exercise 1.5.3.6. Derive Theorem 1.5.2.1 for m = 2 from Theorem 1.5.3.5 and Example 1.4.3.28.
92
S 1.6: T V GIT-Q 92
1.6 The Variation of GIT-Quotients umford’s construction depends on the choice of a linearization, and different choices may lead to different results. We have thoroughly studied this phenomenon in the case of an action by the group '. . In this section, we will have a look at the general situation. First, we will establish a crucial finiteness result, and second, we give a rough sketch of how quotients to different linearizations are interrelated.
,
1.6.1 The Finiteness of the Number of GIT-Quotients Let X be a projective algebraic variety, σ: G x X −→ X an action of the reductive group G on X, and σ: G x L −→ L a linearization of the action in the ample line bundle L. The main result of this section reads as follows: Theorem 1.6.1.1. There are only finitely many open subsets U ⊆ X, such that there exists a linearization σ as above with U = Xσss . This theorem was proved in a more general framework by Białynicki-Birula [21]. Independently, Dolgachev and Hu [57] derived the above result with methods of Geometric Invariant Theory. Here, we present the short proof from [185] which simplifies the arguments of [21] in the GIT-setting.10 Note that, in our proof, unlike [21] and [57], we do not need to assume that X be normal. Proof. We proceed in several steps: First, we reduce to the case that G is a torus, by suitably interpreting the Hilbert–Mumford criterion. Then, we show how the case of a torus follows from several observations which we have already made. Step 1. — Fix a maximal torus T ⊆ G. We then find the action σ|T x X of T on X together with the linearization σ|T x L in L. By the Hilbert–Mumford criterion, a point x ∈ X is semistable, if and only if, for every one parameter subgroup λ: '. −→ G, the condition µ(x, λ) ≥ 0 is verified. The image of λ lies in a maximal torus T B of G. According to [30], 11.13, there is an element g ∈ G with g · T B · g−1 = T , i.e., g · λ · g−1 is a one parameter subgroup of our fixed torus T . As a consequence, we find the characterization x ∈ Xσss
⇐⇒
µ(x, g · λ · g−1 ) ≥ 0,
for all λ: '. −→ T and all g ∈ G.
Making use of Exercise 1.5.1.1, we arrive at the following result: x ∈ Xσss that is,
⇐⇒
µ(g · x, λ) ≥ 0, Xσss =
S^ g∈G
for all λ: '. −→ T and all g ∈ G, 4 g · Xσss|T x L .
Thus, it suffices to prove the assertion for G = T . Suppose T ! theorem for T by induction on n. 10 It
'. n .
We prove the
was not properly credited in [185] that the strategy of proof belongs to Białynicki-Birula.
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93
Step 2. — For n = 1, we look at the action σ: '. x X −→ X together with its linearization σ: '. x L −→ L in the ample line bundle L. Denote by F1 , . . . , Fm the connected components of the fix point locus of σ. Suppose that P := (P1 , . . . , P s ) is a collection of disjoint subsets of { 1, . . . , m }, such that P1 J · · · J P s = { 1, . . . , m }. = Define Γi := j∈Pi F j , i = 1, . . . , s. In analogy to Remark 1.4.3.26, we declare, for i = 1, . . . , s, L F Xi+ := x ∈ X | lim z · x ∈ Γi z→0 L F − x ∈ X | lim z · x ∈ Γi , Xi := z→∞
and, for i < j, Ci j := (Xi+ \ Γi ) ∩ (X −j \ Γ j ). Finally, for κ = 1, . . . , s, we set U(P, κ) :=
)
Ci j
1≤i≤κ κ+1≤ j≤s
and U 0 (P, κ) :=
Xκ− ∪ Xκ+ ∪
)
Ci j .
1≤i≤κ−1 κ+1≤ j≤s
Let U be the finite family of open subsets of X which are of the shape U(P, κ) or U 0 (P, κ). With Remark 1.4.3.26, we easily see that Xσss ∈ U . This sets up the induction. Step 3. — We write the n-dimensional torus T as a product T = '. x T B , T B being an (n − 1)-dimensional torus. Set σ1 := σ|.. x L . For Xσ1 and the categorical quotient π: Xσss1 −→ X//σ1 '. , there are—up to canonical isomorphy—only finitely many possibilities, by Step 2. Furthermore, σ and σ lead to an action σB of T B on X//σ1 '. and a linearization σB of this action in an ample line bundle. The open subset (X//σ1 '. )ss σB belongs, by the induction hypothesis, to a finite list of open subsets. Therefore, the same is true for ^ 4 Xσss = π−1 (X//σ1 '. )ss σB . This equality follows from Theorem 1.5.3.1. Hence, we are done.
"
1.6.2 Variation of the GIT-Quotients Let X be a projective algebraic variety, G a reductive group, and σ: G x X −→ X an action of G on X. In the last section, we have seen that the freedom of linearization leads only to finitely many categorical quotients of open subsets of X. Now, we would like to compare quotients to different linearizations as we have done in the case G = '. . Indeed, we will reduce the general case to that one. Let us recall what we already know. For this, look at two representations *1 : G −→ GL(V1 ) and *2 : G −→ GL(V2 ) of the reductive group G. These representations give an action of G on !(V1 ) x !(V2 ) and, for every pair (m, n) of positive integers, a linearization σm,n of this action in O(m, n). On the other hand, we also have the representation τ := *1 - *2 : G −→ GL(V1 - V2 ) which supplies us with the action τ of G
94 S 1.6: T V GIT-Q S 1.6: T V GIT-Q 94 on !(V1 - V2 ) and a linearization in O(1). Furthermore, we may introduce the representation λ: '. −→ GL(V1 - V2 ), z 1−→ z−1 · idV1 -z · idV2 which gives a '. -action λ on !(V1 - V2 ). This '. -action commutes with τ, so that we arrive at the action λ x τ: ('. x G) x !(V1 - V2 ), (z, g, x) 1−→ z · (g · x). For k > 0 and d ∈ (, this action possesses the linearization (λkd x τk ) in O(k). (The linearizations λkd were introduced before Theorem 1.4.3.22.) By Example 1.4.3.28 and Theorem 1.5.3.1, the following holds true, for m = k − d and n = k + d, 4 4 ^ ^ !(V1 ) x !(V2 ) //σm,n G = !(V1 - V2 )//λkd '. //σm,n G =
!(V1
-
V2 )//(λkd x τk ) ('. x G).
Theorem 1.5.3.1 also shows that any quotient (!(V1 ) x !(V2 ))//σm,n G, m, n ∈ (>0 , is a '. -quotient of !(V1 - V2 )//τG. Here, the '. -action comes from λ, and the used linearizations are of the form λB kd , k > 0, d ∈ (, λB being the linearization that is induced by λ. All the structural results now follow from those for '. -actions. Next, we return to the situation considered at the beginning of this section. Let σ1 : G x L1 −→ L1 and σ2 : G x L2 −→ L2 be two linearizations of σ. Here, we assume that L1 and L2 are very ample and define Vi := H 0 (Li ), i = 1, 2. The linearizations σ1 and σ2 lead to representations *1 : G −→ GL(V1 ) and *2 : G −→ GL(V2 ) as well as to the equivariant embeddings ι1 : X '→ !(V1 ) and ι2 : X '→ !(V2 ). Finally, there is also the equivariant embedding ι: X '→ !(V1 ) x !(V2 ), x 1−→ (ι1 (x), ι2 (x)). Pulling back the m #n with the * L linearized line bundle O(m, n) via ι yields the line bundle Lm,n := L# 1 2 # m # n linearization σm,n := σ1 * σ2 . For m > 0 and n > 0, we set η := n/m. Observe that Xσ(s)s = Xσ(s)s , if n/m = nB /mB . For this reason, we will also write Xη(s)s instead of Xσ(s)s . m,n m,n mB ,nB (s)s (s)s . and X := X Introduce also X0(s)s := Xσ(s)s σ2 ∞ 1 Let M := !(L1 - L2 ). The given linearizations σ1 and σ2 induce a G-action on M and an equivariant embedding κ: M '→ !(V1 - V2 ). As before, we may introduce an additional linearized '. -action on M and conclude that all the quotients Xη(s)s −→ X//σm,n G, η := n/m ∈ [0, ∞], are '. -quotients of the variety M//G. Remark 1.6.2.1. The space M//G was introduced in the article [214] where it was baptized master space. Interesting applications of the master space construction are discussed in the papers [214], [166], and [162]. The results of Section 1.4.3 imply the following statement: Theorem 1.6.2.2. There are finitely many critical values η1 , . . . , ηm ∈ (0, ∞) ∩ 3, such that, setting η0 := 0 and ηm+1 := ∞, the following properties hold true. i) For i = 0, . . . , m and η, ηB ∈ (ηi , ηi+1 ), one has Xη(s)s = Xη(s)s B . ii) For i = 0, . . . , m and η ∈ (ηi , ηi+1 ), there are the inclusions Xηs
Xηss
⊃ ⊂
Xηsi ∪ Xηsi+1 ,
Xηssi ∩ Xηssi+1 .
S 1.7: U P
95
*i := Xηss , for η ∈ (ηi , ηi+1 ), i = 0, . . . , m. Set Qi := Xηssi //G, i = 0, . . . , m + 1, and Q These quotients fit into a diagram as follows:
Q0
$$ $$ $ $ $$$
*0 Q 22 22 22 22 2
Q1
Qm
%% %% % % $%%
*m Q ,, ,, ,, ,, Qm+1 .
*i $ Q *i+1 is, for any i = 0, . . . , m − 1, the birational blow down of a weighted Here, Q projective bundle followed by a birational weighted blow up. Remark 1.6.2.3 (Alternative Proof of Theorem 1.6.2.2). There is another proof which rests on the Hilbert–Mumford criterion, more precisely, on the long-winded Example 1.5.1.18. Let us briefly discuss it. Again, we confine ourselves to the case G = SLr ('). The intersection of two rational polyhedral cones is again a rational polyhedral cone. We look at all the edges k of cones of the shape K(*1 , I, χ) ∩ K(*2 , J, χ. ), I ⊂ ST(*1 ), χ ∈ I, J ⊂ ST(*2 ), and χ. ∈ J, as well as the corresponding elements γ ∈ K ∩ (n . Let Γ be the entirety of all elements γ thus obtained. k k The critical values η are the solutions of the equations ^ 4 ^ 4 (.) x,γ : η · µ ι1 (x), λ(w, γ) + µ ι2 (x), λ(w, γ) = 0, x ∈ X, γ ∈ Γ, such that (.) x,γ has only finitely many solutions (in fact, one solution). Since the number µ(ιi (x), λ(w, γ)) depends only on the set ST(*i , ιi (x)), i = 1, 2, and Γ is finite, this is a finite set of equations. All the other assertions now follow easily. "
1.7 The Analysis of Unstable Points n this section, we will sketch a refined study of the Hilbert–Mumford criterion. There are two related questions which are studied. Suppose we are given a linear representation *: G −→ GL(W) of the reductive group G and suppose that w ∈ W is an unstable point. How may one define a canonical or optimal one parameter subgroup λ: '. −→ G, such that λ(z) · w converges in a certain sense fastest to zero? What are the uniqueness properties of such a construction? Recall that a one parameter subgroup λ defines a parabolic subgroup QG (λ). The question is, for instance, whether the parabolic subgroup QG (λ) is unique. The second question is whether the Hilbert–Mumford criterion does hold over non-algebraically closed fields (of characteristic zero). The exact problems were formulated in [155], Chapter 2.2. At a first glance, the second question seems very natural and interesting while the first one looks a bit technical from our viewpoint. Nevertheless, while answering the first question one finds a solution to the second question. Let K ⊂ ' be a non-algebraically closed subfield, e.g., K = 3 or K = 1. We may study a representation *: SLr (K) −→ GLn (K), i.e., a homomorphism which is defined by an (n x n)-matrix of polynomials with coefficients in K in the entries of (r x r)-matrices. Still, we look at the resulting group action over the complex numbers
"
96
S 1.7: U P 96
(or the algebraic closure * of K in '). Assume that we are given a point w ∈ K n which is unstable for the SLr (')-action. We would like to know whether there is a one parameter subgroup λ: 4m (K) −→ SLr (K), such that µ(w, λ) < 0. Note that V• (λ) will then consist of K-rational subspaces of 'r . The solution of the first question assigns to w a certain one parameter subgroup λ which will be defined over some finite Galois extension K B of K. The uniqueness properties of the assignment “unstable w ∈ W 1−→ λ” basically imply that λ is invariant under the action of Gal(K B /K) and thus descends to K. Therefore, we see that the Hilbert–Mumford criterion for semistability holds over non-algebraically closed fields K of characteristic zero. This property will be very interesting for us in view of the applications in Chapter 2 which we have in mind. The canonical or optimal one parameter subgroups which one assigns to unstable points are usually called instability one parameter subgroups. The general theory of the instability one parameter subgroups was developed independently by four authors: Bogomolov [29] (see also [89], §7, p. 43ff), Hesselink [105], [106], Kempf [118], and Rousseau [181]. The paper [173] contains a very readable account of this theory and several applications which are relevant to our work. (Therefore, we may confine ourselves to a mere outline of the theory.) Analytic versions of the theory were recently studied by Bruasse and Teleman [41]. Note that over non-algebraically closed fields of positive characteristic, the Hilbert– Mumford criterion may fail. Indeed, the instability flag may be defined over an inseparable extension of the ground field and then the Galois descent argument will not work anymore. An explicit counterexample from [105] will be presented below.
1.7.1 A Few Words about GIT on Non-Algebraically Closed Fields So far, the ground field has always been the field of complex numbers. It should be no surprise that everything works the same way for any algebraically closed field * of characteristic zero. (Although one has to rewrite the proofs which made use of real Lie groups.) The theory also works for (geometrically) reductive groups over algebraically closed ground fields of arbitrary characteristic (see Remark 1.1.4.7). More generally, one can develop Geometric Invariant Theory also over non-algebraically closed fields or even over base varieties or schemes. For these questions, the reader may consult [155] and [198]. The reader who has some experience with algebraic geometry over arbitrary base fields will probably foresee how the basic definitions and statements will look in this context. As an illustration, let us return to the example of a representation *: SLr (K) −→ GLn (K). As before, it leads to an action σ of SLr (K) on K n and to an action σ of SLr (K) on !n−1;K . We may also define the invariant ring R := K[x1 , . . . , xn ]SLr (K) which is a again a finitely generated K-algebra. Thus, it leads to the K-varieties K n //* SLr (K) := Spec(R) and !n−1;K //* SLr (K) := Proj(R) which are the categorical quotients in the category of varieties or schemes over K. In fact, !n−1;K //* SLr (K) is only the categorical quotient of the open subset of *-semistable points in !n−1;K . Again, a point [v] ∈ !n−1;K is *-semistable, if there exist d > 0 and r (K) with f (v) % 0. The interesting question is whether these conf ∈ K[x1 , . . . , xn ]SL d structions are compatible with extensions of scalars. Given a field extension K ⊂ K B , let *B , σB , and σB be the representations and actions defined over K B which are ob-
S 1.7: U P
97
tained by base change from *, σ, and σB . We ask how the *B -semistable points in !n−1;KB relate to the *-semistable points in !n−1;K and the quotients (K B )n //*B SLr (K B ) and !n−1;K B //*B SLr (K B ) to K n //* SLr (K) and !n−1;K //* SLr (K), respectively. Theorem 1.7.1.1. A point [v] ∈ !n−1;K is *-semistable, if and only if the corresponding point [v xSpec(K) Spec(K B )] ∈ !n−1;K B is *B -semistable. Proof. This is [155], Proposition 1.14.
"
Thus, the notion of semistability is compatible with field extensions. (The corresponding result holds also for polystable but not for stable points.) The quotients also behave well under base change: ^ 4 (K B )n //*B SLr (K B ) ! K n //* SLr (K) x Spec(K B ) Spec(K) ^ 4 B !n−1,KB //*B SLr (K ) ! !n−1;K //* SLr (K) x Spec(K B ). Spec(K)
Let us call a point [v] ∈ !n−1;K *-HM-semistable, if µ(v, λ) ≥ 0 holds for any one parameter subgroup λ: 4m (K) −→ SLr (K). Then, with the above results and the Hilbert– Mumford criterion, we find for an algebraically closed field K B : [v] is *-semistable
⇐⇒ ⇐⇒ =⇒
[v xSpec(K) Spec(K B )] is *B -semistable [v xSpec(K) Spec(K B )] is *B -HM-semistable [v] is *-HM-semistable
In the following, we would like to explain why the converse direction to the last assertion is true in characteristic zero but may fail in positive characteristic. This gives us a clear picture of the theory of semistable points over non-algebraically closed ground fields.
1.7.2 The Theory of the Instability Flag In this section, * will be an algebraically closed field of characteristic zero. We start with the group GLn (*). Let T be the maximal torus of diagonal matrices. The characters ei : diag(l1 , . . . , ln ) 1−→ li , i = 1, . . . , n, form a basis for the character group X . (T ), and . . 71 (., .).: X/ (T ) x X/ (T ) Z &' 'n / n 7 ' n xi · yi i=1 i=1 xi · ei , i=1 yi · ei . defines a scalar product on X/ (T ) := X . (T ) *# 1 which is invariant under the action of the Weyl group W(T ) := N (T )/T . This yields isomorphisms . . (T ) ! Hom/ (X/ (T ), 1) ! X.,/ (T ) := X. (T ) *# 1. X/ . For the second identification, we use the duality pairing F., .4/ : X.,/ (T ) x X/ (T ) −→ 1 . which is the 1-linear extension of the canonical pairing F., .4: X. (T ) x X (T ) −→ ( (see Example 1.1.1.3, vii). Since the pairing (., .). is W(T )-invariant, the norm 0. We look at the representation *: SL2 (K) −→ GL2 (K) X J X p J a b a bp 1−→ . c d cp d p Note that any point in *2 is unstable for the resulting action of SL2 (*). Now, take an element a ∈ K . which does not possess a pth root and set v := (1, a)t ∈ K 2 . Then,
102 S 1.7: U P
S 1.7: U P 102
one easily sees that there cannot exist a one parameter subgroup λ: 4m (K) −→ SL2 (K) with µ(v, λ) < 0.
1.7.3 The Instability Flag in a Product This time, we fix two representations *1 : G −→ GL(W1 ) and *2 : G −→ GL(W1 ) of the reductive group G defined over the field K of characteristic zero. These representations yield an action of G on the product !(W1 ) x !(W2 ), and, for any pair m, n of positive integers, a linearization of the action in the ample line bundle O(m, n). The resulting notion of (semi)stability depends only on the ratio n/m, and we will write O(1, η) for the polarization that is represented by any line bundle of the form O(m, n) with n/m = η. We call the points which are (semi)stable with respect to the linearization in O(1, η) η-(semi)stable. Proposition 1.7.3.1. There is a positive rational number η∞ , such that, for every η > η∞ , a point (x1 , x2 ) ∈ !(W1 ) x !(W2 ) is η-(semi)stable, if and only if a) x2 is semistable (with respect to the linearization in O((W2 ) (1) induced by *2 ) and b) for any non-trivial one parameter subgroup λ: 4m (K) −→ G with µ*2 (x2 , λ) = 0 one has µ*1 (x1 , λ)(≥)0. Proof. The reader can do this as an exercise for the group G = SLn (K), using the formalism suggested in Remark 1.6.2.3. The case of an arbitrary reductive group is similar. The details may be found in [189], Proposition 2.9. " Exercise 1.7.3.2. Assume G = SLn (K). Show that there is an η.∞ , such that for every pair a1 , a2 of positive integers with a2 /a1 > η.∞ , every point (x1 , x2 ) ∈ !(W1 ) x !(W2 ) which satisfies a) and b) in Proposition 1.7.3.1, and every one parameter subgroup λ: 4m (K) −→ G, the condition a1 · µ*1 (x1 , λ) + a2 · µ*2 (x2 , λ) = 0 is equivalent to
µ*1 (x1 , λ) = 0 = µ*2 (x2 , λ).
Verify the same assertion for the group Xti=1 GLri (K). This proposition suggests the following result: Theorem 1.7.3.3. There is a value ηB∞ ≥ η∞ , such that the following assertion holds for any number η > ηB∞ : Let (x1 , x2 ) ∈ !(W1 ) x !(W2 ) be a point for which x2 ∈ !(W2 ) is semistable (with respect to the linearization in O((W2 ) (1) induced by *2 ). If (x1 , x2 ) is not η-semistable, then an instability one parameter subgroup λ: 4m (K) −→ G for it satisfies µ*2 (x2 , λ) = 0. Proof. Although the result is intuitively clear, one has to be rather careful in proving it. The lengthy proof may be found in [189], Theorem 2.10. "
Chapter 2
Decorated Principal Bundles 2.1 Statement of the Classification Problem
"
n this section, we will state the main classification problems which will keep us busy during the rest of the chapter. In order to make them accessible, we will discuss the background from principal bundles.
2.1.1 Principal Bundles — Definitions and First Properties There does not seem to be a suitable textbook which systematically develops the theory of principal bundles in Algebraic Geometry. Therefore, we will give a basic account. We will mainly follow Serre’s article [195] which is still an excellent reference (especially the TEX-ed and annotated reedition in the Documents Math´ematiques). There are also Behrend’s thesis [15] and Sorger’s notes [206] which focus more on the global aspects of the moduli stack and brief introductions by G´omez [78] and by Balaji [7]. Unramified and Galois Coverings Let π: Y −→ X be a morphism between algebraic varieties. We say that π is an unramified covering (of degree n), if π is a finite morphism, such that the induced map %X,π(y) −→ O %Y,y between the completed local rings at π(y) and y, respectively, is for % π#y : O all points y ∈ Y an isomorphism, and n = ['(Y) : '(X)] is the degree of the extension of the function fields. Remark 2.1.1.1. If π: Y −→ X is an unramified covering of degree n (and therefore an affine and flat morphism), then π. (OY ) is a locally free sheaf of rank n on X. Recall that finite groups are examples for linearly reductive algebraic groups. If Y is a quasi-projective algebraic variety on which the finite group Γ acts via σ: Γ x Y −→ Y, then the geometric quotient π: Y −→ X := Y/Γ does exist: If !(V) is the projectivization of a vector space and Γ acts via a representation *: Γ −→ GL(V), then all points y ∈ !(V) are stable. In fact, if y ∈ !(V) is a point, then there exist an n > 0 and a section sB ∈ H 0 (L ), L := O((V) (n), which does not vanish in any point of the orbit Γ · y.
104
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The section s := γ∈Γ γ · sB ∈ H 0 (L ##Γ ) is Γ-invariant and does not vanish in y. If Y is any quasi-projective variety, we may linearize the Γ-action in an ample line bundle . B L . E.g., if L B is an ample line bundle, then L := γ∈Γ σγ (L ) is a Γ-linearized ample line bundle. Theorem 1.4.3.7 from Chapter 1 and the above remarks establish our claim.
-
Remark 2.1.1.2. Note that the quotient map π: Y −→ X is a finite morphism. We call the action σ free, if the stabilizer Γy is trivial for all points y ∈ Y. Proposition 2.1.1.3. If the action σ is free, then the quotient map π: Y −→ X is an unramified covering of degree #Γ. Proof. See [195], Section 1.4
"
We say that π: Y −→ X is a Galois covering (with Galois group Γ), if there is a free group action σ: Γ x Y −→ Y, such that (X, π) is the geometric quotient. Next, we explain how one may associate to an arbitrary unramified covering a Galois covering. Given an unramified covering π: Y −→ X of degree n, the morphism πx n : Y x n −→ X x n is an unramified covering of degree nn . There is the cartesian diagram Y xX n = Y xX · · · xX Y (n factors) % π
3 X
diagonal
f
7 Yx n 3
πx n
7 Xx n.
Let Z be the preimage under f of the (open) set of points (y1 , . . . , yn ), such that y1 ,. . . ,yn are distinct. The induced map * π: Z −→ X is then an unramified covering of degree n!. (The only thing which might not be obvious is properness. For this, one needs that π: Y −→ X is an unramified covering: n distinct points in a fiber of π lie in distinct “sheets” of the covering, so that they cannot degenerate into a configuration of n points in a fiber of π where two or more points agree.) Observe that the symmetric group Σn acts freely on Z and that * π is Σn -invariant. This suffices to conclude that * π is a Galois covering with Galois group Σn . Define pi : Z −→ Y, (y1 , . . . , yn ) 1−→ yi , i = 1, . . . , n. Then, we find * π = π ◦ pi , i = 1, . . . , n. The Galois covering * π therefore factorizes (in various ways) over the original covering π. (The reader who wishes to fill in some more details in the above arguments is referred to [195], Section 1.3 and 1.5.) Fiber Spaces and Principal Bundles Let F and X be algebraic varieties. A fiber space (with fiber F) over X is a pair (P, π) (usually written simply as P) which consists of an algebraic variety P and a morphism π: P −→ X, such that, for every point x ∈ X, there exist an open neighborhood U, an unramified covering W −→ U, and an isomorphism f over W: PW := P x#X W ### ### # π xX idW ### )
f
W.
7 WxF ) )) ))π ) ) W )( )
(2.1)
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Next, suppose that G is an algebraic group and that β: F x G −→ F is an action (from the right) of G on F. A fibered system (with fiber F) over X is a triple (P, βP , π) where P is an algebraic variety, βP : P x G −→ G is an action of G (from the right) on P, and π: P −→ X is a morphism, such that any x possesses U, W −→ U, and f as before with the extra condition that f in (2.1) may be chosen to be equivariant with respect to the G-actions induced by βP and β, respectively. In the special case that F = G and that β is the group law, a fibered system with fiber G over X will be called a principal G-bundle or a principal bundle with structure group G. Again, we will write P instead of (P, βP , π). We leave it to the reader to define the appropriate notion of an isomorphism between fibered spaces or fibered systems. Example 2.1.1.4. i) The first example of a principal G-bundle on a variety X is the product πX : X x G −→ X with the G-action on the second factor by right multiplication. It is called the trivial principal G-bundle. ii) If H '→ +N is an affine algebraic group and G is a closed subgroup, acting on H by right multiplication, then the quotient H/G exists (see [30], §6) and the quotient map π: H −→ H/G is a principal G-bundle over H/G. We will first show that any point possesses and open neighborhood U, such that there exist an e´ tale morphism η: W −→ U and a morphism ϕ: W −→ H with π ◦ η = idU . We have the locally closed embedding H '→ +N ⊂ !N . By Bertini’s theorem ([96], Theorem II.8.18), a general intersection Z of N − dim(G) hyperplanes in !N will intersect G transversely in finitely many points and not meet G \ G. Let W B be an irreducible component of Z ∩ H. Then, π|W B : W B −→ H/G is a dominant morphism between varieties of the same dimension and, therefore, leads to the finite field extension '(W B )/'(H/G). Let η1 : W1 −→ H/G be the normalization of H/G in '(W B ). Then, η1 is a finite morphism, and we have a rational map ϕ: W1 $ H with π ◦ ϕ = η1 where defined. Let V1 be the closed subset where W1 is singular and V2 be the closed subset where ϕ is not defined. Then, U2 := (H/G) \ (η1 (V1 ) ∪ η1 (V2 )) is a non-empty open subset of H/G. −1 Note that η2 := η1|η−1 (U1 ) : W2 := η1 (U 2 ) −→ U 2 is a finite morphism between smooth 1 varieties and that ϕ is defined in W2 . The differential of η2 is the homomorphism dη2 : T W2 −→ η.2 (T U2 ) between vector bundles of the same rank. The closed subset V3 where dη2 is not surjective is a proper subset of H. Set U := W2 \ η2 (U3 ), W := η−1 2 (U), and η := η2|η−1 (U). By construction, these objects satisfy the properties we have asked 2 for. Next, η and ϕ give a section W −→ PW := W xH/G H. Using the right multiplication on H, we obtain a morphism W x G −→ PW . It is immediate that this is a bijective morphism. Since W x G and PW are smooth, this is an isomorphism (see Chapter 1, Proposition 1.2.1.12 and Remark 1.2.1.14). Fix a point x ∈ U. Let xB ∈ H/G be an arbitrary point. Then, there exists an element h ∈ H with h · x = xB . Set U B := h · U, W B := h · W, and ηB := (h·− ) ◦ η ◦ (h−1 ·− ). Then, we also find a G-equivariant isomorphism W B x G −→ W B xH/G H over W B . This concludes the proof that π: H −→ H/G is a principal G-bundle. iii) If Y is an algebraic variety equipped with a free right action β: Y x G −→ Y by the reductive group G and the geometric quotient π: Y −→ Y/G exists, then (Y, β, π) is a principal G-bundle over Y/G. This is a consequence of Luna’s slice theorem (see [139], Corollaire 1, p. 98), observing the following remark.
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iv) Let π: P −→ X be a principal G-bundle over X and f : Y −→ X any morphism. Then, πY : f . (P) := P xX Y −→ Y is a principal G-bundle over Y, the pull-back of P via f . Exercise 2.1.1.5. Let π: P −→ X and πB : P B −→ X be two principal G-bundles. A homomorphism between P and P B is a G-equivariant morphism f : P −→ P B with π = πB ◦ f . Show that any homomorphism between principal G-bundles is actually an isomorphism. Remark 2.1.1.6. i) As we have already mentioned above, we have used the terminology and definitions of Serre’s article [195]. Nowadays, it is more common to define a principal G-bundle as a variety π: P −→ X with a π-invariant G-action P x G −→ P, such that local triviality does hold in the e´ tale or even fppf-topology (see [150], Chapter III, §4). To be locally trivial in the e´ tale topology means, e.g., that we find for every point x ∈ X an e´ tale morphism η: W −→ X, such that the image U = η(W) (which is an open subset of X) contains x and there is an isomorphism f as in Diagram 2.1. The difference is that we do not require that we may choose η to be finite. Thus, our definition seems more restrictive. Local triviality in the fppf-topology appears to be even weaker. However, we will explain in Remark 2.1.1.19 that for the groups which we do consider here, all these definitions are equivalent. This results easily from Example 2.1.1.4. ii) As we have just stressed, we have required the principal G-bundle to be locally trivial in the so-called e´ tale topology. One could ask the bundle to be locally trivial in the Zariski topology, i.e., that we may take the trivialization over a Zariski open neighborhood of any point x ∈ X. This requirement is too restrictive for many purposes (just think of finite groups and Galois covers). There are, however, a few exceptions. To this end, we say that an algebraic group G is special, if any principal G-bundle over an algebraic variety may be trivialized in the Zariski topology, i.e., given a principal G-bundle over a variety X, any point x ∈ X possesses a Zariski open neighborhood U over which there is a G-equivariant trivialization P|U ! U x G. Here are some examples of special groups: • The additive group 4a (') of the field article ([195], Section 4.4, a).
' is special.
This is proved in Serre’s
• The general linear group GLr (') is special (see [150], p. 134, second paragraph). • It follows that any connected solvable linear algebraic group is special (because it admits a filtration whose subquotients are either isomorphic to 4a (') or to 4m (') (see [30], Section 10.1, for the filtration and Lemma 6 in Section 4.4 of [195] for the conclusion on specialty). • The symplectic group Sp2r (') is special (see [195], Section 4.4, c). Grothendieck managed to classify all special groups in [91]. First of all, Serre showed in [195] that a special group is a connected linear algebraic group. Let G be a connected linear algebraic group. Then, the largest connected solvable normal subgroup R(G) exists and is called the radical of G. The quotient Gss := G/R(G) is then a semisimple algebraic group, i.e., a connected reductive group with finite center.
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Grothendieck’s theorem asserts that G is special, if and only if Gss is a product of groups of the form SLr (') or Sp2r ('). In particular, SOr (') is not a special group, if r ≥ 3. Another instance where one may achieve local triviality is in the case of curves. If G is a connected reductive group and X is a smooth projective curve, then any principal G-bundle over X may be trivialized in the Zariski topology. This is because any such bundle “comes from a B-bundle”, B ⊂ G being a solvable subgroup (see Theorem 2.1.1.17). iii) Let ( fi : Wi −→ Ui ), i ∈ I, be an e´ tale covering of X, i.e., the fi are surjective e´ tale morphisms ([96], Exercise III.10.3), such that the open subsets Ui , i ∈ I, form a Zariski covering of X. (For our purposes, we can and will assume that the fi are finite and e´ tale, i.e., unramified coverings). Assume that the principal G-bundle P may be trivialized with respect to this e´ tale covering. Then, it is defined by morphisms ϕi j : Wi xX W j −→ G, i, j ∈ I, obeying the cocycle rule. We infer that any principal G-bundle on X is determined by a cocycle with values in G with respect to a suitable e´ tale covering of X. Therefore, the abstract classification of principal G-bundles on X is performed by the pointed cohomology set Hˇ et1 (X, G) (see [150], Proposition 4.6). iv) Serre’s GAGA theorems assert that a projective algebraic manifold may be treated equivalently as an algebraic variety or as a complex manifold (see [194] and [96], Appendix B). In general, let X be a complex analytic space with its usual strong topology, and let G be a complex Lie group. Then, a holomorphic principal G-bundle (over X) is a complex space P which is equipped with a right action βP : P x G −→ G and a map π: P −→ X, such that every point x ∈ X possesses an open neighborhood (in the strong topology) over which there exists a G-equivariant trivialization. Any algebraic group G may be viewed as a complex Lie group, a projective algebraic variety X may be viewed as a complex space, and a(n) (algebraic) principal G-bundle over X may be naturally interpreted as a holomorphic principal G-bundle over X. More exactly, the category of algebraic principal G-bundles over X with G-equivariant X-morphisms is equivalent to the category of holomorphic principal G-bundles with G-equivariant holomorphic maps over X. The details are explained in §6 of [195]. Thus, if you are willing to work in the complex analytic category, you may trivialize any principal G-bundle in the strong topology. The classification of principal G-bundles up to isomorphy is an interesting topic in its own right. We will treat the problem of constructing moduli spaces in Section 2.4. For some motivation for studying this problem, we refer the reader to the introduction of this book. If the base curve is !1 , we have Grothendieck’s splitting theorem (cf. Example 2.1.1.22). Associated Fiber Spaces Let π: P −→ X be a principal G-bundle over X and F a quasi-projective algebraic variety together with a G-action α: G x F −→ F. Altogether, we find the right action (P x F) x G −→ P x F ^ 4 (p, f ), g 1−→ (p · g, g−1 · f ).
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Proposition 2.1.1.7. In the above setting, the categorical quotient G
P x F := P(F) := (P x F)/G −→ X exists. It is an orbit space and a fiber space with fiber F over X. Proof. We subdivide the proof into several steps. First, we look at the case of a trivial principal bundle, second, at the case of a principal bundle which can be trivialized by means of a single unramified covering, and finally, at the general case. Step 1. — Suppose that P ! X x G is a trivial principal G-bundle. Then, (X x G) x F −→ X x F (x, g, f ) 1−→ (x, g · f ). is the requested categorical quotient, as one easily checks. Step 2. — Suppose that P is a principal G-bundle over the algebraic variety U, such that there are an unramified covering f : W −→ U and a G-equivariant trivialization ψ: P B := P x W −→ W x G. U
As we have seen above, we may assume that f is a Galois covering with Galois group, say, Γ. Denote the Galois action by σ: W x Γ −→ W. The principal G-bundle P B is linearized with respect to the Γ-action, i.e., there is an isomorphism ϕ: π.W (P B ) ! P B x Γ −→ σ. (P B ) of principal G-bundles on W x Γ. The map ϕ
σB : P B x Γ −→ σ. (P B ) −→ P B defines an action of the group Γ on P B (via G-equivariant automorphisms), such that the diagram σB 7 PB PB x Γ 3 W xΓ
σ
3 7W
commutes. One easily checks that the principal G-bundle P on U identifies with the geometric quotient P B /Γ. We use the isomorphism ψ to replace P B by the trivial principal G-bundle W x G. The linearization ϕ or, equivalently, the action σB are given by a map s: W x Γ −→ G. Then, we define the action αB : (W x F) x Γ −→ W x F ^ 4 ^ 4 (w, f ), γ 1−→ σ(w, γ), α(s(w, γ), f ) .
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(Check that this is an action from the right.) We know that the geometric quotient P(F) := (W x F)/Γ does exist. On the other hand, we have the (G x Γ)-action (W x G x F) x(G x Γ) −→ W x G x F ^ 4 ^ 4 (w, h, f ), (g, γ) 1−→ σ(w, γ), s(w, γ) · h · g, α(g−1 , f ) . By Step 1, we thus have
^ 4 P(F) = (P B x F)/G /Γ.
Standard arguments (see Chapter 1, Remark 1.5.3.4) show ^ 4 ^ 4 (P B x F)/G /Γ = (P B x F)/(G x Γ) = (P B x F)/Γ /G = (P x F)/G. Therefore, P(F) is indeed the quotient which we wanted to obtain. Step 3. — In general, we may cover X by open subsets Ui , i ∈ I, such that there exist unramified coverings fi : Wi −→ Ui with trivializations ψi : P xX Wi −→ Wi x G. Step 2 shows that the quotients P|Ui (F), i ∈ I, do exist. Using the universal property of the categorical quotient, these “local quotients” may be glued to a scheme P(F) which comes with a map P(F) −→ X. It is easily verified that this map is separated, so that P(U) is, in fact, a variety. One checks that it is the categorical quotient we were looking for. (Beware that the group G was nowhere required to be linear let alone to be reductive.) " Exercise 2.1.1.8 (Compare [195], §2.3). i) Formulate the properties which the map s: W x Γ −→ G must have in order to define a group action. ii) Given a map s: W x Γ −→ G as in i), we obtain an action of Γ on the trivial principal G-bundle W x G. The quotient P(s) := (W x G)/Γ is a principal G-bundle on U. When do two maps s, sB as in i) give rise to isomorphic principal G-bundles on U?
Example 2.1.1.9. The group GLr (') is the group of linear automorphisms of the vector space 'r . This implies that a principal GLr (')-bundle may be identified with a vector bundle of rank r. In fact, given a principal GLr (')-bundle P, then P('r ) is a vector bundle of rank r, and any isomorphism ϕ: P1 −→ P2 of principal GLr (')bundles leads in a natural way to an isomorphism ψ: P1 ('r ) −→ P2 ('r ) between the associated vector bundles. Conversely, given a vector bundle E of rank r, its frame bundle R Isom('r , E|{x} ), I som(OX$r , E) = x∈X
consisting of the local isomorphisms between the trivial bundle of rank r and E, is a principal GLr (')-bundle. Again, a vector bundle isomorphism ψ: E1 −→ E2 induces in a canonical way an isomorphism ϕ: I som(OX$r , E1 ) −→ I som(OX$r , E2 ) of principal GLr (')-bundles. These constructions establish an equivalence between the groupoid of principal GLr (')-bundles on X with isomorphisms and the groupoid of vector bundles of rank r on X with isomorphisms. Vector bundles are much easier to handle than principal bundles, because they are linear objects.
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Exercise 2.1.1.10. i) The projective linear group PGLr (') = GLr (')/'. · $r is the automorphism group of the projective space !r−1 ([96], Example II.7.1.1). Use this fact to derive an interpretation of PGLr (')-bundles. ii) The group PGLr (') is also the group of automorphisms of the matrix algebra Mr (') ([150], Chapter IV, Corollary 1.5). How can one describe PGLr (')-bundles with this interpretation of PGLr (')? (Note that PGLr (') is not a special group, so that principal PGLr (')-bundles and the objects found in i) and ii) will, in general, not be locally trivial in the Zariski topology.) Example 2.1.1.11. Look at the action c: G x G −→ G (g, h) 1−→ g · h · g−1 of G on itself by inner automorphisms. We discuss the properties of this special case in the following set of exercises. Exercise 2.1.1.12. Let P be a principal G-bundle, and write G for the fiber space associated to it by means of c. i) Show that G −→ X is a group scheme over X whose fibers are isomorphic to G. (This means that there are a section eX : X −→ G , a group law µX : G xX G −→ G , an inversion invX : G −→ G , µX and invX being morphisms of varieties over X, such that (G , eX , µX , invX ) is a group object in the category of varieties over X, i.e., the diagrams describing the group axioms (see [45], Chapter III, §2, or [30], 1.5) are commutative in the category of varieties over X.) ii) Show that G is the group of those automorphisms of (the variety) G that do commute with the right action G x G −→ G, (g, h) 1−→ g · h. iii) Infer from ii) that G identifies with the bundle of local automorphisms of the principal G-bundle P. Thus, we will also write A ut(P) instead of G . (We can also look at the group Aut(P) of automorphisms of the principal G-bundle P (2.1.1.5). The elements of Aut(P) identify with sections X −→ A ut(P).) iv) Suppose there is also a G-action α: G x F −→ F on the variety F. Show that there is an induced action αX : G xX P(F) −→ P(F), i.e., a morphism of varieties over X, such that the diagrams that define an action do commute in the category of varieties over X. Remark 2.1.1.13. If we define principal G-bundles as objects which are locally trivial in the e´ tale or fppf-topology (see Remark 2.1.1.6, i), then the construction of associated fiber spaces is less elementary, but requires the theory of descent ([150], p. 16ff). Extension and Reduction of the Structure Group We discuss two special cases of the construction of associated fiber spaces which provide us with an approach for attacking the classification of principal bundles with vector bundle techniques. Let κ: G −→ H be a homomorphism of algebraic groups. Then, G acts on H by α: G x H −→ H, (g, h) 1−→ κ(g) · h. Note that this action commutes with the right action
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H x H −→ H, (h, hB ) 1−→ h · hB . If P is a principal G-bundle, then we write κ. (P) for P(H). Since, as we have emphasized before, the action α commutes with the multiplication law on H, viewed as a right action of H on itself, the fiber space κ. (P) comes with an action βP : κ. (P) x H −→ κ. (P). One checks that κ. (P) becomes in this way a principal H-bundle. We say that κ. (P) is obtained from P by extension of the structure group (via κ). Example 2.1.1.14. Let κ: G −→ GL(V) be a representation of G. Then, we obtain the vector bundle P(V) and the principal GL(V)-bundle κ. (P). Using Exercise 2.1.1.9, we may identify 4 ^ κ. (P) ! I som V * OX , P(V) . Next, suppose that G ⊂ H is a closed subgroup. Recall that the quotient H −→ H/G exists and is a principal G-bundle (2.1.1.4, ii). The group H acts on H/G via α: H x(H/G) −→ H/G, (h, [hB]) 1−→ [h · hB ]. Let P be a principal H-bundle. Then, G acts in the obvious way from the right on P. The quotient for this right action is given as P/G := P(H/G). The quotient morphism P −→ P/G turns P into a principal G-bundle over the base variety P/G. A reduction of the structure group1 (of P to G) is a section σ: X −→ P/G. Given the pair (P, σ), we form the cartesian diagram 7P
P(σ) 3 X
G- bundle
σ
3 7 P/G.
Obviously, P(σ) is a principal G-bundle. Exercise 2.1.1.15. Verify that κ. (P(σ)) is canonically isomorphic to P, κ: G −→ H standing for the inclusion. (Observe Exercise 2.1.1.5.) Remark 2.1.1.16. Let (P, σ) be as above. By Remark 2.1.1.6, iii), and Exercise 2.1.1.15, we find an e´ tale covering ( fi : Wi −→ Ui ), i ∈ I, of X and a cocycle of the form ϕi j : Wi xX W j −→ G, i, j ∈ I, for P, i.e., a cocycle with values in the smaller group G. Theorem 2.1.1.17. Let G be a connected reductive linear algebraic group, B ⊂ G a Borel subgroup, and X a smooth projective curve. Any principal G-bundle on X admits a reduction of the structure group to B. Therefore, it is locally trivial in the Zariski topology. Sketch of proof. Let '(X) be the function field of X and Y := Spec('(X)). Steinberg [209] proved that Hˇ et1 (Y, G) = H 1 ('(X), G) is trivial (a conjecture of Serre’s). This 1 We
slightly deviate from standard terminology here.
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means that the fiber space (P/B)|Y is also trivial. In particular, there is a '(X)-rational point σY : Y −→ P/B. Since G/B is projective ([30], 11.1), the morphism P/B −→ X is proper, so that σY extends to a section σ: X −→ P/B. According to Exercise 2.1.1.15, there is a principal B-bundle B, such that P ! κ. (B), κ: B −→ G being the inclusion. Since B is solvable, the principal B-bundle B is locally trivial in the Zariski topology. The same applies, of course, to κ. (B). The reader may refer to the paper [61] for more details, generalizations, and an account of the literature on the subject. " Exercise 2.1.1.18. Let κ: G −→ H be the inclusion as above and let P be a principal G-bundle. Then, we find the G-equivariant embedding G-equivariant 7 κ. (P) P2 22 ** 22 ** * 22 22 (**** X
relative to X. If we take the G-quotient, then this embedding becomes a section σ: X = P/G −→ κ. (P)/G. Verify that P(σ) is isomorphic to P. (Again, Exercise 2.1.1.5 may be helpful.) This exercise shows that the procedures of extending and reducing the structure group are inverse to each other and that we may describe principal G-bundles by means of principal H-bundles together with a reduction of the structure group. Since for any linear algebraic group G, we find a faithful representation κ: G −→ GL(V) (Chapter 1, Theorem 1.1.3.3), we may describe principal G-bundles as vector bundles with certain extra structures. We will fully elaborate on this in Section 2.4.1. Below, we state an illuminating example. Remark 2.1.1.19. Now, let (P, βP , π) be a triple which consists of an algebraic variety P, a right action βP : P x G −→ P by the affine algebraic group G, and a G-invariant morphism π: P −→ X. Assume that we can find for every point x ∈ X an fppf-morphism η: W −→ X whose image contains x and a G-equivariant isomorphism f : W x G −→ P xX W over W. We want to show that (P, βP , π) is a principal G-bundle in the sense of our definition. As in Exercise 2.1.1.5, i), we have to show that for every point x ∈ X there exist an open neighborhood U and an unramified covering η: W −→ U, such that P xX W −→ W admits a section. In the sequel, we will refer to this property of a morphism π: Y −→ X, by saying that π locally admits sections over unramified coverings. We may choose a faithful representation *: G −→ GL(V). By Example 2.1.1.4, ii), GL(V) −→ GL(V)/G is a principal G-bundle, so that it locally admits sections over unramified coverings. By Remark 2.1.1.13, we may form the quotients ^ 4 ^ 4 κ. (P) := P x GL(V) /G and κ. (P)/G ! P x(GL(V)/G) /G. Now, κ. (P) is locally trivial in the Zariski topology ([150], Lemma III.4.10) and therefore also a principal GL(V)-bundle in the sense of our definition. Obviously, the fiber space κ. (P)/G −→ X is also locally trivial in the Zariski topology. Since
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113
GL(V) −→ GL(V)/G locally admits sections over unramified coverings by Example 2.1.1.4, the same is true for κ. (P) −→ κ. (P)/G. As before, we may find a section σ: X −→ κ. (P)/G and a cartesian diagram: 7 κ. (P)
P 3 X
σ
3 7 κ. (P)/G.
This shows that P −→ X locally admits sections over unramified coverings, too. Example 2.1.1.20. i) Let (V, ϕ) be a symplectic vector space, i.e., a vector space together with a non-degenerate anti-symmetric bilinear form ϕ: V x V −→ '. Write ψ: V ∨ −→ V for the induced isomorphism. Let Sp(V, ϕ) ! Spr (') be the isometry group of (V, ϕ), r := dim(V). Then, Sp(V, ϕ) acts from the right on Isom(V, 'r ). Let ASr (') ⊂ GLr (') be the closed subvariety of anti-symmetric automorphisms, i.e., invertible linear maps f : 'r −→ 'r with f ∨ = − f . The morphism Isom(V, 'r ) f
−→ ASr (') 1−→ f ◦ ψ ◦ f ∨
exhibits ASr (') as the quotient Isom(V, 'r )/Sp(V, ϕ) (compare Lemma 1.3.3.13 in Chapter 1). Now, let κ: Sp(V, ϕ) −→ SL(V) be the inclusion. According to our previous discussions, we may describe a principal G-bundle as a vector bundle E of rank dim. (V) together with a section σ: X −→ I som(V * OX , E)/Sp(V, ϕ) =: A S (E). By our description of Isom(V, 'r )/Sp(V, ϕ), the bundle A S (E) is the bundle whose ∨ sections over an open subset U ⊂ X are given by anti-symmetric isomorphisms E|U −→ E|U . Thus, a principal bundle with structure group Sp(V, ϕ) (or Spr (')) is essentially the same as a symplectic bundle (E, η), namely a pair consisting of a vector bundle E and a symplectic form η: E ∨ −→ E, i.e., an isomorphism with η∨ = −η. Note that there is the natural notion of an isomorphism of symplectic bundles. The precise statement which follows from the above considerations is that the groupoid of principal Spr (')bundles with isomorphisms is equivalent to the groupoid of symplectic bundles with isomorphisms (compare Lemma 2.4.1.4). ii) Similarly, one shows that the groupoid of principal Or (')-bundles with isomorphisms is equivalent to the groupoid of orthogonal bundles with isomorphisms. Here, an orthogonal bundle is a pair (E, η) which is composed of a vector bundle E and an orthogonal form η: E ∨ −→ E, i.e., an isomorphism with η∨ = η. The notion of an isomorphism of orthogonal bundles is the obvious one. Exercise 2.1.1.21. i) Let (V, ϕ) be a vector space together with a non-degenerate symmetric bilinear form ϕ: V x V −→ '. Let O(V, ϕ) be the isometry group of (V, ϕ) and SO(V, ϕ) := O(V, ϕ) ∩ SL(V). Note that SO(V, ϕ) is isomorphic to the special orthogonal group SOr ('). Finally, let Symr (') ⊂ GLr (') be the space of symmetric invertible
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matrices. Show that the quotient Isom(V, 'r )/SO(V, ϕ) is isomorphic to the normal(!) variety F L Sym0r (') := (m, z) ∈ Symr (') x '. | det(m) = z2 .
Derive a description of principal SOr (')-bundles as vector bundles with additional structure (compare [82]). ii) Let E be a vector bundle of rank r and P = I som(OX$r , E) its frame bundle. Let W• : {0} ! W1 ! · · · ! W s ! 'r be a flag in 'r and Q ⊂ GLr (') its stabilizer. Show that giving a reduction of the structure group of P to Q is the same as giving a filtration E• : {0} ! E1 ! · · · ! E s ! E of E by subbundles with rk(Ei ) = dim. (Wi ), i = 1, . . . , s. Example 2.1.1.22 (Grothendieck’s splitting theorem). Suppose X = !1 . In this case, the classification of principal G-bundles with reductive structure group was carried out by Grothendieck [90] (see also [7]). Let T ⊂ G be a maximal torus. Then, any principal G-bundle P on X possesses a reduction of the structure group to T . Fixing an isomorphism T ! 4m (k)x n , we see that the classification of principal T -bundles is equivalent to the classification of tuples (L1 , . . . , Ln ) of line bundles on X. Thus, isomorphy classes of principal T -bundles are in bijection to tuples (d1 , . . . , dn ) of integers. The Weyl group W := W(T, G) := (Normalizer of T in G)/T acts on T and therefore on the set of principal T -bundles. It is easy to see that two principal T -bundles give isomorphic principal G-bundles, if and only if they belong to the same W-orbit. Hence, there is a bijection between the set of isomorphy classes of principal G-bundles on !1 and the set ($n /W. In the special case G = GLn ('), W is the symmetric group in n letters and acts on ($n by permuting the summands. Thus, we conclude that, for any vector bundle E on !1 , there is a unique tuple (d1, . . . , dn ) of integers, satisfying d1 ≤ · · · ≤ dn , such that E ! O(1 (d1 ) - · · · - O(1 (dn ) (see [165], Theorem 2.1.1, for a proof with vector bundle techniques). This special case was known long before Grothendieck (it is implicit in the work of Dedekind and Weber [48]). Remark 2.1.1.23. If κ: G −→ H is an injective homomorphism, then we have the map 0 0 7 7 Isomorphy classes of Isomorphy classes of −→ principal H-bundles principal G-bundles 6 _ 6 _ P 1−→ κ. (P) .
The above example shows that this map is, in general, not injective. Grothendieck showed in [90] that it is injective for the inclusion Or (') ⊂ GLr (') (cf. also [7]). The Topology of Principal Bundles If not otherwise specified, X will be a connected smooth projective curve over '. The reader who prefers to think in terms of complex analytic geometry may view X as a connected compact Riemann surface. Let G be a reductive group over the complex numbers. As we know, this defines a real Lie group, and, in particular, a topological manifold. Therefore, we may define the fundamental group π1 (G), the reference point being the neutral element in G. (The
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115
fundamental group may also be described intrinsically in terms of the algebraic group G [113].) Proposition 2.1.1.24. Assume that G is connected. Then, there is a bijection 7 0 Isomorphy classes of topological π1 (G) ←→ . principal G-bundles on X Proof. Let us give a sketch of this result. The reader may consult [206], Section 4, or [174], Proposition 5.1, for more details. We remind the reader that there is a topological space BG, such that giving the isomorphy class of a principal G-bundle on the manifold M is the same as giving the homotopy class of a continuous map f : M −→ BG (cf. [208], §19). We see that every topological principal G-bundle on a contractible space, such as a disc, is trivial. Let us verify that the same holds for S 1 : Take two copies D+ and D− of I = [0, 1] and denote by x±0 and x±1 the point in D± corresponding to 0 and 1, respectively. We think of S 1 as being obtained from D+ and D− , by identifying x+0 with x−0 and x+1 with x−1 . Hence, a principal G-bundle on S 1 is obtained by gluing D+ x G with D− x G via gluing maps fi : {x+i } x G −→ {x−i } x G, i = 0, 1. These maps are of the form fi (h) = gi · h, h ∈ G, i = 0, 1, for appropriate g0 , g1 ∈ G (Exercise 2.1.1.12, ii). Let P be the principal G-bundle obtained by the above gluing procedure from g0 , g1 ∈ G. We want to show that P is trivial. Since we assume that G is connected, there are − − paths γ± : I −→ G with γ+ (0) = eG , γ+ (1) = g−1 1 , γ (0) = g0 , and γ (1) = eG . Define the isomorphisms Γ ± : P|D± x G = D± x G
−→
D± x G
(t, h) 1−→ (t, γ± (t) · h). Then, Γ + and Γ − glue to an isomorphism Γ: P −→ S 1 x G. Now, a similar construction works for the manifold X. For this, let x0 ∈ X be any point and choose a small disc D centered at x0 . Set X . := X \ Int(D). Then, one knows that X . has the homotopy type of a wedge product of circles. Therefore, any principal G-bundle is trivial on X . . The same is true for principal G-bundles over D. A similar argument as above shows that the isomorphy class of a principal G-bundle on X is determined by the homotopy class of the gluing map f : S 1 = X . ∩ D −→ G. " Exercise 2.1.1.25. How does one classify principal G-bundles over an n-dimensional sphere with the above approach? (You may consult the book [208] for the solution.)
2.1.2 The Classification Problem for Decorated Principal Bundles Let F be a quasi-projective algebraic variety on which the group G acts (from the left). The objects we would like to consider are pairs (P, β) which consist of a principal G-bundle P and a section β: X −→ P(F). Two such pairs (P1 , β1 ) and (P2 , β2 ) are said to be isomorphic, if there is an isomorphism ψ: P1 −→ P2 with β2 = ψ(F) ◦ β1 ,
ψ(F): P1 (F) −→ P2 (F) being the induced isomorphism.
In this book, we will start with a representation *: G −→ GL(V) and look at the induced actions α: G x !(V) −→ !(V) and κ: G x V −→ V, i.e., we treat the above problem for the fibers F = !(V) and F = V with linear actions.
116 S 2.1: T C P
116 S 2.1: T C P
The Case of Projective Spaces as Fibers Abbreviate P* := P(V). To give a section ^ 4 β: X −→ P !(V) = !(P* ), one has to give a line bundle L on X and a surjection ϕ: P* −→ L, and two pairs (L, ϕ) and (LB , ϕB ) give the same section, if and only if there is an isomorphism χ: L −→ LB , such that ϕB = χ ◦ ϕ ([96], Proposition II.7.12).
Remark 2.1.2.1. A pair (P, β: X −→ !(P* )) is a relative version of a point x in the G-variety !(V): Let G := A ut(P) −→ X be the group scheme over X from Exercise 2.1.1.12. It comes with the action αX : G x !(P* ) −→ !(P* ). X
The section β: X −→ !(P* ) is then a family of points in the G|{x} -varieties !(P*|{x} ), x ∈ X. In the above definition, we cannot replace a principal G-bundle over X by a group scheme G −→ X with fiber G: Such a group scheme is an Aut(G)-bundle, and G −→ Aut(G) is, in general, neither injective nor surjective. The condition that a morphism between vector bundles be surjective is an open condition in a suitable parameter space. On the other hand, we would like to obtain projective moduli spaces. For this reason, we introduce more general objects. A projective bump2 is a triple (P, L, ϕ), consisting of a principal G-bundle P, a line bundle L, and a non-zero homomorphism ϕ: P* −→ L. Let Π(G) be the set of isomorphy classes of topological principal G-bundles on X (compare Proposition 2.1.1.24). The type of the projective *-bump (P, L, ϕ) is the pair (ϑ, deg(L)) where ϑ ∈ Π(G) classifies P as a topological principal G-bundle. We say that two *-bumps (P1 , L1 , ϕ1 ) and (P2 , L2 , ϕ2 ) are isomorphic, if there are isomorphisms ψ: P1 −→ P2 and χ: L1 −→ L2 , such that ϕ2 = χ ◦ ϕ1 ◦ ψ−1 * ,
ψ* : P1,* −→ P2,* being the induced isomorphism.
Problem 2.1.2.2 (The Classification Problem). Fix the type (ϑ, l), ϑ ∈ Π(G), l ∈ and classify projective *-bumps of type (ϑ, l) up to isomorphy.
(,
The problem is that, even if we fix the type of the *-bumps under consideration, they cannot be parameterized in a reasonable way by an algebraic variety (see Section 2.2.3 for the exact meaning of this statement). Thus, we will have to define a priori a concept of semistability which meets the following requirements: • There exist a projective variety P and an open subset U ⊆ P which (over)parameterizes the semistable *-bumps of given type. 2 “Bump”=“bundle
and map”; a generalization of Langer’s “swamp” (see below).
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117
• There is a GL(Y)-action on P which leaves U invariant, such that two points p1 , p2 ∈ U lie in the same orbit, if and only if they correspond to isomorphic *-bumps. • There is a linearization σ of the GL(Y)-action on P, such that the set of σ-semistable points is U. If we can achieve this, the projective variety M (*, ϑ, l) := P//σ GL(Y) will be the moduli space for semistable *-bumps of type (ϑ, l). Thus, our project is both a generalization of GIT (on projective spaces) to a relative setting and an application of GIT. The Case of Affine Spaces as Fibers Here, we take V as the fiber. Thus, the objects we would like to study are pairs (P, ϕ) where P is a principal G-bundle and ϕ: P* −→ OX is a homomorphism (which might be zero). Such an object will be called an affine *-bump. Two affine *-bumps (P1 , ϕ1 ) and (P2 , ϕ2 ) are isomorphic, if there is an isomorphism ψ: P1 −→ P2 , such that the associated isomorphism ψ: P1,* −→ P2,* satisfies ϕ1 = ϕ2 ◦ ψ. For many applications, e.g., in the theory of Higgs bundles ([38], Section 3.2, and [40], Table 3), one needs more general objects. For this, let * := *1 - · · · - *u be the decomposition of * into irreducible representations. Fix a tuple L = (L1 , . . . , Lu ) of line bundles on X and look at tuples (P, ϕ) which are composed of a principal Gbundle P on X and homomorphisms ϕ j : P* j −→ L j , j = 1, . . . , u. If ϑ ∈ Π(G) is the topological type of P, we refer to (P, ϕ) as a twisted affine *-bump of (type (ϑ, L)). Two such twisted affine *-bumps (P1 , ϕ ) and (P2 , ϕ ) are isomorphic, if there is an 1
2
isomorphism ψ: P1 −→ P2 , inducing the isomorphisms ψ j : P1,* j −→ P2,* j , such that ϕ1, j = ϕ2, j ◦ ψ j , j = 1, . . . , u. Problem 2.1.2.3 (The Classification Problem). Fix the type (ϑ, L), ϑ ∈ Π(G), L = (L1 , . . . , Lu ), and classify twisted affine *-bumps of type (ϑ, L) up to isomorphy. Remark 2.1.2.4. Note that this time the line bundles L1 , . . . , Lu are fixed and not moving as the line bundle in the setting of projective bumps. The strategy for obtaining the appropriate moduli spaces (no longer expected to be projective) for these objects is similar to the one outlined for projective *-bumps. This time we aim at a relative version of GIT on affine spaces. In the following sections, we will solve the above classification problems in greater and greater generality, where each step builds on the previous one, before we obtain the new and very general results in Sections 2.7 and 2.8. The respective sections will also include more specific examples and references to the literature. Remark 2.1.2.5 (Affine bumps vs. projective bumps). The distinction between projective *-bumps and affine *-bumps is rather obsolete. (The details will be presented in Section 2.8.4, p. 318ff.) Indeed, assume that we have solved the problem for affine *-bumps.
118
S 2.2: T T V B 118
Now, let *: G −→ GL(V) be a finite dimensional representation of the reductive group G. We look at GB := G x '. with the induced representation *B on Hom(', V). We let all twisting sheaves agree with OX . Identifying a principal '. -bundle with a line bundle, we see that an affine *B -bump is a triple (P, L, ϕ) which consists of a principal G-bundle P, a line bundle L, and a homomorphism ϕ: P* −→ L. (The non-triviality of ϕ will be a consequence of semistability, see p. 320.) Furthermore, (P1 , L1 , ϕ1 ) is isomorphic to (P2 , L2 , ϕ2 ), if and only if there are isomorphisms ψ: P1 −→ P2 and B χ: L1 −→ L2 with ϕ2 = χ ◦ ϕ1 ◦ ψ−1 * . Hence, the theory of affine * -bumps is the same as the theory of projective *-bumps. Our strategy of constructing the moduli spaces of affine *-bumps is, however, to divide it into smaller steps in order to make it more digestable. One of those steps is the construction of moduli spaces of projective bumps with respect to a homogeneous representation, whence the formalism.
2.2 Rudiments of the Theory of Vector Bundles n this section, we will collect some material concerning the classification of vector bundles on the curve X. On the one hand, this will give another, more advanced illustration of how Geometric Invariant Theory is used for constructing moduli spaces in Algebraic Geometry. On the other hand, we obtain some necessary background material for the later sections.
"
2.2.1 The Topology of Vector Bundles Let L be a line bundle on X. Then, we may write it as OX (D) for some divisor D on X ([96], Section II.6). Define the degree of L as deg(L) := deg(D). Since linearly equivalent divisors have the same degree, this definition makes sense. We may view X as a compact Riemann surface and L as a holomorphic line bundle on X. Forgetting a part of this structure, we get an oriented differentiable or topological manifold, called again X, and a differentiable or topological line bundle, respectively, called again L. If [X] ∈ H 2 (X, () stands for the fundamental class of X, then the formula c1 (L) = deg(L) · [X] holds true. Thus, the degree of L is an invariant of the underlying topological line bundle, and it is well-known that two line bundles L and LB define isomorphic topological line bundles, if and only if the equality deg(L) = deg(LB ) is satisfied.3 If E is a vector bundle of rank r on X, then its determinant is the line bundle r 8 det(E) := E, 3 One
uses the exponential sequence {0} −−−−−−−→
!X
exp
−−−−−−−→ CX0/∞ −−−−−−−→ CX0/∞. −−−−−−−→ {0}
with continuous/differentiable functions to infer H 1 (X, CX0/∞. ) ! H 2 (X, !). Being more algebraically minded, one can argue as follows: The condition deg(L) = deg(LB ) is certainly necessary. If it holds, one might easily construct a connected curve C with points c, cB ∈ C and a line bundle LC on C x X with LC|{c} x X ! L and LC|{cB } x X ! LB . Therefore, L and LB are deformations of each other, whence they are topologically and even differentiably isomorphic ([5], Lemma 1.4.3).
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119
and the degree of E is deg(E) := deg(det(E)). Again, we have ^ 4 c1 (E) = c1 det(E) = deg(E) · [X]. Lemma 2.2.1.1. Two vector bundles E and E B on X are isomorphic as differentiable or topological '-vector bundles, if and only if rk(E) = rk(E B ) and deg(E) = deg(E B ). Proof. The stated condition is certainly necessary. Suppose, conversely, that rk(E) = rk(E B ) and deg(E) = deg(E B ). We may find a line bundle L of high degree such that both E * L and E B * L are globally generated. Thus, we may assume without loss of generality that E and E B are globally generated. Then, by a result of Serre ([4], Theorem 2, or [135], Lemma 8.5.3), we may write the vector bundles E and E B as extensions {0} −−−−−→ OX$(rk(E)−1) −−−−−→ E −−−−−→ det(E) −−−−−→ {0}
{0} −−−−−→ OX$(rk(E )−1) −−−−−→ E B −−−−−→ det(E B ) −−−−−→ {0}. B
Any extension of differentiable or topological vector bundles does split4 , so that we obtain the isomorphisms E ∼ OX$(rk(E)−1) - det(E) and E B ∼ OX$(rk(E)−1) - det(E B ) of differentiable or topological vector bundles. The result, therefore, follows from the one for line bundles. " Remark 2.2.1.2. A more topological argument, in the spirit of the proof of Proposition 2.1.1.24, is given in Section 3 of [135].
2.2.2 The Riemann–Roch Theorem In this section, we will review the Riemann–Roch theorem and the Hilbert polynomial. Coherent OX -Modules A torsion sheaf on X is a coherent OX -module T whose support supp(T ) is a finite set of points. For each point x ∈ supp(D), the stalk T x of T at x is a finite dimensional '-vector space. Hence, we may define the degree of T as deg(T ) :=
%
x∈supp(D)
dim. (T x ).
Note that the local rings of X are principal ideal domains. From the structure theory of modules over a principal ideal domain, it follows easily that any coherent OX -module F may be decomposed in a unique way as F := E - T 4 The
(
space of extensions of the vector bundle G by the vector bundle F is, as in Algebraic Geometry 1 ([96], Section III.6), Ext1top/diff (G, F) ! Htop/diff (G ∨ F) in the cohomology of topological or differentiable sheaves. The sheaf of topological or differentiable sections of a topological or differentiable vector bundle is 1 (G ∨ F) = {0} ([108], Theorem 2.11.1), i.e., any extension of G fine ([108], Section I.3.5) so that Htop/diff by F splits.
(
120 S 2.2: T T V B S 2.2: T T V B 120 where E is a locally free sheaf and T is a torsion sheaf. So, we define the degree of F as deg(F ) := deg(E) + deg(T ). The rank of F, written as rk(F ), is the rank of E. Exercise 2.2.2.1. Let {0} −−−−−→ G −−−−−→ F −−−−−→ Q −−−−−→ {0} be a short exact sequence of coherent OX -modules. Then, rk(F ) = rk(G ) + rk(Q) and
deg(F ) = deg(G ) + deg(Q).
The Riemann–Roch Theorem Theorem 2.2.2.2. Let F be a coherent OX -module. Then, χ(F ) = h0 (F ) − h1 (F ) = deg(F ) + rk(F ) · (1 − g). In the statement of the theorem, g stands for the genus of the curve X. Proof. For a general OX -module, we write F = E - T as above and note that χ(F ) = χ(E) + χ(T ). Thus, the Riemann–Roch theorem may be proved separately for torsion sheaves and for locally free sheaves. For a torsion sheaf T , we have h1 (T ) = 0, because T lives on a zero-dimensional scheme. In this case, the theorem is trivial. If L is an invertible sheaf and x ∈ X is a point, the result for torsion sheaves and the cohomology sequence to the exact sequence {0} −−−−−→ L(−x) −−−−−→ L −−−−−→ OX|{x} −−−−−→ {0} show χ(L) = χ(L(−x)) + 1. Since we also have deg(L) = deg(L(−x)) + 1, the theorem holds for L, if and only if it holds for L(−x). Any line bundle on X is of the form OX (D) for a divisor D on X. Our argument therefore reduces the theorem for line bundles to the case L = OX . This case amounts to the equality g = h1 (OX ) which is settled in [96], IV, Proposition 1.1. Any locally free sheaf E of rank r on a curve can be written as an extension {0} −−−−−→ L −−−−−→ E −−−−−→ Q −−−−−→ {0} where L is an invertible sheaf and Q is a locally free sheaf of rank r − 1. Then, we have χ(E) = χ(L) + χ(Q) and, by Exercise 2.2.2.1, also rk(E) = rk(L) + rk(Q) and deg(E) = deg(L) + deg(Q). Thus, we conclude by the induction hypothesis. " Remark 2.2.2.3. It is remarkable that the Riemann–Roch theorem shows that the Euler characteristic of a vector bundle depends only on the topology of the base curve X and the topology of the vector bundle itself, so that it is also a topological invariant.
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121
The Hilbert Polynomial We fix a point x0 ∈ X, write OX (n) for OX (nx0 ), and, given an OX -module F , F (n) for F *OX OX (n), n ∈ (. Then, the Riemann–Roch theorem shows ^ 4 ^ 4 χ F (n) = n · rk(F ) + deg(F ) + rk(F ) · (1 − g) . Hence, we define the polynomial
^ 4 P(F ) := t · rk(F ) + deg(F ) + rk(F ) · (1 − g) ∈ ([t].
It has the property that
^ 4 P(F )(n) = χ F (n)
holds for any integer n. The polynomial P(F ) is the Hilbert polynomial of F .
2.2.3 Bounded Families of Vector Bundles In this section, we study under which conditions families of vector bundles on the curve X may be parameterized by a scheme or a variety. Once one knows how to deal with families of vector bundles, it is no big issue to pass to decorated vector bundles and even principal bundles, with or without decorations. Let P be a property which vector bundles on X may or may not have. Let S be the family5 of isomorphy classes of those vector bundles on X that have property P. An example is the family S(r, d) of isomorphy classes of vector bundles on X of rank r and degree d. We say that the family S is bounded, if there are a scheme S of finite type over ' and a vector bundle ES on S x ', such that, for every vector bundle E on X with [E] ∈ S, there exists a closed point s ∈ S with E ! ES |{s} x X . In other words, every vector bundle from the family S occurs at least once in the family ES . (Here, we view ES as the family (ES |{s} x X , s ∈ S ).) Remark 2.2.3.1. i) Note that, in the definition of boundedness, we may replace S by S red , so that we could have required S to be a—possibly reducible—variety. However, some of the parameter spaces which we shall construct come with a natural scheme structure which need not be reduced. ii) The rank and the degree of the vector bundle ES |{s} x X are constant on each connected component of S ([96], Theorem III.9.9). Therefore, it suffices to deal with families S ⊆ S(r, d), for appropriate numbers r and d. By Serre’s vanishing theorem, for every coherent OX -module F , there exists a positive integer n with h1 (F (n)) = 0. Suppose F is locally free. Then, the exact sequence {0}
7 F (n)
7 F (n + 1)
7 F (n + 1)|{x0}
7 {0}
implies h1 (F (n+1)) = 0. If h1 (F (n)) = 0, then F (n) is globally generated at the point y ∈ X, if and only if h1 (F (n)(−y)) = 0. The observation we have made before and the 5 “family” is a rather imprecise term for either a set of isomorphy classes of vector bundles on X or— later—a vector bundle ES on S x X, S being a “parameter scheme”.
S 2.2: T T V B S 2.2: T T V B 122 122 openness of the condition of being globally generated at a point imply that there is a number nB , such that F (nB ) is globally generated. Again, if h1 (F (m)) = 0 and F (m) is globally generated, then also h1 (F (m + 1)) = 0 and F (m + 1) is globally generated as well. Now, let S be a scheme of finite type (or, more generally, a noetherian scheme) and ES a vector bundle on S x X. Then, the semicontinuity theorem (more exactly, [96], Theorem 12.11) and our remarks above show that there exists an n0 , such that, for every n ≥ n0 and every point s ∈ S , • ES |{s} x X (n) is globally generated and ^ 4 • h1 ES |{s} x X (n) = 0. Altogether, we have shown: Proposition 2.2.3.2. If S is bounded, then there is a natural number n0 , such that, for every vector bundle E with [E] ∈ S and every n ≥ n0 , the following conditions are verified: • E(n) is globally generated and • h1 (E(n)) = 0. Remark 2.2.3.3. i) For r ≥ 2 and d ∈ (, the set of isomorphy classes of vector bundles of rank r and degree d is not bounded. Indeed, looking at the vector bundles E s := OX (−s) - OX (d + s) - OX$(r−2) ,
s ∈ (,
of rank r and degree d, we see that, given n ≥ 0, En+1 (n) fails to be globally generated. ii) By the Riemann–Roch theorem, ^ 4 ^ 4 h0 E(n) − h1 E(n) = rn + d + r(1 − g). Thus, if h1 (E(n)) = 0, then H 0 (E(n)) is a complex vector space of dimension rn + d + r(1 − g). Let us fix n ≥ n0 and a complex vector space Y of dimension rn + d + r(1 − g). Our discussions imply the following assertion: Corollary 2.2.3.4. If S is bounded, then every vector bundle E with [E] ∈ S may be written as a quotient q: Y * OX (−n) −→ E, such that H 0 (q(n)): Y −→ H 0 (E(n)) is an isomorphism. Now, we state a crucial theorem of Grothendieck’s which is indispensable for all the GIT constructions of moduli spaces which we wish to perform. Theorem 2.2.3.5 (Grothendieck’s quot scheme). Fix d and r > 0, and let G be any coherent OX -module. Then, there are a projective scheme Q, a Q-flat family FQ , and a universal quotient qQ : π.X (G ) −→ FQ
S 2.2: T T V B
123
on Q x X, such that, for every sheaf F of rank r and degree d on X and every quotient q: G −→ F , there is a point t ∈ Q with
q ∼ qQ|{t} x X .
(There is an obvious equivalence relation on quotients which amounts to “q1 ∼ q2 , if and only if ker(q1 ) = ker(q2 )”. We will come back to this in Example 2.2.4.12 , i).) Proof. The reader may consult [135], Section 4, [116], Section 2.2, or [215], Section 1.5. " Remark 2.2.3.6. The quot scheme is an example for a fine moduli space, i.e., it fulfills a certain universal property (see Remark 2.2.4.11 and Example 2.2.4.12). In particular, the condition found in Proposition 2.2.3.2 is equivalent to the boundedness of S. To summarize, we have found a necessary and sufficient criterion for a family of isomorphy classes of vector bundles to be bounded. For the applications which we have in mind, we will put it into another form which is easier to handle. Proposition 2.2.3.7. The family S is bounded, if and only if there exists a constant C, such that, for every vector bundle E with [E] ∈ S, A > deg(F) !! ! {0} ! F ⊆ E a subbundle ≤ µ(E) + C. µmax (E) := max µ(F) := rk(F) Proof. We begin with the direction “=⇒”. Fix an n0 , such that h1 (E(n0 )) = 0, for every E with [E] ∈ S. If the constant C did not exist, we would find an extension {0} −−−−−→ F −−−−−→ E −−−−−→ Q −−−−−→ {0} with [E] ∈ S and µ(Q) < −n0 + g − 1. Then, h1 (E(n0 )) rk(Q)
= ≥
h0 (E ∨ (−n0 ) * ωX ) h0 (Q∨ (−n0 ) * ωX ) ≥ ≥ rk(Q) rk(Q) −µ(Q) − n0 + g − 1 > 0,
a contradiction to our assumptions. Next, we prove the direction “⇐=”. Let n be such that 4∨ 4∨ ^ ^ 4 ^ H 1 E(n) = H 0 E ∨ (−n) * ωX = Hom E(n), ωX % {0}, and let ϕ: E(n) −→ ωX be a non-trivial homomorphism. Then, we get the extension ^ 4 7 L := ϕ E(n) 7 {0} 7 F := ker(ϕ) 7 E(n) {0} and r·n+d
= ≤
deg(E(n)) = deg(F) + deg(L) = (r − 1) · µ(F) + deg(L) d (r − 1) · + (r − 1) · n + (r − 1) · C + 2g − 2, r
≤
S 2.2: T T V B S 2.2: T T V B 124 124 i.e.,
d n ≤ − + (r − 1) · C + 2g − 2. r
In other words, for any n ≥ −µ + (r − 1)C + 2g − 1, the conclusion h1 (E(n)) = 0 holds true. Similarly, we conclude that, for every n ≥ −µ + (r − 1)C + 2g, every vector bundle E with [E] ∈ S, and every y ∈ X, one has ^ 4 h1 E(n)(−y) = 0. We derive the exact sequence ^ 4 H 0 E(n)(−y)
^ 4 7 H 0 E(n)
7 E(n)|{y}
7 {0}.
This shows that E(n) is globally generated at y ∈ X. Since y was allowed to be any point on X, we conclude that E(n) is globally generated. "
2.2.4 The Moduli Space of Semistable Vector Bundles In this section, we give a brief account on the classification theory of vector bundles on a smooth projective curve. The reader is advised to consult the more comprehensive sources [153] (for rank two only), [160], [135], and, for the case of an arbitrary projective base variety, [116]. The general problem we would like to solve is the classification of vector bundles on the curve X. The fundamental invariants are the rank and the degree of a vector bundle. In fact, these invariants classify the topological or differentiable '-vector bundle underlying an algebraic vector bundle (Lemma 2.2.1.1). Therefore, we have to classify vector bundles of fixed rank r and fixed degree d for every positive integer r and every integer d. If r = 1 and d = 0, then the solution is provided by the Jacobian variety Jac of the curve (see Chapter VII of [45] or Chapter 11 of [23]). It is an Abelian variety of dimension g = g(X). Thus, except for the case X ! !1 , it is a positive dimensional variety. It solves the classification problem via a bijection between the set of closed points of Jac and the family of isomorphy classes of degree zero line bundles on X which is characterized by a universal property (compare Remark 2.2.4.11 and Example 2.2.4.12). Since the assignment L 1−→ L(−d) provides a bijection between the family of isomorphy classes of line bundles of degree d and the family of isomorphy classes of line bundles of degree zero, Jac is also the answer to the classification problem of degree d line bundles. Leaving aside the case X ! !1 , where vector bundles may be classified by discrete data, thanks to Grothendieck’s splitting theorem (Example 2.1.1.22, [90], or [165]), we find positive dimensional families of pairwise non-isomorphic vector bundles, simply by taking direct sums of line bundles. Thus, an answer to our classification problem will be again a certain variety, called moduli space, together with a bijection of its set of closed points onto the set of isomorphy classes of vector bundles (satisfying a certain open property P). As moduli spaces, we want to allow only schemes of finite type over ', e.g., varieties. Then, an obvious precondition for the existence of a moduli space is that the family of isomorphy classes of vector bundles (with property P) is bounded. As we have seen in Remark 2.2.3.3, the
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125
family S(r, d) is unbounded. Therefore, we may not expect a moduli space for vector bundles of rank r and degree d, so that we will have to impose extra conditions. Remark 2.2.4.1. There is an answer to the classification problem for vector bundles of rank r and degree d by an algebraic stack ([100], [77], [64]). Proposition 2.2.3.7 suggests how the additional conditions may look like. A vector bundle E on X is called (semi)stable, if, for every non-trivial proper subbundle {0} ! F ! E, the inequality µ(F)(≤)µ(E) is satisfied. Remark 2.2.4.2. i) The introduction of the notion of semistability for vector bundles has several reasons. The first and most obvious one is to obtain a bounded family. The second one is the Harder–Narasimhan filtration (see Theorem 2.2.4.4) which tells us that semistable vector bundles are the natural building blocks for all vector bundles on X. The third—and for us most important—reason is that the GIT construction works out very well for the semistable and stable bundles (see Theorem 2.2.4.8 and Remark 2.2.4.11, iii). Another very interesting result is the one of Narasimhan and Seshadri that we have discussed in the introduction. Finally, semistability of a vector bundle E can also be stated in terms of the geometry of the ruled manifold !(E) (see [96], Section V.2, and [133], Section 6.4.B) ii) Note that, if gcd(r, d) = 1, then a vector bundle of rank r and degree d is stable, if and only if it is semistable. Exercise 2.2.4.3. i) Prove that a vector bundle E is (semi)stable, if and only if, for every surjective homomorphism π: E −→ Q onto a non-trivial vector bundle of rank at most rk(E) − 1, the inequality µ(E)(≤)µ(Q) holds true. ii) Prove that any map ϕ: E −→ F between semistable vector bundles is trivial, if µ(E) > µ(F). Theorem 2.2.4.4 (The Harder–Narasimhan filtration). Let E be any vector bundle on the curve X. There is a unique filtration {0} =: E0 ! E1 ! · · · ! E s ! E s+1 := E of E by subbundles, such that the successive quotients E i := Ei+1 /Ei , i = 0, . . . , s, are semistable vector bundles and µ(E 0 ) > · · · > µ(E s ). The bundle E1 in the Harder–Narasimhan filtration is called the maximal destabilizing subbundle and the quotient E s the minimal destabilizing quotient (bundle). Proof. We choose E1 as a subbundle, such that µ(E1 ) = µmax (E) (see Proposition 2.2.3.7) and rk(E1 ) ≥ rk(F) for every subbundle F with µ(F) = µmax (E). Then, we let E2 be a subbundle, such that E2 /E1 has the same properties with respect to E/E1 ,
126 S 2.2: T T V B 126 S 2.2: T T V B and so on. This gives the filtration, and one checks the claimed properties. The reader may also refer to [160], proof of Lemma 5.6.4, page 162, [135], Proposition 5.4.2, or [116], Theorem 1.3.4. " Exercise 2.2.4.5. i) Show that µ(Ei ) > µ(E), i = 1, . . . , s. ii) Set µmin (E) := µ(E s ). Show that, for every surjective map E −→ Q onto a nontrivial vector bundle Q, the inequality µmin (E) ≤ µ(Q) holds true (observe Exercise 2.2.4.3). iii) Let E and F be vector bundles, such that µmin (E) > µmax (F). Verify that there is no non-trivial homomorphism ϕ: E −→ F. iv) If E1 and E2 are two semistable vector bundles, then E1 * E2 is semistable, too ([116], Theorem 3.1.4, [133], Corollary 6.4.14). Infer that, for any two vector bundles, µmax (E1 * E2 ) = µmax (E1 ) + µmax (E2 ) and µmin (E1 * E2 ) = µmin (E1 ) + µmin (E2 ). v) Formulate and prove an analog to Proposition 2.2.3.7, using µmin instead of µmax . vi) Let E and F be two vector bundles. Explain how to construct the Harder– Narasimhan filtration of E - F from those of E and F. Use your results to show: µmax (E - F) = max{ µmax (E), µmax (F) } and µmin (E - F) = min{ µmin (E), µmin (F) }. Let Sss (r, d) be the family of isomorphy classes of semistable vector bundles of rank r and degree d. By Proposition 2.2.3.7, it is bounded. Therefore, we may choose an n, such that, for every semistable vector bundle E, we have: • E(n) is globally generated and • h1 (E(n)) = 0. Fix a complex vector space Y of dimension p := rn + d + r(1 − g), and let Q be the quot scheme parameterizing the quotients of Y * OX (−n) of rank r and degree d (see Theorem 2.2.3.5). On the scheme Q, we have the action6 α: GL(Y) x Q −→ Q _ 6 _ g−1 #id q g · q: Y * OX (−n) −→ Q := Y * OX (−n) −→ Y * OX (−n) −→ Q . 6
There is an open subscheme Q ⊂ Q, such that a quotient [q: Y belongs to Q, if and only if
*
OX (−n) −→ Q]
• Q is locally free and • H 0 (q(n)): Y −→ H 0 (Q(n)) is an isomorphism. The set Q is obviously invariant under the GL(Y)-action. The relationship between the study of the GL(Y)-action and our classification problem is stated as the following result: Proposition 2.2.4.6. Let [qi : Y * OX (−n) −→ Ei ], i = 1, 2, be two quotients in the open subset Q. Then, the following conditions are equivalent: 6
The precise definition of the group action may be easily given, using the universal property of Q.
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127
1. The vector bundles E1 and E2 are isomorphic. 2. The quotients [q1 ] and [q2 ] lie in the same GL(Y)-orbit. Proof. The implication “2. =⇒ 1.” is completely trivial. In order to establish the implication “1. =⇒ 2.”, we choose an isomorphism ψ: E1 −→ E2 . Since the two quotients belong to the open subset Q, we may define the isomorphism f:Y
H 0 (q1 (n))
^ 4 7 H 0 E1 (n)
^ 4 7 H 0 E2 (n)
H 0 (ψ(n))
H 0 (q2 (n))−1
7 Y.
There is an element g ∈ GL(Y), such that f is given as left multiplication by g−1 . Thus, we find the commutative diagram Y * OX (−n)
#
g−1 idOX (−n)
7 Y * OX (−n)
q1
3 E1
3
ψ
q1
q2
7 E2 . ψ
Define the quotient qB1 : Y * OX (−n) −→ E1 −→ E2 . The equivalence relation on quotients is such that [q1 ] = [qB1 ]. The above diagram clearly shows [qB1 ] = g · [q2 ], and we are done. " Remark 2.2.4.7. Since the action of the center restrict to the induced action of SL(Y).
'. · idY
⊂ GL(Y) is trivial, we may
The solution of the moduli problem with Geometric Invariant Theory is nowadays provided by the following result of Simpson’s that will be discussed below. Theorem 2.2.4.8 (Simpson). There is a linearization σ of the SL(Y)-action on Q, such that for a point [q] ∈ R := Closure of Q in Q (which will not be the whole quot scheme Q), the following conditions are equivalent: 1. The point [q] is (semi)stable with respect to the given linearization. 2. The point [q] belongs to Q, i.e., q is of the form q: Y * OX (−n) −→ E where E is a vector bundle and H 0 (q(n)): Y −→ H 0 (E(n)) is an isomorphism, and the vector bundle E is (semi)stable. Thus, we may define M ss (r, d) := R//σ SL(Y) = U ss // SL(Y)
and M s (r, d) := U s / SL(Y).
Here, U (s)s ⊂ R is the set of σ-(semi)stable points. As we have seen in Section 1.4, the scheme M s (r, d) = U s / SL(Y) is an orbit space. By Proposition 2.2.4.6, its set of closed points is in bijection to the family of isomorphy classes of stable vector bundles of rank r and degree d on X. The moduli space M ss (r, d), in turn, parameterizes Sequivalence classes of semistable vector bundles of rank r and degree d on X. This concept has to be explained. We start with the following result:
128 S 2.2: T T V B S 2.2: T T V B 128 Proposition 2.2.4.9 (Jordan–H¨older filtration). Every semistable vector bundle possesses a filtration {0} =: E0 ! E1 ! · · · ! E s ! E s+1 := E by subbundles with µ(Ei ) = µ(E), i = 1, . . . , s, such that Ei+1 /Ei is stable, i = 0, . . . ., s. The associated graded object gr(E) :=
%E s
i=0
i+1 /E i
is—up to isomorphism—independent of the filtration. Proof. This is again fairly easy to see. We may proceed by induction on the rank. For r = 1, there is nothing to show. If E is stable, there is also nothing to be checked. Otherwise, we find a proper subbundle {0} ! F ! E with µ(F) = µ(E). The bundle F is itself semistable, and so is Q = E/F. Applying the induction hypothesis to F and Q, we find our filtration. The uniqueness of the associated graded object results from the fact that any homomorphism between two stable vector bundles of the same slope is either zero or an isomorphism. More information is contained in the references [135], Proposition 5.3.7, and [116], Proposition 1.5.2. " Remark 2.2.4.10. Unlike the Harder–Narasimhan filtration, the Jordan–H¨older filtration is not uniquely determined. To see this, the reader may describe the different Jordan–H¨older filtrations of the associated graded object. Finally, two semistable vector bundles E1 and E2 are said to be S-equivalent, if gr(E1 ) ! gr(E2 ). Remark 2.2.4.11. i) The phenomenon that two semistable vector bundles of rank r and degree d define the same point in the moduli space, if and only if they are S-equivalent, is analogous to the fact that two matrices define the same point in the quotient space Mn (')// SLn ('), if and only if their semisimple parts agree (Chapter 1, Section 1.3.3, p. 43f). ii) So far, we have just identified the closed points of the moduli spaces. In order to characterize the moduli spaces by a universal property, we need a functor. Define, for every scheme S of finite type over ', Vector bundles ES on S x X, such that, M for every closed point s ∈ S , the bundle (s)s M (r, d)(S ) := !. ES |{s} x X is a (semi)stable vector bundle of rank r and degree d on X The relation “!” is just isomorphy of vector bundles.7 For any morphism f : S −→ T , we define the map M(s)s (r, d)( f ): M(s)s (r, d)(T ) −→ M(s)s (r, d)(S ) 6 _ [ES ] 1−→ ( f x idX ). (ES ) . 7 There
are other possible equivalence relations on families of vector bundles. The choice of the equivalence relation matters, if one asks whether one obtains a fine or a coarse moduli space ([160], Chapter 1, §2). Since we will not pursue this fine point here, we have opted for the simplest possible equivalence relation. Further information will be given in Remark 2.2.4.11.
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129
Thus, M(s)s (r, d) is a contravariant functor from the category of schemes of finite type over ' to the category of sets. Recall that a scheme M defines the contravariant functor h M , its functor of points, with h M (S ) := Mor(S , M). If F is any contravariant functor from the category of schemes of finite type over ' to the category of sets, we say that the scheme M corepresents F, if there is a natural transformation τ: F −→ h M , enjoying the following universal property: If M B is another scheme and τB : F −→ h MB is another natural transformation, then there is a unique morphism f : M −→ M B , defining the natural transformation h f : h M −→ h MB , such that τB = h f ◦ τ. Of course, M is determined up to canonical isomorphy by the universal property. Now, a scheme M ss is called a coarse moduli space for the functors M(s)s (r, d), if there are an open subscheme M s and natural transformations τ(s)s : M(s)s (r, d) −→ h M(s)s , such that the following properties hold true: • The scheme M (s)s corepresents the functor M(s)s (r, d). • The map ^ 4 Ms (r, d) Spec(') ^ ^ 4 4 = 0 −→ M s Spec(') τs Spec(') : 7 Isomorphy classes of stable vector bundles of rank r and degree d on X is a bijection. • The map
^ 4 4 ^ 4 ^ τss Spec(') : Mss (r, d) Spec(') −→ M ss Spec(')
is surjective and every fiber consists of exactly one S-equivalence class. The universal property that distinguishes M ss (r, d) and M s (r, d) is that M ss (r, d), with the open subscheme M s (r, d), is a coarse moduli space for the functors M(s)s (r, d). This property is derived from the universal properties of the quot scheme Q and of the categorical quotient. Again, we recommend that the curious reader refers to [160], Chapter 5, [135], Section 7, or [116], Chapter 4, for a more precise account of these facts. iii) We stress that the moduli space M ss (r, d) is projective. Therefore, the family of semistable vector bundles of rank r and degree d is a maximal subfamily of the family of all vector bundles of rank r and degree d for which such a coarse moduli space exists. In particular, if we did enlarge the constant C in Proposition 2.2.3.7, we would still get
130 S 2.2: T T V B S 2.2: T T V B 130 a bounded family of vector bundles but no nice moduli space (i.e., no moduli space in the category of schemes of finite type over '). So far, the family of semistable vector bundles seems to be the only one with such a nice moduli space. Note that, if one passes to higher dimensional base manifolds, to singular base varieties, or to decorated vector bundles, this is no longer the case, because then the semistability condition will depend on several choices (see [116] and Section 2.9.2, p. 325, for vector bundles on higher dimensional base manifolds and the subsequent paragraphs for the decorated vector bundles). iv) Assume g = g(X) ≥ 2. The scheme M ss (r, d) is a normal, irreducible projective variety of dimension r2 (g − 1) + 1. The open subvariety M s (r, d) is smooth. Thus, if gcd(r, d) = 1 (see Remark 2.2.4.2, ii), then M s (r, d) is a connected projective manifold. More information may be found in [135], Section 8. v) If we are given a contravariant functor F as above, we say that the scheme M represents F, if there is an isomorphism τ: F −→ h M , of functors. In this case, M is called a fine moduli space for F. If the scheme M represents the functor F, then it also corepresents it. Example 2.2.4.12. Q(r, d)(S ) :=
i) For every scheme S of finite type over ', we set
Quotients qS : π.X (G ) −→ FS on S x X, such that FS is flat over S and, for every closed point s ∈ S , the sheaf FS |{s} x X has rank r and degree d on X
M !.
Two families qS : π.X (G ) −→ FS and qBS : π.X (G ) −→ FSB are isomorphic, if there exists an isomorphism ψS : FS −→ FSB with qBS = ψS ◦ qS or, equivalently, ker(qS ) = ker(qBS ). Theorem 2.2.3.5 is better stated as “There is a projective scheme Q which represents the functor Q(r, d)”. This clearly describes the universal property of Q. ii) Let S be a scheme. We say that two families ES and ESB of vector bundles on S x X are equivalent, if there is a line bundle LS on S , such that ES and ESB * π.S (LS ) are isomorphic in the usual sense. We obtain new moduli functors M(s)s (r, d)B , if we assign to a scheme equivalence classes rather than isomorphy classes of families of (semi)stable vector bundles of rank r and degree d parameterized by it. Then, M ss (r, d) is also a coarse moduli space for the functors Mss (r, d)B . If r and d are coprime, then M s (r, d) = M ss (r, d) will be a fine moduli space for Ms (r, d)B = Mss (r, d)B . These issues are discussed in Newstead’s book ([160], Chapter 1, §2). However, if r and d are not coprime, then M s (r, d) will not be a fine moduli space for Ms (r, d)B . This result is due to Ramanan (compare the following exercise). References are [172], Theorem 2, [60], Th´eor`eme 5.5, and [112], Example 6.8. Exercise 2.2.4.13. Show that M s (r, d) is a fine moduli space for Ms (r, d)B , if and only if there is a universal family EM s (r,d) of stable vector bundles of rank r and degree d on M s (r, d) x X, such that, for any scheme S and any family ES of stable vector bundles of rank r and degree d parameterized by S , there is one and only one morphism f : S −→ M s (r, d), such that ES is equivalent to the pullback of the universal family EM s (r,d) under f x idX .
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131
Illustration Simpson’s linearization is obtained as follows: For m = 0, we look at the universal quotient twisted by π.X (OX (m)), i.e., at ^ 4 ^ 4 m qQ : Y * π.X OX (m − n) −→ FQ * π.X OX (m) . The pushforward
& ^ 4Z Vm := πQ. FQ * π.X OX (m)
will be locally free of dimension pm := d + r(m + 1 − g), and we have the quotient ^ 4 ^ 4 m 0 kQ := πQ. qm : Y * H *O O (m − n) . −→ V , H := H X m Q Q This quotient yields the GL(Y)-equivariant embedding ιm : Q '→ Gr(Y * H, pm ) '→ !
pm ^ &8 4Z Y*H ,
and, thus, a linearization σm of the GL(Y)- and the SL(Y)-action on Q. Now, as long as m is large enough, we may use σ := σm in Theorem 2.2.4.8. Remark 2.2.4.14. An interesting connection between Simpson’s construction of the moduli space and the theory of quiver varieties (see Chapter 1, Section 1.5.1) has been ´ recently discovered by Alvarez-C´ onsul and King [3]. Any moduli space of semistable sheaves on a projective variety (in particular, the moduli space M ss (r, d)) can be realized inside a projective variety M. (Q, n). Here, Q is the quiver with vertices 1 and 2 and h arrows from 1 to 2 and h and n are chosen appropriately. The proof of Theorem 2.2.4.8 is quite long and involved. It is delineated in detail in the references [135], Section 7, and [116], Chapter 4. Below, we will give a full account of the GIT construction of the moduli spaces for decorated vector bundles which includes the most important ideas of Simpson. Hence, we omit the full proof of Theorem 2.2.4.8. Nevertheless, we would like to illustrate the relationship between the notion of a semistable vector bundle and the Hilbert–Mumford criterion. In order to do so, we give a sample computation, using an older approach by Gieseker [75]. Note that there is the GL(Y)-invariant morphism det: Q −→ Jac [q: Y * OX (−n) −→ E]
1−→ det(E)(−d).
On the product Jac x X, there exists a so-called Poincar´e line bundle P.8 It has the property that, for any line bundle N on X, giving the point [N] ∈ Jac, the restricted line bundle P|{[N]} x X is isomorphic to N. The Poincar´e line bundle is not uniquely determined by this property (see Example 2.2.4.12, ii), but given a Poincar´e line bundle 8 Recall that, for all matters related to Jacobian varieties, Poincar´ e bundles, and the functorial properties, you may consult Chapter VII of [45] or Chapter 11 of [23].
132 S 2.2: T T V B S 2.2: T T V B 132 P on Jac x X, another line bundle P B will be a Poincar´e line bundle, if and only if there exists a line bundle L on Jac with P B ! P * π.Jac (L ). For a Poincar´e line bundle P, we may view the line bundle ^ 4 Pd := P * π.X OX (d) as the universal family of degree d line bundles parameterized by Jac. We let ^ 4 qQ : Y * π.X OX (−n) −→ EQ be the restriction of the universal quotient on Q x X to Q x X. The distinguished universal property of the Jacobian variety implies that there exists a line bundle A on Q, such that det(EQ ) ! (det x idX ). (Pd ) * π.Q (A ), for an appropriate line bundle A on Q. Remark 2.2.4.15. The line bundle A is naturally linearized with respect to the SL(Y)action on Q, because the morphism det is invariant under the group action. In principle, it is the line bundle with respect to which we perform the GIT construction according to Chapter 1, Theorem 1.4.3.8. Since Q is not projective, we may not directly apply the Hilbert–Mumford criterion to find out about the stable and the semistable points. Moreover, we wouldn’t know whether the resulting quotient is projective or not. To resolve these additional problems, we will introduce the auxiliary Gieseker space and map (see, e.g., (2.2)). Let us form H := H om
r &8
^ 4Z V * OJac , πJac. Pd * π.X (OX (rn)) .
It is clear that, choosing n large enough, we may assume that H is a vector bundle. We define the Gieseker space 4 as !(H∨ ). If we replace Pd by Pd * π.Jac (L ) for a line bundle L on Jac, the projection formula ([96], Exercise II.5.1.(d)) shows that H transforms into H * L , and thus, by [96], Lemma II.7.9, O& (1) into O& (1) * P. (L ∨ ), P: 4 −→ Jac being the bundle projection. If L is a sufficiently negative power of an ample line bundle on Jac, then O& (1) will be an ample line bundle on 4. The line bundle O& (1) is naturally linearized with respect to the obvious SL(Y)-action on 4. On the Gieseker space, we have therefore realized the necessary input data for a successful application of Geometric Invariant Theory. Next, note that, by general base change properties (see [96], Theorem III.12.11) and the construction of H, the homomorphism πQ.
r ^ &8
r & ^ 4Z 4Z 8 V * OQ −→ πQ. det EQ * π.X (OX (n)) qQ * idπ.X (OX (n)) :
gives rise to an SL(Y)-invariant and injective morphism, the Gieseker map, Gies: Q −→ 4,
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133
. such that det(EQ ) ! (Gies x idX ). (P. (Pd ) * π& (O& (1))), i.e., 4 ^ Gies. O& (1) ! A (as linearized line bundles).
(2.2)
Likewise, the fiber over a point [N] ∈ Jac is given by the projective space !(/∨N ),
/N
r &8 ^ 4Z := Hom Y, H 0 N(rn) .
The Gieseker morphism maps a quotient [q: Y * OX (−n) −→ E] with det(E) ! N to a point in !(/∨N ). Since !(/∨N ) is an SL(Y)-invariant closed subscheme of 4 with O& (1)|(("∨N ) = O(("∨N ) (1) as SL(Y)-linearized line bundles and the Hilbert–Mumford criterion is clearly compatible with such closed embeddings, we have to evaluate the Hilbert–Mumford criterion for points in !(/∨N ) on which SL(Y) acts via a representation. Now, we are in familiar terrain. Let us denote by QN the fiber of det over N(−d). This is the closed subscheme of Q that corresponds to quotients [q: Y * OX (−n) −→ E] with det(E) ! N. The computations which follow are very similar to those on a Graßmannian (see Chapter 1, Exercise 1.5.1.14). Therefore, we assume that the reader will be able to easily go through them. We also use the notation of Chapter 1, Example 1.5.1.12, for the one parameter subgroups of SL(Y). So, let λ = λ(y, γ) be a one parameter subgroup of SL(Y). Set L F I := i = (i1 , . . . , ir ) ∈ { 1, . . . , p }x r | i1 < · · · < ir . Then, the elements
yi = yi1 ∧ · · · ∧ yir ,
5r
i ∈ I,
Y. For a point [ f ] ∈ !(/∨N ), one checks ^ 4 F L µ [ f ], λ(y, γ) = −min γi1 + · · · + γir | f (yi ) % 0, i ∈ I .
form a basis for
For q: Y * OX (−n) −→ E and i ∈ { 1, . . . , p − 1 }, let Fi be the subbundle generated by 4 ^ q F y1 , . . . , yi 4 * OX (−n) . With the above formula, one may verify the following equations: Lemma 2.2.4.16. For a point [q] ∈ QN and a basis y of Y, define i0 = (i01 , . . . , i0r ) with F L i0j := min k = 1, . . . , p | rk(Fi ) = j , j = 1, . . . , r. Then, for any weight vector γ as above, ^ 4 µ Gies[q], λ(y, γ) = −γi01 − · · · − γi0r . In particular, for γ =
' p−1 i=1
αi · γ(i) p ,
p−1 ^ 4 T 4 ^ µ Gies[q], λ(y, γ) = αi · µ Gies[q], λ(y, γ(i) p ) . i=1
134 S 2.2: T T V B S 2.2: T T V B 134 As a consequence of this lemma, we have to work only with the basic weight vectors. For those, we find ^ 4 µ Gies[q], λ(y, γ(i) (2.3) p ) = rk(F i ) · p − r · i. Since i ≤ h0 (Fi (n)), we arrive at the following result: Proposition 2.2.4.17. For [q: Y * OX (−n) −→ E] ∈ QN , the point Gies[q] ∈ !(/∨N ) is (semi)stable, if and only if h0 (F(n)) h0 (E(n)) (≤) rk(F) rk(E) holds for every non-trivial subbundle {0} ! F ! E. The same conclusion holds, of course, for any point [q] ∈ Q with associated Gieseker point Gies([q]) ∈ 4. A difficult argument shows that one may restrict to subbundles with h1 (F(n)) = 0. Then, the condition from the proposition becomes h0 (F(n)) h0 (E(n)) = µ(F) + n + 1 − g(≤)µ(E) + n + 1 − g = . rk(F) rk(E) Corollary 2.2.4.18. For [q] ∈ Q, the point Gies[q] ∈ E is a (semi)stable vector bundle.
4 is (semi)stable, if and only if
Denote by 4(s)s the open set of the points in 4 that are (semi)stable with respect to the linearization of the given SL(Y)-action in O& (1). In the final step, one has to show: Proposition 2.2.4.19. The morphism Gies|Gies−1 (&ss ) : Gies−1 (4ss ) −→ 4ss is proper. Sketch of proof. We will use the valuative criterion of properness in the form [96], Exercise II.4.11. Let (C, 0) be the spectrum of a discrete valuation ring R with quotient field K. Suppose we are given a morphism f : C −→ 4ss which lifts over C . := Spec(K) to a morphism f . : C . −→ Q. The morphism f . may be extended to a morphism . f : C −→ Q, because the latter scheme is projective. By the universal property of Q, this morphism is associated to a family ^ 4 qC : Y * π.X OX (−n) −→ FC on C x X. Define
^ 4 * qC : Y * π.X OX (−n) −→ FC −→ EC := FC∨∨ .
Note that EC is a reflexive sheaf ([97], Corollary 1.3) on the regular two-dimensional scheme C x X, so that it is locally free ([97], Corollary 1.4) and therefore also flat over C (see also [116], Proposition 4.4.2). It is easy to check that the kernel of the
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homomorphism FC|{0} x X −→ EC|{0} x X is exactly the torsion subsheaf T of FC|{0} x X . Observe that * qC|{0} x X is a priori only generically surjective (the potential finite cokernel being the sheaf EC|{0} x X / Im(FC|{0} x X ) of length dim. (T )). By the same construction as before, the family * qC defines a morphism f B : C −→ 4. It agrees with f on C . , whence it equals f . In particular, f (0) ∈ 4ss . From the latter fact, one derives: • H 0 (* qC|{0} x X (n)) is injective (easy) and • EC|{0} x X is semistable (more involved). But then H 1 (EC|{0} x X (n)) = {0}, so that, a posteriori, H 0 (* qC|{0} x X (n)) is an isomorphism, . * qC is a quotient and must agree with qC , because it does so over C . , so that f is a map . from C to Q with image in the preimage of 4ss . Finally, we realize that f extends the map f . and lifts f as required. " According to Chapter 1, Exercise 1.4.3.11, the quotient M ss (r, d) := Gies−1 (4ss)// SL(Y) exists as a projective scheme, containing the orbit space M s (r, d) := Gies−1 (4s )/ SL(Y) as an open subvariety.
"
Exercise 2.2.4.20. i) Fix a line bundle N of degree d and look at the classification problem of (semi)stable vector bundles E of rank r on X, such that det(E) ! N. Show that the moduli spaces M (s)s (r, N) exist as closed subschemes of M (s)s (r, d) and that ; M (s)s (r, d) = [N]∈Jac M (s)s (r, N). ii) Given two line bundles N, N B of degree d, show that M (s)s (r, N) ! M (s)s (r, N B ).
2.3 Decorated Vector Bundles: Projective Fibers
"
n this section, we study results for the case G = GLr ('). According to Example 2.1.1.9, i), we formulate the theory for GLr (') in the language of vector bundles.
2.3.1 Set-Up of the Moduli Problem As before, we fix a representation *: GLr (') −→ GL(V). For technical reasons, we have to assume that it is homogeneous. Recall that this means that there is an integer α, the degree, such that *(z · $n ) = zα · idV ,
∀z ∈ '. .
Any vector bundle E of rank r defines a vector bundle E* with fiber V, by means of the construction from Proposition 2.1.1.7 and the transition between vector bundles and principal GL(')-bundles from Example 2.1.1.9. An isomorphism ψ: E −→ E B between vector bundles of rank r induces a canonical isomorphism ψ* : E* −→ E*B .
136
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The objects which we would like to study in this section are pairs (E, β) which consist of a vector bundle E of rank r and a section β: X −→ !(E* ). Two such pairs (E, β) and (E B , βB ) are isomorphic, if there exists an isomorphism ψ: E −→ E B , such that βB = ψ* ◦ β, ψ* : !(E* ) −→ !(E*B ) being the isomorphism induced by ψ* . Recall that a section β: X −→ !(E* ) is specified by a surjection ϕ: E* −→ L onto a line bundle L on X and that two surjections ϕ: E* −→ L and ϕB : E* −→ LB give rise to the same section β: X −→ !(E* ), if and only if there is an isomorphism χ: L −→ LB with ϕB = χ ◦ ϕ. For this reason, we look at triples (E, L, ϕ) which consist of a vector bundle E of rank r, a line bundle L, and a non-zero homomorphism ϕ: E* −→ L. Such a triple is called a *-swamp.9 The pair (deg(E), deg(L)) is referred to as the type of (E, L, ϕ). Next, two *-swamps (E, L, ϕ) and (E B , LB , ϕB ) are called isomorphic, if there are isomorphisms ψ: E −→ E B and χ: L −→ LB with ϕB = χ ◦ ϕ ◦ ψ* −1 . We fix integers d, l, and make it our business to classify *-swamps of type (d, l) up to isomorphy. Remark 2.3.1.1. i) The datum * fixes the rank of the vector bundles involved. Hence, the integer d determines the topological type of the participating vector bundles (see Lemma 2.2.1.1). If ϕ happens to be surjective, then l determines the cohomology class of β(X) in the manifold !(E* ) (which, as a topological manifold, does not depend on the *-swamp of type (d, l)). So, fixing the type amounts again to fixing the topological background data. ii) Recall from Chapter 1, Corollary 1.1.5.4, that we may find non-negative integers a, b, c, such that the homogeneous representation * is a direct summand of the representation *a,b,c of GLr (') on the vector space r ^ 4$b ^8 4 Wa,b,c = ('r )#a * 'r #−c ,
i.e., there is a representation *B of GLr ('), such that *a,b,c = * - *B . Therefore, it will be sufficient to study the case of the representation *a,b,c . Indeed, for a vector bundle E of rank r, E*a,b,c = E* - E*B , so that we recover the *-swamps as the *a,b,c -swamps (E, L, ϕ) with ϕ|E*B ≡ 0. Note that everything is compatible with isomorphisms. iii) The moduli problem for *-swamps on a curve as we have described it, so far, was first solved in [187]. Later, it was studied and solved along similar lines on higher dimensional manifolds in [82] (which is now part of [79]). Although it looks like a very special case of our general program, it is indeed the fundamental result to which all the other results will be reduced via some technical tricks. 9 “Swamp”=“sheaf
with a map”. This terminology was suggested by Adrian Langer. In [187], we gave an object as above the more boring name of a *-pair.
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An Example We take G = GLr (') and *: G −→ Symd ('r ). Let (E, L, ϕ) be a *-swamp. This defines a geometric object: For this, let P(E) := !(E ∨ ) −→ X π
be the projectivization of E in the classical sense, i.e., P(E) parameterizes the lines in the fibers of E. Let D ⊂ P(E) be an effective divisor which is relatively ample, i.e., it projects surjectively onto the base curve X. Then, OP(E) (D) = OP(E) (d) * π. (L) for a unique positive integer d and a unique line bundle L on X, and D is the zero divisor of a section s: OP(E) −→ OP(E) (d) * π. (L). Project this onto X in order to obtain π. (s): OX −→ Symd (E ∨ ) * L. Now, as representations, Symd ('n∨ ) ! Symd ('n )∨ ,
by Chapter 1, (1.3).
Thus, s is the same as a non-trivial homomorphism ϕ: Symd (E) −→ L. There is the induced morphism πD : D −→ X for the fibers of which we have 7 hypersurface of degree d, if ϕ is surjective in x −1 πD (x) = . ∨ !(E|{x} ), else Hence, a *-swamp (E, L, ϕ) basically describes a family of hypersurfaces of degree d (inside P(E)), and isomorphy is a relative version of projective equivalence. In Figure 2.1 (generated with Polyray10) and the cover illustration, we have displayed the real part of an affine piece of a family of cubic curves. The total space is a smooth variety, but two singular cubic curves occur. One of them is the union of three lines and is clearly discernible in the picture. The other one is a triple line which has disappeared into infinity. This example should illustrate that the theory of decorated vector bundles may be used to deal with the classification of some interesting projective algebraic varieties. 10
http://wims.unice.fr/wims/en−tool∼geometry∼polyray.en.html
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Figure 2.1: The surface { (t − 1)(x2 y − xy2 ) + t = 0 }
2.3.2 Semistability of Swamps If we look at the GLr (')-action on !(V) that is induced by the representation *, then the homogeneity of * implies that the center '. · $r acts trivially. Thus, it suffices to look at the SLr (')-action. A similar thing happens for *-swamps. The abstract GIT setting gives us an idea what the testing objects for semistability and stability should be. For this, recall that the Hilbert–Mumford criterion for an action of SLr (') had, at the end, to be tested against weighted flags in the vector space 'r (Chapter 1, Example 1.5.1.36). In analogy, we define a weighted filtration of the vector bundle E on X to be a pair (E• , α• ) which consists of a filtration E• : {0} ! E1 ! · · · ! E s ! E of E by subbundles and a vector α• = (α1 , . . . , α s ),
αi ∈ 3>0 , i = 1, . . . , s.
Remark 2.3.2.1. In the weighted flag (W• , α• ) associated to a one parameter subgroup λ: '. −→ GLr ('), the numbers αi lie in ([1/r]>0 . For some computations in the GIT process and some arguments, it is, however, more convenient to have arbitrary positive rational numbers at our disposal. The “µ-function” which we do have to define for a *-swamp (E, L, ϕ) on weighted filtrations of E will be composed of two ingredients. The first ingredient will come
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from the “moving vector bundle”. Here, our considerations from Section 2.2.4, in particular Lemma 2.2.4.16 and (2.3), help. For a weighted filtration (E• , α• ) of E, we hence set s T ^ 4 M(E• , α• ) := αi · deg(E) · rk(Ei ) − deg(Ei ) · rk(E) . i=1
Next, suppose we are given a *-swamp (E, L, ϕ) and a weighted filtration (E• , α• ) of E. The second ingredient will be the quantity µ(E• , α• , ϕ) which comes from the GIT along the fibers of !(E* ) −→ X. Let η ∈ X be the generic point of the curve X and K := '(X) the function field. Set (2.4) $ := E|{η} , .* := E*|{η} . Then, $ is a finite dimensional K-vector space and .* is a finite dimensional SL($)module. The restriction ϕη of ϕ to the generic point is an element of !(.* ). First, suppose that α• consists of elements in ([1/r]>0. Then, by restriction, the weighted filtration (E• , α• ) defines a weighted flag ($• , α• ) inside $. This weighted flag comes from some one parameter subgroup λK : 4m (K) −→ SL($), and we set µ(E• , α• , ϕ) := µ(ϕη , λK ). According to Chapter 1, Example 1.5.1.36, this is well-defined. If the αi are arbitrary positive rational numbers, then we choose an integer m with m·αi ∈ ([1/r], i = 1, . . . , s, and set 1 µ(E• , α• , ϕ) := · µ(E• , (m · α• ), ϕ). m It is easy to see that this is well-defined, too. Exercise 2.3.2.2. i) Check that the above definition agrees with the one given in [187]. To make things precise, choose a basis w = (w1 , . . . , wr ) for W := 'r , define Wi := F w1 , . . . , wrk(Ei ) 4, i = 1, . . . , s, and fix an open subset ∅ ! U ⊆ X with the following properties: • The map ϕ|U : E*|U −→ L|U is surjective. • There is a trivialization ψ: E|U −→ W * OU with ψ(Ei|U ) = Wi * OU ,
i = 1, . . . , s.
(Check that such a trivialization does exist.) You will get the following morphism “ψ”
ϕ|U
β: U −→ !(E*|U ) ! Finally, define γ :=
s T i=1
!(V) x U −→ !(V).
αi · γr(rk(Ei )) .
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With these data, verify that
A ^ > 4! µ(E• , α• , ϕ) := max µ β(x), λ(w, γ) !! x ∈ U .
(2.5)
ii) Here, we will assume that V = Wa,b,c , so that a *-swamp is a triple (E, L, ϕ) with ϕ: (E #a )$b −→ det(E)#c * L. Let (E• , α• ) be a weighted filtration and define the associated weight vector s 4 T ^ (rk(E )) α j · ·γr j . γ1 , . . . , γ1 , γ2 , . . . , γ2 , . . . , γ s+1 , . . . , γ s+1 := D!!!!!!!!!!WB!!!!!!!!!!\ D!!!!!WB!!!!!\ D!!!!!WB!!!!!\ rk(E1 )
rk(E)−rk(E s )
rk(E2 )−rk(E1 )
(2.6)
j=1
Prove that, using E s+1 := E, > A ! µ(E• , α• , ϕ) = − min γι1 +· · ·+γιa !! (ι1 , . . . , ιa ) ∈ { 1, . . . , s+1 }x a : ϕ|(Eι1 #···#Eιa )"b & 0 . (2.7) Thus, our definition of µ(E• , α• , ϕ) agrees with the one used in [82]. iii) For an index tuple ι = (ι1 , . . . , ιa ) ∈ { 1, . . . , s + 1 }x a , we declare F L (2.8) ν j (ι) := # ιi ≤ j | i = 1, . . . , a . If ι ∈ { 1, . . . , s + 1 }x a is any index tuple, check that −(γι1 + · · · + γιa ) =
s T
^ 4 α j · ν j (ι) · r − a · rk(E j ) .
(2.9)
j=1
The quantity µ(E• , α• , ϕ) has been defined with respect to the linearization in the line bundle O((V) (1) (or O(('* ) (1)). However, we may also define it with respect to the linearized 3-line bundle O((V) (δ), δ ∈ 3>0 . This gives the parameter in our theory. In the GIT construction of the moduli spaces, we will see that δ corresponds, indeed, to the choice of an ample line bundle on the parameter space. So, let δ be a positive rational number. A *-swamp (E, L, ϕ) is said to be δ(semi)stable, if M(E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0 holds for every weighted filtration (E• , α• ) of E. There is also a notion of S-equivalence which one needs in order to describe the closed points of the moduli space. The structure of this equivalence relation is the following: Given a δ-semistable *-swamp (E, L, ϕ) and a weighted filtration (E• , α• ) with M(E• , α• ) + δ · µ(E• , α• , ϕ) = 0, one constructs a new δ-semistable *-swamp df (E• ,α• ) (E, L, ϕ) = (Egr , L, ϕgr ), the admissible deformation associated to (E• , α• ). To begin with, let us fix a flag W• : {0} ! W1 ! · · · ! W s ! W := 'r
with
dim. (Wi ) = rk(Ei ), i = 1, . . . , s,
and a one parameter subgroup λ: '. −→ GLr (') whose weighted flag is (W• , α• ). The one parameter subgroup λ also defines a filtration V• : {0} =: V0 ! V1 ! · · · ! Vt ! Vt+1 := V.
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Exercise 2.3.2.3. i) Show that the parabolic subgroup QG (λ) from Chapter 1, Proposition 1.5.1.32, fixes the flag V• . Consequently, Vgr :=
% V /V t+1 i=1
i
i−1
inherits the structure of a QG (λ)-module. ii) Check that the Levi-subgroup LG (λ) (Chapter 1, (1.17)) fixes the λ-eigenspaces in V. Conclude that the LG (λ)-modules V and Vgr are canonically isomorphic.
!
s+1 First, we set Egr := i=1 E i /E i−1 , E s+1 := E. To simplify our arguments, we will work with cocycles. Let (U j ) j∈I be a covering, such that, over U j , there is a trivialization ψ j : E|U j −→ W * OU j with ψ(Ei|U j ) = Wi * OU j , i = 1, . . . , s, j ∈ I. The resulting cocycle takes values in QG (λ) and is thus of the form
U j ∩ Uk x
−→
QG (λ)
1 M jk (x) 1−→ C jk (x) = 0
..
. .
M s+1 jk (x)
,
j, k ∈ I.
The vector bundle Egr is given with respect to the open covering (U j ) j∈I by the cocycle U j ∩ Uk x
−→
LG (λ)
1 M jk (x) 1−→ Cgr, jk (x) = 0
..
0 .
M s+1 jk (x)
,
j, k ∈ I.
The vector bundle E* is thus given by the cocycle *(C jk ) j,k∈I and Egr,* by the cocycle *(Cgr, jk ) j,k∈I . Exercise 2.3.2.3, i), implies that E* has a filtration F• : {0} =: F0 ! F1 ! · · · ! Ft ! Ft+1 := E* . Part ii) of that exercise demonstrates that Egr,* !
% F /F t+1
i=1
i
i−1 .
(2.10)
We denote by V i ⊂ V the eigenspaces of λ. Let i0 be the index of the eigenspace to the weight −µ(E• , α• , ϕ). Then, by definition, i0 is the first index for which ϕ|Fi0 is non-trivial. Thus, ϕ induces a non-trivial homomorphism ϕ0 : Fi0 /Fi0 −1 −→ L. Using the description of Egr,* in (2.10), we may define ϕgr as ϕ0 on Fi0 /Fi0 −1 and as zero on the other components. This concludes the construction of the admissible deformation. (The reader who feels uncomfortable with the use of cocycles may give an intrinsic definition, using principal bundles and extensions and reductions of structure groups, taking into account Exercise 2.1.1.21.)
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Formally, the relation of S-equivalence is generated by (E, L, ϕ) ∼ df (E• ,α• ) (E, L, ϕ), for every weighted filtration (E• , α• ) with µ(E• , α• , ϕ) = 0. If (E, L, ϕ) is isomorphic to all of its admissible deformations, we call it δ-polystable. Then, for every δ-semistable *-swamp (E, L, ϕ), any sequence of non-trivial11 admissible deformations will lead after finitely many steps to a δ-polystable *-swamp, and two δ-polystable *-swamps are S-equivalent, if and only if they are isomorphic. In order to introduce a functor, we first fix a Poincar´e line bundle L on Jacl x X. (We write Jacl to indicate that we view the Jacobian as the moduli space of line bundles that have degree l.) For every scheme S and every morphism κ: S −→ Jacl , we define L [κ] := (κ x idX ). (L ). Now, let S be a scheme of finite type over '. Then, a family of *-swamps of type (d, l) (parameterized by S ) is a tuple (ES , κS , NS , ϕS ) which has the following entries: • a vector bundle ES of rank r on S x X, having degree d on {s} x X for all s ∈ S , • a morphism κS : S −→ Jacl , • a line bundle NS on S , • a homomorphism ϕS : ES ,* −→ L [κS ] * π.S (NS ) whose restriction to {s} x X is non-zero for every closed point s ∈ S . Two families (ES1 , κ1S , NS1 , ϕ1S ) and (ES2 , κ2S , NS2 , ϕ2S ) are called isomorphic, if κ1S = κ2S =: κS and there exist isomorphisms ψS : ES1 −→ ES2 and χS : NS1 −→ NS2 with ^ 4−1 ϕ1S = idL [κS ] * π.S (χS ) ◦ ϕ2S ◦ ψS ,* . We define the functors Mδ-(s)s (*, d, l): Sch. S
−→ Set Isomorphy classes of families of δ-(semi)stable *-swamps 1−→ of type (d, l) parameterized by S
.
Remark 2.3.2.4. The definition of the moduli functor involves the choice of the Poincar´e line bundle L . Nevertheless, the above moduli functor is independent of that choice. Indeed, choosing another Poincar´e line bundle L B on Jacl x X, there is a line bundle NJacl on Jacl with L ! L B * π.Jacl (NJacl ). Therefore, assigning to a family (ES , κS , NS , ϕS ) defined via L the family (ES , κS , NS * κ.S (NJacl ), ϕS ) defined via L B identifies the functor which is defined with respect to L with the one defined with respect to L B . Using terminology analogous to the one in Remark 2.2.4.11, ii), we state the main result of this section:
Theorem 2.3.2.5. Let *: GLr (') −→ GL(V) be a homogeneous representation. Fix integers d, l, and a positive rational number δ. Then, the coarse projective moduli space M δ-ss (*, d, l) for the functors Mδ-(s)s (*, d, l) does exist. 11 i.e.,
leading to a non-isomorphic swamp
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2.3.3 Examples The definition of δ-semistability resembles the Hilbert–Mumford criterion in GIT. In Chapter 1, Example 1.5.1.18, we have seen how the Hilbert–Mumford criterion for a given action of the group SLr (') can be systematically simplified. There are several devices to simplify the concept of δ-semistability in terms of the representation *, too. In the examples below, we state the condition of δ-semistability in the simplified form. The reader may refer to [187], Section 3, or to [37] for more examples. Bradlow Pairs Here, we choose * = idGLr (.) . Thus, a *-swamp is a triple (E, L, ϕ) where E is a vector bundle of rank r, L is a line bundle, and ϕ: E −→ L is a non-trivial homomorphism. The simplified12 semistability concept reads as follows: A *-swamp (E, L, ϕ) is δ-(semi)stable, if and only if, for every subbundle {0} ! F ! E, µ(F) µ(F) −
δ rk(F)
δ , rk(E) δ (≤) µ(E) − , rk(E)
(≤) µ(E) −
if F ⊆ ker(ϕ) if F $ ker(ϕ).
The semistability concept for the above pairs was introduced by Bradlow in [36]. It was the first vector bundle problem where the notion of semistability depended on a parameter. Bradlow pairs can also be related to certain vector bundles on the surface X x !1 . Using this relation, Garc´ıa-Prada constructed the moduli space for Bradlow pairs inside the moduli space of sheaves on X x !1 (see [71]). An interesting application of Bradlow pairs was given by Thaddeus in [213] (see also below). Exercise 2.3.3.1. The reader should check that the above conditions are those which arise from weighted filtrations of the form ({0} ! F ! E, (1)). Conic Bundles Here, we take *: GL3 (') −→ GL(Sym2 ('3 )), i.e., a *-swamp consists of a vector bundle E of rank 3, a line bundle L, and a non-zero homomorphism ϕ: Sym2 (E) −→ L. These objects were studied in [81]. If ϕ is everywhere surjective, then such a *-swamp describes a conic bundle π: C −→ X, i.e., a projective surface which is fibered over X in plane conics. If F and G are subbundles of E, we let F · G be the subbundle of Sym2 (E) that is the image of F * G ⊆ E * E under the projection E * E −→ Sym2 (E). Let (E, L, ϕ) be a *-swamp as above. For a subbundle {0} ! F ! E, we set 2, if ϕ|F·F & 0 1, if ϕ|F·F ≡ 0 and ϕF·E & 0 . cϕ (F) := 0, if ϕ ≡0 |F·E
12 We
will explain this in Section 2.8.4, p. 307ff.
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A filtration E• : {0} ! E1 ! E2 ! E is called critical, if ϕ|E1 ·E2 ≡ 0,
ϕ|E1 ·E & 0,
and ϕ|E2 ·E2 & 0.
Then, a *-swamp is δ-(semi)stable, if and only if it satisfies the following conditions: • For every subbundle {0} ! F ! E, µ(F) − δ ·
cϕ (F) 2 (≤)µ(E) − δ · . rk(F) 3
• For every critical filtration E• : {0} ! E1 ! E2 ! E, deg(E1 ) + deg(E2 )(≤)deg(E). Remark 2.3.3.2. i) The reader should refer to Chapter 1, Example 1.5.1.19. Then, he or she will probably recognize how the simplification takes place. Otherwise, he or she may consult [187], Section 3.7. ii) This example is interesting, because it is probably the first and most elementary example where the stability condition is not just a condition formulated on subbundles. In fact, we need also filtrations of length two to state it. iii) The reader is invited to translate conditions such as ϕ|F·F ≡ 0 into geometric ones. iv) The generalization of this example to arbitrary rank will be given in Section 2.8.4. There, we will also give more detailed justification for the simplification process.
2.3.4 Boundedness According to Remark 2.3.1.1, ii), we assume that * agrees with the representation *a,b,c of GLr (') on the vector space r ^ 4$b ^8 4#−c Wa,b,c = ('r )#a * 'r
for suitable non-negative integers a, b, and c. Fix integers d and l and the stability parameter δ ∈ 3>0 . As a first step, we would like to show that the family S of isomorphy classes of vector bundles E, such that there exists a δ-semistable *a,b,c -swamp (E, L, ϕ) of type (d, l), is bounded. To this end, we apply Criterion 2.2.3.7. In fact, we find the following slightly more precise result: Theorem 2.3.4.1. There is a non-negative constant C1 , depending only on r, a, and δ, such that, for every δ-semistable *a,b,c -swamp (E, L, ϕ) with deg(E) = d and every non-trivial proper subbundle F of E, µ(F) ≤
d + C1 . r
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Proof. Let E• : {0} ! F ! E be any subbundle. By Chapter 1, Lemma 1.5.1.41, µ*a,b,c (E• , (1), ϕ) ≤ a · (r − 1), so that δ-semistability gives d rk(F) − deg(F)r + δa(r − 1) ≥ d rk(F) − deg(F)r + δµ*a,b,c (E• , (1), ϕ) ≥ 0, i.e., µ(F) ≤
d δ · a · (r − 1) d δ · a · (r − 1) + ≤ + , r r · rk(F) r r
so that the theorem holds for C1 := δ · a · (r − 1)/r.
"
Remark 2.3.4.2. It is important to note that the constant C1 obtained above does not depend on the degree of L. A much stronger boundedness result is true: Theorem 2.3.4.3. There is a non-negative constant C2 , depending only on r, a, c, d, and l, such that, for every δ ∈ 3>0 , every δ-semistable *a,b,c -swamp (E, L, ϕ) with deg(E) = d and deg(L) = l, and every non-trivial proper subbundle F of E, µ(F) ≤
d + C2 . r
Proof. Suppose (E, L, ϕ) is a *a,b,c -swamp of type (d, l) which is δ-semistable for some δ > 0. Assume E is not semistable as a vector bundle and consider its Harder– Narasimhan filtration (Theorem 2.2.4.4) E• : {0} = E0 ! E1 ! E2 ! · · · ! E s ! E s+1 = E. We use the notation E i = Ei /Ei−1 , ri := rk(Ei ), ri := rk(E i ), and µi := µ(E i ), i = 1, . . . , s + 1. Define s+1 > A T ! γi · ri = 0 . C(E• ) = γ = (γ1 , . . . , γ s+1 ) ∈ 1 s+1 !! γ1 ≤ γ2 ≤ · · · ≤ γ s+1 , i=1
We equip 1 s+1 with the maximum norm 0, we write again F (n) instead of F * OA (1)#n . Suppose we are given two vector bundles ES and FS on S x A, as well as a homomorphism ϕS : ES −→ FS .
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Proposition 2.3.5.1. There is a closed subscheme Z ⊆ S , satisfying the following universal property: A morphism f : T −→ S factorizes over Z, if and only if the homomorphism ( f x idA ). (ϕS ): ( f x idA ). (ES ) −→ ( f x idA ). (FS ) is trivial. Proof. Let l and m be the ranks of E and F, respectively. First, assume that A = {pt} and that E and F are trivial vector bundles. After choosing trivializations of E and F, the homomorphism is given by an (m x l)-matrix M with values in OS . Let I be the ideal sheaf which is generated by the entries of M. The subscheme Z ⊆ S defined by I has the desired property. By the universal property, the theorem follows also, if ES and FS are only locally trivial. Now, let A be an arbitrary projective scheme. Then, for all n = 0, the sheaves & & ^ 4Z ^ 4Z Vn := πS . ES * π.A OA (n) and Wn := πS . FS * π.A OA (n) will be locally free, and we have the homomorphism ψn := πS . (ϕS
*
idπ.A (OA (n)) ): Vn −→ Wn .
The construction given for A = {pt} applies to ψn and provides us with the subscheme Z. To see that it has the right universal property, we note that, for every morphism f : T −→ S , we have the following commutative diagram: π.T f . (Vn ) ( f x idA ). π.S (Vn ) ^ 3 4 ( f x idA ). ES * π.A (OA (n)) ^ 4 ^ 4 ( f x idA ). (ES ) * π.A OA (n)
π.T f . (ψn )
7 π. f . (Wn ) T
( f x idA ). π.S (ψn )
7 ( f x idA ). π. (Wn ) S
( f x id A ). (ϕS
#π.A (idπ.A (OA (n)) )) #
(( f x idA ). (ϕS )) idπ. OA (n) A
^ 3 4 7 ( f x idA ). FS * π. (OA (n)) A
^ 4 ^ 4 7 ( f x idA ). (FS ) * π. OA (n) . A
The vertical maps between the second and the third row come from the natural evaluation map & ^ 4Z ^ 4 π.S πS . ES * π.A OA (n) −→ ES * π.A OA (n) and the corresponding map for FS . This diagram shows that the vanishing of f . (ψn ), i.e., the factorization of f over Z, is equivalent to the vanishing of ( f x idA ). (ϕS ), as required. " Remark 2.3.5.2. Note that we do not have any reason to expect that Z will be reduced, even if S is so. This construction will, therefore, lead us out of the realm of varieties.
148 S 2.3: D V B
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Exercise 2.3.5.3. Generalize Proposition 2.3.5.1 to the case when FS is a not necessarily S -flat coherent OS x A -module and ES is an S -flat coherent OS x A -module (see [82], Lemma 3.1, and [186], Proposition 2.1.) Back to the Parameter Space The vector bundles occurring in δ-semistable *a,b,c -swamps of type (d, l) can, by Theorem 2.3.4.1, be parameterized by some quot scheme. Let Q be the quot scheme that parameterizes all quotients of Y * OX (−n) which have rank r and degree d. Recall that we have the universal quotient qQ : Y * OX (−n) −→ FQ on Q x X (Theorem 2.2.3.5 and Example 2.2.4.12, i). As before, we let Q be the open part of the quot scheme which parameterizes quotients q: Y * OX (−n) −→ E, such that E is locally free and H 0 (q(n)): Y −→ H 0 (E(n)) is an isomorphism. Define EQ := FQ|Q x X . Let Jacl be the Jacobian variety of line bundles of degree l on the curve X. We set QB := Q x Jacl and qQB : Y * OX (−n) −→ EQB and LQB to be the pullback to QB x X of the universal quotient and the Poincar´e line bundle, respectively. In the following, we write Aa,b instead of (A#a )$b for any vector bundle A on a scheme S . For m = 0, & ^ 4Z Fm := πQB . Ya,b * π.X OX (a(m − n)) will be a vector bundle on QB , and so will be ^ Gm := πQB . det(EQB )#c * LQB
*
4 π.X (OX (am)) .
We form the projective bundle
4 ^ π: P := ! H om(Fm , Gm )∨ −→ QB .
On P x X, we have the universal quotient ^ 4 qP := (π x id X ). (qQB ): Y * π.X OX (−n) −→ EP := (π x id X ). (EQB ), the pullback LP of LQB , and the tautological homomorphism ^ 4 ^ 4 ^ 4 * fP : Ya,b * π.X OX (a(m − n)) −→ det(EP )#c * LP * π.X OX (am) * π.P OP (1) . fP * idOX (−am) . Set fP := * By Proposition 2.3.5.1, there is a closed subscheme T ⊆ P where the restriction of fP to & Z ^ 4 ker Ya,b * π.X OX (a(m − n)) −→ EP,a,b vanishes, so that fP|T x X factorizes over ET,a,b , ET := EP|T x X . Thus, on T x X, we have the universal quotient ^ 4 qT : Y * π.X OX (−n) −→ ET
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and the universal homomorphism
ϕP : ET,a,b −→ det(ET )#c * LT * π.T (NT ),
LT := LP|T x X , NT := OP (1)|T .
Let πJacl : T −→ Jacl be the projection onto the Jacobian variety. Then, LT = L [πJacl ]. We call (ET , πJacl , NT , ϕT ) the universal family. If t ∈ T is a closed point, then we may restrict the universal family to {t} x X. The resulting *a,b,c -swamp will be written as (Et , Lt , ϕt ). Remark 2.3.5.4. By construction, if (E, L, ϕ) is any δ-semistable *a,b,c -swamp of type (d, l), then we will find a point t ∈ T, such that (E, L, ϕ) is isomorphic to (Et , Lt , ϕt ). This explains the name “universal family”. Let S be a scheme. A quotient family of *a,b,c -swamps (parameterized by S ) is a tuple (qS : Y * π.X (OX (−n)) −→ ES , κS , NS , ϕS ) where (ES , κS , NS , ϕS ) is a family of *a,b,c -swamps parameterized by S and qS : Y * OS x X −→ ES is a surjective map, such that πS . (qS * idOX (n) ): Y * OS −→ πS . (ES * π.X (OX (n))) is an isomorphism. We call two such quotient families (qS : Y * π.X (OX (−n)) −→ ES , κS , NS , ϕS ) and (qBS : Y * π.X (OX (−n)) −→ ESB , κBS , NSB , ϕBS ) isomorphic, if κS = κBS and there are isomorphisms ψS : ES −→ ESB and χS : NS −→ NSB , such that ^ 4 qBS = ψS ◦ qS and ϕBS = idL [κS ] * π.S (χS ) ◦ ϕS ◦ ψ−1 S ,* . We obviously have a universal quotient family on T x X. The next proposition gives the precise universal property of the parameter scheme T. Proposition 2.3.5.5. Any quotient family parameterized by the scheme S gives rise to a unique morphism f : S −→ T, such that it is isomorphic to the pullback of the universal quotient family by ( f x idX ). Proof. Let (qS : Y * π.X (OX (−n)) −→ ES , κS , NS , ϕS ) be a quotient family as before. By the universal property of the quot scheme, the quotient qS gives rise to a unique morphism f0 : S −→ Q, such that qS is isomorphic to the pullback of the universal quotient. Since ES is locally free and πS . (qS * idOX (n) ) is an isomorphism, this morphism lands in the open subscheme Q. Together with κS , we obtain a morphism f0 : S −→ QB . The universal property of the scheme P says that giving a morphism f1 : S −→ P to it which covers f0 amounts to giving a line bundle NS and a homomorphism ΦS : f0. (Fm ) −→ f0. (Gm ) * NS . By construction, L [κS ] = ( f0 x idX ). (LQB ). The base change theorem implies that & ^ 4Z f0. (Fm ) = πS . Ya,b * π.X OX (a(m − n)) and
4 ^ f0. (Gm ) = πS . det(ES )#c * L [κS ] * π.X (OX (am)) .
Thus, we define ^ 4 ΦS : Ya,b * πS . π.X (OX (am)) πS . (ϕS
#idπ.X (OX (am)) )
^ 4 7 Ea,b * π. OX (a(m − n)) X
πS . (ϕS
#idπ.X (OX (am)) ) 7
^ 4 7 πS . det(ES )#c * L [κS ] * π. (OX (am)) * NS . X
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By the construction of ΦS and the universal property of T (Proposition 2.3.5.1), it is evident that the morphism f1 given by ΦS factorizes over T ⊂ P. " Exercise 2.3.5.6. Derive the following local universal property: Proposition. Let S be a scheme of finite type over ', and (ES , κS , NS , ϕS ) a family of δ-semistable *a,b,c -swamps parameterized by S . Then, there exist an open covering S i , i ∈ I, of S , and morphisms βi : S i −→ T, i ∈ I, such that the restriction of the family (ES , κS , NS , ϕS ) to S i x X is equivalent to the pullback of (ET , κT , NT , ϕT ) via βi x idX , for all i ∈ I. There is also a natural GL(Y)-action Γ on P which leaves T invariant.
Exercise 2.3.5.7. i) Show that '. · idY acts trivially on T. ii) Show that for two points t and tB in the parameter space T, there exists an element g ∈ GL(Y), such that tB = g · t, if and only if (Et , Lt , ϕt ) and (EtB , LtB , ϕtB ) are isomorphic. More exactly, the following gluing property is verified: Proposition. Let S be a scheme of finite type over ' and β1,2 : S −→ T two morphisms, such that the pullbacks of (ET , κT , NT , ϕT ) via β1 x idX and β2 x idX are isomorphic. Then, there exists a morphism Ξ: T −→ GL(Y), such that β2 equals the morphism Ξ x β1
Γ
T −→ GL(Y) x T −→ T. (In [187], Proposition 2.1, we have verified the analogous statement for the SL(Y)action.) Remark 2.3.5.8. Recall that we assume that the underlying representation is *a,b,c . If * and *B are representations with *a,b,c := * - *B , then ET,a,b = ET,* - ET,*B . Hence, we may define T* by the condition that fT vanishes on ET,*B . By Proposition 2.3.5.1, this is a closed subscheme of T. It is invariant under the group action and parameterizes *-swamps. Since forming good quotients is—over '—compatible with closed embeddings, we see here on a more functorial level why it suffices to look at *a,b,c . Exercise 2.3.5.9. State the universal and the local universal property of the parameter scheme T* . If we have chosen n large enough, the sheaf r &8 ^ 4Z Y * OJacd , πJacd . P * π.X (OX (rn)) G1 := H om
is locally free. We set 41 := !(G1∨ ). By replacing P with P * π.Jacd (dual of sufficiently ample), we may assume that O&1 (1) is very ample. Let d: T −→ Jacd be the 5 morphism associated to the line bundle r ET on T x X, and let AT be a line bundle on 5r T with ET ! (d x idX ). (P) * π.T (AT ). Then r ^ r 8 ^ 4 4 8 qT * idπ.X (OX (n)) : Y * OT −→ (d x idX ). P * π.X (OX (rn)) * π.T (AT)
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defines a morphism ι1 : T −→ 41 with ι.1 (O&1 (1)) = AT . Set J d,l := Jacd x Jacl . The sheaf & ^ 4Z G2 := H om Ya,b * O J d,l , π J d,l. π.Jacd x X (P)#c * π.Jacl x X (L ) * π.X (OX (na)) on J d,l is also locally free. Set 42 := !(G2∨ ). Making use of Remark 2.3.2.4, it is clear that we can assume O&2 (1) to be very ample. The homomorphism ^ 4 −→ Ya,b * OT −→ ET,a,b * π.X OX (na) 4 ^ 4 4 ^ ^ −→ (d x idX ). P #c * L [κT ] * π.X OX (na) * π.T AT#c * NT provides a morphism ι2 : T −→ 42 with ι.2 (O&2 (1)) = AT#c * NT . Altogether, defining the Gieseker space 4 := 41 x 42 , the Gieseker map Gies := ι1 x ι2 : T −→ 4 is an injective and GL(Y)-equivariant morphism. Linearize the GL(Y)-action on 4 in O& (s1 , s2 ) with p − aδ s1 . (2.12) = ε := s2 rδ Remark 2.3.5.10. i) This is exactly the point where we see that the semistability parameter δ corresponds to the choice of a linearization in the GIT construction. ii) Let N and L0 be two line bundles on the curve X. The Gieseker space comes, by construction, with a map to J d,m . Its fiber over ([N], [L0 ]) is !(/) x !(*) with r &8
/ := /N := Hom and
* := *N,L
0
^ 4Z∨ Y, H 0 N(rn)
& 4Z∨ ^ := Hom Ya,b , H 0 N #c * L0 (an) .
There are the representations *1 : GL(Y) −→ GL(/) and *2 : GL(Y) −→ GL(*) which give the GL(Y)-actions on !(/) and !(*) together with linearizations in O((") (1) and O((0) (1), respectively. Thus, we have an action of GL(Y) on !(/) x !(*), and, for positive integers s1 and s2 , the linearization σ s1 ,s2 of that action in the line bundle & 4Z & 4Z ^ ^ . O(s1 , s2 ) := π.((") O((") (s1 ) * π( (0) O((0) (s2 ) . Since the evaluation of the Hilbert–Mumford criterion clearly commutes with closed embeddings, we are lead to evaluate it on the spaces !(/N ) x !(*N,L0 ) where N varies over Jacd and L0 over Jacm . This brings us to the terrain with which Chapter 1 has familiarized us. For the following considerations, we stress that, by Part i) of Exercise 2.3.5.7, we have to investigate the SL(Y)-action. The first major step in the construction of the moduli spaces is the following result:
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Theorem 2.3.5.11. There exists n6 ∈ (>0 , such that for all n ≥ n6 the following property is verified: For a point t ∈ T, the point Gies(t) ∈ 4 is (semi)stable with respect to the linearization from (2.12), if and only if (Et , Lt , ϕt ) is a δ-(semi)stable *a,b,c -swamp of type (d, l). The proof of this theorem is a quite involved evaluation of the Hilbert–Mumford criterion, using techniques of Simpson and Huybrechts and Lehn. We will prove it in several stages. As the first step, we establish: Proposition 2.3.5.12. There is an n1 > 0, such that the following holds true: The set S of isomorphy classes of vector bundles E of rank r and degree d for which there exist an n ≥ n1 and a point t ∈ T, such that E ! Et and Gies(t) is semistable with respect to the above linearization, is bounded. Recall that the parameter space T and the Gieseker space 4 both depend on n, although, for simplicity, we have not indicated this in the notation. Proof. We would like to find a lower bound for µmin (E) for a bundle E as in the proposition. Then, we derive an upper bound on µmax (E) and may conclude by Proposition 2.2.3.7. Let Q = E/F be a quotient bundle of E. We have the exact sequence ^ 4 ^ 4 ^ 4 {0} −−−−−→ H 0 F(n) −−−−−→ H 0 E(n) −−−−−→ H 0 Q(n) . Let λ: '. −→ SL(Y) be a one parameter subgroup with weighted flag & Z ^ 4−1 ^ 4 Y• (λ) : {0} ! Y1 := H 0 q(n) H 0 (F(n)) ! Y, α• (λ) = (1) . Define F B := q(Y1 * OX (−n)). If Gies(t) = ([M], [L]), then µ([M], λ) = p · rk(F B ) − h0 (F(n)) · r ≤ p · rk(F) − h0 (F(n)) · r. (Recall that p = d + r(n + 1 − g) depends on n.) Lemma 1.5.1.41 from Chapter 1 provides us with the estimate ^ 4 µ([L], λ) ≤ a · p − h0 (F(n)) . The assumption that Gies(t) is semistable thus gives 0 ≤ ≤ =
s1 · µ([M], λ) + µ([L], λ) s2 4 ^ 4 p−a·δ ^ · p · rk(F) − h0 (F(n)) · r + a · p − h0 (F(n)) r·δ p2 · rk(F) p · h0 (F(n)) a · p · rk(F) − − + a · p. r·δ δ r
We multiply this by r · δ/p and find p · rk(F) − r · h0 (F(n)) + δ · a · (r − 1) ≥ p · rk(F) − r · h0 (F(n)) + δ · a · (r − rk(F)) ≥ 0.
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The first exact sequence implies h0 (F(n)) ≥ p − h0 (Q(n)). This enables us to transform the above inequality into h0 (Q(n)) p δ · a · (r − 1) p δ · a ≥ − ≥ − . r r rk(Q) · r r r
(2.13)
For a semistable vector E with µ(E) ≥ 0, we have the estimate h0 (E) ≤ µ(E) + 1. rk(E)
(2.14)
This is Lemma 7.1.2 in [135]. If µ(E) < 0, we have of course h0 (E) = 0. If we take Q to be the minimal destabilizing quotient of E, then (2.13) and (2.14) yield the inequality d − δa δa . µmin (E) + n + 1 = µ(Q(n)) + 1 ≥ µ(E) + n + 1 − g − = n + 1 − g + D!!!!!!!!!!!!!!!WB!!!!!!!!!!!!!!!\ r r ≤p/r
This gives the required bound from below.
"
Theorem 2.3.5.13. There is an n2 , such that for every n ≥ n2 and every point t ∈ T with (semi)stable Gieseker point Gies(t) ∈ 4, the *a,b,c -swamp (Et , Lt , ϕt ) is δ-(semi)stable. Proof. As in [187], Theorem 3.3, one may show that there is a finite set A ! T = (r•j , α•j ) !! r•j = (r1j , . . . , r sj j ) : 0 < r1j < · · · < r sj j < r,
> α•j = (α1j , . . . , α sj j ) : αij ∈ 3>0 , i = 1, . . . , s j , j = 1, . . . , t ,
depending only on the GLr (')-module Wa,b,c , such that δ-(semi)stability of a *a,b,c swamp (E, L, ϕ) of type (d, l) has to be verified only for weighted filtrations (E• , α• ) with 4 ^ (rk(E1 ), . . . , rk(E s )), α• ∈ T . (In fact, this is a consequence of Example 1.5.1.18 in Chapter 1.) We may prescribe a constant C B . Then, there exists a constant C BB , such that for every *a,b,c -swamp (E, L, ϕ) of type (d, l) with [E] ∈ S (see Proposition 2.3.5.12) and every weighted filtration (E• , α• ), such that ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T and µ(Ei ) ≤ C BB , one has
for one index i ∈ { 1, . . . , s },
(2.15)
M(E• , α• ) > C B .
This follows from the fact that µ(Ei ) < µmax (E) and there is a constant which uniformly bounds µmax (E) for all vector bundles E with [E] ∈ S, by Proposition 2.2.3.7. Using Chapter 1, Lemma 1.5.1.41, it is easy to determine a constant C BBB which depends only on Wa,b,c with µ(E• , α• , ϕ) ≥ −C BBB
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for any weighted filtration (E• , α• ) of a vector bundle E as above with ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T . We choose C B ≥ δ · C BBB . Then, for a *a,b,c -swamp (E, L, ϕ) of type (d, l) with [E] ∈ S and a weighted filtration (E• , α• ), such that ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T and (2.15) holds, one has M(E• , α• ) + δ · µ(E• , α• , ϕ) > C B − δ · C BBB ≥ 0. Thus, the condition of δ-(semi)stability has to be verified only for weighted filtrations (E• , α• ) with ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T for which (2.15) fails. But these weighted filtrations live in bounded families, by a slightly refined application of Proposition 2.2.3.7. We infer from these considerations: Corollary 2.3.5.14. There is a positive integer n3 ≥ n2 , such that any n ≥ n3 has the following property: For every *a,b,c -swamp (E, L, ϕ) of type (d, l) for which [E] belongs to the bounded family S, the conditions stated below are equivalent. 1. (E, L, ϕ) is δ-(semi)stable. 2. For any weighted filtration (E• , α• ) with ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T , such that E j (n) is globally generated and h1 (E j (n)) = 0, j = 1, . . . , s, one has s T
^ 4 α j · h0 (E(n)) · rk(E j ) − h0 (E j (n)) · rk(E) + δ · µ(E• , α• , ϕ)(≥)0.
j=1
We assume that n ≥ n3 . Now, let t ∈ T be a point with (semi)stable Gieseker point Gies(t). Then, [E] belongs to the bounded family S. Therefore, it suffices to check Criterion 2. in Corollary 2.3.5.14 for establishing the δ-(semi)stability of (E, L, ϕ). Let, more generally, (E• , α• ) be a weighted filtration of E, such that E j (n) is globally generated and h1 (E j (n)) = 0, j = 1, . . . , s, and no condition on ((rk(E1 ), . . . , rk(E s )), α• ) is imposed. Since H 0 (q(n)) is an isomorphism, we define the subspaces ^ 4−1 ^ 4 Y j := H 0 q(n) H 0 (E j (n)) ! Y, The basic weight vectors had been defined as ^ 4 γ(i) − p, . . . , i − p, i, . . . , i , p := iD!!!!!!!!!!!!WB!!!!!!!!!!!!\ D!WB!\ ix
j = 1, . . . , s.
i = 1, . . . , p − 1.
(p−i) x
Choose a basis y = (y1 , . . . , y p ) of Y, such that F y1 , . . . , yl j 4 = Y j ,
l j = dim(Y j ) = h0 (E j (n)), j = 1, . . . , s.
These data yield the weight vector γ = (γ1 , . . . , γ p ) :=
s T j=1
(l )
α j · γp j
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155
and the one parameter subgroup λ := λ(y, γ): '. −→ SL(Y) with λ(z) ·
p T
ci · yi =
i=1
p T
zγi · ci · yi ,
i=1
z ∈ '. . (l )
Similarly, we define the one parameter subgroups λ j := λ(y, γ p j ), j = 1, . . . , t. Let ^ 4 L: Ya,b −→ H 0 det(E)#c * Lt * OX (an) be a linear map which represents the second component of Gies(t). We wish to compute µ([L], λ). First, we note that the choice of the basis y provides an identification gr(Y) :=
%Y s+1
j
! Y,
j=1
Y j := Y j /Y j−1 , j = 1, . . . , s + 1,
which we will use without further mentioning in the following. Define I := { 1, . . . , s + 1 }x a and abbreviate Yi1 ,...,ia := (Yi1
*
· · · * Yia )$b ⊂ Ya,b ,
(i1 , . . . , ia ) ∈ I.
All weight spaces of the one parameter subgroup λ inside Ya,b are direct sums of some of these subspaces. In addition, the subspaces Yi1 ,...,ia are eigenspaces for the one parameter subgroups λ1 , . . . , λ s . More precisely, λ j acts on Yi1 ,...,ia with the weight a · l j − ν j (i1 , . . . , ia ) · p,
(i1 , . . . , ia ) ∈ I, j = 1, . . . , s.
In that formula, we have used
F L ν j (i1 , . . . , ia ) := # ik ≤ j | k = 1, . . . , a .
Thus, we find s > AT ^ 4! α j · a · l j − ν j (i1 , . . . , ia ) · p !! Yi1 ,...,ia $ ker(L), (i1 , . . . , ia ) ∈ I . µ([L], λ) = − min j=1
Fix an index tuple (i01 , . . . , i0a ) ∈ I for which the minimum is achieved. Let r 8 ^ 4 M: Y −→ H 0 det(E)(rn)
(2.16)
represent the first component of Gies(t). We have explained before (see Formula 2.3) that µ([M], λ) = =
s T j=1 s T j=1
^ 4 α j · h0 (E(n)) · rk(E j ) − h0 (E j (n)) · rk(E) ^ 4 α j · p · rk(E j ) − h0 (E j (n)) · r .
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Since we assume Gies(t) to be (semi)stable, we have 0
s1 · µ([M], λ) + µ([L], λ) s2 s s 4 ^ 4 T s1 T ^ α j al j − ν j (i01 , . . . , i0a )p α j p rk(E j ) − h0 (E j (n))r − s2 j=1 j=1
(≤) =
s ^ 4 p−a·δ T · α j · p · rk(E j ) − h0 (E j (n)) · r − r·δ j=1
=
− l j =h0 (E j (n))
=
s T
j=1 s T
^ 4 α j · a · l j − ν j (i01 , . . . , i0a ) · p
αj ·
j=1
+
s T j=1
& p2 · rk(E ) p · a · rk(E ) p · h0 (E (n)) Z j j j + − − r·δ r δ
α j · ν j (i01 , . . . , i0a ) · p.
We multiply this inequality by r · δ/p. This leads to the inequality s T j=1
s & T ^ 4 ^ 4Z α j · p · rk(E j ) − h0 (E j (n)) · r + δ · − α j · a · rk(E j ) − ν j (i01 , . . . , i0a ) · r (≥)0. j=1
To conclude, we have to verify µ(E• , α• , ϕ) ≥ −
s T j=1
^ 4 α j · a · rk(E j ) − ν j (i01 , . . . , i0a ) · r .
(2.17)
If γ is the weight vector with the distinct weights γ1 < · · · < γ s+1 that is associated to (E• , α• ) as in Exercise 2.3.2.2, then we have checked (2.9), that is γi01 + · · · + γi0a =
s T j=1
^ 4 α j · a · rk(E j ) − ν j (i01 , . . . , i0a ) · r .
In view of (2.7), it remains to show that ϕ|(Ei0 #···#Ei0 )"b & 0. 1
a
(2.18)
To this end, note that, up to a scalar, L is given as ^ 4 H 0 (ϕt #idOX (an) ) ^ 4 H 0 (((qt (n))#a )"b ) Ya,b −−−−−−−−−−−−→ H 0 Ea,b (an) −−−−−−−−−−−→ H 0 det(E)#c * Lt * OX (an) . The image of Yi01 ,...,i0a lies in the subspace H 0 ((Ei01 (2.18).
*
· · · * Ei0a )$b (an)) and that shows "
We now turn to the converse direction in the proof of Theorem 2.3.5.11. Again, we need a preparatory result.
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Proposition 2.3.5.15. There is a positive integer n4 , such that any δ-(semi)stable *a,b,c swamp (E, L, ϕ) of type (d, l) satisfies s T
^ 4 α j · p · rk(E j ) − h0 (E j (n)) · r + δ · µ(E• , α• , ϕ)(≥)0
j=1
for every weighted filtration (E• , α• ) of E and every n ≥ n4 . Proof. By Theorem 2.3.4.1, we know that there is a bounded family SB of vector bundles of rank r and degree d, such that [E] ∈ SB for any δ-semistable *a,b,c -swamp (E, L, ϕ) of type (d, l). We choose a constant C, such that µmax (E) ≤ C for every vector bundle E on X with [E] ∈ SB . Given an additional positive constant C B , we subdivide the class of vector bundles F which might occur as subbundles of a vector bundle E with [E] ∈ S into two classes: A. µ(F) ≥ −C B B. µ(F) < −C B . By Proposition 2.2.3.7, the vector bundles F falling into Class A live again in bounded families, because there are obviously only finitely many options for their rank and their degree. So, we may always assume that our n is large enough, such that any such vector bundle F satisfies h1 (F(n)) = 0 and is globally generated. If E is a vector bundle on X with Harder–Narasimhan filtration {0} =: E0 ! E1 ! · · · ! E s ! E s+1 := E, then h0 (E) ≤
s+1 T
h0 (Ei /Ei−1 ),
i=1
so that (2.14) gives h0 (E) ≤ (rk(E) − 1) · (µmax (E) + 1) + (µ(E) + 1), if the right hand side is positive. Otherwise, h0 (E) = 0. For any bundle E with [E] ∈ SB and any subbundle F of E which belongs to the Class B, we thus find h0 (F(n)) ≤ (rk(F) − 1) · C − C B + rk(F)(n + 1) =: R(rk(F), C B ). We choose C B so large that p · rk(F) − h0 (F(n)) · r ≥ p · rk(F) − R(rk(F), C B ) · r > δ · a · (r − 1).
(2.19)
Now, let (E• , α• ) be any weighted filtration of E. Write { 1, . . . , s } = IA J IB with < · · · < iA/B i ∈ IA/B if and only if Ei belongs to the Class A/B. Let 1 ≤ iA/B sA/B ≤ s be the 1 A/B A/B indices in IA/B and define the weighted filtrations (E• , α• ) with E•A/B α•A/B
:
{0} ! E1A/B := EiA/B ! · · · ! E A/B ! E, sA/B := E iA/B s
=
(α1A/B , . . . , αA/B sA/B )
1
A/B
:= (αiA/B , . . . , αiA/B ). s 1
A/B
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According to Lemma 1.5.1.41 µ(E• , α• , ϕ) ≥ µ(E•A , αA • , ϕ) − a · (r − 1) ·
sB T j=1
αBj
(2.20)
holds true. Now, we compute s T
^ 4 α j · p · rk(E j ) − h0 (E j (n)) · r + δ · µ(E• , α• , ϕ)
j=1 (2.20)
≥
sA T j=1
+
^ 4 αAj · p · rk(E Aj ) − h0 (E Aj (n)) · r + δ · µ(E•A , αA • , ϕ) +
sB T j=1
=
(2.19)
≥
sB T 4 ^ αBj αBj · p · rk(E Bj ) − h0 (E Bj (n)) · r − δ · a · (r − 1) · j=1
A A M(E•A , αA • ) + δ · µ(E • , α• , ϕ) + sB &^ T αBj · p · rk(E Bj ) − h0 (E Bj (n)) · + j=1 A A M(E•A , αA • ) + δ · µ(E • , α• , ϕ)
Z 4 r − δ · a · (r − 1)
(≥) 0.
The last estimate results from the condition of δ-(semi)stability, applied to the weighted filtration (E•A , αA " • ). Theorem 2.3.5.16. There exists a positive integer n5 , enjoying the following property: If n ≥ n5 and (E, L, ϕ) is a δ-(semi)stable *a,b,c -swamp of type (d, l), then, for a point t ∈ T of the form t = ([q: Y * OX (−n) −→ E], L, ϕ), the associated Gieseker point Gies(t) is (semi)stable for the given linearization. Proof. Let λ: '. −→ SL(Y) be a one parameter subgroup and suppose Gies(t) = ([M], [L]). Then, we have to verify that s1 · µ([M], λ) + µ([L], λ)(≥)0. s2 The one parameter subgroup λ provides the weighted flag (Y• (λ), β• (λ)) with Y• (λ) : {0} =: Y0 ! Y1 ! · · · ! Yτ ! Yτ+1 := Y,
β• (λ) = (β1 , . . . , βτ ).
*h be the subbundle that is generically generated by For each h ∈ { 1, . . . , τ }, we let E *h ’s. After clearing q(Yh * OX (−n)). There may be improper inclusions among the E these, we obtain the filtration E• : {0} =: E0 ! E1 ! · · · ! E s ! E s+1 := E. For j = 1, . . . , s, we define
L F *h = E j T ( j) := h ∈ { 1, . . . , τ } | E
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α j :=
159 T
βh .
h∈T ( j)
This gives the weighted filtration (E• , α• ) of E. By Proposition 2.3.5.15, s T
^ 4 α j · p · rk(E j ) − h0 (E j (n)) · r + δ · µ(E• , α• , ϕ)(≥)0.
(2.21)
j=1
Recall from (2.7) that A > ! µ(E• , α• , ϕ) = − min γi1 + · · · + γia !! (i1 , . . . , ia ) ∈ I : ϕ|(Ei1 #···#Eia )"b & 0 .
(2.22)
Let (i01 , . . . , i0a ) ∈ I = { 1, . . . , t + 1 }x a be an index tuple which computes the minimum. With F L ν j (i1 , . . . , ia ) := # ik ≤ j | k = 1, . . . , a , one calculates (2.9) that γi01 + · · · + γi0a =
s T j=1
^ 4 α j · a · rk(E j ) − ν j (i01 , . . . , i0a ) · r .
Thus, (2.21) transforms into s T j=1
s & T ^ 4 ^ 4Z α j · p · rk(E j ) − h0 (E j (n)) · r + δ · − α j · a · rk(E j ) − ν j (i01 , . . . , i0a ) · r (≥)0. j=1
A computation as in the proof of Theorem 2.3.5.13, but performed backwards, shows that this implies 4 4Z T ^ s1 &T ^ α j ah0 (E j (n)) − ν j (i01 , . . . , i0a )p (≥)0. (2.23) α j p rk(E j ) − h0 (E j (n))r − s2 j=1 j=1 s
s
First, we see that µ([M], λ) =
τ T
s ^ 4 T ^ 4 *h ) − dim(Yh )r ≥ βh p rk(E α j p rk(E j ) − h0 (E j (n))r .
h=1
j=1
To continue, we need a little more notation. For j = 0, . . . , s + 1, we introduce L F *h = E j , Y := Yh( j) h( j) := min h = 1, . . . , τ | E j L F * h( j) := max h = 1, . . . , τ | Eh = E j , Y j := Yh( j) , as well as
*j := Y /Y j−1 , Y j
j = 1, . . . , s + 1.
(2.24)
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(Note that the image of Y j * OX (−n) under qt generically generates the vector bundle E j .) For an index tuple (i1 , . . . , ia ) ∈ I, we find the vector space ^ *i1 ,...,ia := Y *i1 Y
*
*ia ···*Y
4$b
.
Using a basis y of Y which consists of eigenvectors for the one parameter subgroup λ, we identify these spaces with subspaces of Ya,b . Note that λ = λ(y, γ)
with
γ=
τ T h=1
h )) βh · γ(dim(Y . p
h )) ), h = 1, . . . , τ. The effect of our definition is that the We define λh := λ(y, γ(dim(Y p *i1 ,...,ia , (i1 , . . . , ia ) ∈ I, are weight spaces for λ as well as for λ1 , . . . , λτ . We spaces Y *h = E j(h) . Then, associate to an index h ∈ { 1, . . . , τ } the index j(h) ∈ { 1, . . . , s } with E *i0 ,...,i0 with the h( j) ≤ h holds if and only if j ≤ j(h), and one verifies that λ acts on Y a 1 weight
−
τ T h=1
s T ^ ^ 4 4 α j ah0 (E j (n)) − ν j (i01 , . . . , i0a )p . (2.25) βh a dim(Yh ) − ν j(h) (i01 , . . . , i0a )p ≥ − j=1
In view of the estimates (2.23), (2.24), and (2.25), it is now sufficient to ascertain that *i0 ,...,i0 is non-trivial. If it were trivial, then there would have to be the restriction of L to Y a 1 B an index tuple (i1 , . . . , iBa ) with iBl ≤ i0l , l = 1, . . . , a, at least one inequality being strict, such that (2.26) L|(Y*iB #···#Y*iB )"b & 0. 1
a
This is because L restricts to a non-zero map on (Y i0 * · · · * Y i0a )$b , as ϕ|(Ei0 #···#Ei0 )"b is 1
1
a
non-trivial. (Recall that the image of that vector space tensorized with OX (−an) under a $b (q# generically generates the vector bundle ϕ|(Ei0 #···#Ei0 )"b .) Now, if (2.26) holds t )
true, then we must also have
1
ϕ|(EiB #···#EiB )"b & 0. 1
a
a
(2.27)
(Compare the arguments at the end of the proof of Theorem 2.3.5.13.) But, then the " tuple (i01 , . . . , i0a ) would not give the minimum in (2.22), a contradiction. By Theorem 2.3.5.11, the subsets Tδ-(s)s parameterizing the δ-(semi)stable *a,b,c swamps are the preimages of the open sets of GIT-(semi)stable points in 4 under the Gieseker morphism. Therefore, they are open subsets. Proposition 2.3.5.17. The restricted Gieseker morphism Gies|Tδ-ss : Tδ-ss −→ 4ss is proper. Since it is also injective, it is finite.
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161
Proof. This is similar to the proof of Proposition 2.2.4.19. Again, we will be a bit sketchy. We apply the valuative criterion of properness. Suppose we are given a discrete valuation ring R with quotient field K and a morphism η: C := Spec(R) −→ 4ss which lifts over C . := Spec(K) to a morphism η. : C . −→ Tδ-ss . The morphism η. is associated to a family 4 ^ qC. : Y * π.X (OX (−n)) −→ EC. , κC. , OC. , ϕC. : EC. ,a,b −→ det(EC. )#c * L [κS ] on C . x X. This follows from the universal property of T, given in Proposition 2.3.5.5. Recall that the quot scheme Q may be compactified to Q (see Theorem 2.2.3.5). Therefore, we may extend qC. to a quotient qC : Y * π.X (OX (−n)) −→ EC where the restriction of EC to the special fiber {0} x X may have torsion. Let Z be the support of that torsion. Then, we define ^ 4 det(EC ) := ι. det(EC|(C x X)\Z ) , ι: (C x X) \ Z −→ C x X being the inclusion. We have used the fact that EC|(C x X)\Z is a vector bundle and that any line bundle on (C x X) \ Z extends uniquely to a line bundle on C x X, because the latter is a regular two-dimensional scheme and the codimension of Z is two. The Jacobian variety is projective, so that we also find an extension κC : C −→ Jacl of the morphism κC. . Claim. The homomorphism ϕC. extends to a homomorphism ϕC : EC,a,b −→ det(EC )#c * L [κS ]. In fact, if we combine the techniques which we have used in the construction of the parameter space T with Exercise 2.3.5.3, we obtain a scheme H together with a projective map H −→ C which parameterizes the homomorphisms from EC,a,b to det(EC )#c * L [κS ]. Next, the family qC may be altered to the family * qC : Y * π.X (OX (−n)) −→ EC −→ *C := E ∨∨ . Since C x X is a regular two-dimensional scheme and E ∨∨ is a reflexive E C C *C is a family of vector bundles sheaf (by [97], Corollary 1.3), it is locally free, i.e., E ([97], Corollary 1.4). Note that * qC agrees with qC on (C x X) \ Z, but * qC|{0} x X may fail to be surjective in points of Z. Let ι: (C x X) \ Z −→ C x X be the inclusion and define *C,a,b * ϕC : E
4 ^ 4 ^ ι. (ϕC|(C x X)\Z ) 7 7 ι. E *C,a,b|(C x X)\Z = ι. EC,a,b|(C x X)\Z ^ 4 7 ι. det(EC|(C x X)\Z )#c * L [κC ]|(C x X)\Z = det(E *C )#c * L [κC ].
The family (* qC , κC , OC , * ϕC ) also defines a morphism to 4 which coincides with η. Let * * (* q: Y * OX (−n) −→ E, L, * ϕ) be the restriction of the new family to {0} x X. One checks the following results: • H 0 (* q(n)) must be injective. * belongs to a • Since (* q, * L, * ϕ) defines a semistable point in the Gieseker space, E bounded family (this is an easy adaptation of the proof of Proposition 2.3.5.12).
162 S 2.3: D V B
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• The techniques of the proof of Theorem 2.3.5.13 may also be used to show that ** * (E, L, * ϕ) must be δ-semistable. In particular, the first cohomology group of E(n) 0 vanishes, so that H (* q(n)) is indeed an isomorphism. By Proposition 2.3.5.5, the family (* qC , κC , OC , * ϕC ) is thus induced by a morphism * η: C −→ Tδ-ss which lifts η and extends η. . This finishes the argument. " Since 4ss possesses a projective quotient, Proposition 2.3.5.17 and Chapter 1, Exercise 1.4.3.11, show that the good quotient M δ-ss (*a,b,c , d, l) := Tδ-ss // SL(Y) exists as a projective scheme. Likewise, the geometric quotient M δ-s (*a,b,c , d, l) := Tδ-s / SL(Y) exists as an open subscheme of M δ-ss (*a,b,c , d, l). By Remark 2.3.5.6, Exercise 2.3.5.7, and the universal property of the categorical quotient (Chapter 1, Lemma 1.4.1.1), the space M δ-ss (*a,b,c , d, l) is indeed a coarse moduli space. " Remark 2.3.5.18 (S-equivalence). Recall from Chapter 1, Section 1.4, that two points in Tδ-ss are mapped to the same point in the quotient if and only if the closures of their orbits intersect in Tδ-ss . Given a point t ∈ Tδ-ss , let tB ∈ Tδ-ss be the point whose orbit is the unique closed orbit in SL(Y) · t(⊆ Tδ-ss ). Then, by the Hilbert–Mumford criterion (Chapter 1, Theorem 1.5.1.4), there exists a one parameter subgroup λ: '. −→ SL(Y) with limz→∞ λ(z) · t ∈ SL(Y) · tB . For this one parameter subgroup, one has of course µ(t, λ) = 0. Thus, the equivalence relation which we have to consider on the points of Tδ-ss is generated by t ∼ limz→∞ λ(z) · t for all one parameter subgroups λ: '. −→ SL(Y) with µ(t, λ) = 0. If one looks carefully at the arguments given in the proofs of Theorem 2.3.5.13 and 2.3.5.16, one sees that, for a point t = ([q: Y * OX (−n) −→ E], L, ϕ) ∈ Tδ-ss , the following observations hold true: • If λ: '. −→ SL(Y) verifies µ(t, λ) = 0, then its weighted flag (Y• (λ), α• (λ)) has the property that the weighted filtration (E• , α• (λ)) with E j := q(Y j * OX (−n)), j = 1, . . . , s, satisfies ^ 4 M E• , α• (λ) + δ · µ(E• , α• (λ), ϕ) = 0. • Given a weighted filtration (E• , α• ) of E with M(E• , α• ) + δ · µ(E• , α• , ϕ) = 0, one can assume that h1 (E j (n)) = 0 and that E j (n) is globally generated, j = 1, . . . , s. Hence, there is a unique flag Y• in Y, such that H 0 (q(n)) maps Y j onto H 0 (E j (n)), j = 1, . . . , s. Then, any one parameter subgroup λ: '. −→ SL(Y) with weighted flag (Y• , α• ) satisfies µ(t, λ) = 0. • For a one parameter subgroup λ with µ(t, λ) = 0, tB := limz→∞ λ(z)·t, and induced weighted filtration (E• , α• ) on E, the *a,b,c -swamp (EtB , LtB , ϕtB ) is isomorphic to df (E• ,α• ) (E, L, ϕ).
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163
This shows that the equivalence relation induced by the GIT process on the closed points of Tδ-ss is just S-equivalence of *a,b,c -swamps as introduced in Section 2.3.2.
2.3.6 On the Geometry of the Moduli Spaces Since we have constructed our moduli spaces in a fairly general setting, it is hard to formulate any general observation on their geometry. There are however two remarkable properties which we shall briefly describe. Asymptotic Irreducibility Fix the representation *, the integer d, and the stability parameter δ ∈ 3>0 . Suppose that * is a direct summand of the representation *a,b,c . As we have stressed in Remark 2.3.4.2, the constant C1 in Theorem 2.3.4.1 does not depend on the degree l of the line bundle L. Therefore, the set S of isomorphy classes of vector bundles E, such that there exist an l ∈ ( and a δ-semistable *-swamp (E, L, ϕ) of type (d, l) is still bounded. The same goes for the set S* of isomorphy classes of vector bundles of the form E* with [E] ∈ S. Thus, there is a constant l0 , such that for every l ≥ l0 and every δ-semistable *-swamp (E, L, ϕ) of type (d, l), one has Ext1 (E* , L) = H 1 (E*∨ * L) = {0}.
(2.28)
Our construction shows that the natural parameter space for δ-semistable *-swamps of type (d, l) is a projective bundle over the product of the quasi-projective quot scheme and the Jacobian of degree l line bundles: By (2.28), the sheaf ∨ H := πQB . (EQ B ,* * LQB )
is locally free. The usual base change properties grant that !(H ∨ ) has the same universal property as T, whence there is a canonical isomorphism between T and !(H ∨ ). Moreover, standard arguments as in [135], §8.5, show that the quot scheme Q is smooth and irreducible. All in all, we see that T is smooth and irreducible. The quotient thus inherits some good properties (see [123], II.3.3, [110], [119]): Theorem 2.3.6.1. Given the data *, d, and δ as above, there exists a constant l0 , such that the moduli space M δ-ss (*, d, l) is a normal, Cohen–Macaulay, and irreducible projective variety for every l ≥ l0 . Remark 2.3.6.2. i) Given lB , l with l − lB > 0 and a point p0 ∈ X, the map (E, L, ϕ) 1−→ (E, L((l − lB )p0 ), ϕB ) with ϕB : E* −→ L ⊂ L((l − lB )p0 ) induces a closed embedding M δ-ss (*, d, lB ) '→ M δ-ss (*, d, l). ii) In the above situation, it is not automatic that the open subscheme M δ-s (*, d, l) which parameterizes the stable swamps is smooth. This is because stable swamps may have more automorphisms than just homotheties (see [164] for an example).
164 S 2.3: D V B
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Stable Rationality Hoffmann [112] investigated the rationality of moduli spaces of vector bundles with an additional structure in a broad context. Let us mention here what his results state about our moduli spaces. Note that we have the morphism f : M δ-ss (*, d, l) −→ Jacd x Jacl ^ 4 [E, L, ϕ] 1−→ [det(E)], [L] .
(2.29)
Given a line bundle L0 on X, we let M δ-ss (*, d, L0 ) be the preimage of Jacd x{[L0 ]} under the above morphism. It is the moduli space for δ-semistable *-swamps of type (d, l) of the shape (E, L0 , ϕ). Theorem 2.3.6.3 ([112], Example 5.13). Let * be a homogeneous representation of degree α and assume that r, d, and α do not share a common factor and that l ≥ l0 as in Theorem 2.3.6.1. Then, the generic fiber of the morphism f : M δ-ss (*, d, L0 ) −→ Jacd 6 _ 6 _ E, L0 , ϕ 1−→ det(E)
(2.30)
is stably rational. A variety V over the ground field, say, K is said to be stably rational, if +NK x V is a rational variety for some N ≥ 0. The theorem thus says that, for some N ≥ 0, the B variety 'N x M δ-ss (*, d, L0 ) is birationally equivalent to the variety 'N x Jacd . Asymptotic Semistability One may give indeed a nicer description of the concept of δ-(semi)stability for those parameters δ which are very large. For this, call a *-swamp (E, L, ϕ) asymptotically (semi)stable, if a) µ(E• , α• , ϕ) ≥ 0, for every weighted filtration (E• , α• ) of E, and b) M(E• , α• )(≥)0, for every weighted filtration (E• , α• ) with µ(E• , α• , ϕ) = 0.
Remark 2.3.6.4. i) Recall the notation from (2.4), and write !(.* )ss for the open subset of points that are semistable for the action of the linear algebraic group SL($). Condition b) is equivalent to Condition bB ) which we set to be ϕη ∈ !(.* )ss . This characterization results from Chapter 1, Exercise 1.7.2.8. ii) If ϕη is even stable, then we will have µ(E• , α• , ϕ) > 0, for every weighted filtration of E, so that (E, L, ϕ) will be δ-stable for all δ = 0. Note that (E, L, ϕ) is then asymptotically stable. Proposition 2.3.6.5. Fix the input data *, d, and l. Then, there exists a positive number δ∞ , such that, for every stability parameter δ > δ∞ , a *-swamp (E, L, ϕ) of type (d, l) is δ-(semi)stable, if and only if it is asymptotically (semi)stable. Proof. Thanks to Theorem 2.3.4.3, we may construct our parameter space T in Section 2.3.5 in such a way that it parameterizes the *-swamps that are semistable for some stability parameter δ > 0.
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165
Given a stability parameter δ > 0, the set Uδ of points t ∈ T for which (Et , Lt , ϕt ) is δ-semistable is open. This follows from Theorem 2.3.5.11. We also define ) ) Uδ . UδB and U ss := U≤δ := 00
These are clearly open subschemes of T, and U≤δ1 ⊆ U≤δ2 , if δ1 ≤ δ2 . Since U ss is a quasi-projective scheme, there exists a positive number δ0 with U≤δ0 = U ss .
(2.31)
Lemma 2.3.6.6. Suppose we are given stability parameters δ0 ≤ δ1 ≤ δ2 , then Uδ1 ⊇ Uδ2 . Proof. Let t ∈ T be a point, such that (Et , Lt , ϕt ) is δ2 -semistable but not δ1 -semistable. Then, there exists a non-trivial filtration (E• , α• ) of Et with and
M(E• , α• ) + δ2 · µ(E• , α• , ϕt )
≥
0
M(E• , α• ) + δ1 · µ(E• , α• , ϕt )
0. For any δ ≤ δ1 (in particular, for any δ ≤ δ0 ), we find M(E• , α• ) + δ · µ(E• , α• , ϕt ) ≤ M(E• , α• ) + δ1 · µ(E• , α• , ϕt ) < 0. This tells us t ∈ U≤δ2 ⊆ U ss but t # U≤δ0 , a contradiction to (2.31). We form
U∞ :=
S
"
Uδ .
δ≥δ0
It will be our task to show that i) U∞ is open and ii) to characterize the points in U∞ by an intrinsic semistability condition. Note that once we have established i), it follows easily that there is a stability parameter δ∞ , such that Uδ = U∞ ,
∀δ > δ∞ .
(2.32)
Lemma 2.3.6.7. Let (E, L, ϕ) be an asymptotically (semi)stable *-swamp of type (d, l). Then, (E, L, ϕ) is δ-(semi)stable for all numbers δ = 0. Proof. As we have noted on several occasions, one can see that there is a finite set T (which is independent of δ), such that the condition of δ-(semi)stability has to be verified only for non-trivial weighted filtrations (E• , α• ) of E, such that ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T . Without loss of generality, we may assume that the weight vectors associated to the elements in T by means of (2.6) are integral. This grants that µ(E• , α• , ϕ) is an integer. Now, there is a negative number ε0 , depending on d, µmax (E), and T , such that M(E• , α• ) > ε0
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holds for all non-trivial weighted filtrations (E• , α• ) of E with ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T . Suppose δ ≥ −ε0 , and let (E• , α• ) be a weighted filtration of E with ((rk(E1 ), . . . , rk(E s )), α• ) ∈ T . If µ(E• , α• , ϕ) = 0, then M(E• , α• )+δ·µ(E• , α• , ϕ)(≥)0, by asymptotic (semi)stability. On the other hand, if µ(E• , α• , ϕ) > 0 (and, thus, ≥ 1, because it is an integer), we have M(E• , α• ) + δ · µ(E• , α• , ϕ) > ε0 + δ ≥ 0. This establishes the δ-(semi)stability of (E, L, ϕ).
"
Let U as be the set of points t ∈ T, such that (Et , Lt , ϕt ) is asymptotically semistable. Lemma 2.3.6.8. The set U as is open. Proof. We first verify that Condition b) is open. By Remark 2.3.6.4, we may verify the openness of Condition bB ). Let U ⊆ T x X be the maximal open subset where ϕT is surjective. Then, ϕT|U yields a section σ: U −→ ! := !(ET,* ). Let NC ⊆ !(V) denote the GLr (')-invariant subset of points which are not SLr (')semistable. Associated to the vector bundle ET|U and the GLr (')-variety NC, we have a closed subscheme N C '→ !. Let B ⊆ U be the preimage of N C under σ and B ⊆ T x X its closure. Then, we have the morphism p: B −→ T. Define the closed subset F ^ 4 L E := t ∈ T | dim p−1 (z) ≥ 1 ⊆ T. The open subset UbB := T \ E is precisely the set of points which satisfy Condition bB ). Obviously, Uδ ∩ UbB ⊆ U as is open for any δ > 0, and, by Lemma 2.3.6.7, ) U as = (Uδ ∩ UbB ). δ>0
This exhibits U as as an open subset.
"
Lemma 2.3.6.9. U∞ = U as . Proof. The inclusion U∞ ⊇ U as follows from Lemma 2.3.6.6 and Lemma 2.3.6.7. The converse inclusion is rather obvious. " We have now achieved the aims formulated before. Note that our proofs show that U∞ = U as = Uδ0 ∩ UbB . " Hitchin Pairs Here, we will look at the representation *: GLr (') −→ GL(Mr (') - '). We will write a *-swamp as (E, L, ϕ - τ) where ϕ: E −→ E * L is a twisted endomorphism and τ: OX −→ L is a section.
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Exercise 2.3.6.10. Let η ∈ X be the generic point. Then, ϕ|{η} defines a point [ f, e] ∈ !(End($) - '(X)). Use Remark 2.3.6.4 and the section on quiver moduli in Chapter 1, p. 75ff, to verify that a *-swamp (E, L, ϕ - τ) is asymptotically (semi)stable, if and only if it meets the following requirements: • Either τ & 0 or ϕ is not nilpotent, i.e., the composition (ϕ * idL#(n−1) ) ◦ · · · ◦ ϕ: E −→ E * L#n is non-zero for all n > 0. • The condition
µ(F)(≤)µ(E)
holds for every subbundle {0} ! F ! E which is ϕ-invariant, i.e., which satisfies ϕ(E) ⊆ E * L. Fix a line bundle N on X. Now, a Hitchin pair (of type (r, d, N)) is a triple (E, ϕ, ε) which consists of a vector bundle E of rank r and degree d, a twisted endomorphism ϕ: E −→ E * N, and a complex number ε. We say that the Hitchin pair (E1 , ϕ1 , ε1 ) is isomorphic to the Hitchin pair (E2 , ϕ2 , ε2 ), if there exist an isomorphism ψ: E1 −→ E2 and a non-zero complex number z, such that z · ϕ2 = (ψ * idN ) ◦ ψ−1
and z · ε2 = ε1 .
Finally, we say that the Hitchin pair (E, ϕ, ε) is (semi)stable, if i) ε % 0 or ϕ is not nilpotent and ii) µ(F)(≤)µ(E) holds for every ϕ-invariant subbundle {0} ! F ! E. Exercise 2.3.6.11. Choose a line bundle L and embeddings ι1 : N −→ L and ι2 : OX −→ L. Finally, let δ be a stability parameter, such that any *-swamp of type (d, deg(L)) is δ-(semi)stable, if and only if it is asymptotically (semi)stable. Show that the moduli space H it(r, d, N) of semistable Hitchin pairs of type (r, d, N) may be realized as a closed subscheme of M δ-ss (*, d, l). The moduli space H it(r, d, N) is a compactification of the so-call Hitchin space that classifies semistable Higgs bundles. Consult the papers [98], [183], and [187] for more details.
2.3.7 The Chain of Moduli Spaces Theorem 2.3.4.3, our discussions in Chapter 1, Section 1.6.2, the last section, and a little extra work yield the following result: Theorem 2.3.7.1. Fix the input data *, d, and l. Then, there is a finite set { % δ1 , . . . , % δm } of rational numbers 0 =: % δ0 < % δ1 < · · · < % δm < % δm+1 := ∞, such that, for a *-swamp (E, L, ϕ) with deg(E) = d and deg(L) = l, the following properties hold true:
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i) Suppose there is an index i ∈ { 0, . . . , m } with % δi < δ1 < δ2 < % δi+1 . Then, (E, L, ϕ) is δ1 -(semi)stable, if and only if it is δ2 -(semi)stable. In particular, there is a canonical isomorphism M δ1 -ss (*, d, l) ! M δ2 -ss (*, d, l). ii) Assume % δi < δ < % δi+1 , for some index i ∈ { 1, . . . , m − 1 }. If (E, L, ϕ) is δsemistable, then (E, L, ϕ) is also % δi - and % δi+1 -semistable, so that there are canonical morphisms M δ-ss (*, d, l) −→ M δi -ss (*, d, l) and M δ-ss (*, d, l) −→ M δi+1 -ss (*, d, l). %
%
Conversely, if (E, L, ϕ) is % δi - or % δi+1 -stable, then (E, L, ϕ) is also δ-stable. % iii) Suppose δ > δm . If (E, L, ϕ) is δ-semistable, it is also % δm -semistable, so that there is a natural morphism M δ-ss (*, d, l) −→ M δm -ss (*, d, l). %
Conversely, if (E, L, ϕ) is % δm -stable, then (E, L, ϕ) is also δ-stable. % iv) Suppose 0 < δ < δ1 . If (E, L, ϕ) is δ-semistable, then E is a semistable vector N bundle. Letting M 0 be the moduli space of semistable vector bundles of rank r and degree d, we find a canonical morphism N0 . M δ-ss (*, d, l) −→ M If E is a stable vector bundle, then (E, L, ϕ) is δ-stable. Ni := M %δi -ss (*, d, l), i = 1, . . . , m, Mi := M δ-ss (*, d, l) for some δ with We set M % δi−1 < δ < % δi , i = 1, . . . , m, and M∞ := M δ-ss (*, d, l) for some δ with δ > % δm . Our theorem is then summarized by the following picture:
N M 0
M1 & 000 & 00 && 00 && 0. & &&
N M 1
Mm M∞ -) ( ) ( -)) (( -)) (( ) ( () (( N Nm . M M m−1
Note that this theorem is a nice analog to the phenomenon which we have encountered in the GIT setting in Chapter 1, Section 1.6. Remark 2.3.7.2. The moduli space M∞ may be empty or not. If one takes * = id: GLr (') −→ GL('r ), so that we are in the realm of Bradlow pairs, then M∞ is indeed empty. If we take N ≥ r and look at the representation * := id$N : GLr (') −→ GL('r $N ), then the moduli space M∞ parameterizes pairs (E, ϕ: E −→ L$N ) where ϕ is an injection. These objects do not have to verify any other semistability condition [164]. The observations i) and ii) in Remark 2.3.6.4 yield natural explanations for these examples: In !($), we never have semistable points, because the kernel will always lead to a destabilizing one parameter subgroup. On the other hand, in !($$N ), the points which correspond to injections $ −→ '(X)N are stable.
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Bradlow Pairs In this section, we will sketch some results from the paper [213]. We look at the representation * = id: GL2 (') = GL('2 ) and let d be an odd integer. Fix a line bundle N on X, and recall the morphism f from (2.29). This time, we will be interested only in the moduli spaces ^ 4 M δ-ss (*, N, OX ) := f −1 [N], [OX ] for δ-semistable *-swamps of type (d, 0) of the shape (E, OX , ϕ) where det(E) ! N (see (2.30)) as δ varies. We begin with the following simple result. (To ease notation, we will write a *-swamp as a pair (E, ϕ).) Lemma 2.3.7.3. i) If δ > −d, then the moduli space M δ-ss (*, N, O ) is empty. X
ii) The critical values of the stability parameter are −d + 2 f , f = ;d/2/, . . . , 0.
Proof. Ad i). Let (E, ϕ) be a δ-semistable *-swamp. There is an effective divisor D, such that ϕ(E) = OX (−D). Therefore, E is given as an extension {0} −−−−−→ N(D) −−−−−→ E −−−−−→ OX (−D) −−−−−→ {0}. Using semistability, we estimate ^ 4 d−δ . d ≤ deg N(D) ≤ 2 This shows δ ≤ −d. Ad ii). Let 0 < δ1 < δ2 be two stability parameters and (E, ϕ) a *-swamp which is δ1 -semistable but not δ2 -semistable. This means that ker(ϕ) desemistabilizes13 (E, ϕ) with respect to the stability parameter δ2 . The critical value where ker(ϕ) becomes destabilizing is ^ 4 δ. := d − 2 · deg ker(ϕ) . If on the other hand, (E, ϕ) is δ2 -semistable but not δ1 -semistable, there is a line subbundle F which is not contained in ker(ϕ), such that it is desemistabilizing for the parameter δ1 . In fact, we find the largest parameter δ.. for which it becomes destabilizing via d − δ.. , deg(F) − δ.. = 2 so that δ.. = 2 · deg(F) − d. (Note that deg(F) ≤ 0, because F maps injectively to OX under ϕ.) " Exercise 2.3.7.4. Show that, for a non-critical stability parameter δ, the notions of δstability and δ-semistability agree on *-swamps (E, ϕ) with det(E) ! N. N Let us describe the extremal moduli spaces. According to Lemma 2.3.7.3, M m is the moduli space to the stability parameter −d. As we have seen in the proof of that lemma, a (−d)-semistable *-pair is given as an extension {0} −−−−−→ N −−−−−→ E −−−−−→ OX −−−−−→ {0}. 13 With
this ugly but precise terminology, we want to express that ker(ϕ) violates the condition of semistability. When we speak of “destabilization”, we mean the fact that the condition of stability fails.
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Now, (E• , α• ) := ({0} ! N ⊂ E, (1)) is a weighted filtration with µ(E• , α• , ϕ) = 0. One N computes that the associated admissible deformation is (N - OX , 0 - idOX ). Thus, M m is just a point. Exercise 2.3.7.5. Let δ ∈ (−d − 2, −d) be a stability parameter. Show that the vector bundle E in a δ-semistable *-swamp (E, ϕ) is given as a non-split extension {0} −−−−−→ N −−−−−→ E −−−−−→ OX −−−−−→ {0}. Let / := Ext1 (OX , N) ! H 1 (N) ! H 0 (N ∨ * ωX )∨ be the vector space that parameterizes such extensions. Show that the moduli space Mm = M δ-ss (*, N, OX ) identifies—at least set-theoretically—with the projective space !(/∨ ). Next, let us look at a stability parameter δ ∈ (0, 2). In that case, a *-swamp (E, ϕ) is δ-semistable, if and only if it is δ-stable, if and only if E is a stable vector bundle. (This results from Theorem 2.3.7.1, iv), and the fact that a rank two vector bundle of N odd degree is stable if and only if it is semistable.) The fiber of M1 −→ M 0 over the stable vector bundle [E] naturally identifies with !(H 0 (E ∨ )∨ ). Proposition 2.3.7.6. Assume that d < 2 − 2g. Then, the moduli space M1 is the N0 . projectivization of a vector bundle of rank −d + 2 − 2g over M Proof. Let E be a stable vector bundle of rank 2 and degree r. Then, h1 (E ∨ ) = h0 (E * ωX ). By assumption, E * ωX is a stable vector bundle of degree d + 2g − 2 < 0, whence N it cannot have a global section. Now, there is a universal bundle EMN0 on M 0 x X ([135], Section 7). The sheaf 4 ^ H = πMN0 . E ∨N M0
is locally free and one may check that M1 is just !(H ∨ ).
"
Remark 2.3.7.7. Note that both the moduli space Mm and the moduli space M1 have dimension −d + g − 2. Next, let us look at the transitions between the moduli spaces at a critical value. So, let δi = −d + 2 f be the ith critical value. We first analyze the map Ni . c−i : Mi −→ M Let δ−i ∈ (−d + 2( f − 1), −d + 2 f ) be a stability parameter and (E, ϕ) a δ−i -stable *swamp (observe Exercise 2.3.7.4) which is not δi -stable. Then, the kernel of ϕ has degree d − f . It is therefore of the form N(D) for the divisor D of degree h := − f with N ϕ(E) = OX (−D). The image of [E, ϕ] in M i is the S-equivalence class of ^ 4 (E D , ϕD ) := N(−D) - OX (−D), 0 - (OX (−D) ⊂ OX ) . (2.33) Conversely, let D be an effective divisor of degree f . We look at non-split extensions {0} −−−−−→ N(D) −−−−−→ E −−−−−→ OX (−D) −−−−−→ {0}.
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For such an extension, (E, E −→ OX (−D) ⊂ OX ) is a *-swamp which is δ−i -stable but properly δi -semistable (check this as an exercise). The projectivized extension space !(Ext1 (OX (−D), N(D))∨) identifies with the fiber of c−i over [ED , ϕD ]. We have ^ 4 ^ 4 Ext1 OX (−D), N(D) ! H 1 N(2D) . Since N(2D) has negative degree, we find h0 (N(2D)) = 0, and it follows that ^ ^ 4 4 dim Ext1 (OX (−D), N(D)) = h0 N ∨ (−2D) * ωX = −d + 2 f + g − 1.
(2.34)
The effective divisors of degree h may be parameterized by a smooth projective algebraic variety X (h) of dimension h, the so-called h-fold symmetric power of the curve X (see §3 in [151]). On X (h) x X, there is the universal divisor Δ, and we may form the sheaf 4 ^ Di− := πX (h). π.X (N * ωX ) * OX (h) x X (−2Δ) . It is locally free, and we set D−i := !(Di−∨ ). There is a universal extension {0} −−−−−→
π.X (N)(Δ)
ϕBD−
4 ^ i −−−−−→ ED−i −−−−−→ π.D− OD−i (−1) (−Δ) −−−−−→ {0} i
D−i
on x X ([127], Corollary 4.5). Then, (ED−i , κD−i , OD−i (−1), ϕD−i := (inclusion) ◦ ϕBD− ), i κD−i being the constant morphism mapping everything to [N], is a family of δ−i -stable *-swamps and thus defines an injective morphism e−i : D−i −→ Mi . Likewise, we have the morphism fi : X (h)
−→
Ni M
D
1−→
[E D , ϕD ]
(2.35)
and the diagram D−i 3 X (h)
e−i
7 Mi c−i
fi
3 7M N i
commutes. By construction, we have: Proposition 2.3.7.8. The induced morphism Ni \ fi (X (h) ) Mi \ e−i (D−i ) −→ M is an isomorphism. Remark 2.3.7.9. Note that the dimension of e−i (D−i ) is −d + g − 2 + f , by (2.34). We see that the maps c−i are, for i = 1, . . . , m − 2, isomorphisms away from a closed subset of codimension at least two. (The map c−m was the contraction of a projective space onto a point. The map c−m−1 will be explained below.)
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Let us now study the morphism Ni . c+i : Mi+1 −→ M Choose a stability parameter δ+i ∈ (−d + 2 f, −d + 2( f + 1)) and assume that (E, ϕ) is a δ+i -stable *-swamp which fails to be δi -stable. Then, there is a subbundle M of E which does not lie in the kernel of ϕ and which has degree f . Therefore, M ! ϕ(M) = OX (−D) N for a unique effective divisor D of degree h := − f . Again, the image of [E, ϕ] in M i is the S-equivalence class of (E D , ϕD ) from (2.33). Note that the restriction ϕ|D : E|D −→ OD vanishes on OD (−D), whence it gives a homomorphism ϕ: N(D)|D −→ OD (note that N(D) ! (E/M)). So, we have associated to (E, ϕ) a pair (D, ϕ) where D is an effective divisor of degree h and ϕ ∈ H 0 (N ∨ (−D)|D ). Conversely, suppose we are given a pair (D, ϕ) as above. Let ^ 4 ^ 4 (e) ∈ H 1 N ∨ (−2D) ! Ext1 N(D), OX (−D) be the image of ϕ under the boundary map H 0 (N ∨ (−D)|D ) −→ H 1 (N ∨ (−2D)). This yields an extension {0} −−−−−→ OX (−D) −−−−−→ E −−−−−→ N(D) −−−−−→ {0}. Furthermore, we have the natural inclusion ι: OX (−D) ⊂ OX . An explicit computation with cocycles ([213], (3.3)) reveals that ι maps to zero under the boundary map H 0 (OX (D)) −→ H 1 (N ∨ (−D)) in the cohomology sequence of the dual of the above extension. Thus, ι gives rise to a non-trivial homomorphism ϕ: E −→ OX . The above constructions establish a bijection between the set of isomorphy classes of δ+i -stable *-swamps which are properly δi -semistable and pairs (D, [ϕ]) which consist of an effective divisor of degree h and [ϕ] ∈ !(H 0 (N ∨ (−D)|D )∨ ). These objects may again be parameterized. To this end, note that ^ 4 Di+ := πX (h) . π.X (N)(−Δ) *OX(h) x X OΔ is a locally free sheaf of rank h on X (h) . Hence, D+i := !(Di+∨ ) is the parameter space which we have been looking for. Again, it is possible to construct a universal family ([213], (3.3)) (ED+i , κD+i , OD+i (1), ϕD+i ) on D+i x X which defines an injective morphism e+i : D+i −→ Mi+1 . With the morphism fi from (2.35), the diagram D+i 3
X (h)
e+i
7 Mi+1 c+i
fi
commutes. We deduce the following result:
3 7M Ni
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Proposition 2.3.7.10. The induced morphism Ni \ fi (X (h) ) Mi+1 \ e+i (D+i ) −→ M is an isomorphism. Remark 2.3.7.11. The dimension of e+i (D+i ) is 2 f − 1. Thus, the morphisms c+i are, for i = 1, . . . , m, isomorphisms away from a closed subset of codimension at least two. In the special case i = m, we see that the dimension of e+i (D+i ) equals the dimension of D+i , so that c+m is bijective. Nm−1 is irreducible and normal under Exercise 2.3.7.12. Show that the moduli space M the assumption d < 2 − 2g. (Adapt the arguments given in the proof of Theorem 2.3.6.1.) Theorem 2.3.7.13. The moduli spaces M1 ,. . . ,Mm are irreducible smooth projective varieties of dimension −d + g − 2. Proof. Everything except for the smoothness follows from the determination of the extremal moduli spaces (Exercise 2.3.7.5 and Proposition 2.3.7.6), the determination of the exceptional loci in the flip diagram (Proposition 2.3.7.8 and 2.3.7.10), and the evaluation of their codimensions (Remark 2.3.7.9 and 2.3.7.11). Using deformation theory, one may compute the tangent spaces of the moduli spaces M δ-ss (*, N, OX ) at stable points ([213], §2). It turns out that they are always (−d + g − 2)-dimensional. " By smoothness and Exercise 2.3.7.12, the final observation in Remark 2.3.7.11 N tells us that the map c+m : Mm −→ M m−1 is an isomorphism. Recall that the image N of fm−1 : X −→ Mm−1 describes those non-split extensions of OX by N which have a maximal line subbundle of degree d + 1. It follows from the geometry of the extension space Ext1 (OX , N) ! H 0 (N ∨ * ωX )∨ ([128], Proposition 1.1) that these are exactly those that lie on the image of X under the embedding X '→ !(H 0 (N ∨ * ωX )) which comes from the evaluation map H 0 (N ∨ * ωX ) * OX −→ N ∨ * ωX . The morphism Nm−1 c−m−1 : Mm−1 −→ M is then the blow-up of !(H 0 (N ∨ * ωX )) in the curve X. Theorem 2.3.7.14. The Picard group of the moduli spaces Mi , i = 1, . . . , m − 1, is isomorphic to ( - (. Proof. For Mm−1 , this follows from the explicit description of the morphism c−m−1 . For the other spaces it follows from the fact that they are isomorphic to Mm−1 away from a closed subset of codimension at least two (Proposition 2.3.7.8 and 2.3.7.10, Remark 2.3.7.9 and 2.3.7.11). " N Corollary 2.3.7.15. The Picard group of the moduli space M 0 is isomorphic to (. N Proof. In Proposition 2.3.7.6, we have seen that M1 is a projective bundle over M 0 , so N ( . We conclude by Theorem 2.3.7.14. " that Pic(M1 ) ! Pic(M ) 0
174
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This result shows how the theory of *-swamps may be used to gather information on the moduli space of vector bundles. The above discussion has a much more interesting application than the determination of the Picard group. Indeed, let L be an ample N0 ). Then, the above birational transformations allow to transfer the generator of Pic(M computation of χ(L #n ), n ∈ (, to a computation on a projective space. As a result, one finds a closed formula for those Euler characteristics. Via a stunning connection with theoretical physics, this formula was predicted by Verlinde. This marvelous application of *-swamps in presented in [213]. The strategy of that paper has found many other applications (see, e.g., [10], [39], [46]).
2.4 Principal Bundles as Swamps n this part, we will explain how principal bundles with structure group G may be described as vector bundles with sections in an associated vector bundle in order to define a notion of semistability for them, and, most importantly, construct moduli spaces, using the techniques which we have developed in the last section.
"
2.4.1 Principal Bundles and Associated Vector Bundles We assume that G is a semisimple linear algebraic group, i.e., a connected reductive linear algebraic group with finite center. The groups SLn ('), PGLn ('), and SOn (') are examples for such groups. Let κ: G '→ GL(W) be a faithful representation. Any semisimple group is equal to its derived group D(G) = [G, G] ([30], 14.2 Corollary). Therefore, a semisimple group doesn’t have any non-trivial character, so that, in particular, det ◦κ: G −→ '. is trivial, i.e., κ takes values in the special linear group SL(W). This will be quite important for our arguments. Remark 2.4.1.1. i) The condition that G be connected is not essential for our arguments. Indeed, it suffices that κ maps the connected component G0 of the neutral element e ∈ G to SL(W) (compare Remark 2.4.2.2). ii) In principle, all the constructions we carry out here apply to any reductive group G and any faithful representation κ: G −→ SL(W) ⊂ GL(W) (which always exists). However, the notion of semistability which comes out of our theory is too restrictive, if the center of G has positive dimension. The technical point is that one parameter subgroups of the center of G will contribute to the semistability concept. But, by the work of Ramanathan [175], they shouldn’t.14 In Section 2.4.8, we will first illustrate this by an example and then introduce a formalism which allows to treat arbitrary connected reductive groups via representations of G whose kernel may be non-trivial but lies in the center. 14 The reason is the following: If P is a principal G-bundle, then any element of the center of G gives rise to an automorphism of P. (This is just like in the case of GLr (").) Moreover, if λ is a one parameter subgroup of the center of G, then the associated parabolic subgroup is the whole group G, whereas the associated parabolic subgroup of SL(W) may be a genuine subgroup. (Give an example for that!)
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175
Given a principal G-bundle P on X, we use the representation κ in order to associate to P the principal GL(W)-bundle κ. (P) := P(GL(W)). Recall from Example 2.1.1.9 that we have an equivalence between the groupoid of principal GL(W)bundles and the groupoid of vector bundles with fiber W via P 1−→ P(W) and E 1−→ P(E) := I som(W * OX , E). Thus, to any principal G-bundle P, we assign the vector bundle E = P(W), such that I som(W * OX , E) = κ. (P). Remark 2.4.1.2. Observe that the vector bundle E has trivial determinant, i.e., det(E) = 5r E ! OX . In fact, the determinant of E is isomorphic to the line bundle associated to P via the representation d: G −→ GL(W) −→ '. . det
Since G is semisimple, this homomorphism is trivial, so that det(E) is trivial. In order to reconstruct the principal G-bundle P, we have to equip E with an additional structure. The representation κ: G '→ GL(W) yields the G-equivariant embedding !" P = P(G) '' '' '' '' '' +
X.
^ 4 7 P GL(W) = κ. (P) !! !!! ! ! !! /!!!
In the above diagram, we may take the G-quotients, so that we also find a section σ: X −→ I som(W * OX , E)/G. Conversely, let (E, σ) be a pair which consists of a vector bundle E with fiber W and a section σ: X −→ I som(W * OX , E)/G. Then, we form the fiber product 7 I som(W * OX , E)
P(E, σ) 3 X
G- bundle
σ
3 7 I som(W * OX , E)/G.
In other words, we pull back the principal G-bundle I som(W *OX , E) −→ I som(W * OX , E)/G to X by means of the section σ. Exercise 2.4.1.3. Show that E has trivial determinant, if a section σ: X −→ I som(W * OX , E)/G does exist. An isomorphism between pairs (E, σ) and (E B , σB ) as above is a vector bundle iso* ◦ σ, ψ *: P(E) −→ P(E B ) being the associated morphism ψ: E −→ E B , such that σB = ψ isomorphism. Lemma 2.4.1.4. The groupoid of principal G-bundles with isomorphisms is equivalent to the groupoid of pairs (E, σ: X −→ I som(W * OX , E)/G) with isomorphisms. Proof. Let P and P B be two principal G-bundles. Any isomorphism α: P −→ P B of principal G-bundles gives rise to a vector bundle isomorphism α: E −→ E B of the
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176 S 2.4: P B
associated vector bundles, and thus to an isomorphism * α: P(E) −→ P(E B ) of principal GL(W)-bundles. Finally, there are G-equivariant embeddings ι: P '→ P(E) and ιB : P B '→ P(E B ), such that the diagram P
!"
ι
α
3 !" PB
7 P(E)
(2.36)
αB
ι
B
3 7 P(E B )
commutes. This follows readily, using the canonical identifications P = (P x G)/G and P(E) = (P x GL(W))/G. Conversely, suppose we are given (E, σ) and (E B , σB ) and an isomorphism ψ: E −→ B * ◦ σ. Let P '→ P(E) be the preimage of σ(X) and P B '→ E , such that σB = ψ B P(E ) the preimage of σB (X). The isomorphism αB : P(E) −→ P(E B ) resulting from ψ induces an isomorphism α: P −→ P B , such that Diagram 2.36 commutes. Note that the induced map P '→ P(E) −→ X equips P with the structure of a principal G-bundle over X. Obviously, it is the one constructed from (E, σ) as described before. The same goes for P B . Finally, the morphism α is an isomorphism of principal G-bundles. " The Notion of Semistability Let us describe our notion of semistability for a pair (E, σ) which consists of a vector bundle E of rank r = dim. (W) and a section σ: X −→ I som(W * OX , E)/G. In order to do so, let λ: '. −→ G be a one parameter subgroup of G. This yields the parabolic subgroup QG (λ) (Chapter 1, Proposition 1.5.1.32) and the weighted flag (W• (λ), α• (λ)) in W (Chapter 1, Example 1.5.1.36). In fact, QG (λ) consists exactly of the elements of G that fix the flag W• (λ). A reduction of (E, σ) to λ is a section β: X −→ P(E, σ)/QG (λ).15 It defines a weighted filtration (E• (β), α• (β)) of E with α• (β) = α• (λ), and the filtration E• (β) : {0} ! E1 ! · · · ! E s ! E is (as in Exercise 2.1.1.21, ii) obtained from the section β
βB : X −→ P(E, σ)/QG (λ) '→ I som(W * OX , E)/QGL(W) (λ). We call a principal G-bundle (E, σ) (semi)stable, if for every one parameter subgroup λ: '. −→ G and every reduction β of (E, σ) to λ, we have ^ 4 M E• (β), α• (β) (≥)0. 15 The formalism which we have explained before applies to any closed subgroup of GL(W). Thus, P(E, σ) and β define—via a suitable fiber product—a principal QG (λ)-bundle Q, such that the principal G-bundle P associated to Q via the inclusion QG (λ) ⊂ G is canonically isomorphic to P(E, σ). Thus, the structure group G of P(E, σ) is “reduced” to the smaller group QG (λ). (Our terminology is a bit abusive with respect to the usual one: A reduction to λ does not mean that the structure group is reduced to ". by means of the homomorphism λ.)
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Remark 2.4.1.5. i) We may describe a principal G-bundle P on X as a pair (E, σ). Here, we use the chosen representation κ. For (E, σ) and, hence, for P, we have defined a notion of (semi)stability. It may seem that this notion depends on the choice of the representation κ. This is, however, not the case. Ramanathan has defined (semi)stability for principal G-bundles in intrinsic terms ([175], Definition 2.13 and Lemma 2.15), and our notion agrees with his (Theorem 2.4.9.3). This is also verified in [79], Section 4.5. ii) In general, there will be unstable semistable principal G-bundles for any topological type t ∈ Π(G) ([174], Proposition 7.8). (For the trivial topological type, the trivial principal G-bundle X x G is such an example.) Polystability and S-Equivalence Let us give a brief hint how the notion of S-equivalence for principal G-bundles looks like. Again, we let λ: '. −→ G be a one parameter subgroup. Recall that a Levi subgroup of QG (λ) is given by F L LG (λ) := g ∈ G | λ(z) · g = g · λ(z), for all z ∈ '. and the unipotent radical of QG (λ) by ^ 4 F L Ru QG (λ) := g ∈ G | lim λ(z) · g · λ(z)−1 = e . z→∞
Then, Ru (QG (λ)) is a normal subgroup of QG (λ), and one may identify QG (λ) with the semidirect product Ru (QG (λ)) " LG (λ). Now, let P be a semistable principal G-bundle defined by means of the pair (E, σ). Suppose that β: X −→ P(E, σ)/QG (λ) is a reduction to the one parameter subgroup λ with ^ 4 M E• (β), α• (β) = 0. Via the cartesian square 7 P(E, σ)
Q(β) 3 X
β
3 7 P(E, σ)/QG (λ),
we define the principal QG (λ)-bundle Q(β). Next, we use the group homomorphism QG (λ) −→ LG (λ) ⊂ G to associate to the principal QG (λ)-bundle the principal G-bundle df β (P) := df β (E, σ), the admissible deformation of P associated to β. A principal G-bundle is called polystable, if it is semistable and isomorphic to each of its admissible deformations.
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Lemma 2.4.1.6. Let P be a semistable principal G-bundle. Every sequence of nontrivial admissible deformations yields after finitely many steps a polystable admissible deformation. The polystable principal G-bundle thus obtained is well-defined up to isomorphy of principal G-bundles. Proof. This will follow from the GIT construction of the moduli space of semistable principal bundles (cf. Remark 2.3.5.18 for the argument in the case of swamps): We will construct a parameter space Y for principal G-bundles with an action of a general group SL(Y), such that the moduli space is the GIT-quotient Y// SL(Y). Let P = (E, σ) be a principal G-bundle. The GIT-computations will show that reductions β: X −→ P(E, σ)/QG (λ) to one parameter subgroups λ with ^ 4 M E• (β), α• (β) = 0 correspond to one parameter subgroups Λ: '. −→ SL(Y) with µ(y, Λ) = 0, y ∈ Y representing (E, σ). Furthermore, the principal G-bundle corresponding to the point limz→∞ Λ(z) · y is df β (P). Thus, our definition and the Hilbert–Mumford criterion (Chapter 1, Section 1.5.1) imply that polystable principal G-bundles correspond to " points with closed SL(Y)-orbit in Y. The polystable bundle defined by P will be denoted by gr(P). Now, we say that the principal G-bundles P1 and P2 are S-equivalent, if gr(P1 ) and gr(P2 ) are isomorphic as principal G-bundles. Moduli Spaces Let Π(G) be the set of isomorphy classes of topological principal G-bundles on the Riemann surface X. This set can be mapped bijectively onto the fundamental group π1 (G) (see Proposition 2.1.1.24). (The fundamental group π1 (G) is finite for semisimple groups.) Fix an element t ∈ Π(G). As for vector bundles, we easily introduce the moduli functors M(s)s (ϑ): Sch. −→ Set
which associate to a scheme over ' the set of isomorphy classes of principal G-bundles PS on S x X whose restriction to {s} x X is (semi)stable, for every geometric point s of S.
Remark 2.4.1.7. Assume that G is not special in the sense of Remark 2.1.1.6. For a parameter scheme S , a principal G-bundle on S x X will, in general, not be trivial in the Zariski topology, even if its restriction to every subscheme {s} x X is. The reader may consult [61] for more information. Theorem 2.4.1.8. There exists a projective moduli space M ss (ϑ) for the moduli functors M(s)s (ϑ). This theorem was first proved by Ramanathan [175]. His construction is very difficult and ingenious. The rough idea is the following. We look at the adjoint representation Ad: G −→ GL(g) of G on its Lie algebra g. Then, the image of G is the
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connected component Aut(g)0 of the identity of the group of Lie algebra automorphisms of g ([113], §14.1). Given a principal G-bundle on X, we get an associated principal Aut(g)-bundle. Now, principal Aut(g)-bundles may be easily identified with pairs (E, τ: E * E −→ E), consisting of a vector bundle E with fiber g and a Lie bracket τ: E * E −→ E which is fiberwise isomorphic to g. With our background on swamps (which Ramanathan did not have at his disposal), we may imagine that the classification problem for pairs (E, τ) is tractable. So, Ramanathan first constructs the moduli space for these Lie bundles and explains how one may work one’s way back to principal G-bundles. As remarked before, the latter is a very involved procedure. Recently, G´omez and Sols gave a construction of moduli spaces for so-called principal G-sheaves on base manifolds of arbitrary dimension which, though different in the details, follows Ramanathan’s philosophy [83]. In Section 2.4.8, we will discuss this approach in order to obtain moduli spaces for semistable principal G-bundles whose structure group G is connected and reductive but has a non-trivial radical. Alternative constructions for the moduli spaces were given by Faltings [63] and Balaji and Seshadri[12]. Balaji and Seshadri also start with a faithful representation κ: G −→ GL(W). Then, they use the fact that a principal G-bundle P is semi- or polystable, if and only if the associated vector bundle P(W) is semi- or polystable, respectively, in order to get a quick construction of the moduli space as a quasi-projective variety. The proof of properness they use requires some intricate arguments related to Bruhat–Tits theory. Balaji applied these techniques (together with the formalism of singular principal bundles from [186] and [188]) to arrive at interesting generalizations on higher dimensional base manifolds. A new and much easier proof for properness which also applies to large extent to positive characteristic is due to Heinloth [101]. Below, we will explain the construction the author gave in [186] and [188], using some simplifications found in [79] and [80]. The advantage of this approach is that it is the one which is technically the most simple and the most faithful to the GIT process. It also enables us to treat decorated principal bundles which does not seem straightforward to do with the other approaches. Remark 2.4.1.9. Again, if one does not want to impose a semistability condition on the principal bundles, then one must construct the moduli space as a stack. We recommend the reader to have a look at [15] and the lecture notes [206] for an account of this theory.
2.4.2 Back to Some GIT Let κ: G −→ GL(W) be the representation which we have fixed before. We have the space - := Isom(W, 'r )/G. Recall that we may describe a principal G-bundle on the curve X by means of a vector bundle E and a section σ: X −→ I som(W * OX , E)/G. Note that there is a canonical GLr (')-action on -. In the formalism of associated fiber spaces, we find I som(W * OX , E)/G ! P(-). Thus, it is important to understand the GLr (')-variety -. First, we look at the representation ^ 4 R: GLr (') x G −→ GL Hom(W, 'r )
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X & ^ 4Z J r −1 (h, g) − 1 → f ∈ Hom(W, ' ) 1−→ w 1−→ h f κ(g )(w) . The representation R provides an action of GLr (') x G on ^ 4 Hom(W, 'r ) and ! Hom(W, 'r )∨ and induces a GLr (')-action on the categorical quotients 4 ^ ^ 4 / := Hom(W, 'r )//G and / := ! Hom(W, 'r )∨ //G = / \ {0} //'.. The latter equality results from Chapter 1, Exercise 1.5.3.3. The coordinate algebra of / is Sym. (W * 'r ∨ )G . For s > 0, we set
,s :=
!
%0 , s
i
i=1
0i :=
&
4G Z∨ ^ , Symi W * 'r∨
i ≥ 0.
s Symi (W * 'r ∨ )G contains a set of generators for the algebra If s is so large that i=0 . r∨ G Sym (W * ' ) , then we have a GLr (')-equivariant surjection of algebras
Sym. (,∨s ) −→ Sym. (W * 'r∨ )G , and, thus, a GLr (')-equivariant embedding ι s : / '→ , s . We choose a basis for W in order to identify it with 'r . This enables us to define the function det: Hom(W, 'r ) f
−→
'
1−→ det( f ).
Now, this function is invariant under the G-action, because the image of κ lies in SL(W), as we have remarked before. Therefore, det descends to a function &: / −→ '. Exercise 2.4.2.1. Verify that F L Isom(W, 'r )/G = [ f ] ∈ / | &[ f ] % 0 . Remark 2.4.2.2. i) The construction of the embedding - ⊂ / is exactly the point where it becomes important that the image of the representation κ lies in SL(W). ii) If G is a non-connected reductive group, such that the connected component G0 of the neutral element is semisimple, some positive power detr will be G-invariant and we may play the same game as before. The next result describes the semistability of points ι s (h), h ∈ the action of the special linear group SLr ('). Lemma 2.4.2.3. i) Every point ι s (i), i ∈ -, is SLr (')-polystable. ii) A point ι s (h), h ∈ / \ -, is not SLr (')-semistable.
/, with respect to
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Proof. Ad i). As before, we choose a basis for W. This choice provides us with the (SLr (') x G)-invariant function det: Hom(W, 'r ) −→ ', f 1−→ det( f ), which descends to the (non-constant) function & on /. For any i ∈ -, we clearly have &(ι s (i)) % 0, so that ι s (i) is SLr (')-semistable. Furthermore, the (SLr (') x G)-orbit of a point f ∈ Isom(W, 'r ) is just a level set &−1 (z) for an appropriate z ∈ '. . In particular, it is closed. The image of this orbit is the SLr (')-orbit of i := [ f ] in / which is, therefore, closed. Since ι s is a closed, SLr (')-equivariant embedding, the orbit of ι s (i) is closed, too. Ad ii). It is obvious from the construction that the ring of SLr (')-invariant functions on / is generated by &. This makes the asserted property evident. " The following is a key result: Proposition 2.4.2.4. Let W −→ 'r be an isomorphism. It induces the isomorphism ψ: GL(W) −→ GLr (') of linear algebraic groups and the isomorphism ϕ: GL(W)/G −→ Isom(W, 'r )/G which is equivariant with respect to the action of GL(W) on the left hand space, of GLr (') on the right hand space, and the isomorphism ψ. Define i = ϕ([e]) as the class of the fixed isomorphism and x := *ι s (i). Then, for a one parameter subgroup λ: '. −→ SLr ('), the following conditions are equivalent: +s. i) µ*κs (x, λ) = 0, * κ s being the representation of SLr (') on , ii) There is a one parameter subgroup λB : '. −→ G(⊂ SLr (') via ψ ◦ κ) with ^ 4 ^ 4 W• (λ), α• (λ) = W• (λB ), α• (λBi ) , W := 'r . For the proof of the proposition, we need another result which was communicated to the author by Kraft and Kuttler. Let H be a reductive algebraic group, G a closed reductive subgroup, and X := H/G the associated affine homogeneous space. (Observe that H/G is the quotient H//G which we constructed in Chapter 1, Section 1.4.2. Since all orbits have the same dimension, they must be all closed, so that H//G is also an orbit space. It comes with an obvious transitive action by H.) Then, the following holds true: Proposition 2.4.2.5. Suppose that x ∈ X is a point and λ: '. −→ H a one parameter subgroup, such that x0 := limz→∞ λ(z) · x exists in X. Then, x ∈ Ru (QH (λ)) · x0 . Proof. We may assume x0 = [e], so that λ is a one parameter subgroup of G. Define L F Y := y ∈ X | lim λ(z) · y = x0 . z→∞
This set is closed and invariant under the action of Ru (QH (λ)). Note that viewing X as a variety with '. -action, x0 is the unique point in Y with a closed '. -orbit. We claim that there is a '. -equivariant morphism f : X −→ T x0 (X) which maps x0 to 0
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and is e´ tale at x0 . The coordinate algebra '[X] of X is a locally finite '. -module. The maximal ideal m of x0 is a submodule, because x0 is a fix point for the '. -action. Thus, '[X] = ' - m as a '. -module. Now, '. acts on '[X] as a group of '-algebra automorphisms. This means that m2 is a submodule of m. Since the module '[X] is completely reducible, there exists a submodule m which is isomorphic to m/m2 as '. -module, so that '[X] = ' - m - m2 as a '. -module. We derive the homomorphism
Sym. (m/m2 ) ! Sym. (m) −→ '[X] of '-algebras and '. -modules. It yields the morphism f . Note that the differential of f at x0 is the identity of T x0 (X), so that f is e´ tale at x0 . (The above construction is the initial step in the proof of Luna’s famous slice theorem [139].) Obviously, f maps Y to F L N := v ∈ T x0 X | lim λ(z) · v = 0 . z→∞
We claim that
N = uH (λ)/uG (λ) ⊂ h/g.
(2.37)
Here, uH (λ) and uG (λ) are the Lie algebras of Ru (QH (λ)) and Ru (QG (λ)), respectively, and h and g are the Lie algebras of H and G, respectively. Note that h and g receive their G-module structures through the adjoint representation of G, and, moreover, by definition, F L uH (λ) = v ∈ h | lim λ(z) · v = 0 . z→∞
This yields the asserted equality in (2.37). Note that the Ru (QH (λ))-orbit of x0 is closed, because X is affine ([30], 4.10 Proposition), and isomorphic to Ru (QH (λ))/Ru(QG (λ)). Its dimension equals the one of uH (λ)/uG (λ). Therefore, Ru (QH (λ)) · x0 is a connected component of Y. But Y is connected, because every element in it is connected to x0 by the image of a morphism from " +1. ! '. ∪ {∞} to X. Hence, Y = Ru (QH (λ)) · x0 , as desired. Exercise 2.4.2.6. Suppose G is a reductive linear algebraic group, σ: G x G −→ G, (g, h) 1−→ g · h · g−1 , is the adjoint action of G on itself, and Ad: G −→ GL(g) is the adjoint representation. Show that there is a G-equivariant morphism G −→ g which maps the neutral element e ∈ G to zero and whose differential at e is idg . Proof of Proposition 2.4.2.4. Let us begin with the implication “ii)=⇒i)”. Since G is the GL(W)-stabilizer of e, it also stabilizes x. We conclude that µ(x, λB ) = 0, for any one parameter subgroup λB : '. −→ G. We turn to the implication “i)=⇒ii)”. By Lemma 2.4.2.3, i), there exists an element gB ∈ SLr (k), such that ^ 4 xB := lim λ(z) · x = ι s ϕ(gB ) . z→∞
By Proposition 2.4.2.5, we may choose gB ∈ Ru (QSLr (k) (λ)). In particular, the element gB fixes the flag W• (λ) (see Exercise 2.3.2.3). Since λ fixes xB , it lies in gB · G · gB −1 .
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Setting λB := gB −1 · λ · gB , we obviously have (W• (λ), α• (λ)) = (W• (λB ), α• (λB )), and λB is a one parameter subgroup of G. " Next, we look at the categorical quotient 4 ^ / = Proj Sym. (W * 'r∨ )G . For any positive integer d, we define Sym(d) (W * 'r ∨ )G :=
% Sym (W ∞
id
*
i=0
'r∨ )G .
Then, by the Veronese embedding (or Segre transformation [154]) 4 ^ 4 ^ Proj Sym. (W * 'r∨ )G ! Proj Sym(d) (W * 'r∨ )G . We can choose s, such that a) Sym. (W * 'r ∨ )G is generated by elements in degree ≤ s. b) Sym(s!) (W * 'r ∨ )G is generated by elements in degree 1, i.e., by the elements in the vector space Syms! (W * 'r ∨ )G ([154], Chapter III, §8, Lemma). Set
,s :=
%
&
(d1 ,...,d s ): ' di ≥0, idi =s!
^ 4 4Z ^ Symd1 (W * 'r ∨ )G * · · · * Symds Sym s (W * 'r∨ )G .
(2.38)
Obviously, there is a natural surjection , s −→ Syms! (W *'r ∨ )G and, thus, a surjection Sym. (, s ) −→ Sym(s!) (W * 'r ∨ )G . This defines a closed and GLr (')-equivariant embedding ι s : / '→ !(, s ). Lemma 2.4.2.7. Let s be a positive integer, such that a) and b) as above are satisfied, and f ∈ Hom(W, 'r ) a G-semistable point. Set h := ι s ([ f ]) and h := ιs ([ f ]), and let * s be the representation of SLr (') on , s and σ s the linearization of the SLr (')-action on / in O" (s!). Then, for any one parameter subgroup λ: '. −→ G, we have µ*s (h, λ) > (= / (= / (= / 0 . Then, we call a pseudo G-bundle δ-(semi)stable, if the associated *a,b,c -swamp (E, ϕτ ) is δ-(semi)stable. Furthermore, we call two δsemistable pseudo G-bundles (E, τ) and (E B , τB ) S-equivalent, if the associated swamps (E, ϕτ ) and (E, ϕτB ) are S-equivalent. The definitions lead to the moduli functors Mδ-(s)s (κ) for the δ-(semi)stable pseudo G-bundles. Theorem 2.4.3.3. There exists a projective moduli space M δ-ss (κ) for the moduli functors Mδ-(s)s (κ).
2.4.4 Semistable Reduction for Principal Bundles and the Proof of Theorem 2.4.1.8 In this section, we will explain how Theorem 2.4.1.8 may be deduced from Theorem 2.4.3.3, using some observations on swamps and GIT which we have already made. Theorem 2.4.4.1 (Semistable reduction). Suppose δ > % δm (see Theorem 2.3.7.1), and let (E, τ) be a pseudo G-bundle. If (E, τ) is δ-semistable, then it is a principal G-bundle. Proof. Let (E, ϕτ ) be the associated *a,b,c -swamp. Denote the generic point of X by η and choose a trivialization E *OX OX,η ! '(X)$r . This trivialization and τ and ϕτ yield the points 4 ^ σ(τ)η ∈ Hom.(X) W *. '(X), '(X)$r //G ^ 4 and σ(ϕτ )η ∈ !(Wa,b,c ) x Spec '(X) , respectively. Spec(.)
If the stability parameter satisfies δ > % δm , then we know, by Remark 2.3.6.4, i), that σ(ϕτ )η ∈ !(Wa,b,c )ss xSpec(.) Spec('(X)), !ss (Wa,b,c ) being the open set of SLr (')semistable points in !(Wa,b,c ). Invoking Lemma 2.4.2.7 and the formulae at the end of Section 2.4.2, this shows 4 ^ σ(τ)η ∈ Isom.(X) W *. '(X), '(X)$r /G. The assertion now results from Lemma 2.4.3.1.
"
Remark 2.4.4.2. i) In the notation of the above proof, we choose δ > % δm . Recall that we have seen in Remark 2.3.4.4 that (E, τ) is δ-semistable, if and only if E is a semistable
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vector bundle. If P = (E, τ) is a principal G-bundle and E is semistable, then P is a semistable principal G-bundle. Corollary 2.4.4.6 will refine this observation. ii) The term “semistable reduction” usually refers to the following statement: Let R be a discrete valuation ring with field of fractions K. For any family PK of semistable principal G-bundles on Spec(K) x X, there exist a finite field extension K B /K and a family PRB of semistable principal G-bundles on Spec(RB ) x X, RB the normalization of R in K B , which extends PK xSpec(K) x X (Spec(K B ) x X). If one has, as in the work of Balaji and Seshadri [12], constructed the moduli space as a quasi-projective variety, then this property is equivalent to the properness of the moduli space. (We will treat this in Exercise 2.4.4.3; it is also discussed in the appendix to [79].) Theorem 2.4.4.1 therefore also implies the semistable reduction theorem. In fact, Theorem 2.4.3.3 grants the existence of a moduli space for δ-semistable pseudo G-bundles in which the semistable G-bundles sit as an open subset. Theorem 2.4.3.3 shows that the moduli space consists only of principal bundles, whence the moduli space of semistable principal bundles is projective. Exercise 2.4.4.3. After having studied the construction of M ss (ϑ) as the GIT quotient of a projective scheme, deduce the semistable reduction theorem in Remark 2.4.4.2 from the following result ([197], Theorem 4.1): Theorem 2.4.4.4 (Seshadri). Let V be a projective scheme on which the reductive group G acts, L an ample line bundle on V, and σ: G x L −→ L a linearization of the G-action on V in L. Then, given a discrete valuation ring R with field of fractions K and a K-valued point v of Vσss , there exist a finite extension K B /K and g ∈ G(K B ), such that g · x is an RB -valued point of Vσss , RB the normalization of R in K B . Proposition 2.4.4.5. Let (E, τ) be a principal G-bundle with associated *a,b,c -swamp (E, ϕτ ). For any weighted filtration (E• , α• ) of E, the following conditions are equivalent: i) µ(E• , α• , ϕτ ) = 0. ii) There exist a one parameter subgroup λ: '. −→ G and a reduction β: X −→ P(E, σ)/QG (λ), σ: X −→ I som(W * OX , E)/G being the section defined by τ, such that (E• , α• ) = (E• (β), α• (β)). Proof. Note that this proposition is just the relative version of Proposition 2.4.2.4. Moreover, the implication “ii)⇒i)” is trivial. For the converse direction, we will proceed as follows: • Find a one parameter subgroup λ: '. −→ G whose weighted flag is (W• , α• ) for a flag W• with dim. (Wi ) = rk(Ei ), i = 1, . . . , s. • The weighted filtration E• of E corresponds to a section βB : X −→ I som(W * OX )/QGL(W) (λ). We have to find a section β: X −→ P(E, τ)/QG (λ), such that βB is the composition of P(E, τ)/QG (λ) '→ I som(W * OX )/QGL(W) (λ) and the map β.
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For the second item, we have to verify that the image of βB lies in the image of P(E, τ)/ QG (λ) under the inclusion morphism. To do so, it suffices to verify that this holds over the generic point of X after possibly extending the field K := '(X). Fix an algebraic closure K of K. Let $ be the vector space E|{η} *K K and GK the extension of P(E, τ)(G)|{η} to K. Recall from Exercise 2.1.1.12 that P(E, τ)(G) is the automorphism group scheme of the principal G-bundle P(E, τ). An extension of an object from the ground field ' or K to K will be indicated by the subscript K . Note that GK is a closed subgroup of GL($). Finally, we choose a one parameter subgroup λBB : 4m (K) −→ GL($) whose weighted flag is (E•|{η} *K K, α• ). Now, we may apply Proposition 2.4.2.4, not over the complex numbers but over K. The conclusion is that there is a one parameter subgroup λB : 4m (K) −→ G K whose weighted flag is (E•|{η} *K K, α• ), too. Finally, GK ! G K and there is a one parameter subgroup λ: '. −→ G, such that λK is conjugate to λB . This finishes the construction of λ. We have seen that we may choose a trivialization P(E, τ)K ! G K as a principal G-bundle, such that the induced isomorphism GK ! G K of algebraic groups over K transforms the one parameter subgroup λB into the one parameter subgroup λK . This trivialization induces an isomorphism $K ! WK which carries the weighted flag (E•|{η} *K K, α• ) into (W•,K , α• ). The point ^ 4 ^ 4 βBK ∈ GL(W)/QGL(W) (λ) = I som(W * OX )/QGL(W) (λ) K
K
that is induced by βB is just [e]K and clearly lies in the image of (G/QG (λ))K = (P(E, τ)/QG (λ))K under the inclusion map. " Corollary 2.4.4.6. Let (E, τ) be a principal G-bundle and δ > % δm . Then, (E, τ) is a δ-(semi)stable pseudo G-bundle, if and only if it is a (semi)stable principal G-bundle in the sense of Section 2.4.1. Furthermore, (E, τ) is a semistable principal G-bundle, if and only if E is a semistable vector bundle. Remark 2.4.4.7. Ramanathan derived the last assertion from the correspondence between semistable principal G-bundles and representations of the fundamental group in a maximal compact subgroup K of G ([175], Proposition 3.17). Later, Ramanan and Ramanathan could give an algebraic proof for that fact based on the theory of the instability flag [173]. In the constructions of Ramanathan [175], Faltings [63], and Balaji and Seshadri [12], this fact is essentially used in the constructions. In our approach, we used the related fact that the tensor product of two semistable vector bundles is again semistable (see proof of Theorem 2.3.4.3). We also point out that the last assertion in the corollary does not hold for stability. Theorem 2.4.1.8 results now directly from Theorem 2.4.3.3, Theorem 2.4.4.1, and Corollary 2.4.4.6. "
2.4.5 The Proof of Theorem 2.4.3.3 By Theorem 2.3.4.1, there is a constant C, such that µmax (E) ≤ C, for every δ-semistable pseudo G-bundle (E, τ), i.e., E lives in a bounded family. (Recall that E has
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trivial determinant, whence degree 0.) We choose an n = 0, such that, for every vector bundle E of rank r with trivial determinant and µmax (E) ≤ C, one has the following: • H 1 (E(n)) = {0} and • E(n) is globally generated. Moreover, we suppose: • The construction of the moduli space of δ-semistable *a,b,c -swamps (E, ϕτ ) with det(E) ! OX can be performed with respect to n. We choose a complex vector space Y of dimension r(n + 1 − g). Let Q be the quot scheme of Theorem 2.2.3.5 for G := Y * OX (−n). We have the universal quotient ^ 4 qQ : Y * π.X OX (−n) −→ FQ . Then, there is the maximal open subset U ⊂ Q x X where FQ is locally free. Let Z be complement of U and QB be the complement of πQ (Z) in Q. By [116], Lemma 2.1.7, QB consists of those points q ∈ Q, such that FQ|{q} x X is locally free. Moreover,
the semicontinuity theorem implies that there is an open subset QBB of Q, consisting of the quotients q: Y * OX (−n) −→ F for which h1 (F (n)) = 0 holds. Now, set QBBB := QB ∩ QBB , and let ^ 4 qQBBB : Y * π.X OX (−n) −→ EQBBB be the restriction of the universal quotient to QBBB x X. We form ^ p := πQBBB . (qQBBB * idπ.X (OX (n)) ): Y * OQBBB −→ πQBBB . EQBBB
*
4 π.X (OX (n)) .
This is a homomorphism of locally free sheaves (of the same rank), using the semicontinuity theorem again. There is an open subset QBBBB ⊆ QBBB of points q where the restriction p|{q} is surjective, whence an isomorphism. Next, there is the morphism QBBBB
−→ Jac0
[q: Y * OX (−n) −→ E] 1−→ [det(E)]. We define Q as its fiber over [OX ]. The quasi-projective scheme Q parameterizes quotients q: Y * OX (−n) −→ E, such that E is a vector bundle of rank r with trivial determinant and H 0 (q(n)) is an isomorphism. Let ^ 4 qQ : Y * π.X OX (−n) −→ EQ be the universal quotient. For the vector bundle EQ , as for any vector bundle of rank r, we have the canonical isomorphism ∨ EQ
!
r−1 8
EQ *
r &8
Z∨ EQ .
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5 Since the restriction of ( r EQ )∨ to any fiber {q} x X, q ∈ Q, is trivial, there is a line bundle A on Q, such that r Z∨ &8 EQ ! π.Q (A ). Gathering all this information, we find a surjection r−1 ^ 8 4G ^ ^ 4 4 ∨ G Y * π.X (OX (−n)) * π.Q (A ) −→ S ym. W * EQ S ym. W * .
For a point [q: Y * OX (−n) −→ E] ∈ Q, any homomorphism τ: S ym. (W * E ∨ )G −→ OX of OX -algebras is determined by the composite homomorphism
% S ym &W s
i
r−1 ^ 8 *
i=1
Y * OX (−n)
4ZG
−→ OX
of OX -modules. Noting that r−1 ^ r−1 ZG & & 8 8 ^ 4 4ZG Y * OX (−n) S ymi W * Y * OX −i(r − 1)n , ! Symi W *
τ is determined by a collection of homomorphisms r−1 ZG & 8 ^ 4 Y * OX −→ OX i(r − 1)n , ϕi : Symi W *
i = 1, . . . , s.
Since ϕi is determined by the induced linear map on global sections, we will construct the parameter space inside Y :=
% H om&S ym &W s
i
r−1 8 *
i=1
Y *A
ZG
Z ^ 4 , H 0 OX (i(r − 1)n) * OQ .
Write π: Y −→ Q for the bundle projection and observe that, over Y x X, there are universal homomorphisms r−1 ZG & 8 ^ 4 * ϕi : S ymi W * Y * π.Q (A ) → H 0 OX (i(r − 1)n) * OY x X ,
i = 1, . . . , s.
Define ϕi = ev ◦ * ϕi as the composition of * ϕi with the evaluation map ^ 4 ^ 4 ev: H 0 OX (i(r − 1)n) * OY x X −→ π.X OX (i(r − 1)n) , i = 1, . . . , s. We twist ϕi by idπ.X (OX (−i(r−1)n)) and put the resulting maps together to the homomorphism ϕ: WY :=
% S ym &W s
i=1
i
r−1 & 8 *
ZG ^ 4Z Y * π.X OX (−n) * π.Q (A ) −→ OY x X .
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Next, ϕ yields a homomorphism of OY x X -algebras * τY : S ym. (WY ) −→ OY x X . On the other hand, there is a surjective homomorphism 4 ^ ∨ G ) β: S ym. (WY ) −→ S ym. W * (π x idX ). (EQ of graded algebras where the left hand algebra is graded by assigning the weight i to the elements in S ymi (· · ·)G . The parameter space Y is defined by the condition that * τY factorize over β, i.e., setting EY := ((π x idX ). (EQ ))|Y x X , there be a homomorphism 4G ^ τY : S ym. W * EY∨ −→ OY x X with * τY|Y x X = τY ◦ β. Formally, Y is defined as the scheme theoretic intersection of the closed subschemes (see Proposition 2.3.5.1) F L τd Yd := y ∈ Y |* : ker(βd|{y} x X ) −→ OX is trivial , d ≥ 0. Y|{y} x X
The family (EY , τY ) is the universal family of pseudo G-bundles parameterized by Y. By its construction, it has the following property: Proposition 2.4.5.1. The set of isomorphy classes A6 > _! (Ey , τy ) := (EY|{y} x X , τY|{y} x X ) !! y ∈ Y contains the isomorphy class of every δ-semistable pseudo G-bundle. There are natural actions of GL(Y) on the quot scheme Q and on Y. The latter action leaves the closed subscheme Y invariant, and therefore yields an action Γ: GL(Y) x Y −→ Y. As an exercise, the reader may again verify: Proposition 2.4.5.2. Two points y1 and y2 lie in the same GL(Y)-orbit, if and only if the pseudo G-bundles (Ey1 , τy1 ) and (Ey2 , τy2 ) are isomorphic. Conclusion of the Proof Suppose we knew that the points ([q: U * OX (−n) −→ E], τ) in the parameter space Y for which (E, τ) is δ-semistable form an open subscheme Yδ-ss . Then, it suffices to show that Yδ-ss possesses a projective categorical quotient by the action of GL(Y). Indeed, Proposition 2.4.5.1 and 2.4.5.2 and the universal property of the categorical quotient then imply that M δ-ss (κ) := Yδ-ss // GL(Y) has the desired properties. We have the natural surjection '. x SL(Y) −→ GL(Y), (z, m) 1−→ z · m, and obviously ^ 4 Yδ-ss // GL(Y) = Yδ-ss // '. x SL(Y) .
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By Chapter 1, Exercise 1.5.3.3, we may first form *δ-ss := Yδ-ss //'. Y and then
*δ-ss // SL(Y). Y
* := Y//'. which is a weighted projective bundle We can easily form the quotient Y * −→ Q is proper. Let T −→ Q be the over Q, so that we note that the morphism Y parameter space for *a,b,c -swamps which was constructed in Section 2.3.5. The map A: Y −→ T 4 ^ 4 ^ [q: Y * OX (−n) −→ E], τ 1−→ [q: Y * OX (−n) −→ E], ϕτ is invariant under the action of '. and equivariant with respect to the action of SL(Y). By the universal property of a categorical quotient, it induces a morphism * −→ T. *Y A: * enjoys the following properties: The morphism A • It is SL(Y)-equivariant, because A was. * −→ Q is a * is a morphism of schemes over Q. Since Y • It is proper. In fact, A * proper morphism, the same is true of A ([96], Corollary II.4.8). • It is injective. This follows from the arguments presented in the proof of Proposition 2.4.3.2. Now, there are open subsets Tδ-(s)s of T which parameterize the δ-(semi)stable * swamps, and we know that the good quotient
a,b,c
M δ-ss (*a,b,c , 0, 0) := Tδ-ss // SL(Y) exists as a projective scheme and that M δ-s (*a,b,c , 0, 0) := Tδ-s // SL(Y) exists as an open subscheme of M δ-ss (*a,b,c , 0, 0). By definition, A−1 (Tδ-ss ) = Yδ-ss , whence
*−1 (Tδ-ss ) = Yδ-ss //'. . A
* is finite (it is proper and quasi-finite), Chapter 1, Exercise 1.5.3.3, implies that Since A the quotient *−1 (Tδ-ss )// SL(Y) M δ-ss (κ) := Yδ-ss // GL(Y) = A exists as a projective scheme. Likewise, the open subscheme *−1 (Tδ-s )/ SL(Y) M δ-s (κ) := Yδ-s / GL(Y) = A is a geometric quotient and an open subscheme of M δ-ss (κ).
"
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2.4.6 The Geometry of the Moduli Spaces — A Guide to the Literature Since the moduli spaces of principal bundles are less familiar in Algebraic Geometry than the moduli spaces of vector bundles, we have compiled below a list of properties of the moduli spaces and references. • The moduli space M ss (ϑ) is an irreducible, normal, and Cohen–Macaulay projective variety of dimension (g(X) − 1) · dim(G) ([175], Theorem 5.9). (Some of these properties and generalizations to Higgs bundles may also be found in Faltings’s paper [63], e.g., Corollary III.2.) Note that M s (ϑ) might be singular ([174], Remark 4.1). We will establish a part of these attributes in Corollary 2.4.7.5. • For simple and simply connected structure groups, the moduli spaces are unirational ([126], Corollary 6.3). • If G is again simple and simply-connected, the Picard group of M ss (ϑ) is isomorphic to (, and explicit generators are known ([125], Theorem 2.4, and [34], Theorem 1.3). • Let ϑ0 ∈ Π(G) be the class of the bundle X x G. If M B (ϑ0 ) ⊂ M s (ϑ0 ) denotes the open subset that consists of stable principal G-bundles whose automorphism group consists of the center16 Z (G) of G, then a universal principal bundle on M B (ϑ0 ) x X exists, if and only if G is an adjoint group ([9], Theorem 1.1). • Biswas and Hoffmann are preparing a paper [26] in which they will determine the Picard group and decide the existence of universal families in all cases. • If X is an elliptic curve and G is simple and simply connected, then the moduli spaces can be explicitly described (see [132], Theorem 4.16, [67], Theorem in the introduction): They are weighted projective spaces where the weights are computed from the root system. • As a strengthening to the result of Grothendieck (see Remark 2.1.1.23), Serman [193] proved that the moduli space of semistable principal G-bundles embeds, for G = SOr ('), r odd, G = Or ('), and G = Spr (') into the moduli space of semistable vector bundles of rank r and degree zero.
2.4.7 Appendix I: Some Remarks Concerning the Moduli Stack of Principal Bundles Let
F: Sch. −→ Sets
be a moduli functor. If we ask for moduli spaces, then the best thing which can happen is that F is representable. This means that there is fine moduli space S , so that we may 16 The group Z (G) is contained in the automorphism group of any principal G-bundle; in general, M B (ϑ ) 0 will be a proper subset of M s (ϑ0 ) ([174], Remark 4.1).
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identify F and Mor(−, S ). By the Yoneda lemma, Mor(−, S ) allows to reconstruct S . Roughly speaking, a representable moduli functor F already knows everything about the moduli scheme S . Unfortunately, we have seen that important moduli functors such as the moduli functor of isomorphy classes of vector bundles of rank r and degree d on the curve X are not representable, even if we restrict to semistable vector bundles or stable bundles with (r, d) % 1. The natural question is therefore, if we can do geometry just using the moduli functor F itself and define a structure which is more general than a scheme. This can be done in a certain sense and leads to the theory of stacks. In order to do it properly, we must also keep track of the automorphisms of the objects under consideration. Let us explain this in the case of vector bundles of rank r and degree d. Our new moduli functor Vectr,d X assigns to every scheme S the groupoid that has vector bundles of rank r and fiberwise degree d on S x X as objects and vector bundle isomorphisms as morphisms rather than just the set of isomorphy classes of vector bundles of rank r and fiberwise degree d on S x X. Example 2.4.7.1. A baby but hopefully still instructive example what happens when one passes from a groupoid to its set of isomorphy classes is the case of finite sets. The objects of the groupoid are finite sets, and the morphisms are bijections. In particular, the set { 1, . . . , n } has attached to it the symmetric group in n letters as automorphism group. If we pass to the set of isomorphy classes in this groupoid, we obtain the set of natural numbers. Clearly, we have given away much structure. Another important example arises when we try to form quotients by group actions. Let H be an algebraic group, Z a scheme, and Z x H −→ Z an action of H on Z from the right. The quotient stack [Z/H] assigns to any scheme T the groupoid whose objects are pairs (P, ϕ) which consist of a principal H-bundle P on T and an H-equivariant morphism ϕ: P −→ Z. A morphism h: (P1 , ϕ1 ) −→ (P2 , ϕ2 ) in that groupoid is an isomorphism h: P1 −→ P2 of principal H-bundles with ϕ1 = ϕ2 ◦ h. Now, [Z/H] is a geometric object: Its geometry is the H-equivariant geometry on Z, e.g., a vector bundle on [Z/H] is an H-linearized vector bundle on Z. This example also suggests how more complicated stacks like Vectr,d X become ge(N) be the stack of vector ometric objects: For every natural number N, we let Vectr,d X bundles of rank r and degree d whose maximal slope is bounded by N. We may write ) Vectr,d Vectr,d X = X (N). N∈+
We know that the set of isomorphy classes in Vectr,d X (N) is bounded. Using the theory of quot schemes and the construction of the moduli spaces of semistable vector bundles, we see that we can find a quasi-projective scheme Q with an action of GL(Y), such that Vectr,d X (N) = [Q/ GL(Y)]. Let us discuss the latter construction in more detail for (semistable) principal Gbundles. As for vector bundles, we have the stack G-BunϑX of principal G-bundles of topological type ϑ ∈ Π(G). We let G-Bunϑ,ss X be the open substack of semistable principal G-bundles. In Section 2.4.5, we have constructed the parameter scheme Yδ-ss for δsemistable pseudo G-bundles. By Theorem 2.4.4.1 and Proposition 2.4.4.5, this scheme
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contains the parameter scheme Yϑ,ss for semistable principal G-bundles of topological type ϑ as a union of connected components. We have equipped this scheme with a group action Γ: GL(Y) x Yϑ,ss −→ Yϑ,ss . We can view this also as a right action, by defining Γ B : Yϑ,ss x GL(Y)
−→ Yϑ,ss
(y, g) 1−→ Γ(g−1 , y). Our moduli spaces were constructed as the GIT quotients Yϑ,ss // GL(Y). Now, we refine this result to: Proposition 2.4.7.2. There is an isomorphism 6 _ ϑ,ss / GL(Y) G-Bunϑ,ss X ! Y of stacks. Proof. By construction, there is a universal family (EYϑ,ss , τYϑ,ss ) of semistable principal G-bundles on Yϑ,ss x X. This family is linearized with respect to the group action. Let us explain this concept in geometric terms: Assume that we are given a scheme S , an algebraic group H, an action A: H x S −→ S of H on S , and a principal G-bundle P on S . A linearization of P is an action B: H x P −→ P, such that the bundle projection P −→ S is H-equivariant and the induced morphism B(h, −)|{s}: P|{s} −→ P|{h·s} is G-equivariant for every h ∈ H and s ∈ S . The analogous definition for right actions is left to the reader. Exercise 2.4.7.3. Show that a linearization B as above gives rise to an isomorphism C: π.S (P) −→ A. (P) of principal G-bundles on H x S . Conversely, what are the conditions on such an isomorphism to be induced by a linearization? Hence, we can think of a linearization of (EYϑ,ss , τYϑ,ss ) as a linearization of the corresponding principal G-bundle on Yϑ,ss x X. If you work out the details of the exercise, then you can define it as an isomorphism between the pull-back of (EYϑ,ss , τYϑ,ss ) (to Yϑ,ss x X x GL(Y)) via Γ B x idX and the pull-back of (EYϑ,ss , τYϑ,ss ) via πYϑ,ss x X , satisfying the respective compatibility conditions. Next, let T be any scheme. We have to establish an equivalence between the ϑ,ss / GL(Y)](T ). Let us start with a family (ET , τT ) of groupoids G-Bunϑ,ss X (T ) and [Y semistable principal G-bundles on T x X. The push-forward 4 ^ πT . ET * π.X (OX (n))
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is a locally free sheaf of rank dim(Y) on T . Its frame bundle H is the principal GL(Y)bundle which we need. Look at the morphism H x X −→ T x X −→ T. On T x X, there is the natural surjection & 4Z ^ qT : π.T πT . ET * π.X (OX (n)) −→ ET
*
^ 4 π.X OX (n) .
Note that the pull-back of πT . (ET * π.X (OX (n))) to H possesses a canonical trivialization. Hence, pulling back qT to H x X gives the surjection ^ 4 qH : Y * π.X OX (−n) −→ EH . Here, EH is the pull-back of ET . Let τH be the pull-back of τT . The family (qH : Y * π.X (OX (−n)) −→ EH , τH ) supplies a morphism k: H −→ Yϑ,ss , such that the pullback of the family (qYϑ,ss : Y * π.X (OX (−n)) −→ E, τYϑ,ss ) is isomorphic to the family (qt : Y * π.X (OX (−n)) −→ ET , τT ). (The notion of isomorphy for quotient families like (qt : Y * π.X (OX (−n)) −→ ET , τT ) is defined as in the setting of swamps.) We have to check that k is equivariant, i.e., if ΓH : H x GL(Y) −→ H is the right action of GL(Y) on H , we have to show that H x GL(Y)
k x idGL(Y)
ΓH
3 H
k
7 Yϑ,ss x GL(Y) 3
ΓB
7 Yϑ,ss
commutes. This amounts to show that the pull-backs of (qYϑ,ss : Y * π.X (OX (−n)) −→ EYϑ,ss , τYϑ,ss ) via Γ B ◦ (k x idGL(Y) ) and k ◦ ΓH are isomorphic. But, this is rather evident. Now, let T be a scheme, H a principal GL(Y)-bundle on T , and ϕ: H −→ Yϑ,ss a GL(Y)-equivariant morphism. Let (EH , τH ) be the pull-back of the universal family via f x idX . Note that (EH , τH ) is linearized with respect to the GL(Y)-action on H . This linearization results from the linearization of the universal family on Yϑ,ss x X which is preserved by pull-back via an equivariant morphism. Since H −→ T is a principal GL(Y)-bundle, it is a standard fact that one can descend (EH , τH ) to T . This means that there exists a family (ET , τT ) whose pull-back to H x X is isomorphic to (EH , τH ). Up to now, we have defined maps between the sets of objects of the two groupoids. The attentive reader will be able to check that everything is compatible with isomorphisms in both categories and gives an equivalence. " Corollary 2.4.7.4. The parameter scheme Yϑ,ss is smooth. Proof. The stack G-BunGϑ,ss is smooth. This is a property of the moduli functor and means that for any Artin ring AB , any ideal I ⊂ AB with I 2 = 0, and A := AB /I, a family of semistable principal G-bundles parameterized by Spec(A) can be extended
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to a family parameterized by Spec(AB ) via Spec(A) '→ Spec(AB ). This is explained in Section 4.5 of [15]. Since the projection morphism Yϑ,ss −→ G-BunGϑ,ss exhibits the parameter scheme as a principal GL(Y)-bundle over the moduli stack G-BunGϑ,ss , the scheme Yϑ,ss is smooth, too. " Corollary 2.4.7.5. The moduli scheme M ss (ϑ) is normal and Cohen–Macaulay. Proof. Since these are local questions, one has to study the invariant subalgebra AG of a finitely generated '-algebra A on which a reductive group G acts as in Chapter 1, Section 1.4.2. It is quite obvious that AG is normal, if A is so ([123], II.3.3, Satz 4). It is a difficult theorem by Hochster and Roberts that AG is Cohen–Macaulay, if A is a regular ring (see [110] and [119]). " For more details on the stack of principal G-bundles on a curve, we refer the reader to [15] and [206]. A nice application of the theory of moduli spaces as developed in this book to the study of moduli stacks of principal G-bundles is contained in the paper [103].
2.4.8 Appendix II: Moduli Spaces for Principal Bundles with Reductive Structure Group via the Ramanathan–G´omez–Sols Method In this section, we will explain how one can use the above methods also for connected reductive structure groups with non-trivial radical. First, let us illustrate by an example why the methods which we have developed so far are not perfectly suited for dealing with that case. For this, look at the structure group G = GL(W1 ) x GL(W2 ). We set W := W1 - W2 - ' and look at the faithful representation κ: G (m1 , m2 )
−→ GL(W) m1 0 1−→ 0 m2 0 0
0 0
1 det(m1 ) det(m2 )
.
(2.40)
Note that the image of κ lies in SL(W). Our techniques can be applied to this situation, too. But then, Corollary 2.4.4.6 tells us that the moduli space which we get classifies pairs (E1 , E2 ) of vector bundles with rk(Ei ) = dim. (Wi ), i = 1, 2, such that F := E1 - E2 - det(E1 )∨ * det(E2 )∨ is a semistable vector bundle. Now, F is a semistable vector bundle, if and only if E1 and E2 are semistable vector bundles and µ(E1 ) = µ(E2 ) = deg(det(E1 )∨ * det(E2 )∨ ) = − deg(E1 ) − deg(E2 ), i.e., deg(E1 ) = 0 = deg(E2 ). On the other hand, given any two integers d1 and d2 , the product of the moduli space of semistable vector bundles of rank dim. (W1 ) and degree d1 with the moduli space of semistable vector bundles of rank dim. (W2 ) and degree d2 certainly is a moduli space for principal G-bundles as well. Thus, our methods give the moduli spaces only for certain topological types. Moreover, a principal G-bundle can never be stable in our definition. The reader may check that, in our set-up, the conditions which violate stability and which imply the conditions on the degrees arise from one parameter subgroups of the center of G (see p. 285).
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Representations with Kernel in the Center An obvious way out is to use a representation κ: G −→ GL(W) with image in SL(W), such that the connected component of the kernel containing the neutral element is the radical of G. The canonical candidate for this is the adjoint representation Ad: G −→ GL(g) of G on its Lie algebra g. The kernel of this representation is the center Z (G) * := G/Z (G) is semisimple and of G ([123], II.2.5, Folgerung 4). The adjoint group G * Ad factorizes over a faithful representation * κ: G −→ GL(g). The approach to get nice moduli spaces for principal G-bundles is to apply our results to the semisimple group * with its faithful representation * G κ. Let us state the result which we get. If P is a principal G-bundle, we define + as the principal G-bundle * P obtained from P by extending the structure group via * * G −→ G. Now, we call a principal G-bundle P (semi)stable, if the principal G-bundle + P is (semi)stable in the sense of Section 2.4.1. * ◦λ λ := (G −→ G) Remark 2.4.8.1. Let λ: '. −→ G be a one parameter subgroup and * * Then, we find that the projection the corresponding one parameter subgroup of G. * maps QG (λ) onto Q *(* * *(* G −→ G λ). G λ) and induces an isomorphism G/QG (λ) ! G/QG * * Note that G = QG*(λ) holds precisely when λ is a one parameter subgroup of the radical of G. * Next, let D(G) be the derived group of G. The induced morphism D(G) −→ G . is still surjective. As before, any one parameter subgroup λ: ' −→ D(G) yields a * Conversely, given a one parameter subgroup * one parameter subgroup * λ of G. λ, then * z 1−→ * * λ(z)m , can be lifted to D(G) for some m > 0. In particular, any λm : '. −→ G, * is the image of a parabolic subgroup of G. parabolic subgroup of G To conclude, let P be a principal G-bundle. By means of Ad, we associate to it the vector bundle Ad(P) with fiber g. Then, I som(g*OX , Ad(P)) is the principal GL(g)bundle associated to P by means of Ad. Our previous digression demonstrates the following: If λ: '. −→ D(G) is a one parameter subgroup, then there is the canonical embedding ^ 4 ι: P/QG (λ) '→ I som g * OX , Ad(P) /QGL(g) (λ). Thus, if β: X −→ P/QG (λ) is a section, we may compose ι and β and define the weighted filtration (Ad(P)• (β), α• (β)) of Ad(P). Our arguments show: * Proposition 2.4.8.2. A principal G-bundle P is (semi)stable, if and only if the inequality ^ 4 M Ad(P)• (β), α• (β) (≥)0 holds for every one parameter subgroup λ: '. −→ D(G) and every reduction β: X −→ P/QG (λ) of P to λ. Below, we will give another interpretation of the number M(Ad(P)• (β), α• (β)). Universal Spaces Via Central Isogenies Motivated by the previous discussions, we would like to build a construction of moduli spaces of principal G-bundles on a representation κ: G −→ GL(W) with non-trivial
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kernel in the center. This leads to new difficulties in setting up the GIT process, i.e., constructing the parameter scheme for principal G-bundles and the group action on that parameter scheme. We will now explain how Ramanathan and G´omez and Sols solved these extra problems. The situation is as follows: Let G and GB be two connected reductive groups and assume that we are given a surjective homomorphism α: G −→ GB whose kernel is finite and contained in the center of the group G. Such a homomorphism is called a central isogeny. We assume that we have a certain universal space for principal GB bundles together with a universal family of principal GB -bundles and would like to obtain a universal space for principal G-bundles (but without a universal family). The method was invented by Ramanathan in [175] and recently simplified and extended to higher dimensions by G´omez and Sols [83] (see also [80]). To carry it out properly, one needs the methods of e´ tale cohomology. We will present here the main ideas, using cohomology with respect to the strong topology of schemes when viewed as complex analytic spaces (see [96], Appendix B). This should allow the reader to follow the argument even without knowing about e´ tale cohomology. On the other hand, the reader who is familiar with e´ tale cohomology will have no problem in transcribing the proof to the algebraic setting to arrive at the proper arguments. Such a reader may also consult [83], Section 3, or [80], Section 5. Let S be a scheme, U = (Ui )i∈I a covering in the strong topology, and H an algebraic group. A 0-cochain for U with values in H is a collection of morphisms ϕ = (ϕi : Ui −→ H), i ∈ I; a 1-cochain for U with values in H is a tuple ψ = (ψi j : Ui j := Ui ∩ U j −→ H, i, j ∈ I); a 1-cocycle is a 1-cochain ψ = (ψi j , i, j ∈ I), such that ψik = ψ jk · ψi j
on Ui jk := Ui ∩ U j ∩ Uk , i, j, k ∈ I.
We let C i (U , H) be the group of i-cochains with values in H, i = 1, 2, and Z 1 (U , H) the group of 1-cocycles. To any 0-cochain ϕ = (ϕi , i ∈ I) we associate its boundary 4 ^ ∂ϕ = (ϕ−1 j · ϕi ): U i j −→ H, i, j ∈ I . ˇ It is a 1-cochain. The first Cech-cohomology set with respect to U is then defined as Hˇ 1 (U , H) := Z 1 (U , H)/∂C 0(U , H), ˇ and the first Cech-cohomology set of S as H 1 (S , H) := lim Hˇ 1 (U , H). −→ U
Note that H 1 (S , H) is pointed by the class of the trivial cochain. Furthermore, H 0 (S , H) is the group of morphisms S −→ H. If H is an abelian group, then H i (S , H) is the singular cohomology group with values in H, i = 1, 2 [76]. If P is a principal H-bundle, then it defines a class hP ∈ H 1 (S , H): To get it, we choose an open covering U with trivializations τi : P|Ui −→ Ui x H. Then, there are morphisms ψi j : Ui j −→ H with ^ 4 1 → (u, ψi j (u) · h) . τ j ◦ τ−1 i = (u, h) −
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Hence, we may define hP as the class defined by the cocycle ψ = (ψi j , i, j ∈ I). It is straightforward to check that hP depends only on the isomorphy class of P and that 7 0 Isomorphy classes of −→ H 1 (S , H) principal H-bundles over S [P] 1−→ hP is a bijection of pointed sets. (The left hand side is pointed by the class of the principal H-bundle S x H.) If β: H −→ H B is a homomorphism, then we may associate to a principal H-bundle P the principal H B -bundle P B := β. (P), by extending the structure group. Moreover, there is also a natural homomorphism H 1 (β): H 1 (S , H) −→ H 1 (S , H B ). Under the above correspondence, this map is hP 1−→ hP B . Let us return to the setting described at the beginning. We have an exact sequence α
{0} −−−−−→ D −−−−−→ G −−−−−→ GB −−−−−→ {1}. It yields the exact sequence H 1 (α)
δ
H 1 (S , D) −−−−−→ H 1 (S , G) −−−−−→ H 1 (S , GB ) −−−−−→ H 2 (S , D) of pointed sets. (This means that the image of one map is the set of elements which maps to the highlighted point under the following map.) The fact that one may define the boundary map in the usual way relies on the fact that D is contained in the center of G. Let S be a scheme and PSB a principal GB -bundle over S x X. The functor Γ(PSB , α) assigns to a scheme f : T −→ S over S the set of isomorphy classes of pairs (PT , βT ) which consist of a principal G-bundle PT on the product T x X and an isomorphism βT : α. (PT ) −→ PTB . Two such pairs (PT,1 , βT,1 ) and (PT,2 , βT,2 ) are isomorphic, if there is an isomorphism ϕ: PT,1 −→ PT,2 , such that βT,1 = βT,2 ◦ α. (ϕT ). As usual πS : S x X −→ X is the projection onto the first factor. The functor Γ(PTB , α) * B , α) as the is in general not a sheaf ([175], Section 4.9). Thus, we introduce Γ(P S sheafification of Γ(PSB , α). We let Ri πS . (H) be the sheaf associated to the presheaf U 1−→ H i (U x X, H) for i = 0, 1, if H ∈ { G, GB }, and for i = 0, 1, 2, if H = D. The construction of the above sequence of pointed sets generalizes to give the exact sequence δS
R1 πS . (D) −−−−−→ R1 πS . (G) −−−−−→ R1 πS . (GB ) −−−−−→ R2 πS . (D) of sheaves of pointed sets. Lemma 2.4.8.3. The sheaf R2 πS . (D) is isomorphic to the constant sheaf associated to the finite abelian group H 2 (X, D). Proof. Any complex space is locally contractible. This result is due to Burghelea and Verona ([43], Theorem 4.4, 1).17 17 If the complex space has only isolated singularities, this is Theorem 2.10 in [152]. A discussion of the result with additional references is also contained in Chapter 1, §5, of [54].
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Therefore, any point s ∈ S possesses a contractible open neighborhood U. In particular, the projection U x X −→ X is a homotopy equivalence, so that it yields the canonical identification H 2 (U x X, D) = H 2 (X, D). The assertion is thus a direct consequence of the definition of the sheaf R2 πS . (D). " Remark 2.4.8.4. i) The proof shows that the assertion holds, in fact, for H i (X, D), i > 0, and all coefficients D allowed in singular cohomology. ii) The scheme to which we will apply the results of this section is the parameter scheme Y for principal GB -bundles from Section 2.4.5 which is smooth (see Lemma 2.4.7.4). If one admits this, then local contractibility is obviously not an issue. Lemma 2.4.8.5. Let s ∈ S and U a contractible neighborhood of s. If Γ(PSB , α)(U ⊂ S ) is non-empty, then the group H 1 (X, D) acts simply transitively on Γ(PSB , α)(U ⊂ S ). Proof. The following arguments involve choices of an open covering, of representing cocycles for cohomology classes with respect to the covering, and trivializations of principal bundles with respect to the covering. Hence, there are lots of routine checks to be made in order to verify that the argument does not depend on those choices. These checks are, of course, omitted. We set PUB := PSB |U x X Then, for any morphism f : T −→ U, one has Γ(PUB , α)( f : T −→ U) = Γ(PSB , α)( f : T −→ U ⊂ S ). First, let us define the action. Suppose σ ∈ H 1 (X, D) = H 1 (U x X, D) and [PU , β] ∈ Γ(PUB , α)(idU ). Then, we may choose an open covering U = (Ui )i∈I of U x X, a cocycle ϕ = (ϕi j : Ui j −→ D, i, j ∈ I) representing σ, a cocycle ψ = (ψi j : Ui j −→ G, i, j ∈ I), and an isomorphism ι: Pψ −→ PU . As before, Pψ is obtained from gluing the trivial pieces Ui x G, i ∈ I, with ψ. We replace (PU , β) by the isomorphic pair (Pψ , βψ := β ◦ α. (ι)). Now, we define the cocycle ψσ := (ψi j · ϕi j : Ui j −→ G, i, j ∈ I). Note that α. (Pψσ ) and α. (Pψ ) are canonically isomorphic. Therefore, it makes sense to set σ · [PU , α] := [Pψσ , βψ ]. The next task is to verify the transitivity of the action. Let [PU,1 , β1 ] and [PU,2 , β2 ] be two elements of Γ(PUB , α)(idU ). We may then choose an open covering U = (Ui )i∈I of U x X, cocycles ψk = (ψkij : Ui j −→ G, i, j ∈ I), k = 1, 2, such that Pψk is isomorphic to Pk , k = 1, 2, and replace (Pk , βk ) by (Pψk , βψk ). The datum of the isomorphism β−1 ◦ βψ1 : α. (Pψ1 ) −→ α. (Pψ2 ) ψ2 * i ∈ I, such that is the datum of maps * βi : Ui −→ G, ^ 4 ^ 4 α ψ1i j (x) = * β j (x)−1 · α ψ2i j (x) · * βi (x),
x ∈ Ui j , i, j ∈ I.
* is a topological covering, we may assume, after possibly refining U , Since G −→ G * that the βi lift to morphisms βi : Ui −→ G, i ∈ I. Now, set ^ 4 ψB := ψBi j : x 1−→ β j (x)−1 · ψ2i j (x) · βi (x), i, j ∈ I . Note that the βi define an isomorphism ϕ: Pψ1 −→ PψB ,
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◦ βψ1 . We replace ψ1 by ψB . Then, we have achieved such that α. (ϕ) = β−1 ψ2 α. (ψ1 ) := (α ◦ ψ1i j , i, j ∈ I) = (α ◦ ψ2i j , i, j ∈ I) =: α. (ψ2 ). In other words, there are maps ϕi j : Ui j −→ D with ψ2i j (x) = ψ1i j (x) · ϕi j (x),
x ∈ Ui j , i, j ∈ I.
It is clear that ϕ = (ϕi j , i, j ∈ I) is a cocycle. Let σ ∈ H 1 (U x X, D) = H 1 (X, D) be its cohomology class. By construction, σ · [Pψ1 , βψ1 ] = [Pψ2 , βψ2 ]. A careful analysis of its construction reveals that the cohomology class σ which we have just constructed is indeed unique, i.e., the action of H 1 (X, D) is also free. So, we have verified that the action is simply transitive. " Remark 2.4.8.6. i) The same construction allows to define an action of H 1 (T x X, D) on Γ(PS , α)( f : T −→ S ) for any f : T −→ S . ii) As J. Heinloth told me, one may define the above action intrinsically and prove the lemma without cocycles. The intrinsic definition of the action comes from the following construction: Let K a principal D-bundle on a variety V and P a principal G-bundle on V. Then, D acts diagonally from the right on K xV P. This action commutes with the G-action on the second factor. One readily verifies that K · P := (K xV P)/D is again a principal G-bundle. So, set V := U x X, let PUB be a principal GB -bundle on U x X and σ ∈ H 1 (X, D) = 1 H (U x X, D) the class of the principal D-bundle K . For [PU , β] ∈ Γ(PUB , α)(idU ), the class σ · [PU , β] is represented by (K · PU , β). The principal GB -bundle PSB defines the class hPSB ∈ H 1 (S x X, GB ) and therefore a global section of the sheaf R1 πS . (GB ). Applying δS , we get a locally constant settheoretic map S −→ H 2 (X, D). If we interpret the finite set H 2 (X, D) as a reduced complex analytic space, this map is therefore a morphism. We define S* as the preimage of 0 under this morphism. This is obviously a union of connected components of S . * B , α)( f : T −→ S ) % ∅. We Suppose that f : T −→ S is a morphism with Γ(P S may then find an open covering V of T , such that there is a principal G-bundle P j on V j x X together with a isomorphism ψ j : α. (P j ) −→ ( f x idX ). (PSB )|V j x X , j ∈ J. So, the principal GB -bundle ( f x idX ). (PSB )|V j x X defines a class in R1 πS . (GB ) which maps to zero in R2 πS . (D). That class in R2 πS . (D) is also the composition f|V j : V j −→ S with the map δS (hPSB ): S −→ H 2 (X, D). This shows that f maps into S*. Conversely, let s ∈ S* be a point. There is a contractible open neighborhood U of s, such that the class hPSB lifts over U to a section of R1 πS . (G) which can be represented by a class in H 1 (U x X, G). The latter class stands for the isomorphy class of a principal G-bundle P on U x X. It is obvious that α. (P) is isomorphic to PSB |V x X , so that * B , α)( f : U −→ S ) % ∅. Γ(P S * B , α) is representable by a complex space f : R −→ Theorem 2.4.8.7. The functor Γ(P S S , such that the image of f is S* and f : R −→ S* is an unramified covering.
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205
* B , α) is a sheaf, it suffices to show that there is an open Proof. Since the functor Γ(P S * B , α|Ui ) is representable for covering U = (Ui )i∈I of S*, such that the functor Γ(P S |Ui every i ∈ I. We have already explained that every point s ∈ S* possesses a contractible open neighborhood U with Γ(PSB , α)(U ⊂ S ) = Γ(PSB |U x X , α)(idU ) % ∅. It follows from 1 * B Lemma 2.4.8.5 that Γ(P S |U x X , α) is representable by U x H (X, D). Indeed, let (PS , β) B represent a non-trivial class in Γ(PS |U x X , α). Then, for any space T and any morphism f = ( f1 , f2 ): T −→ U x H 1 (X, D), we obtain the pull-back f . ([PS , β]). To define it, we immediately reduce to the case that T is connected. Then, the morphism f2 : T −→ H 1 (X, D) just picks a class σ ∈ H 1 (X, D) = H 1 (U x X, D). We pull it back to f2. (σ) ∈ H 1 (T x X, D). Then, making use of Remark 2.4.8.6, we define ^ 4 6 _ f . [PS , β] := f2. (σ) · ( f1 x idX ). (PS ), ( f1 x idX ). (β) . * B The assignment f 1−→ f . ([PS , β]) identifies Γ(P S |U x X , α) and the functor Mor(−, U x 1 H (X, D)). " Remark 2.4.8.8. If we assume that S is a smooth quasi-projective scheme, then it follows from the generalized Riemann existence theorem ([96], Theorem 3.2, Appendix B) that R is also a smooth quasi-projective scheme. Now, assume further that we are given an action A: H x S −→ S of the algebraic group H on S and a linearization B of PSB , giving the isomorphism C: π.S x X (PSB ) −→ (A x idX ). (PSB ) of principal GB -bundles on H x S x X (Exercise 2.4.7.3), such that, for any two points s1 , s2 ∈ S and any isomorphism ϕ: P sB1 −→ P sB2 , there is an element h ∈ H, such that h · s1 = s2 and ϕ agrees with the restriction of C to (h, s1 ). By the above construction, we may assign to every point r ∈ R an isomorphy class Ir of principal G-bundles on X. Then, for every principal G-bundle P on X, there is a point r ∈ R with P ∈ Ir . In that way, we can view R as a parameter space for principal G-bundles on X. Furthermore: * R −→ S , such that two Proposition 2.4.8.9. The group action A lifts to an action A: points r1 and r2 from R lie in the same H-orbit, if and only if Ir1 = Ir2 . * B , α), the morphism idR : R −→ R correProof. Since R represents the functor Γ(P S* * B , α)( f : R −→ S*), and we define the class sponds to the tautological class KR ∈ Γ(P S* * B , α)(H x R −→ S*) as its pull-back to H x R via πR . By definition, we KH x R ∈ Γ(P S*
may cover S* by open subsets Ui , i ∈ I, such that the restriction of KR to Vi := f −1 (Ui ) is represented by a pair (PVi , βVi ), i ∈ I. Denote by (PH x Vi , βH x Vi ) the pullback of (PVi , βVi ) to H x Vi via πVi x idX , i ∈ I. On the other hand, there is the morphism h := (idH x f ): H x R −→ H x S*. Hence, we have the isomorphism 4 ^ 4. ^ Ch := h. (C): ( f ◦ πR ) x idX (PSB ) −→ h. (A x idX ). (PSB ) .
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One checks that the elements 4 ^ PH x Vi , Ch|Vi x X ◦ βH x Vi ,
i ∈ I,
glue to a class
^ 4 h A * SB , α) H x R −→ KHB x R ∈ Γ(P H x S* −→ S* .
The element KHB x R supplies the morphism * H x R −→ R, A: such that the diagram * A
HxR h
3
A
H x S*
7R 3 7 S*
f
* is indeed a group action. commutes. We leave it to the reader to verify that A We still have to show that two points r1 and r2 in the scheme R correspond to the same isomorphy class, if and only if they lie in the same H-orbit. The “if”-direction is clear from the construction of the group action. If, conversely, we have Ir1 = Ir2 , then * B , α)( f (ri ) ∈ S*) by a pair (Pi , βi ), we may represent the corresponding element of Γ(P S i = 1, 2. There are isomorphisms ϕ: P1 −→ P2 and ψ: P Bf (r1 ) −→ P Bf (r2 ) , such that the diagram ϕ 7 P2 P1 /D
/D
3 α. (P1 ) β1
α. (ϕ)
3
ψ
P Bf (r1 )
3 7 α. (P2 ) 3
β2
7 P Bf (r ) 2
is commutative. By our assumptions on the action A and the linearization of PSB , the isomorphism ψ gives rise to a certain element h ∈ H with h · f (r1 ) = f (r2 ). The above diagram reveals that h · r1 = r2 holds, too. " Construction of the Moduli Spaces * its adjoint Let G be a connected linear algebraic group. As above, we denote by G group, by D(G) its derived group, and by R(G) its radical. Note that the homomorphism R(G) x D(G) (r, t)
−→ G 1−→ r · t
is surjective. Hence, we have a surjection αB : R(G) −→ T := G/D(G).
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From this, we see that T is a connected, commutative, and diagonalizable algebraic group, i.e., a torus ([30], III.8.5). Fix an isomorphism T ! '. n . We find the homomorphism *x T ! G * x '. n =: GB . α: G −→ G It is evident that α is a central isogeny. Hence, we may try out the strategy which we have outlined in the preceding sections. Assume that we would like to classify principal G-bundles of topological type ϑ ∈ * and a tuple d = (d1 , . . . , dn ) of integers, Π(G). This determines an element * ϑ ∈ Π(G) such that, for any principal G-bundle P of topological type ϑ, the associated principal + has topological type * * G-bundle P ϑ and the associated tuple L = (L1 , . . . , Ln ) of line bundles features deg(Li ) = di , i = 1, . . . , n. We first have to construct the parameter * −→ scheme for principal GB -bundles. To do so, we choose a faithful representation κ: G * ϑ,ss GL(W). This gives us a parameter scheme Y := Y with an action of GL(Y) and a +Y on Y x X together with a linearization * universal principal G-bundle P +Y −→ P +Y . BY : GL(Y) x P Next, we set
n
J := X Jacdi . i=1
We equip J with the trivial ' -action, choose Poincar´e line bundles L i on Jacdi x X, i = 1, . . . , n, and pull them back to J x X in order to find the tuple L J = (L J1 , . . . , L Jn ) of line bundles on J x X. The line bundle L Ji determines a principal '. -bundle T Ji , i = 1, . . . , n, and the tuple L J the principal '. n -bundle .n
T J := T J1 x · · · x T Jn . JxX
JxX
We equip this bundle with the linearization ^
B J : '. x T J −→ T J 4 (z1 , . . . , zn ), (t1 , . . . , tn ) −→ (z1 · t1 , . . . , zn · tn ). n
Finally, we define S := Y x J, H := GL(Y) x '. n , and +Y ) x π. (T J ). PSB := π.Y x X (P JxX S xX
Then, the group H acts on
PSB ,
and we have a linearization B: H x PSB −→ PSB ,
such that the assumptions stated before Proposition 2.4.8.9 are satisfied. By the construction of the last section and that proposition, we have a parameter scheme R for principal G-bundles which comes with an action H x R −→ R and a finite H-equivariant morphism f : R −→ S . We know that the H-quotient of S exists as a projective scheme. Indeed, n ϑ) x J. S //H = (Y// GL(Y)) x(J//'. ) = M ss (*
By Chapter 1, Exercise 1.5.3.3, the categorical quotient M ss (ϑ) := R//H also exists as a projective scheme and is our moduli space. "
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2.4.9 Appendix III: Anti-Dominant Characters First, let G be a semisimple algebraic group and Q a proper parabolic subgroup of G. Then, there is the principal G-bundle Q := Q −→ G/Q over the projective manifold G/Q. If χ: Q −→ '. is a character of Q, we get an action of Q on ' and may form the associated fiber space Lχ := Q('). This is a line bundle on G/Q. We say that χ is an anti-dominant character, if Lχ is an ample line bundle. * := Ad(G) its If G is an arbitrary connected reductive linear algebraic group, G * ⊂ G, * then an antiadjoint group, and Q a proper parabolic subgroup with image Q dominant character on Q is a character χ on Q which factorizes over an anti-dominant * character * χ on Q. Let us consider a one parameter subgroup λ: '. −→ SLr ('). Without loss of generality, we may assume that it is of the form ^ 4 λ(z) = diag zγ1 , . . . , zγ1 , . . . , zγs+1 , . . . , zγs+1 . D!!!!!!WB!!!!!!\ D!!!!!!!!!!WB!!!!!!!!!!\ r1 x
(r−r s ) x
The associated parabolic subgroup in GLr (') is . !! m1 . ! ! . . ! QGLr (.) (λ) = ' ), i = 1, . . . , s + 1 ( m ∈ GL i r −r . i i−1 ! 0 . ! 0 0 m s+1 4 ^ 4 ^ = Ru QGLr (.) (λ) " GLr1 (') x · · · x GLr−rs (') . In Chapter 1, Example 1.7.2.4, we have associated to λ the character χλ : QGLr (.) (λ) −→
'.
(u; m1 , . . . , m s+1 ) 1−→ det(m1 )γ1 · · · · · det(m s+1 )γs+1 . We claim that this is an anti-dominant character of QGLr (.) (λ). First of all, we note that it is trivial on '. · $r , so that it factorizes indeed over QPGLr (.) (λ). Next, F := GLr (')/QGLr (.) (λ) = PGLr (')/QPGLr (.) (λ) parameterizes flags {0} =: W0 ! W1 ! · · · ! W s ! W s+1 := i = 1, . . . , s. Furthermore, there is the universal flag
'r where dim
.
(Wi ) = ri ,
{0} =: F0 ! F1 ! · · · ! F s ! F s+1 := OF$r on F. For any choice ε := (ε1 , . . . , ε s ) of positive rational numbers, the 3-line bundle Lε :=
) det^O $ /F 4# = ) det(F )# s
i=1
F
r
i
εi
s
i=1
i
−εi
is ample. Indeed, the quotient OF$r −→ OF$r /Fi defines a morphism F −→ Gi to the Graßmannian of (r − ri )-dimensional quotients of 'r . The resulting map F −→ G :=
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209
s Gi is an embedding. Our line bundle Lε is just the pull-back of the ample line Xi=1
bundle OG (ε1 , . . . , ε s ). Now, Lχλ =
for
) det(F /F s+1 i=1
i
i−1 )
γi
=
) det(F )# s
i=1
i
(γi −γi+1 )
= Lε
ε = (γ2 − γ1 , . . . , γ s+1 − γ s ).
This settles our claim. If G is now a semisimple algebraic group and κ: G −→ GL(W) is a faithful representation, then any one parameter subgroup λ: '. −→ G supplies the parabolic subgroup QG (λ) of G and the parabolic subgroup QGL(W) (λ) of GL(W). The above construction provides an anti-dominant character χGL(W),λ on QGL(W) (λ). It restricts to an anti-dominant character χλ on QG (λ). Proposition 2.4.9.1. Given a pair (Q, χ) which consists of a proper parabolic subgroup Q of G and an anti-dominant character χ on Q, there is a one parameter subgroup λ: '. −→ G, such that QG (λ) = Q and χλ = l · χ with l ∈ 3>0 . Proof. We give a sketch of the argument. Since any two pairs (B, T ) consisting of a Borel subgroup B of G and a maximal torus T ⊂ B are conjugate inside G ([30], 11.19), we may fix such a pair. Then, the one parameter subgroups λ: '. −→ G, such that B ⊆ QG (λ) form a convex rational polyhedral cone inside X.,% (T ). (It is, in fact, the closure of the Weyl chamber associated to B.) The edges of that cone identify with the maximal parabolic subgroups that contain B. A moment of thought now shows that one has to verify the assertion only for maximal parabolic subgroups. For a maximal parabolic subgroup Q, it is, however, clear, because there is one antidominant character χ0 , such that any other anti-dominant character on Q is a positive multiple of χ0 . The details of the above argument can be easily filled in, if one uses the correspondence between the parabolic subgroups that contain B and subsets of the simple roots defined with respect to B and T . This formalism allows to read off the anti-dominant characters on the parabolic subgroup from those roots. The reader should consult the book [207] and the paper [175]. " Exercise 2.4.9.2. Let G be a semisimple linear algebraic group with faithful representation κ: G −→ GL(W), P = (E, τ) a principal G-bundle, and λ: '. −→ G a one parameter subgroup. This time, we look at the principal QG (λ)-bundle P −→ P/QG (λ). This principal QG (λ)-bundle and the anti-dominant character χλ define a line bundle L on P/QG (λ). Next, let β: X −→ P/QG (λ) be a reduction of P to λ. Define L (β) as the pull-back of L to X via β. On the other hand, the section β
βB : X −→ P/QG (λ) '→ I som(W * OX , E)/QGL(W) (λ)
210
S 2.5: D T V B 210
provides us with a weighted filtration (E• (β), α• (β)) of E. Prove that ^ 4 ^ 4 deg L (β) = M E• (β), α• (β) . We may now apply Proposition 2.4.9.1 and Exercise 2.4.9.2 to reformulate our notion of (semi)stability in an intrinsic way. Recall that we use a faithful representation κ: G −→ GL(W), if G is semisimple, and Ad: G −→ GL(g), if G is connected and reductive. The latter representation factorizes over a faithful representation of the * = Ad(G). semisimple group G Theorem 2.4.9.3. A principal G-bundle is (semi)stable in the sense defined above, if and only if ^ 4 deg L (Q, χ, β) (≥)0 holds for every proper parabolic subgroup Q of G, every anti-dominant character χ on Q, and every section β: X −→ P/Q. Here, the line bundle L (Q, χ, β) is the pull-back via β of the line bundle on P/Q that is associated to the principal Q-bundle P −→ P/Q and the character χ. Note that this theorem shows that neither the notion of semistability nor the moduli space in Theorem 2.4.1.8 depends on the choice of the representation κ.
2.5 Decorated Tuples of Vector Bundles: Projective Fibers n this section, we will discuss the projective moduli problem for the structure group G := GLr1 (') x · · · x GLrt ('). The formalism which we shall develop to treat this moduli problem will be crucial for the solution of the moduli problems associated to arbitrary reductive groups. The reader should pay special attention on how an embedding G '→ GL(W) is used for defining semistability and constructing the moduli spaces and how the resulting notion of semistability, in fact, does depend on the choice of that embedding. Giving a principal bundle for the structure group GLr1 (') x · · · x GLrt (') is equivalent to giving a tuple P = (P1 , . . . , Pt ) where Pi is a principal GLri (')-bundle, i = 1, . . . , t. Recall from Example 2.1.1.9 that giving a principal GLri (')-bundle is equivalent to giving a vector bundle of rank ri , i = 1, . . . , t. All in all, we may describe a principal G-bundle as a tuple E = (E1 , . . . , Et ) of vector bundles where Ei has rank ri , i = 1, . . . , t.
"
2.5.1 Homogeneous Representations For technical reasons, we will have to restrict our attention again to a certain class of representations of G which we will now introduce. Let V = { v1 , . . . , vt } be a finite index set, r := (rv , v ∈ V) a tuple of positive integers, and define GL(V, r) := X GLrv ('). v∈V
S 2.5: D T V B
211
A (finite dimensional, rational) representation *: GL(V, r) −→ GL(A) is said to be homogeneous (of degree α ∈ (), if *(z, . . . , z) = zα · idA ,
for all z ∈ '. .
Example 2.5.1.1. i) Every irreducible representation is homogeneous. ii) Let Q = (V, A, t, h) be a quiver with vertex set V = { v1 , . . . , vt }, arrow set A = { a1 , . . . , an }, tail map t: A −→ V, and head map h: A −→ V (see Chapter 1, Section 1.3.3). Fix a dimension vector r = (rv , v ∈ V) as well as another tuple α• = (αv , v ∈ V) of positive integers with αh(a) − αt(a) = αh(aB ) − αt(aB ) =: α for all a, aB ∈ A. Then, the GL(V, r)-module ^ 4 Hom ('rt(a) )#αt(a) , ('rh(a) )#αh(a)
% a∈A
is homogeneous of degree α. iii) For any tuple κ = (κ1 , . . . , κt ) of positive integers and a, b, c ∈ (≥0 , we define W(κ, r) := '
't
i=1
κi ri
(2.41)
and the GL(V, r)-module ^
W(κ, r)a,b,c := W(κ, r)#a
4$b
'
& ti=1 8κi ri *
W(κ, r)
Z#−c
.
The corresponding representation ^ 4 *a,b,c : GL(V, r) −→ GL W(κ, r)a,b,c is homogeneous. Proposition 2.5.1.2. Fix a tuple κ = (κ1 , . . . , κt ) of positive integers, and consider the homogeneous representation *: GL(V, r) −→ GL(A). Then, there are non-negative integers a, b, and c, such that A is a direct summand of the module W(κ, r)a,b,c . Proof. By Chapter 1, Theorem 1.1.6.1, there is a representation * *: GL(W(κ, r)) −→ GL(U), such that * is a direct summand of * * ◦ (GL(V, r) ⊂ GL(W(κ, r))). It is easy to see that we may choose * * to be homogeneous of the same degree as *. The assertion follows now from Chapter 1, Corollary 1.1.5.4. " Remark 2.5.1.3. The tuple κ will be a natural parameter in our theory.
2.5.2 More on One Parameter Subgroups In the case of decorated vector bundles, we have used the weighted flags associated to one parameter subgroups of SLr (') in order to motivate the testing objects for semistability in the relative setting. Here, we will explain similar concepts for the group GL(V, r).
S 2.5: D T V SB 2.5: D T V B 212 212 Weighted Flags First, let W be a finite dimensional '-vector space. In the sequel, we will call a weighted flag in W a pair (W• , γ• ) with W• : {0} ! W1 ! · · · ! W s ! W a—not necessarily complete—flag in W and γ• = (γ1 , . . . , γ s+1 ) a vector of integers with γ1 < · · · < γ s+1 . Example 2.5.2.1. In our context, weighted flags arise from one parameter subgroups. If λ: '. −→ GL(W) is such a one parameter subgroup and χ1 , . . . , χ s+1 , s ≥ 0, are the characters of '. with non-trivial eigenspace in W, we write χi (z) = zγi with γi ∈ (, i = 1, . . . , s + 1. Let Wχi ⊂ W be the corresponding eigenspace. We number the characters in such a way that γ1 < · · · < γ s+1 and set γ• (λ) := (γ1 , . . . , γ s+1 ). The flag W• (λ) is obtained by setting Wi :=
%W i
χj
,
i = 1, . . . , s.
j=1
Remark 2.5.2.2. In Chapter 1, Example 1.5.1.36, we associated to a one parameter subgroup λ: '. −→ SL(W) a weighted flag (W• (λ), α• (λ)). Now, we have associated to it another weighted flag (W• (λ), γ• (λ)). This abuse of language is tolerable, because we know how to compute α• (λ) from γ• (λ) and conversely (see Chapter 1, Example 1.5.1.36, and Chapter 2, Example 2.3.2.2). V-Split Vector Spaces Let V = { v1 , . . . , vt } be an index set. A V-split vector space is a collection (Wv , v ∈ V) of vector spaces indexed by V. Note that V-split vector spaces form in a natural way an abelian category. A weighted flag (W• , γ• ) in the V-split vector space (Wv , v ∈ V) is a pair with W• : {0} ! (W1v , v ∈ V) ! · · · ! (W sv , v ∈ V) ! (Wv , v ∈ V) a filtration of (Wv , v ∈ V) by V-split subspaces and γ• = (γ1 , . . . , γ s+1 ) a vector of integers. We have the following equivalent notions: %•v , γ•v ), v ∈ V) of weighted flags in the Wv , v ∈ V, a) Tuples ((W %•v : {0} ! W %v ! · · · ! W % sv ! Wv , W 1 v
γ•v = (γ1v , . . . , γvsv +1 ),
Here, sv = 0 is permitted. b) Weighted flags (W• , γ• ) in the V-split vector space (Wv , v ∈ V).
v ∈ V.
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213
Indeed, suppose we are given a tuple as in a). Let γ1 < · · · < γ s+1 be the different weights occurring among the γvj , v ∈ V, i = 1, . . . , sv + 1. Then, we define (W vj , v ∈ V) % v with by W v := W j
ιv ( j)
L F ιv ( j) := max ι = 1, . . . , sv + 1 | γιv ≤ γ j ,
v ∈ V, i = 1, . . . , s + 1. Conversely, given a weighted flag (W• , γ• ) in the V-split vector space (Wv , v ∈ V), we get the filtration *v ⊆ · · · ⊆ W * sv ⊆ W * v = Wv {0} ⊆ W 1 s+1 of Wv , by just projecting onto Wv , v ∈ V. Eliminating all improper inclusions, we arrive at the flags % sv +1 = Wv , v ∈ V. %1v ! · · · ! W % sv ! W {0} ! W i i Finally, we set
L F % iv , j = 1, . . . , s + 1 , * vj = W γiv := min γ j | W
i = 1, . . . , sv + 1, v ∈ V.
%•v , γ•v ) of weighted flags in the Wv , v ∈ V. These two operaThis provides the tuple (W tions are clearly inverse to each other. Weighted Flags and Group Embeddings Again, we fix a tuple κ = (κ1 , . . . , κt ) of positive integers and define the vector space W(κ, r) as in (2.41). A one parameter subgroup λ: '. −→ GL(V, r) corresponds to a tuple λ = (λv , v ∈ V) where λv is a one parameter subgroup of GLrv ('), v ∈ V. The one parameter subgroup λv provides a weighted flag (W•v (λ), γ•v (λ)) in Wv := 'rv , v ∈ V. As we have seen before, this tuple of weighted filtrations corresponds to a weighted filtration (W• , γ• ) of the V-split vector space (Wv , v ∈ V). Now, define Witotal := and the flag
%W $ v∈V
v κv , i
i = 1, . . . , s,
W•total : {0} ! W1total ! · · · ! W stotal ! W(κ, r).
Then, one immediately checks that (W•total , γ• ) is the weighted flag of λ as a one parameter subgroup of GL(W(κ, r)). Convention 2.5.2.3. We will deal only with homogeneous representations, so that we have to take into account only one parameter subgroups λ: '. −→ GL(V, r) ∩ SL(W(κ, r)). Therefore, we will use weighted flags of the V-split vector space (Wv , v ∈ V) of the form (W• , α• ) where α• = (α1 , . . . , α s ) is a tuple of positive rational numbers. To this weighted flag, we associate the weighted flag (W•total , α• ) in the vector ' space W(κ, r). Then, using Ri := dim(Witotal ) = v∈V κv · dim Wiv , i = 1, . . . , s, and ' R := dim(W total ) = v∈V κv · rv , we may decode the weights as s 4 T ^ γ1 , . . . , γ1 , . . . , γ s+1 , . . . , γ s+1 = αi · γR(Ri ) . D!!!!!WB!!!!!\ D!!!!!!!!!!WB!!!!!!!!!!\ R1 x
(R s+1 −R s ) x
i=1
(2.42)
S 2.5: D T V SB 2.5: D T V B 214 214 Characters and One Parameter Subgroups Any character of the group GL(V, r) is of the form 2 ^ 4 mv ∈ GLrv ('), v ∈ V 1−→ det(mv )χv v∈V
for suitable integers χv , v ∈ V. If we are given also a one parameter subgroup λ: '. −→ GL(V, r), then we obtain the composite homomorphism χ ◦ λ: '. −→ '. which is of the form z 1−→ zγ , z ∈ '. , for some well-defined integer γ. We remind the reader that we have set F λ, χ 4 := γ. Suppose that λ is a one parameter subgroup of GL(V, r) ∩ SL(W(κ, r)). Then, we have assigned to it a weighted flag (W• , α• ) where the rational numbers α1 , . . . , α s are related to the weights of the one parameter subgroup λ by means of Formula 2.42. Recall that the weighted flag equips every vector space Wv with a filtration v = Wv . {0} ⊆ W1v ⊆ · · · ⊆ W sv ⊆ W s+1 One easily computes F λ, χ 4 =
T
χv ·
v∈V
s+1 T i=1
^ 4 v γi · dim(Wiv ) − dim(Wi−1 ).
Exercise 2.5.2.4. Using Formula 2.42, verify that γ s+1 = r ·
s T
αi ,
T
r :=
^ 4 κv · rv = dim W(κ, r)
v∈V
i=1
and that s+1 T i=1
γi ·
^
dim(Wiv )
−
v dim(Wi−1 )
4
= γ s+1 · dim(Wv ) − r ·
s T i=1
αi · dim(Wiv ).
With this exercise, we finally compute F λ, χ 4 = r ·
s T i=1
αi ·
&T v∈V
χv · rv −
T v∈V
Z χv · dim(Wiv ) .
(2.43)
A Weight Formula The following data should be given: An index set V and tuples r = (rv , v ∈ V) and κ = (κv , v ∈ V) of positive integers. Then, we get the V-split vector space (Wv := 'rv , v ∈ V) and W(κ, r) as in (2.41). Let (W• , γ• ) be a weighted flag in (Wv , v ∈ V). As before, we set Witotal := v∈V (Wiv )$κv , i = 1, . . . , s, in order to obtain a weighted flag (W•total , γ• ) in W(κ, r). Assume, furthermore, that we are given quotients kv : Wv −→ 'tv , ' v ∈ V, and set k := v∈V kv$κv : W(κ, r) −→ 't , t := v∈V κv · tv . We will next derive a formula which will be crucial for the determination of semistable points during our construction of moduli spaces.
!
!
S 2.5: D T V B
215
%•v , Proposition 2.5.2.5. Suppose that, in the above situation, we are given a tuple ((W v γ• ), v ∈ V) of weighted flags in the Wv , v ∈ V. Let (W• , γ• ) be the corresponding weighted flag in (Wv , v ∈ V) and (W•total , γ• ) the resulting weighted flag in W(κ, r). Then, the following identity holds true,
=
s T 4 γi+1 − γi ^ · r · dim k(Witotal ) − t · dim Witotal r i=1 sv &T v T 4Z γi+1 − γiv ^ %iv %iv ) − tv · dim W κv · · rv · dim kv (W rv v∈V i=1
− Here, r =
'
T
κv ·
v∈V
&t
v
rv
v ^ 4Z t Z &T v % i−1 % iv − dim W . · γiv · dim W r i=1
s +1
−
κv · rv .
v∈V
Proof. From the definitions, the formula T
κv ·
sv +1 &T
v∈V
i=1
s+1 ^ 4Z T 4 ^ total %iv − dim W %v γiv · dim W = γi · dim Witotal − dim Wi−1 i−1 i=1
follows immediately. Therefore, the assertion is equivalent to the following equation s T 4 γi+1 − γi ^ · r · dim k(Witotal ) − t · dim Witotal r i=1
=
s+1 ^ 4 t T total − · γi · dim Witotal − dim Wi−1 r i=1 sv &T v T 4Z γi+1 − γiv ^ %iv ) − tv · dim W %iv · rv · dim kv (W κv · rv v∈V i=1 sv +1 ^ 4Z tv &T v % iv − dim W % i−1 − κv · · γiv · dim W . rv i=1 v∈V
T
Now, t T (γi+1 − γi ) · dim Witotal − · r i=1 s s Z &t T Z &t T · γi · dim Witotal − · γi+1 · dim Witotal r i=1 r i=1 s
= =
s s+1 &t T Z &t T Z total · γi · dim Witotal − · γi · dim Wi−1 r i=1 r i=1
=
T ^ 4 t total γi · dim Witotal − dim Wi−1 . − · γ s+1 · dim W(κ, r) + r i=1
W0total ={0}
s+1
216 S 2.5: D T V SB 2.5: D T V B 216 Therefore, the left hand side simplifies to T t − · γ s+1 · dim W(κ, r) + (γi+1 − γi ) · dim k(Witotal ). r i=1 s
With the same argument as before, we see s T 4 ^ (γi+1 − γi ) · dim k(Witotal ) = γ s+1 · dim k W(κ, r) − i=1
−
s+1 T i=1
4 ^ total ). γi · dim k(Witotal ) − dim k(Wi−1
Since dim W(κ, r) = r and dim k(W(κ, r)) = t, the left hand side finally takes the form −
s+1 T i=1
^ 4 total γi · dim k(Witotal ) − dim k(Wi−1 ).
(2.44)
Likewise, the right hand side becomes sv +1 &T
^ 4Z %iv ) − dim k(W %v ) . γiv · dim kv (W i−1
(2.45)
The equality of (2.44) and (2.45) is now clear from the definitions.
"
−
T
κv ·
v∈V
i=1
Remark 2.5.2.6. The conceptual way to see the above formula which explains how it will arise later is the following: Denote by 4v the Graßmannian of tv -dimensional quotients of Wv , v ∈ V. Let kv : Wv * O&v −→ Qv be the universal quotient and O&v (1) := det(Qv ), v ∈ V. Likewise, we let 4 be the Graßmannian of t-dimensional quotients of W(κ, r), k: W(κ, r) * O& −→ Q the universal quotient, and O& (1) := det(Q). On Xv∈V 4v , we have the quotient kB :=
%π v∈V
.
&v
κv (k)$ v : W(κ, r) * OXv∈V &v −→
%π v∈V
.
&v
(Qv )$κv .
This quotient defines a (Xv∈V GL(Wv ))-equivariant embedding h: X
v∈V
such that
4v '→ 4,
^ 4 h. O& (1) = OXv∈V &v (κv1 , . . . , κvt ).
Let λv : '. −→ GL(Wv ) be a one parameter subgroup which induces the weighted flag %•v , γ•v ) in Wv , v ∈ V (see Remark 2.5.2.1). Then, Expression 2.45 is just (W ^ 4 µ (kv1 , . . . , kvt ), λ , λ := (λv1 , . . . , λvt ), with respect to the linearization of the GL(V, r)-action in OXv∈V &v (κv1 , . . . , κvt ). Now, we can view λ as a one parameter subgroup of GL(V, r). The induced weighted filtration of W(κ, r) is then (W•total , γ• ), and (2.44) agrees with µ(k, λ) with respect to the linearization in O& (1). Obviously, we must have ^ 4 µ(k, λ) = µ (kv1 , . . . , kvt ), λ .
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217
2.5.3 The Moduli Problem of Tumps We will now formulate the classification problem for tumps.18 V-Split Sheaves The category of V-split vector spaces readily generalizes to give the category of V-split sheaves. Let us now introduce the relevant notation. We fix a finite index set V = { v1 , . . . , vt }. A V-split sheaf is simply a tuple (Fv , v ∈ V) of coherent sheaves on X. Likewise, a homomorphism between V-split sheaves (Fv , v ∈ V) and (FvB , v ∈ V) is a collection ( fv , v ∈ V) of homomorphisms fv : Fv −→ FvB , v ∈ V. In this way, the V-split sheaves on X form an abelian category. Remark 2.5.3.1. Recall that the datum of a V-split vector bundle on X is equivalent to the datum of a principal GL(V, r)-bundle.
Now, we fix additional parameters κ = (κv ∈ (>0 , v ∈ V) and χ = (χv ∈ Then, we define, for any V-split sheaf (Fv , v ∈ V), the (κ, χ)-degree as degκ,χ (Fv , v ∈ V) :=
3, v ∈ V).
T^ 4 κv · deg Fv + χv · rk Fv , v∈V
the κ-rank as rkκ (Fv , v ∈ V) :=
T
κv · rk Fv ,
v∈V
and the (κ, χ)-slope as µκ,χ (Fv , v ∈ V) :=
degκ,χ (Fv , v ∈ V) rkκ (Fv , v ∈ V)
.
Note that the (κ, χ)-degree and the κ-rank behave additively on short exact sequences and that the degree of a non-trivial V-split sheaf in which all entries are torsion sheaves is positive. Thus, the (κ, χ)-slope will have all the formal properties of the usual slope. The Weight Formula for Split Sheaves Let (Fv , v ∈ V) be a V-split sheaf. As before, the following data are equivalent: %v , γv ), v ∈ V) of weighted filtrations of the Fv , v ∈ V, a) Tuples ((F • • %v : {0} ! F%v ! · · · ! F %v ! Fv F • sv 1
γ•v = (γ1v , . . . , γvsv +1 ),
b) Weighted filtrations (F• , γ• ) of the V-split sheaf (Fv , v ∈ V). 18 “tump”=“tuple
with a map”.
v ∈ V.
218 S 2.5: D T V SB 2.5: D T V B 218
!
Moreover, given κ = (κv ∈ (>0 , v ∈ V), a V-split sheaf (Fv , v ∈ V), and a weighted filtration (F• , γ• ), we set F total := v∈V Fv$κv and F•total : {0} ! F1total :=
%F v∈V
1
$ ! · · · ! F total := s
v, κv
%F v∈V
s
$ ! F total .
v, κv
Now, we fix a point x0 ∈ X and write OX (n) for the line bundle associated to the divisor n · x0 , and, given an OX -module F , the symbol F (n) stands for the sheaf F *OX OX (n). Recall that the Hilbert polynomial P(F ) of the coherent OX -module F is defined by ^ 4 P(F )(l) = χ F (l) = l · rk(F ) + deg(F ) + rk(F )(1 − g), l ∈ (. The next result generalizes our weight formula from Proposition 2.5.2.5 to the setting of V-split sheaves. %v , Proposition 2.5.3.2. Suppose that, in the above situation, we are given a tuple ((F • v total γ• ), v ∈ V) of weighted filtrations of the Fv , v ∈ V. Let (F• , γ• ) be the resulting weighted filtration of F total . Then, for all l = 0,
=
s T 4 γi+1 − γi ^ · P(F total )(l) · rk Fitotal − P(Fitotal )(l) · rk F total total P(F )(l) i=1 sv &T v T 4Z γi+1 − γiv ^ %v )(l) · rk Fv %v − P(F κv · · P(Fv )(l) · rk F i i P(Fv )(l) v∈V i=1
−
T v∈V
κv ·
sv +1 & rk F ^ 4Z rk F total Z &T v v %v )(l) − P(F %v )(l) . − · γ · P( F i i i−1 P(Fv )(l) P(F total )(l) i=1
Proof. For l = 0, we have: • F%iv (l) is globally generated, v ∈ V, i = 1, . . . , sv + 1, ^ 4 %v (l) = {0}, v ∈ V, i = 1, . . . , sv + 1. • H1 F i Then, we may write Fv (l) as a quotient qv : OX$P(Fv )(l) −→ Fv (l), such that H 0 (qv ) is an isomorphism, v ∈ V. Restricting this to a general point x ∈ X yields kv : 'P(Fv )(l) −→ 'rk Fv , v ∈ V. Now, apply Proposition 2.5.2.5 to the tuple ((H 0(F%•v (l)), γ•v ), v ∈ V) under the identification of 'P(Fv )(l) with H 0 (Fv (l)). " Tumps Now, we are ready to introduce the objects which we would like to classify. We fix a dimension vector r = (rv , v ∈ V) and a homogeneous representation *: GL(V, r) −→ GL(A). Note that a V-split vector bundle (Ev , v ∈ V) with rk Ev = rv , v ∈ V, then gives rise to a vector bundle E* with fiber A (compare Proposition 2.1.1.7). A *-tump of type (d, l) is a tuple (Ev , v ∈ V, L, ϕ) where (Ev , v ∈ V) is a V-split vector bundle, L is a line bundle, ϕ: E* −→ L is a non-trivial homomorphism, and
S 2.5: D T V B
219
(d, l) = (deg(Ev ), v ∈ V, deg(L)). Two *-tumps (Ev , v ∈ V, L, ϕ) and (EvB , v ∈ V, LB , ϕB ) will be considered isomorphic, if there are isomorphisms ψv : Ev −→ EvB , v ∈ V, and z: L −→ LB , such that ϕB = z ◦ ϕ ◦ ψ−1 * , with ψ* : E* −→ E*B the isomorphism induced by the ψv , v ∈ V. Semistable Tumps In order to define the semistability concept, we introduce additional parameters: • a tuple κ = (κv , v ∈ V) of positive integers, • a tuple χ = (χv , v ∈ V) of rational numbers, such that
'
v∈V
χv · rv = 0,
• a positive rational number δ. The test objects for the semistability concept will be weighted filtrations (E• , γ• ) of the V-split vector bundle (Ev , v ∈ V) where each (E vj , v ∈ V) consists of subbundles E vj ⊂ Ev , v ∈ V, i = 1, . . . , s. For such a weighted filtration, we define α• = (α1 , . . . , α s+1 ) by αi := (γi+1 − γi )/r, r := rk( v∈V Ev$κv ), i = 1, . . . , s. We now set
!
Mκ,χ (E• , α• )
:=
s T i=1
^ α j · degκ,χ (Ev , v ∈ V) · rkκ (E vj , v ∈ V) − 4 − degκ,χ (E vj , v ∈ V) · rkκ (Ev , v ∈ V) .
If we are also given a non-trivial homomorphism ϕ: E* −→ L, we have to define the quantity µ(E• , α• , ϕ).
!
We adapt the definition from Section 2.3.2. For appropriate a, b, c, the module A will be a submodule of W(κ, r)a,b,c , by Lemma 2.5.1.2. We introduce Eitotal := v∈V (Eiv )$κv , i = 1, . . . , s. There is the weighted filtration (E•total , α• )
of E total =
!
v∈V
E•total : {0} ! E1total ! · · · ! E total ! E total s Ev$κv . Restricting to the generic point, we get the weighted flag
$• : {0} ! $1 ! · · · ! $s ! $ of K-vector spaces, K := '(X). We also write .* for the restriction of E* to the generic point of X. We may decompose
$a,b,c := ($#a )$b
*
det($)#−c = .* - .B ,
220 S 2.5: D T V SB 2.5: D T V B 220 so that ϕ yields a point ϕη ∈ !($a,b,c ). After the choice of a one parameter subgroup λK : 4m (K) −→ SL($) which induces the weighted filtration ($• , γ•B ) with γ•B = (γ1B , . . . , γBs+1 ) =
s T i=1
^ 4 αi · ri − r, . . . , ri − r , ri , . . . , ri , D!!!!!!!!!!!!!WB!!!!!!!!!!!!!\ D!!!WB!!!\ ri x
we set
ri := rk Eitotal , i = 1, . . . , s,
(r−ri ) x
(2.46)
µ(E• , α• , ϕ) := µ(ϕη , λK ).
Remark 2.5.3.3. An easier, more elegant, and equivalent definition is as in Exercise 2.3.2.2 F L µ(E• , α• , ϕ) := − min γiB1 + · · · + γiBa | τ|(Eitotal #···#Eitotal )"b is non-trivial . (i1 ,...,ia )∈ { 1,...,s+1 }x a
Here,
a
1
τ: Ea,b,c := (E #a )$b * det(E)#−c −→ E* −→ L. ϕ
Convention 2.5.3.4. Since the quantities introduced above depend only on α• , we will refer to a pair (E• , α• ), composed of a filtration E• : {0} ! (E1v , v ∈ V) ! · · · ! (E vs , v ∈ V) ! (Ev , v ∈ V) of (Ev , v ∈ V) by non-trivial and proper V-split subbundles and a tuple α• = (α1 , . . . , α s ) of positive rational numbers, as a weighted filtration in the future. A *-tump (Ev , v ∈ V, L, ϕ) will be called (κ, χ, δ)-(semi)stable or just (semi)stable, if for every weighted filtration (E• , α• ) of (Ev , v ∈ V) Mκ,χ (E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0. A few comments are in order. ' Remark 2.5.3.5. i) Since v∈V χv · rv = 0, we may write Mκ,χ (E• , α• ) = − rk +
s T i=1
X & αi · deg
% E$
&
% E$ v∈V
v∈V
v
κv
v
Z
κv
s Z T &T Z · αi · χv · rk Eiv +
%E $ · rk &
i=1
v∈V
v, κv i
Z
v∈V
%E $
& − deg
v∈V
v, κv i
Z
· rk
% E$
&
v∈V
v
κv
ZJ
.
D!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!WB!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\ =:Mκ (E• ,α• )
A bounded family of V-split vector bundles (Ev , v ∈ V) may be parameterized by a product of Q = Xv∈V Qv . Assigning to a point ([qv ], v ∈ V) ∈ Q the 6 quot schemes _ $ κv quotient induces an injective and proper morphism from Q to some other v∈V qv * quot scheme Q. In this way, we can induce linearizations on Q by linearizations on * and this shows how the quantity Mκ (E• , α• ) is obtained. The linearization of the Q, GL(V, r)-action on the space !(W(κ, r)a,b,c ) in O((W(κ,r)a,b,c ) (1) induced by *a,b,c may be modified by a character, such that the determinant on W(κ, r)a,b,c induces the trivial
!
S 2.5: D T V B
221
character on the center Z of GL(V, r), and the quantity µ(E• , α• , ϕ) has been defined with respect to such a linearization. The parameter δ reflects the fact that the given linearization may be raised to some tensor power. Finally, any linearization might be altered by a character χ of GL(V, r). The choice of such a character is encoded by the rational numbers χv , v ∈ V (see (2.43)). These considerations explain how the semistability concept we have introduced naturally mixes the semistability concept for vector bundles and the invariant theory of the representation *. ' ii) The condition v∈V χv · rv = 0 is used to simplify some computations. It can, however, be assumed without loss of generality. For this, note that for arbitrary parameters κv and χv , v ∈ V: • Mκ,χ (E• , α• ) is defined the same way, • µ(E• , α• , ϕ) does not depend on the χv , v ∈ V. In particular, we may define (semi)stability with respect to these parameters. Suppose we are given arbitrary rational numbers χv , v ∈ V, and d ∈ 3. Define χBv := χv + d · κv , v ∈ V. Then, &T Z κv · rk(Fv ) , degκ,χB (Fv , v ∈ V) = degκ,χ (Fv , v ∈ V) + d · v∈V
for any V-split sheaf (Fv , v ∈ V). For every V-split vector bundle (Ev , v ∈ V) and every V-split subbundle (Fv , v ∈ V), it follows that degκ,χ (Ev , v ∈ V) · rkκ (Fv , v ∈ V) − degκ,χ (Fv , v ∈ V) · rkκ (Ev , v ∈ V) =
degκ,χB (Ev , v ∈ V) · rkκ (Fv , v ∈ V) − degκ,χB (Fv , v ∈ V) · rkκ (Ev , v ∈ V),
so that always
Mκ,χ (E• , α• ) = Mκ,χB (E• , α• ),
and the concept of (semi)stability defined with respect to the parameters κv , χv , v ∈ V, and δ equals the one defined with respect to the parameters κv , χBv , v ∈ V, and δ. If we now set ' χv · rv , d := − 'v∈V v∈V κv · rv then
T v∈V
χBv · rv =
&T v∈V
Z Z &T κv · rv = 0. χv · rv − d · v∈V
S-Equivalence Assume that (Ev , v ∈ V, L, ϕ) is a (κ, χ, δ)-semistable *a,b,c -tump of type (d, l) and that (E• , α• ) is a weighted filtration with Mκ,χ (E• , α• ) + δ · µ(E• , α• , ϕ) = 0.
S 2.5: D T V SB 2.5: D T V B 222 222 As before, we would like to associate to this *a,b,c -tump and this weighted filtration an admissible deformation df (E• ,α• ) (Ev , v ∈ V, L, ϕ). Recall that the weighted filtration (E• , α• ) leads, in particular, to filtrations {0} =: E0v ! E1v ! · · · ! E vsv ! E vsv +1 := Ev , So, we set gr
Ev :=
% E /E sv +1 i=1
v i
v i−1 ,
v ∈ V.
v ∈ V.
As in Section 2.3.2, one checks that one has a filtration F• : {0} =: F0 ! F1 ! · · · ! Ft ! Ft+1 := E*a,b,c , such that Egr,*a,b,c !
% F /F t+1
i=1
i
i−1 .
Then, we have the index i0 , such that ϕ|Fi0 & 0, but ϕ|Fi ≡ 0 for i = 1, . . . , i0 − 1. Thus, ϕ induces a non-trivial homomorphism ϕ0 : Fi0 /Fi0 −1 −→ L. Finally, we define ϕgr : Egr,*a,b,c −→ L as the homomorphism that agrees with ϕ0 on Fi0 /Fi0 −1 and vanishes on the other direct summands, so that we arrive at gr
df (E• ,α• ) (Ev , v ∈ V, L, ϕ) := (Ev , v ∈ V, L, ϕgr ). As in Section 2.3.2, we let S-equivalence on (κ, χ, δ)-semistable *a,b,c -tumps of type (d, l) be the equivalence relation that is generated by (E, v ∈ V, L, ϕ) ∼ df (E• ,α• ) (E, v ∈ V, L, ϕ), for every (κ, χ, δ)-destabilizing weighted filtration (E• , α• ). We say that (E, v ∈ V, L, ϕ) is (κ, χ, δ)-polystable, if it is isomorphic to all of its admissible deformations. The Moduli Functors and the Moduli Spaces We proceed as in Section 2.3.2 and use similar notation. Again, we first fix a Poincar´e line bundle L on Jacl x X. Let S be a scheme of finite type over '. Then, a family of *-tumps of type (d, l) (parameterized by S ) is a tuple (ES ,v , v ∈ V, κS , NS , ϕS ) with the following ingredients: • vector bundles ES ,v of rank rv and degree dv on the fibers {s} x X, s ∈ S , v ∈ V, • a morphism κS : S −→ Jacl , • a line bundle NS on S , • a homomorphism ϕS : ES ,* −→ L [κS ] * π.S (NS ) whose restriction to {s} x X is non-zero for every closed point s ∈ S .
S 2.5: D T V B
223
Two families (ES1 ,v , v ∈ V, κ1S , NS1 , ϕ1S ) and (ES2 ,v , v ∈ V, κ2S , NS2 , ϕ2S ) are said to be isomorphic, if κ1S = κ2S =: κS and there exist isomorphisms ψS ,v : ES1 ,v −→ ES2 ,v , v ∈ V, and χS : NS1 −→ NS2 with 4−1 ^ ϕ1S = idL [κS ] * π.S (χS ) ◦ ϕ2S ◦ ψS ,* ,
ψS ,* : ES1 ,* −→ ES2 ,* the induced isomorphism.
For a tuple (κ, χ, δ) of stability parameters, these concepts give rise to the moduli functors M(κ,χ,δ)-(s)s (*, d, l): Sch. S
−→ Set Equivalence classes of families of (κ, χ, δ)-(semi)stable *-tumps 1−→ of type (d, l) parameterized by S
.
Exercise 2.5.3.6. Check that the definition of the moduli functor is independent of the choice of the Poincar´e line bundle L . Using the terminology from Remark 2.2.4.11, ii), the main result in the classification theory of *-tumps reads: Theorem 2.5.3.7. Let *: GL(V, r) −→ GL(A) be a homogeneous representation. Fix the stability parameters (κ, χ, δ) and the type (d, l). Then, the coarse projective moduli space M (κ,χ,δ)-ss (*, d, l) for the functors M(κ,χ,δ)-(s)s (*, d, l) does exist. Again, it is obvious that the theorem has to be proved only for representations of the type *a,b,c , a, b, c ∈ (≥0 .
2.5.4 Proof of Theorem 2.5.3.7 The proof of Theorem 2.5.3.7 is very similar to the proof of Theorem 2.3.2.5. The only point which has to be given special attention and which is indeed rather tricky is the correct choice of a linearization on the parameter space. For this reason, we will give most of the ingredients of the construction which are straightforward generalizations of their counterparts in the setting of swamps as exercises to the reader and explain by means of a sample computation that the linearization which we will introduce is the correct one. For the whole section, we fix the index set V, the dimension vector r, non-negative integers a, b, and c, giving rise to the homogeneous representation *a,b,c : GL(V, r) −→ GL(W(κ, r)a,b,c ), the stability parameters (κ, χ, δ), and the type (d, l). We also introduce ' r := v∈V κv · rv . Boundedness Exercise 2.5.4.1. Prove the following statement:
S 2.5: D T V SB 2.5: D T V B 224 224 Proposition 2.5.4.2. For a (κ, χ, δ)-semistable *a,b,c -tump (Ev , v ∈ V, L, ϕ) of type (d, l), an index v0 ∈ V, and a subbundle {0} ! F ! Ev0 , one has ' a · (r − 1) a · (r − 1) (κv · dv + χv · rv ) +δ· = v∈V ' . µ(F) ≤ µκ,χ (Ev , v ∈ V) + δ · r r v∈V κv · rv Deduce the following: Corollary 2.5.4.3. Fix v0 ∈ V. Then, the set of isomorphism classes of vector bundles E of rank rv0 and degree dv0 for which there exists a (κ, χ, δ)-semistable *a,b,c -tump (Ev , v ∈ V, L, ϕ) of type (d, l) with Ev0 ! E is bounded. The Parameter Space For any integer e, we let Jace be the Jacobian variety which parameterizes line bundles of degree e on X. We write Jacd := X Jacdv . v∈V
-
By Corollary 2.5.4.3, we can find an n0 , such that for all n ≥ n0 , all (κ, χ, δ)κv semistable *a,b,c -tumps (Ev , v ∈ V, L, ϕ) of type (d, l), all line bundles N = v∈V L# v e dv with [Lv ] ∈ Jac , v ∈ V, and all line bundles L with [L] ∈ Jac the following holds: • Ev (n) is globally generated and H 1 (Ev (n)) = {0}, v ∈ V, • N #c * L * OX (a · n) is globally generated and H 1 (N #c * L * OX (a · n)) = {0}. ' We fix such an n, and set pv := dv +rv (n+1−g), v ∈ V, and p := v∈V κv · pv . Moreover, we choose vector spaces Yv of dimension pv and let Q0v be the quasi-projective quot scheme parameterizing quotients q: Yv * OX (−n) −→ F with F a vector bundle of rank rv and degree dv , such that H 0 (q(n)) is an isomorphism, v ∈ V (see Theorem 2.2.3.5). Let Ev be the universal quotient on Q0v x X, v ∈ V, and ^ 4 κv π.Q0 x X E$ EQB := v
% v∈V
v
be the resulting vector bundle on QB x X, QB := (Xv∈V Q0v ) x Jacl . Let LQB be a line bundle on QB x X which is the pullback of a Poincar´e sheaf on Jacl x X. Define Y := $κv and Y := (Y #a )$b . For large m, we form the vector bundles a,b v∈V Yv & ^ 4Z Fm := πQB . Ya,b * π.X OX (a(m − n)) , & ^ 4Z Gm := πQB . det(EQB )#c * LQB * π.X OX (am) ,
!
and the projective bundle 4 ^ π: P := ! H om(Fm , Gm )∨ −→ QB .
S 2.5: D T V B
225
Exercise 2.5.4.4. i) Formulate the generalizations of Proposition 2.3.5.5 and Remark 2.3.5.6. ii) Show that there are a parameter scheme T '→ P and a universal family (ET,v , v ∈ V, κT , NT , ϕT ) which satisfy these properties. (Note that the parameter scheme T is projective over QB .) iii) Show that the group Xv∈V GL(Yv ) acts naturally on T in such a way that a) '. acts trivially, '. being diagonally embedded into Xv∈V GL(Yv ), and b) the analog of the proposition stated in Exercise 2.3.5.7 holds true. By Part iii), a), of the above exercise, the group (Xv∈V GL(Yv ))/'. , '. being diagonally embedded, acts on the parameter scheme T. We define 4 ^ * := G X GL(Yv ) ∩ SL(Y) v∈V L F = (h1 , . . . , ht ) ∈ X GL(Yv ) | det(h1 )κv1 · · · · · det(ht )κvt = 1 . v∈V
* maps with finite kernel onto (Xv∈V GL(Yv ))/'. , whence we may restrict The group G * our attention to the action of G. Next, we will introduce the corresponding Gieseker space. Note that we may also assume: • H 1 (H(n)) = {0} and H(n) is globally generated for every line bundle H on X with deg(H) ∈ { dv | v ∈ V }. Fix a Poincar´e sheaf Pv on Jacdv , v ∈ V. The sheaves Gv1 := H om
r &8
^ 4Z Yv * OJacdv , πJacdv . Pv * π.X (OX (rv n)) ,
v ∈ V,
are locally free. Set 41v := !((Gv1 )∨ ), v ∈ V. Choosing Pv appropriately, we may assume that O&1v (1) is very ample for all v ∈ V. Next, we set 1 41 := v∈V X 4v .
Abbreviate J d,l := Jacd x Jacl . Let L be a Poincar´e sheaf on Jacl x X. Also set ^ 4 π.Jacdv x X Pv#κv . P :=
) v∈V
Then, the sheaf
& ^ 4Z G2 := H om Ya,b * O J d,l , π J d,l . π.Jacd x X (P)#c * π.Jacl x X (L ) * π.X (OX (na))
on J d,l is locally free, too. Set 42 := !(G2∨ ). Again, O&2 (1) can be assumed to be very ample. * Exercise 2.5.4.5. Show that there is a G-equivariant and injective morphism Gies: T −→ 4 := 41 x 42 .
226 S 2.5: D T V SB 2.5: D T V B 226 For given ηv ∈ (>0 , v ∈ V, and β ∈ (>0 , there is a natural linearization of the * G-action on 4 in the ample line bundle O(ηv , v ∈ V, β). This may be altered by any character of Xv∈V GL(Yv ). Let xv := χv /δ, xBv := rv · xv /pv , v ∈ V, ε :=
p−a·δ , r·δ
and xBBν
εv := κv −
xv r · χv = κv − , ε p−a·δ
J rv r − , := ε · κν · p pv
v ∈ V,
X
v ∈ V.
Now, we choose ηv ∈ (>0 and β ∈ (>0 , such that ηv = ε · εv , β
v ∈ V.
Remark 2.5.4.6. To be very precise, the quantities ε and εv , v ∈ V, are functions in n. Since p = P(n), P being the Hilbert polynomial of the vector bundle E total , is a positive polynomial of degree 1 and both δ and χv , v ∈ V, are constants, it is clear that ε and εv , v ∈ V, will be positive for n = 0, i.e., the line bundle in which the action is linearized is really ample. * We modify the linearization of the G-action on 42 in O(ηv , v ∈ V) by the character (mv , v ∈ V) 1−→
2
det(mv )ev
v∈V
with
ev := β · (xBv + xBBv ),
v ∈ V.
I (Note that Xv∈V '. · idYv (! '. t ) then acts via (zv , v ∈ V) − 1 → v∈V zvpv ·ev ). We work * with the resulting linearization of the G-action on 4 in O(ηv , v ∈ V, β). A Sample Computation In order to illustrate that our choice of the linearization is accurate, we go through a part of the calculations which are analogous to those in the proof of Theorem 2.3.5.13 of Section 2.3.5. More precisely, we show the following: Let m = (qv : Yv * OX (−n) −→ Ev , v ∈ V, ϕ) be a point in the parameter space T, such that Gies(m) is (semi)stable with respect to the chosen linearization in O(ηv , v ∈ V, β), then (Ev , v ∈ V, ϕ) is a (κ, χ, δ)(semi)stable *-tump of type (d, l). First, as in Corollary 2.3.5.14, one verifies that the (semi)stability condition has to be checked only for those weighted filtrations (E• , α• ) which satisfy ^ 4 Eiv (n) is globally generated and H 1 Eiv (n) = {0}, i = 1, . . . , s, v ∈ V. For weighted filtrations of that type, we have to prove that Mκ,χ (E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0.
(2.47)
S 2.5: D T V B
227
Define γ• = (γ1 , . . . , γ s+1 ) by the conditions
and, setting Eitotal :=
γi+1 − γi = αi , p
!
v∈V
s+1 T i=1
i = 1, . . . , s,
Eiv,$κv , i = 1, . . . , s + 1, & ^ 4 ^ 4Z total (n) = 0. γi · h0 Eitotal (n) − h0 Ei−1
%•v , γ•v ) of Then, we obtain a weighted filtration (E• , γ• ) and, thus, weighted filtrations (E v v v the Ev , v ∈ V. Next, we choose bases y = (y1 , . . . , y pv ) of the Yv with 1
4 ^ P %iv (n) , yv1 , . . . , yvh0 (E%v (n)) = H 0 E i
and set
i = 1, . . . , sv , v ∈ V,
^ 4 γ*• v := γ1v , . . . , γ1v , . . . , γvsv +1 , . . . , γvsv +1 . D!!!!!WB!!!!!\ D!!!!!!!!!!!!WB!!!!!!!!!!!!\ %v (n)) x h0 (E i
%vsv (n))) x (pv −h0 (E
This yields the one parameter subgroup 4 ^ λ := λ(yv1 , γ*• v1 ), . . . , λ(yvt , γ*• vt ) * Now, with Gies(m) = ([mB ], [mv ], v ∈ V), of G.
=
µ(Gies(m), λ) β ^ 4 B µ [m ], λ + XT sv +1 ^ 4ZJ ^ 4 T & rv r Z&T v 0 %v 0 %v − γ h (Ei (n)) − h (Ei−1 (n)) +ε εv µ [mv ], λ − κv pv p i=1 i v∈V v∈V D!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!WB!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\ +
T& v∈V
=:A
xBv
sv +1 T
^ 4Z %iv (n)) − h0 (E %v (n)) . γiv h0 (E i−1
i=1 D!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!WB!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!\ =:B
Observe sv +1 T i=1
γiv
sv T ^ 4 ^ 4 v 0 %v 0 %v v %iv (n) , (γi+1 − γiv ) · h0 E · h (Ei (n)) − h (Ei−1 (n)) = γ sv +1 · pv − i=1
so that we find B=−
T& v∈V
xBv ·
sv Z T T v %iv )(n) + rv · xv · γvsv +1 . (γi+1 − γiv ) · P(E i=1
v∈V
D!!!!!!!!!!!!!!WB!!!!!!!!!!!!!!\ =:C
228 S 2.5: D T V SB 2.5: D T V B 228 Next, ^ 4 µ [mv ], λ = =
sv v T ^ 4 Z γi+1 − γiv & %iv − h0 E %iv (n) · rv · pv · rk E pv i=1 s Z v v T γi+1 − γiv & %iv − P(E %iv )(n) · rv . · pv · rk E pv i=1
Thus, sv T 4 ^ ^ 4 v %iv (n) + C (γi+1 − γiv ) · h0 E ε · εv · µ [mv ], λ − xBv · i=1
= = = =
sv Z v T 4 ^ − γiv & γi+1 %iv )(n) + C %iv − P(E %iv )(n) · rv − xBv · pv · P(E · ε · εv · pv · rk E pv i=1 sv Z v T 4 ^ γi+1 − γiv & %iv )(n) + C %iv )(n) · rv − xv · rv · P(E %iv − P(E · ε · εv · pv · rk E pv i=1 s Z v v T γ − γv & 4 ^ i i+1 %iv + C %iv − P(E %iv )(n) · rv − xv · pv · rk E · ε · κv · pv · rk E pv i=1 sv & v sv ^ T 4Z T 4 γi+1 − γiv ^ v %iv + C. %iv − P(E %iv )(n)rv − εκv − γiv ) rk E xv (γi+1 pv rk E pv i=1 i=1
Using Proposition 2.5.3.2, we discover that ε · A + B equals ε· =
s T
s T
i=1
αi ·
i=1
+
s & T^ T ^ 4 4Z αi · p · rk Eitotal − P(Eitotal )(n) · r − p · αi · xv · rk Eiv
T v∈V
X
2
p ·
rk Eitotal
r·δ
rv · xv · γvsv +1 −
−
p·a·
sv ^ T i=1
rk Eitotal
−
r
p·
v∈V i=1 total P(Ei )(n)
δ
+a·
P(Eitotal )(n)
J +
4 v %iv . xv · (γi+1 − γiv ) · rk E
In order to conclude, we have to compute µ([mB ], λ). Under the identification of Y with the space H 0 (E total (n)), we define 4 ^ total )(n) , i = 1, . . . , s + 1. gri (Y) = H 0 (Eitotal /Ei−1 The basis y of Y induced by the bases yv for the Yv , v ∈ V, yields an isomorphism Y!
% gr (Y). s+1 i=1
i
For an index tuple ι ∈ I a := { 1, . . . , s + 1 }x a , we define Yι := grι1 (Y) * · · · * grιa (Y), and for k ∈ { 1, . . . , b }, we let Yιk be Yι embedded into the k-th copy of Y #a in Ya,b . If
S 2.5: D T V B
229
we denote P(Eitotal (n)) = h0 (Eitotal (n)) by mi , i = 1, . . . , s, then λ = a one parameter subgroup of SL(Y). Therefore,
's
i=1
i) αi · λ(y, γ(m p ) as
s L ^ 4 FT αi · (a · mi − νi (ι) · p) | k ∈ { 1, . . . , b }, ι ∈ I a : Yιk " ker(mB ) . µ [mB ], λ = − min i=1
Here,
F L νi (ι) = # ι j ≤ i | ι = (ι1 , . . . , ιa ), j = 1, . . . , a .
Let ι0 ∈ I a be an index which realizes the precise value of µ([mB ], λ). Altogether, we then find s & p2 · rk E total p · a · rk E total p · P(E total )(n) Z T i i i αi · − − + r · δ r δ i=1 +
p·
s T
αi · νi (ι0 ) − p ·
i=1
s T
αi ·
^T v∈V
i=1
(xv · rk Eiv )
4
as the value for µ(Gies(m), λ)/β. We multiply this by r · δ/p and get s T i=1
s &T 4Z ^ ^ 4 αi · νi (ι0 ) · r − a · rk Eitotal + αi · p · rk Eitotal − r · P(Eitotal )(n) + δ ·
+
T
rv · χv ·
r · γvsv +1
v∈V
p
i=1
sv & Z & r · γv T r · γiv Z i+1 %iv . · rk E χv · − − p p i=1
Setting γiB := (r/p) · γi , i = 1, . . . , s + 1, we have αi =
B γi+1 − γiB , r
i = 1, . . . , s.
The computations leading to (2.43) therefore show T v∈V
B
rv · χv · γ svv +1 −
sv T i=1
s &T Z T ^ B 4 B v %iv = −r · χv · γi+1 − γi v · rk E αi · χv · rk Eiv . i=1
v∈V
One verifies (see the argument for (2.17)) that µ(E• , α• , ϕ) ≥
s T i=1
4 ^ αi · νi (ι0 ) · r − a · rk Eitotal ,
so that, by Remark 2.5.3.5, µ(Gies(m), λ)(≥)0 implies Inequality 2.47.
"
2.5.5 Properties of the Semistability Concept We will now derive the properties of the semistability concept for tumps which are analogous to those of the semistability concept for swamps which were described in Section 2.3.6, p. 164ff, and Theorem 2.3.4.3. Since the author does not see how the arguments used in Theorem 2.3.4.3 may be adapted to the setting of tumps, the logic order of the arguments will be different.
230 S 2.5: D T V SB 2.5: D T V B 230 Asymptotic Semistability We fix, for this section, the stability parameters κ, χ, and δ. It will be assumed that ' v∈V χv · rv = 0, and the representation is *a,b,c for some non-negative integers a, b, and c. Recall from Remark 2.5.3.5 that the choice of the tuple χ reflects the choice of a ' character of the group GL(V, r) ∩ SL(W(κ, r)), r := v∈V κv · rv . This freedom of choice also implies that we will have various notions of “asymptotic semistability” which we may look at. To explain these notions of asymptotic semistability, we write χv = χBv + χBBv , v ∈ V, ' where χB = (χBv , v ∈ V) is another tuple of rational numbers which satisfies v∈V χBv · rv = 0. We set out to describe the behavior of semistability when looking at (κ, χδ , δ)semistable objects, χδ := (χBv + δ · χBBv , v ∈ V), and making the parameter δ very large. We say that the *-tump (Ev , v ∈ V, L, ϕ) is (κ, χB , χBB )-asymptotically (semi)stable, if a) µ(E• , α• , ϕ) − rk
% E$
^
v∈V
v
κv
s &T Z 4 T · αi · χBBv · rk Eiv ≥ 0
(2.48)
v∈V
i=1
holds for every weighted filtration (E• , α• ) and b)
Mκ,χB (E• , α• )(≥)0
holds for every weighted filtration (E• , α• ) for which there is equality in (2.48). Remark 2.5.5.1. There is a similar interpretation to Condition a) as in Remark 2.3.6.4. To explain it, we make the simplifying assumption that χBB is a tuple of integers and leave it to the reader to work out the correct interpretation in the general case. We write $v for the restriction of Ev to the generic point η of X, $ for the restriction of $κv to η, and . for the restriction of ((E #a )$b ) * det(E)#−c . The action of a,b,c v∈V E v 4 ^ *K := X GL($v ) ∩ SL($) G
!
v∈V
on .a,b,c is canonically linearized in the ample line bundle O(('a,b,c ) (1). Call this linearization σ. Moreover, with K := '(X), we have the character *K χBB : G
−→
(mv , v ∈ V)
1−→
4 m (K), 2
BB
det(mv )χv .
v∈V
We twist the linearization σ by the character χBB (see Chapter 1, Lemma 1.4.3.16 and before), in order to obtain the linearization σχBB . Now, Condition a) is equivalent to the fact that the point ϕη ∈ .a,b,c defined by ϕ is semistable with respect to the action of *K and the linearization ση . This is a consequence of Chapter 1, Exercise 1.7.2.8, and G Formula 2.43. Theorem 2.5.5.2. Fix the input data a, b, and c, the type (d, l), and the tuples κ, χB , and χBB . Then, there exists a positive rational number δ∞ , such that for any stability parameter (κ, χδ , δ), such that δ > δ∞ , any (κ, χδ , δ)-(semi)stable *a,b,c -tump (Ev , v ∈ V, ϕ) of type (d, l) is (κ, χB , χBB )-asymptotically (semi)stable.
S 2.5: D T V B
231
Proof. We assume that (Ev , v ∈ V, ϕ) is a (κ, χδ , δ)-(semi)stable *a,b,c -tump of type (d, l) for which Condition a) stated above fails. By Remark 2.5.5.1, this means that the point *K -action. ϕη ∈ !(.a,b,c) defined by ϕ is not semistable for the linearization σχBB of the G Thus, by Chapter 1, Theorem 1.7.2.7, we find an instability one parameter subgroup *K . λK : 4m (K) −→ G This one parameter subgroup gives rise to a weighted filtration ($• , γ• ) of the V-split vector space ($v , v ∈ V). For i ∈ { 1, . . . , s } and v ∈ V, we let ri,v be the dimension of the K-vector space $vi . Then, we have the Graßmann bundle G r(Ev , ri,v ) of subbundles of Ev of rank ri,v . It has the universal property that, given a variety π: U −→ X, a morphism U −→ G r(Ev , ri,v ) over X is uniquely specified by the datum of a subbundle F ⊂ π. (Ev ) of rank ri,v . The Graßmann bundle is constructed from the principal GLrv (')-bundle associated to Ev and the Graßmann variety of ri,v -dimensional subspaces of 'rv , on which the group GLrv (') acts, by means of the construction outlined in the proof of Proposition 2.1.1.7. The subspace $vi yields the map Spec(K) −→ G r(Ev , ri,v ),
i = 1, . . . , s, v ∈ V.
Since X is a smooth projective curve, this map extends to a morphism X −→ G r(Ev , ri,v ) which, in turn, corresponds to a subbundle Eiv ⊂ Ev of rank ri,v , i = 1, . . . , s, v ∈ V. Altogether, we find the weighted filtration (E• , α• ) of the V-split vector bundle (Ev , v ∈ V). Here, we have defined α• = (α1 , . . . , α s ) via αi := (γi+1 − γi )/r, i = 1, . . . , s. By construction, we find s &T Z 4 T ^ $ κv · µ(E• , α• , ϕ) − rk αi · χBBv · rk Eiv < 0. Ev
% v∈V
v∈V
i=1
Now, we have to be a little careful. To simplify matters, we may assume that χBB is integral. The general case is again left to the reader. The fact that we study the linearization σχBB means that we rather work with the representation χBB
*a,b,c := *a,b,c * (−χBB ) than with *a,b,c . Therefore, we also have to work with the vector bundle that is associBB ated to the tuple (Ev , v ∈ V) by means of the representation *χa,b,c , that is with BB
E*χa,b,c := E*χBB = E*a,b,c a,b,c
We also define
BB
Lχ := L *
*
) det(E )# v∈V
) det(E )# v∈V
v
so that we write the decoration as BB
v
ϕ: E*χa,b,c −→ Lχ . BB
−χBBv
,
−χBBv
.
232 S 2.5: D T V SB 2.5: D T V B 232 BB
Let Lχ (−D) be the image of ϕ. Note that D is an effective divisor. Then, the surjection BB χBB ϕ: E*a,b,c −→ Lχ (−D) gives rise to the morphism 4 ^ BB f : X −→ !(E*χa,b,c ) = !(E*a,b,c ) with
^ f . O((EχBB
*a,b,c )
4 BB (1) ! Lχ (−D).
As in the definition of S-equivalence, we have the filtration BB
F• : {0} =: F0 ! F1 ! · · · ! F s ! F s+1 := E*χa,b,c whose restriction to the generic point is just the filtration of .a,b,c which is induced by the one parameter subgroup λK . As before, we define the index i0 ∈ { 1, . . . , s + 1 } to be the least index for which ϕ|Fi0 is a non-trivial homomorphism. Then, the mor-
phism f factorizes over !(E*χa,b,c /Fi0 ). Let Lχ (−D − DB ) be the image of the inBB duced homomorphism ϕ0 : Fi0 /Fi0 −1 −→ Lχ (−D). Again, DB is an effective divisor. BB The induced surjection ϕ0 : Fi0 /Fi0 −1 −→ Lχ (−D − DB ) corresponds to the morphism f : X −→ !(Fi0 /Fi0 −1 ) which extends the rational map BB
f
BB
X −→ !(E*χa,b,c /Fi0 ) $ !(Fi0 /Fi0 −1 ). Of course, we find f
.^
BB
4 BB O((Fi0 /Fi0 −1 ) (1) ! Lχ (−D − DB ).
Let ϕη be the image of the generic point η in !("i0 /"i0 −1 ). Here, "i0 and "i0 −1 are the restrictions of Fi0 and Fi0 −1 to η. Now, we have to apply the theory of the instability flag of Chapter 1, Section 1.7.2. Indeed, if we start with the point wK := ϕη ∈ ., we see that ϕη ∈ !("i0 /"i0 −1 ) is just the point xK,∞ obtained before Proposition 1.7.2.5 in Chapter 1. Now, the instability one parameter subgroup λK provides a character χK,. of the group HK := HG*K (wK ). The group HK acts on !("i0 /"i0 −1 ), and this action is canonically linearized in O(()i0 /)i0 −1 ) (1). Call this linearization σB . We twist this linearization by the character −χK,. in order to obtain the linearization σB−χK,. . Chapter 1, Proposition 1.7.2.5, now tells us that the point ϕη is σB−χK,. -semistable. We have to explain how we can make use of this result. So far, we have worked mainly over the field K = '(X). We will have to come back to the ground field '. In order to do so, we choose bases wv = (wv1 , . . . , wvrv ) for 'rv and trivializations ψv : Ev|U ! OU$rv , such that v ) = F wv1 , . . . , wvri,v 4 * OU , ψv (Ei|U
i = 1, . . . , sv , v ∈ V,
over a sufficiently small non-empty open subset U of X. If we set ^ 4 * := GL(V, r) ∩ SL W(κ, r) , G then these trivializations induce an isomorphism *K ! G * G
x
Spec(.)
Spec(K).
S 2.5: D T V B
233
* whose extension to K Furthermore, we may find a one parameter subgroup λ: '. −→ G yields λK under the above isomorphism. Next, we have the group H := QG (λ)/LG (λ). Again, there is an induced isomorphism HK ! H xSpec(.) Spec(K). The one parameter subgroup λ induces, by the procedure outlined in Chapter 1, Section 1.7.2, a character χ. of H. This character extends to HGL(W(κ,r)) := QGL(W(κ,r)) (λ)/LGL(W(κ,r)) (λ), and this extension was described in Example 1.7.2.4 in Chapter 1. Using the choices of bases, the group H identifies with 4 ^ ^ 4 X Hv ∩ SL W(κ, r) . v∈V
Here,
4 4 ^ ^ Hv := GL F wv1 , . . . , wvr1,v 4 x · · · x GL F wvrsv ,v +1 , . . . , wvrv 4 ,
Then, the tuple (Egr,v , v ∈ V) with Egr,v =
% E /E sv +1 i=1
v i
v ∈ V.
v i−1
corresponds to a principal H-bundle. Using the filtration V• : {0} =: V0 ! V1 ! · · · ! Vt ! Vt+1 := W(κ, r)a,b,c which is induced by λ, the associated graded vector space
% V /V t+1
i=1
i
i−1
is in a natural way an H-module. Thus, H acts on the projective space !(Vi0 /Vi0 −1 ), and the fiber space that is associated to (Egr,v , v ∈ V) and !(Vi0 /Vi0 −1 ) is simply the projective bundle !(Fi0 /Fi0 −1 ). The H-action on !(Vi0 /Vi0 −1 ) comes with a natural linearization σBB in O((Vi0 /Vi0−1 ) (1), and the line bundle on !(Fi0 /Fi0 −1 ) that is constructed from (Egr,v , v ∈ V) and O((Vi0 /Vi0 −1 ) (1) is, of course, O((Fi0 /Fi0 −1 ) (1). We may also alter the linearization σBB by the character −χ. in order to construct the linearization σBB−χ. . Let O(−χ(F.i /Fi −1 ) (1) be the resulting line bundle on !(Fi0 /Fi0 −1 ). On the other hand, 0 0 (Egr,v , v ∈ V) and the character χ. define a line bundle Lχ. on X. Our conventions imply 4 .^ BB f O(−χ(F.i /Fi −1 ) (1) ! Lχ (−D − DB ) * L∨χ. . 0
0
The bundle of local automorphisms of the principal bundle associated to (Egr,v , v ∈ V) is a reductive group H over X with typical fiber H. It acts on !(Fi0 /Fi0 −1 ) and this action is linearized in O(−χ(F.i /Fi −1 ) (1). If we use the trivialization which we introduced 0 0 above and restrict everything to the generic point, then we just get the action of HK on !("i0 /"i0−1 ) together with the linearization σ−χK,. . Let Y := !(Vi0 /Vi0 −1 )//σ−χ. H be the quotient, and denote by !(Fi0 /Fi0 −1 )ss σBB−χ. the open subset of !(Fi0 /Fi0 −1 ) that is associated to (Egr,v , v ∈ V) and the open subset
234 S 2.5: D T V SB 2.5: D T V B 234
!(Vi /Vi −1 )ssσ
of σBB−χ. -(semi)stable points in there is a canonical morphism 0
0
BB −χ.
!(Vi /Vi −1 ). 0
0
It is easy to check that
π: !(Fi0 /Fi0 −1 )ss σBB−χ −→ Y. .
The composite rational map f
X −→ !(Fi0 /Fi0 −1 ) $ Y π
is, by our construction, defined at the generic point of X. Thus, it induces a morphism * f : X −→ Y. Some tensor power of O(−χ(V.i /Vi −1 ) (1) descends to an ample line bundle M on Y, and it 0 0 follows readily that 4#m ^ BB * (−DBB ) f . (M) ! Lχ (−D − DB ) * L∨χ. for some positive integer m and some effective divisor DBB . It is clear that the degree of * f . (M) is non-negative. Next, we will compute the degree of Lχ. . Recall that we have an induced filtration E•total : {0} ! E1 ! · · · ! E s ! E total . As a one parameter subgroup of SL(W(κ, r)), we may decompose λ as λ=
s T i=1
αi · λ(w, γr(rk Ei ) ),
where w is the basis obtained from the previously fixed bases 'rv , v ∈ V. Then, viewing χ. as a character of GL(W(κ, r)), we also have χ. =
s T
αi · χi
i=1
with χi the character corresponding to λ(w, γr(rk Ei ) ). We therefore have deg(Lχ. ) =
s T
αi · deg(Lχi ).
i=1
Now, Chapter 1, Example 1.7.2.4, shows that the degree of Lχi is deg(E total ) rk(Ei ) − deg(Ei ) rk(E total ), so that deg(Lχ. ) = Mκ (E• , α• ). In particular, we have derived the estimate Mκ (E• , α• )
= ≤
^ BB 4 1 ≤ deg Lχ (−D − DB ) − · deg(DBB ) m T χBB BB χBB deg(L ) = l − χv · dv =: l .
− deg(L∨χ. )
v∈V
≤
S 2.5: D T V B
235
Next, we point out that, for given a, b, and c, there are only finitely many conjugacy * which may occur as instability one parameter classes of one parameter subgroups in G subgroups of an element in W(κ, r)a,b,c . In particular, there are only finitely many possibilities for the tuple (rk(Eiv ), i = 1, . . . , sv , v ∈ V, α• ) coming with the weighted filtration (E• , α• ) which we have constructed (compare Chapter 1, Remark 1.7.2.3). Therefore, we find a constant C which depends on a, b, c, and the tuple χB , such that − rk
% E$
^
v
v∈V
κv
s 4 ^T 4 T χBv · rk Eiv ≤ C. αi · · v∈V
i=1
Now, the condition of (κ, χδ , δ)-semistability requires 0 ≤ =
Mκ,χδ (E• , α• ) + δ · µ(E• , α• , ϕ) ^ Mκ (E• , α• ) − rk &
^ +δ · − rk ≤
χBB
l
% E$
% E$ v
v∈V
v∈V
v
κv
s ^T 4 4 T αi · χBv · rk Eiv + · v∈V
i=1
s Z 4 T ^T 4 κv · αi · χBBv · rk Eiv + µ(E• , α• , ϕ) v∈V
i=1
+ C − δ.
We see
BB
δ ≤ lχ + C =: δ∞ .
So, for δ > δ∞ , Condition a) is satisfied for any weighted filtration (E• , α• ) of (Ev , v ∈ V). Condition b) is a trivial consequence of the semistability condition. " Proposition 2.5.5.3. Fix the input data a, b, and c, the type (d, l), and the tuples κ, χB , and χBB as in Theorem 2.5.5.2. There is a constant K which depends only on the above data, such that, for any (κ, χB , χBB )-asymptotically semistable *-tump (Ev , v ∈ V, ϕ) of type (d, l), any index v0 ∈ V, and any subbundle {0} ! F ! Ev0 , one finds µ(F) ≤ K. Proof. The argument will resemble the one used in the proof of Theorem 2.5.5.2. For 0 < s < rv0 , we look at the Graßmannian Gr(rv0 , s) of s-dimensional sub vector spaces 5 of 'rv0 . Recall that this Graßmannian is embedded into !( rv0 −s 'rv0 ) by means of the Pl¨ucker embedding. For any e > 0, we look at the Veronese embedding followed by the Segre embedding v0 −s &r8
!
'
rv0
Z
x !(A) '→
!
v0 −s &^r8
4 'r #e v0
*
Z A.
Likewise, we have the embedding v0 −s &r8
!
Z
E v0 x !
^
BB E*χa,b,c
4
'→ !e := !
v0 −s &^r8
E v0
4#e
*
Giving a subbundle F of Ev0 is equivalent to giving a section X −→ G r(Ev0 , s)
Z BB E*χa,b,c ,
e > 0.
236 S 2.5: D T V SB 2.5: D T V B 236 to the Graßmann bundle of subbundles of rank s of Ev0 . Moreover, we have the Pl¨ucker embedding v0 −s Z &r8 ^ 4 G r Ev0 , s '→ ! E v0 . BB
Therefore, the subbundle F of Ev0 and the decoration ϕ: E*χa,b,c −→ Lχ provide us, for any e > 0, with a section ιe : X −→ !e . We note that
BB
4#e ^ 4 ^ χBB . * L ι.e O(e (1) = det(Ev0 ) * det(F)∨
(2.49)
If we knew that the image of the generic point of X under the morphism ιe were contained in the semistable locus for the obvious group action, the argument used in the proof of Theorem 2.5.5.2 would yield & ^ 4Z BB 0 ≤ deg ι.e O(e (1) = e · deg(Ev0 ) − e · deg(F) + deg(Lχ ), so that
BB
lχ BB deg(F) ≤ dv0 + (2.50) ≤ dv0 + max{ 0, lχ }, e and we would be done. If the image of the generic point of X is not semistable, we choose an e = e(v0 , s) which so large that the conclusion of Chapter 1, Theorem 1.7.3.3, holds true. (Note that e depends only on the situation at the generic point and thus only on the tuple (rv , v ∈ V), a, b, c, and χBB .) As in the proof of Theorem 2.5.5.2, we construct from the instability flag a weighted filtration (E• , α• ), such that 4 ^ BB Mκ (E• , α• ) ≤ e · deg(Ev0 ) − deg(F) + lχ . By an argument as before, we find a constant C(e), such that Mκ,χB (E• , α• ) ≤ Mκ (E• , α• ) + C(e), so that the estimate
^ 4 BB Mκ,χB (E• , α• ) ≤ C(e) + e · deg(Ev0 ) − deg(F) + lχ
holds true. Our choice of e and Chapter 1, Theorem 1.7.3.3, grant that s &T Z 4 T ^ $ κv Ev · αi · χBBv · rk Eiv = 0. µ(E• , α• , ϕ) − rk
% v∈V
i=1
v∈V
The condition of asymptotic semistability therefore gives Mκ,χB (E• , α• ) ≥ 0. Altogether, we find the inequality BB F L C(e) + lχ BB ≤ dv0 + max 0, C(e) + lχ . e Our assertion is now an obvious consequence of (2.50) and (2.51).
deg(F) ≤ dv0 +
(2.51) "
S 2.5: D T V B
237
This proposition and arguments parallel to those used in the proof of Lemma 2.3.6.7 give now the converse to Theorem 2.5.5.2, i.e.: Corollary 2.5.5.4. Suppose the data a, b, and c, the type (d, l), and the tuples κ, χB , and χBB have been fixed. Then, there exists a positive rational number δB∞ , such that a (κ, χB , χBB )-asymptotically (semi)stable *a,b,c -tump (Ev , v ∈ V, ϕ) of type (d, l) is (κ, χδ , δ)-(semi)stable, for every δ > δB∞ .
2.5.6 Quiver Representations In this section, we will discuss the example of the formalism of *-tumps which is at the moment the most interesting and useful one. It is the relative version of the GIT problem of quiver representations which was studied in Chapter 1, Section 1.3.3, p. 45ff, and Section 1.5.1, p. 75ff. Twisted Quiver Representations For this and the following sections, we fix a quiver Q = (V, A, t, h). In principle, we may study representations of Q in any abelian category. In Chapter 1, we studied representations of Q in the category of finite dimensional '-vector spaces. Here, we would like to look at representations of Q in the category of vector bundles on the curve X and some more general objects. Let us state explicitly what we mean by that. We fix a tuple M := (Ma , a ∈ A) of vector bundles on X. An M-twisted representation (Ev , v ∈ V, ϕa , a ∈ A) of the quiver Q consists of vector bundles Ev , v ∈ V, on X and homomorphisms ϕa : Et(a) −→ Eh(a) * Ma , a ∈ A. We refer to (r, d) := (rv := rk(Ev ), dv := deg(Ev ), v ∈ V) as the type of the representation. Remark 2.5.6.1. i) If one specializes the general concept of a quiver representation in an abelian category to the setting of vector bundles on X, one arrives at the “untwisted representations”, i.e., at the representations as above where Ma = OX , for every arrow a ∈ A. ii) In the setting of representations of Q in the category of complex vector spaces, the twisting is obsolete. Indeed, the “twisting objects” would be vector spaces, and twisting by the vector space Ma of dimension ma just means that we replace the arrow a by ma identical copies of it, a ∈ A. On the curve X, however, the twisting by vector bundles allows us to introduce arrows and multiple arrows in a twisted way. A famous example is provided by Higgs bundles which are associated to the Jordan quiver with one vertex and one arrow and the twisting sheaf ωX , i.e., a Higgs bundle is a pair (E, ϕ) which consists of a vector bundle E and a twisted endomorphism ϕ: E −→ E * ωX . We will discuss in Example 2.5.6.7 why the untwisted objects for this quiver are rather uninteresting. Higgs bundles were introduced by Hitchin in [109] and are a crucial tool for analyzing representation spaces of π1 (X) ([204], [205]). iii) Since we have the freedom of choosing twisting bundles, we may assume without loss of generality that our quiver has no multiple arrows. A homomorphism from the M-twisted quiver representation (Ev , v ∈ V, ϕa , a ∈ A) to the M-twisted representation (EvB , v ∈ V, ϕBa , a ∈ A) is the datum (ψv , v ∈ V) of vector
238 S 2.5: D T V SB 2.5: D T V B 238 bundle maps ψv : Ev −→ EvB , v ∈ V, such that the diagram ϕa
Et(a) −−−−−→ Eh(a) * Ma ψt(a) U Uψh(a) #idMa ϕBa
B B * Ma Et(a) −−−−−→ Eh(a)
commutes for every arrow a ∈ A. A subrepresentation of the M-twisted quiver representation (Ev , v ∈ V, ϕa , a ∈ A) is an M-twisted representation (Fv , v ∈ V, σa , a ∈ A) of Q where Fv is a subbundle of Ev , v ∈ V, such that ϕa (Ft(a) ) ⊂ Fh(a) * Ma and σa = ϕa|Fa , a ∈ A. Likewise, we say that (Qv , v ∈ V, τa , a ∈ A) is a quotient representation of the M-twisted representation (Ev , v ∈ V, ϕa , a ∈ A) of Q, if there are surjective maps qv : Ev −→ Qv , v ∈ V, such that (qv , v ∈ V) is a homomorphism of representations. With these concepts, the M-twisted representations of Q form the abelian category Rep(X, Q, M). Exercise 2.5.6.2. Check that all the notions which have been introduced so far make sense in the category Coh(X) of coherent sheaves on X, too. The Twisted Path Algebra We have already encountered the notion of a path in the quiver Q in Chapter 1, Section 1.3.3. We also define the path ev of length zero which connects the vertex v to itself, v ∈ V. Now, suppose that M = (Ma , a ∈ A) is a collection of twisting bundles. Then, the M-twisted path algebra of Q is the following sheaf A (Q, M) of OX -algebras: • As an OX -module
A (Q, M) :=
%
p a path in Q
Mp
with M p := Mas and
*
· · · * Ma1 ,
if p = (a1 , . . . , a s ) is a path of positive length,
M p := OX ,
if p = ev , v ∈ V.
• If p = (a1 , . . . , a s ) and q = (b1 , . . . , bt ) are two paths of positive length and v, vB ∈ V, then, for x p ∈ M p , xq ∈ Mq , yv ∈ Mev and yvB ∈ MevB , we define 7 0, if h(a s) % t(b1 ) . xq · x p := , xq * x p ∈ M(a1 ,...,as ,b1 ,...,bt ) , otherwise 7 0, if h(a s) % v . yv · x p := , yv · x p ∈ M p , otherwise 7 0, if t(b1 ) % vB . xq · yvB := , xq · yvB ∈ Mq , otherwise 7 0, if v % vB . yv · yvB := . yv · yvB ∈ Mev , otherwise
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Of course, we made use of the structure of M p as left OX -module, of Mq as right OX module, and of OX as algebra, respectively. This definition is taken from the paper [86]. A more elegant definition of A (Q, M) may be found in Section 5.1.1 of [1]. It is not hard to show that the category of M-twisted representations of Q is equivalent to the category of vector bundles endowed with the structure of a right A (Q, M)module (see Proposition 5.1 in [1], and [86] for the corresponding assertion about left A (Q, M)-modules). In the set-up of vector bundles on curves (or higher dimensional varieties), this interesting observation has not yet been exploited. The analogous statement for representations of Q in the category of finite dimensional '-vector spaces is one of the main motivations to study quivers (see, e.g., [179]). Semistability Since M-twisted representations of Q form an abelian category, one has stability and semistability concepts for them which very much resemble the corresponding notions of vector bundles and share the formal properties of these. Indeed, one just has to say what the rank and the degree of an M-twisted quiver representation should be. Then, one has immediately the slope and can just formally define (semi)stability of quiver representations. We have already introduced these quantities in Section 2.5.3. Let us briefly recall them. We have the following parameters: A tuple κ = (κv , v ∈ V) of positive integers and a tuple χ = (χv , v ∈ V) of rational numbers.19 Then, the κ-rank of the M-twisted representation R = (Ev , v ∈ V, ϕa , a ∈ A) is T κv · rk(Ev ) rkκ (R) := v∈V
and its (κ, χ)-degree is degκ,χ (R) :=
T^
4 κv · deg(Ev ) + χv · rk(Ev ) .
v∈V
Consequently, we define the (κ, χ)-slope of R as µκ,χ (R) :=
degκ,χ (R) rkκ (R)
and say that R is (κ, χ)-(semi)stable, if µκ,χ (S )(≤)µκ,χ (R) holds for any subrepresentation {0} ! S = (Fv , v ∈ V, σa , a ∈ A) ! R. Finally, we say that the representation R is (κ, χ)-polystable, if there are α-stable representations R1 ,. . . ,R s of Q, such that µκ,χ (R1 ) = · · · = µκ,χ (R s ) 19 So far, we have not fixed the type of the representations, so the condition “' v∈V χv · rv = 0” does not make sense. We will come back to this in Remark 2.5.6.3.
240 S 2.5: D T V SB 2.5: D T V B 240 and
R ! R1 - · · · - R s .
Remark 2.5.6.3. Let c be a rational number. Given (κ, χ), define (κ, χB ) by χBv := χv + c · κv , v ∈ V. Then, one checks that the conditions of (κ, χ)-(semi)stability and (κ, χB )-(semi)stability on M-twisted representations of Q are equivalent. If we fix the type (r, d) of the representations under consideration, we may assume without loss of ' generality that v∈V χv · rv = 0. In some situations, however, other normalizations, such as χv0 = 0 for some fixed vertex v0 ∈ V, seem more useful (see the section on holomorphic chains, p. 251ff). Exercise 2.5.6.4. In this exercise, we look at M-twisted representations of Q in the category Coh(X) of coherent sheaves on X (see Exercise 2.5.6.2). i) Let {0} −−−−−→ S −−−−−→ R −−−−−→ Q −−−−−→ {0} be a short exact sequence of M-twisted representations of Q. Then, rkκ (R) = rkκ (S ) + rkκ (Q)
and degκ,χ (R) = degκ,χ (S ) + degκ,χ (Q).
ii) If R = (Ev , v ∈ V, ϕa , a ∈ A) is a non-trivial M-twisted representation of Q which consists only of torsion sheaves, then rkκ (R) = 0 and degκ,χ (R) > 0. iii) We say that the M-twisted representation R is (κ, χ)-semistable, if degκ,χ (S ) · rkκ (R) ≤ degκ,χ (R) · rkκ (S ) holds for any subrepresentation {0} ⊆ S ⊆ R. Conclude that, in an (κ, χ)-semistable representation R = (Ev , v ∈ V, ϕa , a ∈ A), all the sheaves Ev must be torsion free. (This explains why we have restricted to M-twisted representations in which all coherent sheaves are vector bundles.) Exercise 2.5.6.5 (The Harder–Narasimhan filtration). We look only at M-twisted representations of Q in the category of vector bundles on X. Fix the stability parameter (κ, χ) and show that any representation R of Q possesses a unique filtration {0} =: R0 ! R1 ! · · · ! R s ! R s+1 := R by subrepresentations, such that Ri := Ri+1 /Ri is (κ, χ)-semistable, i = 0, . . . , s, and µκ,χ (R0 ) > · · · > µκ,χ (R s ). Exercise 2.5.6.6 (The Jordan–H¨older filtration). In this exercise, we work again exclusively with M-twisted representations of Q in the category of vector bundles on X. We fix the stability parameter (κ, χ). i) Prove that any (κ, χ)-semistable representation R of Q possesses a filtration R• : {0} =: R0 ! R1 ! · · · ! R s ! R s+1 := R by subrepresentations, such that Ri := Ri+1 /Ri is (κ, χ)-stable, i = 0, . . . , s. ii) Give an example which illustrates that this filtration needs not be unique.
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iii) Show that the associated graded object gr(R• ) :=
%R s
i=0
i+1 /Ri
is well defined up to isomorphy, i.e., it does not depend on the choice of the filtration R• with the asserted property. Hence, we may denote it simply by gr(R). Motivated by the last observation in this exercise, we call two (κ, χ)-semistable M-twisted representations R and RB of Q S-equivalent, if gr(R) ! gr(RB ). Example 2.5.6.7 (Higgs bundles). Let Q be the quiver • 0 . Then, we have to specify one twisting bundle M. We choose it to be a line bundle. As observed in Remark 2.5.6.3, we have to look only at stability parameters of the form (κ, 0). It is apparent from the definition that the dependence on κ drops out. Therefore, we have a parameter independent (semi)stability concept for the M-twisted representations of Q which we will call Higgs bundles from now on. So, a Higgs bundle (E, ϕ: E −→ E * M) is (semi)stable, if the inequality µ(F)(≤)µ(E) is satisfied for any subbundle {0} ! F ! E with ϕ(F) ⊆ F * M.20 Assume that (E, ϕ) is a semistable Higgs bundle. If E is not semistable as a vector bundle, then we look at its Harder–Narasimhan filtration {0} =: E0 ! E1 ! · · · ! E s ! E. It follows from the defining properties of the Harder–Narasimhan filtration (see Exercise 2.2.4.5, i), that µ(Ei ) > µ(E), i = 1, . . . , s. Thus, none of the subbundles Ei , i = 1, . . . , s, is invariant under ϕ. This means that the homomorphisms ϕi : Ei −→ E/Ei * M, i = 1, . . . , s, which are induced by ϕ are all non-trivial. For i ∈ { 1, . . . , s }, set L F i := max j ∈ { 1, . . . , i } | ϕi|E j−1 ≡ 0 F L ı := min j ∈ { i + 1, , . . . , s + 1 } | Im(ϕ) ⊆ E j /Ei * M . Then, we have the non-trivial induced homomorphism ϕi : Ei /Ei−1 −→ Eı /Eı−1 * M. By definition of the Harder–Narasimhan filtration, Ei /Ei−1 and Eı /Eı−1 are semistable vector bundles. Hence, Eı /Eı−1 * M is also semistable. Using Exercise 2.2.4.5, ii), we deduce µ(Ei /Ei−1 ) ≤ µ(Eı /Eı−1 * M) = µ(Eı /Eı−1 ) + deg(M). On the other hand, i < ı, so that µ(Ei /Ei−1 ) > µ(Eı /Eı−1 ). 20 Note that this is the same condition as the semistability of the associated Hitchin pair (E, ϕ, 1) discussed in Section 2.3.6.
S 2.5: D T V SB 2.5: D T V B 242 242 If we assume deg(M) ≤ 0, these two inequalities are incompatible and E must be a semistable vector bundle. Finally, if E is a stable vector bundle and M is a line bundle of degree ≤ 0, then Hom(E, E * M) = {0} unless M ! OX . In that case, Hom(E, E) = ' · id M . These considerations show that the theory of Higgs bundles with the trivial twisting bundle M = OX does not bring much new. On the other hand, for twisting bundles of sufficiently positive degree, in particular, for M = ωX , one gets a very rich theory as remarked before. We point out that the computations presented in this example are due to Nitsure [163]. The formalism of quiver representations, in particular, the notions of semistability, which we have presented up to now is taken from the paper [1]. That paper also relates polystable quiver representations to solutions of certain vortex equations by a so-called Kobayashi–Hitchin correspondence. Augmented Representations In order to apply Theorem 2.5.3.7 on the existence of moduli spaces of tumps, we have to generalize the notion of an M-twisted representation of the quiver Q a little further. We fix the quiver Q and the family M of twisting vector bundles as before. Then, an augmented M-twisted representation (Ev , v ∈ V, ϕa , a ∈ A, ε) of the quiver Q is composed of an M-twisted representation (Ev , v ∈ V, ϕa , a ∈ A) of Q and a complex number ε. The tuple (r, d) := (rv := rk(Ev ), dv := deg(Ev ), v ∈ V) is again called the type of the augmented representation. A homomorphism from the augmented M-twisted quiver representation (Ev , v ∈ V, ϕa , a ∈ A, ε) to the augmented M-twisted representation (Ev , v ∈ V, ϕa , a ∈ A, ε) is the datum (ψv , v ∈ V) of vector bundle maps ψv : Ev −→ EvB , v ∈ V, and a non-zero complex number λ, such that εB = λ · ε and the diagram
ϕa
Et(a) −−−−−→ Eh(a) * Ma ψt(a) U Uψh(a) #(λ·idMa ) ϕBa
B B * Ma Et(a) −−−−−→ Eh(a)
commutes for every arrow a ∈ A. Remark 2.5.6.8. We see that we recover the category of M-twisted representations of Q as the full subcategory of the category of augmented M-twisted representations of Q which are of the form (Ev , v ∈ V, ϕa , a ∈ A, 1). In particular, the classification problem of augmented M-twisted representations of Q comprises the one of M-twisted representations. Let Q(M) = (V, A(M), t(M), h(M)) be the quiver that is obtained from Q by replacing any arrow a ∈ A by rk(Ma ) copies of it. Denote by η the generic point of the curve X and by K its function field '(X). The formalism of quiver representations in the
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243
category of vector spaces which was introduced in Chapter 1, p. 45ff and p. 75ff, for the base field ' makes, of course, sense for any other base field as well, in particular, for the (non-algebraically closed) base field K. We set 4 ^ Hom(K rt(M)(a) , K rh(M)(a) ) Rep Q(M), r :=
% %
a∈A(M)
=
Hom(K rt(a) , K rh(a) )$ rk(Ma ) .
a∈A
For any augmented M-twisted representation R = (Ev , v ∈ V, ϕa , a ∈ A, ε), we let 4 ^ ^ 4∨ rR ∈ ! Rep Q(M), r - K be the point defined by the restriction of R to η. Recall from Chapter 1, Section 1.5.1, ' that rR induces an endomorphism fR : K r −→ K r where r = b · v∈V rv .21 Now, an augmented M-twisted representation (Ev , v ∈ V, ϕa , a ∈ A, ε) is (κ, χ)(semi)stable, if • (Ev , v ∈ V, ϕa , a ∈ A) is a (κ, χ)-(semi)stable M-twisted representation of Q, • ε % 0, or the endomorphism f : K r −→ K r defined above is not nilpotent. An augmented representation (R, ε) is (κ, χ)-polystable, if R is (κ, χ)-polystable. Likewise, we call two augmented (κ, χ)-semistable M-twisted representations (R, ε) and (RB , εB ) of Q S-equivalent, if (gr(R), ε) and (gr(RB ), εB ) are isomorphic. To complete our set-up, we have to introduce the moduli functors. A family of augmented M-twisted representations of Q of type (r, d) is a tuple (ES ,v , v ∈ V, ϕS ,a , a ∈ A, NS , εS ) with the following ingredients: • vector bundles ES ,v of rank rv and degree dv on the fibers {s} x X, s ∈ S , v ∈ V, • a line bundle NS on S , • homomorphisms ϕS ,a : ES ,t(a) −→ ES ,h(a) * π.S (NS ) * π.X (Ma ), a ∈ A, • a homomorphism ε: OS x X −→ π.S (NS ). Two families (ES1 ,v , v ∈ V, ϕ1S ,a , a ∈ A, NS1 , ε1S ) and (ES2 ,v , v ∈ V, ϕ2S ,a , a ∈ A, NS2 , ε2S ) are said to be isomorphic, if there exist isomorphisms ψS ,v : ES1 ,v −→ ES2 ,v , v ∈ V, and χS : NS1 −→ NS2 with ε2S = π.S (χS ) ◦ ε1S and 21 The
^ 4 ϕ2S ,a = π.S (χS ) * ψS ,h(a) ◦ ϕ1S ,a ◦ ψ−1 S ,t(a) ,
a ∈ A.
number b was defined in Chapter 1 as the maximal number of arrows with identical tail and head in the quiver Q(M). Since we assume that the quiver Q has no multiple arrows (Remark 2.5.6.1), b is just max{ rk(Ma ) | a ∈ A }.
S 2.5: D T V SB 2.5: D T V B 244 244 For a tuple (κ, χ) of stability parameters, these concepts yield the moduli functors (κ,χ)-(s)s
R
(Q, M, r, d): Sch.
−→
S
1−→
Set Equivalence classes of families of (κ, χ)-(semi)stable augmented M-twisted representations of Q of type (r, d) parameterized by S
.
Theorem 2.5.6.9. Fix the type (r, d) and the stability parameter (κ, χ). Then, there (κ,χ)-(s)s exists a projective coarse moduli space R (Q, M, r, d) for the moduli functors (κ,χ)-(s)s (Q, M, r, d). R Proof. Let x0 be a point of X and write OX (1) := OX (x0 ). We may then choose m = 0, such that Ma∨ (m) is globally generated, a ∈ A, and OX (m) has a non-zero global section σ0 : OX −→ OX (m). Hence, if we choose F L β ≥ max h0 (Ma∨ (m)) | a ∈ A , we may find surjections qa : OX (−m)$β −→ Ma∨ , whence injections
ιa : Ma −→ OX (m)$β,
a ∈ A, a ∈ A.
Therefore, any family (ES ,v , v ∈ V, ϕS ,a , a ∈ A, NS , εS ) of augmented M-twisted representations of type (r, d) gives rise to the following data: • the homomorphisms (idES ,h(a) # idπ. (N ) #π.X (ιa ))◦ϕS ,a ^ 4 S S * ϕS ,a : ES ,t(a) −−−−−−−−−−−−−−−−−−−−−−−−→ ES ,h(a) * π.S (NS ) * π.X OX (m)$β ,
a ∈ A, • the homomorphism idπ. (NS ) #π.X (σ0 ) ^ 4 εS X * εS : OS x X −−−−−→ π.S (NS ) −−−−−−−−−−−−→ π.S (NS ) * π.X OX (m) .
By filling up with some zero maps, we may assign to the collection * ϕS ,a , a ∈ A, a twisted endomorphism * ϕS :
!
%E v∈V
v
−→
^
% E 4$ v∈V
v
β
*
^ 4 π.S (NS ) * π.X OX (m) .
!
We view * ϕS also as a homomorphism from E nd( v∈V Ev )$β to π.S (NS ) * π.X (OX (m)). Set B := End( v∈V 'rv )$β - '. Then, we have the standard representation * of Xv∈V GLrv (') on B. The above construction assigns to a family (ES ,v , v ∈ V, ϕS ,a , a ∈
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245
A, NS , εS ) of augmented M-twisted representations of Q of type (r, d) the family (ES ,v , "S , * εS ) of *-tumps of type (d, m). Here, κS : S −→ Jacm (X), s 1−→ v ∈ V, κS , N ϕS - * [OX (m)], and N+ S is a line bundle on S , such that ^ 4 . . π.S (N+ S ) * πX OX (m) ! πS (NS ) * L [κS ]. Conversely, given any family (ES ,v , v ∈ V, κS , NS , ϕS ) of *-tumps of type (d, m), we can construct a (possibly empty) subscheme S Q of S , such that the restriction of the family to S Q x X is induced by the above construction from a family of augmented M-twisted representations of Q of type (r, d) parameterized by S Q . First, we have the morphism κS : S −→ Jacm (X), and we let S B be the fiber of κS over [OX (m)]. Then, we have a line bundle NS B on S B with ^ 4 ^ 4 π.S B (NS B ) * π.X OX (m) ! π.S (NS ) * L [κS ] B . |S x X
Let (ES B ,v , v ∈ V, NS B , ϕS B ) be the family where ES ,v , v ∈ V, and ϕS B are obtained by restriction from ES ,v , v ∈ V, and ϕS , respectively. We have ϕS B = * ϕS B - * εS B where * ϕS B :
% E −→ ^% E 4$ v∈V
and
v
v∈V
v
β
*
^ 4 π.S B (NS B ) * π.X OX (m)
^ 4 * εS B : OS x X −→ π.S B (NS B ) * π.X OX (m) .
We may view * ϕS B as a collection of homomorphisms ^ 4 * ϕS B ,(v,vB ) : ES ,v −→ ES ,vB * π.X OX (m)$β ,
(v, vB ) ∈ V x V.
According to Proposition 2.3.5.1, we may define the closed subscheme S BB of S B by the condition (* ϕS B ,(v,vB ) )|S BB x X ≡ 0, whenever (v, vB ) # A.22 ϕS B ,(v,vB ) , (v, vB ) ∈ A, to S BB x X (or S BB ) and Then, we may restrict ES B ,v , v ∈ V, NS B , and * ϕS BB ,a , a ∈ A. Using Proposition 2.3.5.1 and Exercise obtain ES BB ,v , v ∈ V, NS BB , and * 2.3.5.3, we can define a closed subscheme S Q of S BB by the following conditions: • The composition ES BB ,t(a)
* ϕS BB ,a
^ 4 7 ES BB ,h(a) * π.BB (NS BB ) * π. OX (m)$β S X 3 4 ^ ES BB ,h(a) * π.S BB (NS BB ) * π.X OX (m)$β /Ma
is zero on S Q x X, a ∈ A. (Thus, Im(* ϕS BB |S Q x X ) ⊆ ES BB ,h(a)|S Q x X . πX (Ma ), a ∈ A.)
*
π.S Q (NS BB |S Q ) *
22 We may write an arrow a ∈ A in the form (t(a), h(a)), because we assume that Q does not contain any multiple arrow.
246 S 2.5: D T V SB 2.5: D T V B 246 • The composition ^ 4 ^ 4 * εS BB OS BB x X −→ π.S BB (NS BB ) * π.X OX (m) −→ π.S BB (NS BB ) * π.X OX (m)/σ0 (OX ) is zero on S Q x X.23 Now, the reader will easily construct the family of augmented M-twisted representations parameterized by S Q whose associated family of *-tumps is the restriction of (ES ,v , v ∈ V, κS , NS , ϕS ) to S Q x X. Finally, we have to worry about semistability. Taking into account Chapter 1, Proposition 1.5.1.22, the reader will have no difficulty in proving that the *-tump associated to a (κ, χ)-(semi)stable augmented M-twisted representation of Q is (κ, χ, 0)asymptotically (semi)stable. Corollary 2.5.5.4 and our previous considerations there(κ,χ)-(s)s fore show that, for δ > δ∞ , the moduli functor R (Q, M, r, d) is a closed subfunctor of the moduli functor M(κ,χ,δ)-(s)s (*, d, l). Hence, our claim on the existence of the (κ,χ)-(s)s " moduli spaces R (Q, M, r, d) is a direct consequence of Theorem 2.5.3.7. The Generalized Hitchin Map We look at a tuple M = (Ma , a ∈ A) of twisting vector bundles with Ma ! OX (m)$β , for all a ∈ A. In this case, the quiver Q(M) is obtained from Q by replacing each arrow by β copies of it. We let ^ ^ 4 ^ ^ 4∨ 4 4 M χ Q(M), r := ! Rep Q(M), r - ' //σχN GL Q(M), r be the projective moduli space which was constructed in Section 1.3.3 of Chapter 1, p. 75ff. We fix a projective embedding ^ 4 '→ ! M ι: M χ Q(M), r 6 _ 6 _ r = fa , a ∈ A, ε 1−→ τ0 (r) : · · · : τ M (r) Here, the τ0 ,. . . ,τ M are homogeneous GL(Q(M), r)-invariant functions on the vector space Rep(Q(M), r) - ' of degree, say, e. Note that ^ 4 ϑ0 : Rep Q(M), r - ' −→ ' [ fa , a ∈ A, ε] 1−→
ε
is an invariant homogeneous function of degree one. Hence, we may assume τ0 = ϑe0 . Next, we introduce &^ 4 Z / := /(m, β) := ! H 0 (OX (e · m))$M+1 ∨ . Moreover, we also fix a non-trivial section σ0 : OX −→ OX (m). Let s0 be the homogeneous coordinate on / which corresponds to the one-dimensional subspace spanned by σe0 , and set F L / := /(m, β) := [s0 , . . . , sD ] ∈ / | s0 % 0 . 23 The
torsion sheaf OX (m)/σ0 (OX ) is flat over Spec("), so its pullback to S BB x X is flat over S BB .
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247
Let (RS , NS , εS ) := (ES ,v , v ∈ V, ϕS ,a , a ∈ A, NS , εS ) be a family of augmented M-twisted representations of Q parameterized by S . We want to assign to this family a morphism hRS ,Ns ,εS : S −→ /. In order to do so, we choose an open covering U = (Ui )i∈I of S x X, such that there are trivializations ψi,v : ES ,v|Ui ! OU$i rv , i ∈ I, and
4 ^ χi : π.S (NS ) * π.X (OX (m))
|Ui
! OUi ,
i ∈ I.
Using these trivializations, the representation RS and idπ. (NS ) #π.X (σ0 ) ^ 4 εS S * εS : OS x X −−−−−→ π.S (NS ) −−−−−−−−−−−−→ π.S (NS ) * π.X OX (m)
provide us with morphisms ^ 4 κi : Ui −→ Rep Q(M), r - ',
i ∈ I.
Combining these maps with the invariant sections which we have fixed before, we obtain sections τi,µ ∈ H 0 (Ui , OS x X ), i ∈ I, µ = 0, . . . , M. *i,v of the vector bundles ES ,v|Ui , i ∈ I, v ∈ V, we still If we pass to other trivializations ψ get the same sections τi,µ , i ∈ I, µ = 0, . . . , M. However, if we change the trivializations χi by sections fi ∈ OS. x X (Ui ), all the sections τi,µ will be modified by the factor fie , i ∈ I, µ = 0, . . . , M. This means that the above functions glue to global sections & ^ 4Z#e τS ,µ : OS x X −→ π.S (NS ) * π.X OX (m) ,
µ = 0, . . . , M.
Therefore, we get the morphism h := hRS ,NS ,εS : S −→ / with
4 ^ h. O" (1) ! π.S (NS )#e .
Similar considerations as before show that the morphism does not depend on the choice of the covering U either. Remark 2.5.6.10. Note that, by construction, τS ,0 = (* εS ) e . In Section 2.5.4, we have introduced the parameter space T for *-tumps of type (d, m). The construction explained in the proof of Theorem 2.5.6.9 supplies a closed subscheme TQ which is invariant under the Xv∈V GL(Yv )-action and a universal family (RTQ , NTQ , εTQ ), such that R
(κ,χ)-ss
(Q, M, r, d) = TQ // X GL(Yv ). v∈V
248 S 2.5: D T V SB 2.5: D T V B 248 The morphism
h := hRTQ ,NTQ ,εTQ : TQ −→ /
is invariant under the action of Xv∈V GL(Yv ), so that it descends to the projective morphism (κ,χ)-ss H: R (Q, M, r, d) −→ /. We call H the (generalized) Hitchin map. Remark 2.5.6.11. The Hitchin map was introduced by Hitchin in [109] in the context of Higgs bundles. Recall from Chapter 1, Section 1.3.3, p. 43ff, that the invariants of an (r x r)-matrix are the coefficients of its characteristic polynomial. Therefore, the Hitchin map assigns to a Higgs bundle a kind of characteristic polynomial. This characteristic polynomial defines an (r : 1)-covering of the curve, the so-called spectral curve. If the spectral curve happens to be smooth, then one can identify the moduli space of Higgs bundles with the given characteristic polynomial with the Jacobian of the spectral curve. Thus, the general fiber of the Hitchin map is an abelian variety. This Hitchin fibration has many interesting aspects up to applications in the Langlands program [161]. One may hope that the Hitchin map will also play an important role in studying the moduli spaces of quiver representations. Here, we will give a modest application to the existence of moduli spaces for non-augmented quiver representations. From now on, we allow again arbitrary tuples M of twisting bundles. A family of M-twisted representations of Q of type (r, d) is a tuple (ES ,v , v ∈ V, ϕS ,a , a ∈ A) with the following components: • vector bundles ES ,v of rank rv and degree dv on the fibers {s} x X, s ∈ S , v ∈ V, • homomorphisms ϕS ,a : ES ,t(a) −→ ES ,h(a) * π.S (NS ) * π.X (Ma ), a ∈ A. Two families (ES1 ,v , v ∈ V, ϕ1S ,a, a ∈ A) and (ES2 ,v , v ∈ V, ϕ2S ,a , a ∈ A) are considered isomorphic, if there exist isomorphisms ψS ,v : ES1 ,v −→ ES2 ,v , v ∈ V, such that ϕ2S ,a = ψS ,h(a) ◦ ϕ1S ,a ◦ ψ−1 S ,t(a) ,
a ∈ A.
For a tuple (κ, χ) of stability parameters, we introduce the following moduli functors: R(κ,χ)-(s)s (Q, M, r, d): Sch. S
−→ Set Equivalence classes of families of (κ, χ)-(semi)stable M-twisted 1−→ representations of Q of type (r, d) parameterized by S
.
Remark 2.5.6.12. Note that the functor R(κ,χ)-(s)s (Q, M, r, d) is an open subfunctor of (κ,χ)-(s)s (Q, M, r, d). In fact, let (ES ,v , v ∈ V, ϕS ,a , NS , εS ) be a family of augmented R M-twisted representations of Q. Let T be the support of the cokernel of εS . Set S . := S \ πS (T ).
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249
This is the open subset of S which parameterizes exactly those augmented quiver representations with ε % 0. By definition, εS . := εS |S . x X : OS . x X −→ π.S . (NS |S . ) is an isomorphism, and the push forward of εS . to S . is a trivialization of NS . . Hence, the restriction of (ES ,v , v ∈ V, ϕS ,a , a ∈ A, NS , εS ) to S . x X can be interpreted in a natural way as a family of M-twisted representations of Q parameterized by S . . Theorem 2.5.6.13. Fix the type (r, d) and the stability parameter (κ, χ). Then, there is a quasi-projective coarse moduli space R (κ,χ)-(s)s (Q, M, r, d) for the moduli functors R(κ,χ)-(s)s (Q, M, r, d). Furthermore, there are an affine variety / and a projective morphism
H: R (κ,χ)-(s)s (Q, M, r, d) −→ /.
Proof. As in the proof of Theorem 2.5.6.9, we can find m > 0, β > 0 and vector bundle inclusions ιa : Ma −→ OX (m)$β, a ∈ A. We define M B = (MaB := OX (m)$β , a ∈ A). (κ,χ)-(s)s Then, the moduli functor R (Q, M, r, d) is a closed subfunctor of the functor (κ,χ)-(s)s B (Q, M , r, d), and there is the corresponding closed embedding R R := R
(κ,χ)-ss
B
(Q, M, r, d) '→ R := R
(κ,χ)-ss
(Q, M B , r, d).
Hence, we have the Hitchin morphism B
H * R '→ R −→ H: /.
The subscheme / of / which was introduced before is clearly an affine space, and *−1 (/) R (κ,χ)-(s)s (Q, M, r, d) := H * is the moduli scheme we have been looking for. We define H as the restriction of H to R (κ,χ)-ss (Q, M, r, d). This morphism is obtained by base change from the projective * so that it is also projective. morphism H, " Remark 2.5.6.14. i) Assume that the quiver Q has no oriented cycles. Then, it follows that / = {pt}, and R := R (κ,χ)-ss (Q, M, r, d) is a projective scheme. (κ,χ)-ss (Q, M, r, d) given as ii) There is a natural '. -action on R := R z · (Ev , v ∈ V, ϕa , a ∈ A) := (Ev , v ∈ V, z · ϕa , a ∈ A),
z ∈ '. .
It is interesting to study the fixed point locus of this '. -action. We will come back to this in the case of Higgs bundles. Let (Ev , v ∈ V, ϕa , a ∈ A) be a fixed point for this action. This means that (Ev , v ∈ V, ϕa , a ∈ A) is isomorphic to (Ev , v ∈ V, z · ϕa , a ∈ A), for every z ∈ '. . Let f : K r −→ K r be the associated endomorphism. Then, f lies in the same (Xv∈V GLrv (K))-orbit as z · f , z ∈ '. , so that f is a nullform and therefore nilpotent. Therefore, all the fixed points of the '. -action lie in the nullfiber of the generalized Hitchin morphism.
S 2.5: D T V SB 2.5: D T V B 250 250 Thus, if (Ev , v ∈ V, z · ϕa , a ∈ A) is a fixed point for the '. -action, then the augmented representation (Ev , v ∈ V, z · ϕa , a ∈ A, 0) is always unstable.24 Note that we may view R as a compactification of R. The compactifying divisor is just the GIT-quotient R//'. . Exercise 2.5.6.15. We suppose that the tuple of twisting bundles is M = (Ma = OX , a ∈ A) and that κ = (κv = 1, v ∈ V). Finally, we assume that V = { 0, . . . , t }. i) Verify that you may assume without loss of generality that χ0 = 0. (Compare Remark 2.5.6.3.) So, the stability parameter is now a tuple χ = (χ1 , . . . , χt ) of rational numbers. To simplify our language and notation, we will call a tuple (Ei , i = 0, . . . , l, ϕa : Et(a) −→ Eh(a) , a ∈ A) a representation of Q of type (r, d) and speak of χ-(semi/poly)stability rather than of (κ, χ)-(semi/poly)stability. The resulting moduli spaces are denoted by R χ-(s)s (Q, r, d). ii) We fix the type (r, d) and two tuples τ = (τ1 , . . . , τt ) and σ = (σ1 , . . . , σt ) of rational numbers and set χλ := τ + λ · σ, λ ∈ 3≥0 .
Give a proof of the following result: Proposition. There is a value λ∞ , such that a representation (Ei , i = 0, . . . , t, ϕa , a ∈ A) is χλ -(semi)stable for λ > λ∞ , if and only if: 1. For every non-trivial proper subrepresentation (Fi , i = 0, . . . , t, ψa , a ∈ A), the inequality 't 't i=1 σi · rk(F i ) i=1 σi · rk(E i ) ≤ 't 't rk(F ) i i=0 i=0 rk(E i ) holds true. 2. If equality occurs above, then one has also µτ (Fi , i = 0, . . . , t, ψa , a ∈ A)
't
' deg(Fi ) + ti=1 τi · rk(Fi ) := 't i=0 rk(F i ) (≤) µτ (Ei , i = 0, . . . , t, ϕa , a ∈ A). i=0
Recall that the first condition just says that the restriction of (Ei , i = 0, . . . , t, ϕa , a ∈ A) to the generic point of X is a σ-semistable representation of Q with dimension vector r in the category of '(X)-vector spaces. Here, we see how the study of semistable representations in the category of vector bundles on X is related to the study of semistable representations of Q in the category of finite dimensional vector spaces over '(X). 24 We have
stressed this to avoid the following misunderstanding: If (Ev , v ∈ V, z · ϕa , a ∈ A) is a fixed point for the ". -action, then we have (Ev , v ∈ V, ϕa , a ∈ A, 1) ! (Ev , v ∈ V, z · ϕa , a ∈ A, 1) ! (Ev , v ∈ V, ϕa , a ∈ A, 1/z),
z ∈ ". .
This might suggest that, by a limiting process, (Ev , v ∈ V, ϕa , a ∈ A, 1) will be identified with (Ev , v ∈ V, ϕa , a ∈ A, 0) in the moduli space of augmented representations, contradicting the properties of the Hitchin map. But, we have explained that the limiting object is unstable, so that this limiting process and the suggested identification do not occur within the moduli space R.
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To conclude this paragraph, we note that we have performed the construction of the moduli space of quiver representations for two reasons: 1) The resulting moduli spaces are very likely to have interesting applications in different areas of mathematics. We will briefly mention one in the following section; 2) The strategy of constructing these moduli spaces is very important. First, we have added some additional objects in order to obtain a projective moduli space. Then, we have introduced the gadget of the Hitchin map in order to verify that the moduli space we are really interested in can be realized as an open subscheme of the projective moduli space which we have obtained. (Note that we made a first acquaintance with this strategy in Exercise 1.5.1.26 of Chapter 1.) In Section 2.8, we will use a refined version of this method in order to solve our very general moduli problem of principal bundles decorated with a section in a twisted associated vector bundle. Holomorphic Chains We work in the setting of Exercise 2.5.6.15 and use the respective notation. Our stability parameter is, therefore, a tuple χ = (χ1 , . . . , χn ) ∈ 3n . Formally, one can define χ-(semi)stability also for real parameters χ ∈ 1n . In fact, even for a real parameter, the moduli spaces R χ-ss (Q, r, d) do exist. We will explain this in an easy example (see Remark 2.5.6.21). We fix the quiver Q and the type (r, d). The most basic question we can ask about the moduli spaces of quiver representations is whether they are empty or not. More exactly, we formulate: Problem 2.5.6.16. Describe the region F L R(r, d) := χ ∈ 1n | R χ := R χ-ss (Q, r, d) % ∅ . This is, in general, a very difficult problem. One can also be more modest and try to find a (proper) subset RB ⊂ 1n , such that R(r, d) ⊆ RB . We now restrict to the quiver Q = (V, A, t, h) with V = { 0, . . . , n }, A = { 1, . . . , n }, t(i) = i, h(i) = i − 1, i = 1, . . . , n. We may depict this quiver as n −−−−−→ n − 1 −−−−−→ · · · −−−−−→ 1 −−−−−→ 0. A representation C = (Ei , i = 0, . . . , n, ϕi : Ei −→ Ei−1 , i = 1, . . . , n) is called a holomorphic (n + 1)-chain. We sometimes write a holomorphic chain in the form ϕn
ϕn−1
ϕ2
ϕ1
C : En −−−−−→ En−1 −−−−−→ · · · −−−−−→ E1 −−−−−→ E0 . Let us try to briefly motivate why holomorphic chains are interesting. We look at the fundamental group π1 (X) of the Riemann surface X. If g is the genus of X, then a possible description of this group is 1 P π1 (X) ! α1 , β1 , . . . , αg , βg | [α1 , β1 ] · · · · · [αg , βg ] = 1 . Of course, [α, β] = αβα−1 β−1 . We might, e.g., wish to study the space of representations of π1 (X) in GLr ('). Note that, for each g ∈ GLr ('), there is the inner automorphism ιg : GLr (') −→ GLr ('), h 1−→ g · h · g−1 , and we identify a representation
S 2.5: D T V SB 2.5: D T V B 252 252 *: π1 (X) −→ GLr (') with ιg ◦ *, g ∈ GLr ('). With our methods from Chapter 1, we can easily construct a moduli space for the S-equivalence classes of representations. Indeed, by the above description of π1 (X), a representation of this group is specified by a tuple (A1 , B1 , . . . , Ag , Bg) in the affine algebraic variety 4 := GLr (')2g . Writing down the equation [A1 , B1 ] · · · · · [Ag , Bg ] = $r yields equations in the entries of A1 , B1, . . . , Ag , Bg which have to be fulfilled, so that the tuple (A1 , B1, . . . , Ag , Bg ) defines a representation of π1 (X). These equations give a closed subscheme Rep(π1 (X), r) of 4. As usual, GLr (') acts on 4 by simultaneous conjugation: GLr (') x 4 −→ 4 4 ^ 4 ^ h, (A1 , B1 , . . . , Ag , Bg) 1−→ h · A1 · h−1 , h · B1 · h−1 , . . . , h · Ag · h−1 , h · Bg · h−1 . Obviously, Rep(π1 (X), r) is invariant under this action, and the moduli space is M := M (π1 (X), r) := Rep(π1 (X), r)// GLr ('). We refer to M also as representation space. Note that π1 (X) is a topological invariant of X and so is M . It is now a natural task to analyze the space M , e.g., we could try to understand its Betti numbers. By the work of Hitchin [109] and Simpson [204], [205], we might study instead the topology of the moduli spaces H := H (r, d) of Higgs bundles (E, ϕ: E −→ E * ωX ) where E has rank r and a certain degree d. We have already discussed the '. -action z · (E, ϕ) := (E, z · ϕ) on H . In certain good situations, one can compute the Betti numbers of a variety with a '. -action from the Betti numbers of the fixed point locus. (For smooth projective varieties, this is due to Białynicki-Birula [20].) Now, fixed points look, e.g., as follows: (E, ϕ) where E = V - W and ϕ: V −→ W * ωX filled up by zero homomorphisms. Thus, such a fixed point is described by the 2-chain (E0 := W * ωX , E1 := V, ϕ1 := ϕ). Another possibility is that E = U - V - W and ϕ is determined by ϕBB : U −→ V * ωX 2 and ϕB : V −→ W * ωX . This supplies the 3-chain (E0 := W * ω# X , E 1 := V * ωX , E 2 := U, ϕ1 := ϕB * idωX , ϕ2 := ϕBB ). In general, we may describe all the fixed points by holomorphic m-chains, m ≤ r + 1. Therefore, we are led to study holomorphic chains. It is interesting to note that the stability parameter which we need is (2g −2, 4g −4, 6g − 6, . . .). We refer to the paper [38] for an example where this whole strategy is put into practice. After these motivating comments, we now study 3-chains for the type (2, 1, 1, d0, d1 , d2 ). Let us first see what we can say about the possible stability parameters. For this, assume that C = (E0 , E1 , E2 , ϕ1 , ϕ2 ) is a χ-semistable 3-chain of type (2, 1, 1, d0, d1 , d2 ). 1. Obviously, C1 := (E0 , 0, 0, 0, 0) is a subchain. The condition of χ-semistability for it reads µχ (C1 ) = µ(E0 ) = i.e.,
d0 d0 + d1 + d2 + χ1 + χ2 ≤ = µχ (C), 2 4
χ1 + χ2 ≥ AI := d0 − d1 − d2 .
2. We also have the subchain C2 := (E0 , E1 , 0, 0, ϕ1 ). We infer the necessary condition −χ1 + 3 · χ2 ≥ AII := d0 + d1 − 3d2 .
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3. We assume that ϕ1 is non-trivial (see Exercise 2.5.6.18). Then, there is the subchain C3 := (Im(ϕ1 ), E1 , E2 , ϕ1 , ϕ2 ). (Note that Im(ϕ1 ) need not be a subbundle.) Since E1 is a line bundle and ϕ1 is non-trivial whence injective, the image of ϕ1 is isomorphic to E1 . So, our test object yields the inequality χ1 + χ2 ≤ AIII := 3d0 − 5d1 − d2 . We stress that the degrees d0 , d1 , and d2 are fixed, so that the above inequalities are conditions on the stability parameter χ which are necessary for the existence of χsemistable 3-chains. We let RB (d) be the subset of 12 that is cut out by the above inequalities. We then have R(2, 1, 1, d) ⊆ RB (d). (See Figure 2.2 for a sketch of RB (d).) Remark 2.5.6.17. In the following, we denote by LI , LII , and LIII the lines where the respective inequality obtained above becomes an equality. i) Note that LI and LIII are parallel. ii) The set RB (d) is non-empty (has non-empty interior), if and only if AI ≤ AIII (AI < AIII ), i.e., d1 ≤ d0 /2 (d1 < d0 /2). In order words, Im(ϕ1 ) must not desemistabilize (destabilize) the vector bundle E0 . iii) For a parameter χ on the line LI and a χ-semistable chain C = (E0 , E1 , E2 , ϕ1 , ϕ2 ), the subchain C1 introduced above will always violate stability, so that C is not χ-stable. Furthermore, by definition of S-equivalence, it is S-equivalent to the chain C1 - C 1 , C 1 = (0, E1 , E2 , 0, ϕ2 ). Note that, in the case ϕ2 & 0, the chain C1 - C 1 will be χ-semistable, if and only if E0 is a semistable vector bundle and (E1 , E2 , ϕ2 ) is a χ-semistable 2-chain. Observe that the map from E1 to E0 is zero in the chain C1 - C 1 . iv) If χ lies on the line LII and C = (E0 , E1 , E2 , ϕ1 , ϕ2 ) is a χ-semistable chain, then C2 will violate the condition of χ-stability and C will be S-equivalent to C2 - C 2 with C 2 = (0, 0, E2, 0, 0). v) Finally, if χ lies on the line LIII and C = (E0 , E1 , E2 , ϕ1 , ϕ2 ) is χ-semistable, then C3 will contradict χ-stability. Moreover, Im(ϕ1 ) must be a subbundle of E0 . Otherwise, the subbundle F0 generated by Im(ϕ1 ) would have degree strictly larger than Im(ϕ1 ) and the subchain C3B := (F0 , E1 , E2 , ϕ1 , ϕ2 ) would violate even χ-semistability, in contradiction to our assumption. This time, C will become S-equivalent to C3 - C 3 with C 3 = (E0 /Im(ϕ1 ), 0, 0, 0, 0).
Exercise 2.5.6.18. i) Suppose C = (E0 , E1 , E2 , ϕ1 , ϕ2 ) is a χ-semistable chain with ϕ1 ≡ 0 (ϕ2 ≡ 0). Then, χ must lie on LI (LII ). ii) Let χ be the point of intersection of LI and LII . Show that the moduli space R χ0 0 can—at least set-theoretically—be identified with M (2, d0) x Jacd1 x Jacd2 . iii) Work out similar descriptions for the moduli spaces belonging a) to parameters on LI \ LII and b) to parameters on LII \ LI . Now, we have a good picture of the parameter region and of the chains which are semistable with respect to a parameter on the boundary. The next step is to understand chains which are semistable with respect to a parameter which lies in the interior of
S 2.5: D T V SB 2.5: D T V B 254 254 RB (d). By Exercise 2.5.6.18, all the maps in the chains which do arise in that way will be non-trivial. First, let us consider the case when the moduli spaces to two different parameters are distinct. So, let χB and χBB be two stability parameters in the interior of RB (d).Assume that C = (E0 , E1 , E2 , ϕ1 , ϕ2 ) is a holomorphic 3-chain of type (2, 1, 1, d0, d1 , d2 ) which is χB -semistable but not χBB -semistable. Then, there is a subchain C B = (F0 , F1 , F2 , σ1 , σ2 ) with µχB (C B ) ≤ µχB (C) and µχBB (C B ) > µχBB (C). Now, define a line L in 12 by the equation µχ (C B ) = µχ (C),
(2.52)
i.e., α1 (3r1B − r0B − r2B ) + α2 (3r2B − r0B − r1B ) = (d0 + d1 + d2 )(r0B + r1B + r2B ) − 4(d0B + d1B + d2B ). Here, riB := rk(Fi ) and diB = deg(Fi ), i = 0, 1, 2. Remark 2.5.6.19. Since we assume that both ϕ1 and ϕ2 are non-trivial maps, the tuple (r0B , r1B , r2B ) is (1, 1, 1), (1, 1, 0), or (1, 0, 0). Either χB lies on L and χBB does not or χB and χBB lie on opposite sides of L. For (1, 1, 1) and (1, 0, 0), the line L will be parallel to LI and LIII . For (1, 1, 0), it is parallel to the line spanned by (1, 1). (We will rule out this case below.) For d ∈ (, define the line Ld by the equation α1 + α2 = d, d α1 − α2 = , 2
if d is odd if d is even.
The two-dimensional chambers are the connected components of &) Z RB (d) \ Ld . d∈#
We label them Ci2 , i ∈ nents of
%.
The one-dimensional chambers are the connected compoR&^ 4 ^ ) 4Z RB (d) ∩ Ld \ LdB . d B %d
d∈#
We label these chambers as Ci1 , i ∈ %. Finally, the zero-dimensional chambers are points of the form Ld ∩ LdB , d odd, d B even, which lie in RB (d). Again, we list the zero-dimensional chambers in the form Ci0 , i ∈ %. Hence, we have the decomposition RB (d) =
2 R R j=0 i∈+
Ci j
(2.53)
of RB (d) into locally closed subsets. Note that this decomposition is locally finite, i.e., any bounded subset of RB (d) meets only finitely many chambers.
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Proposition 2.5.6.20. i) Let C be a chamber in the decomposition (2.53). For any two stability parameters χ and χB in the chamber C , a holomorphic 3-chain C of type (2, 1, 1, d0, d1 , d2 ) will be χ-(semi)stable, if and only if it is χB -(semi)stable. ii) Let C be any chamber as above and C B ⊂ C another chamber which is contained in the closure of C . Suppose χ ∈ C and χB ∈ C B . Then, a holomorphic 3-chain C of type (2, 1, 1, d0, d1 , d2 ) which is χ-semistable will also be χB -semistable. Conversely, if C is χB -stable, then it will be χ-stable, too. Proof. These assertions are easy consequences of the construction of the chambers and are, therefore, left to the the reader. " Remark 2.5.6.21. It follows from the construction of the chambers that any chamber contains rational points. Hence, if χ ∈ RB (d) is any stability parameter and C is the unique chamber which contains it, then we find a parameter χB ∈ C ∩ 3n . By PropoB
sition 2.5.6.20, the moduli space R χ for χB -semistable holomorphic 3-chains of type (2, 1, 1, d0, d1 , d2 ) is also the moduli space for χ-semistable holomorphic 3-chains of type (2, 1, 1, d0, d1 , d2 ). Exercise 2.5.6.22. Show that, for a stability parameter χ in a two-dimensional chamber Ci20 , the notions of χ-stability and χ-semistability agree. We can use Proposition 2.5.6.20 also to simplify the chamber decomposition. To do so, let χB and χBB be two stability parameters in the interior of RB (d). Let l be the line segment joining the two parameters. For analyzing the chamber decomposition, we may assume without loss of generality that there is exactly one chamber C0 which intersects l in a single point χ . (If this point happens to be one of the stability param0 eters, we choose it to be χB .) Let L be the line defined by the subchain C B via Equation 2.52. Then, it is clear that L∩l = {χ }. Now, it follows from Proposition 2.5.6.20 that C 0 is χ -semistable with C B as χ -destabilizing subchain. The chain C is then S-equivalent 0 0 to the χ -semistable chain C B - C/C B . Since χ is a stability parameter in the interior 0 0 of RB (d), Exercise 2.5.6.18, i), implies that the homomorphisms in the chain C B - C/C B must be non-zero. This can only occur, if (rk(F0 ), rk(F1 ), rk(F2 )) ∈ { (1, 0, 0), (1, 1, 1) }. This means that L is a line of the form Ld for odd d, i.e., parallel to LI and LIII . Therefore, we can define a new chamber decomposition, using only the lines Ld , d odd, such that the analog of Proposition 2.5.6.20 still holds. This new chamber decomposition is depicted in Figure 2.2. Remark 2.5.6.23. Note that the new chamber decomposition has only finitely many chambers. (Thus, there are also only finitely many different moduli spaces.) Having fixed the type (2, 1, 1, d0, d1 , d2 ), this implies that the set of isomorphy classes of holomorphic 3-chains C of type (2, 1, 1, d0, d1 , d2 ) for which there exists some stability parameter χ ∈ 1n for which C becomes χ-semistable is bounded. We have deduced this important fact here with relatively simple arguments. For arbitrary 3-chains, the analogous observation was obtained in [2] with great effort. For m-chains with m > 3, it is unknown whether this result still holds. In Figure 2.2, we have highlighted a chamber C∞ . The reason is that we can easily describe the corresponding moduli space. Note that this chamber is adjacent to the line LIII .
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•••
!
C∞
LIII
LI LII Figure 2.2: The chamber structure for 3-chains of type (2, 1, 1, d0, d1 , d2 ). Proposition 2.5.6.24. Let χ ∈ C∞ and C = (E0 , E1 , E2 , ϕ1 , ϕ2 ) a χ-semistable chain of type (2, 1, 1, d0, d1 , d2 ). Then: i) The vector bundle E0 is given as a non-split extension ϕ1
{0} −−−−−→ E1 −−−−−→ E0 −−−−−→ Q0 −−−−−→ {0}. ii) One has
^ 4 dim. Ext1 (Q0 , E1 ) = d0 − 2d1 + g − 1.
Proof. Let χB ∈ LIII be a stability parameter. By Proposition 2.5.6.20, ii), we know that C is also χB -semistable. Therefore, Im(ϕ1 ) is a subbundle of E0 , by Remark 2.5.6.17, v). Hence, E0 is an extension of a line bundle Q0 of degree d0 −d1 by E1 . This extension is non-split, because otherwise C = (Im(ϕ1 ), E1 , E2 , ϕ1 , ϕ2 ) - (Q0 , 0, 0, 0, 0) could not be χ-semistable. This settles i). We have Ext1 (Q0 , E1 ) = H 1 (Q∨0 * E1 ). Now, H 0 (Q∨0 * E1 ) = {0}, because deg(Q∨0 * E1 ) = 2d1 −d0 < 0. Indeed, our assumption implies that RB (d) has a non-empty interior. By Remark 2.5.6.17, ii), this is equivalent to 2d1 < d0 . Part ii) is thus a mere restatement of the theorem of Riemann–Roch. " Exercise 2.5.6.25. Let E2 be a line bundle of degree d2 , D an effective divisor of degree d1 − d2 , E1 := E2 (D), and ϕ2 : E2 −→ E2 (D) the homomorphism defined by D. Assume that Q0 is a line bundle of degree d0 − d1 and that ϕ1
{0} −−−−−→ E1 −−−−−→ E0 −−−−−→ Q0 −−−−−→ {0} is a non-split extension. Then, (E0 , E1 , E2 , ϕ1 , ϕ2 ) is χ-stable for any stability parameter χ ∈ C∞ .
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By Exercise 2.5.6.22, the notions of χ-stability and χ-semistability are equivalent for any stability parameter χ in the chamber C∞ . By Proposition 2.5.6.24 and Exercise 2.5.6.25, given χ ∈ C∞ , a χ-stable holomorphic 3-chain of type (2, 1, 1, d0, d1 , d2 ) is specified by a line bundle E0 of degree d2 , an effective divisor D of degree d1 − d2 , a line bundle Q0 of degree d0 − d1 , and an extension class [e] ∈ !(Ext1 (Q0 , E2 (D))∨ ). Setting N := d0 −2d1 +g −2, one may conclude that the moduli space R χ is isomorphic to a !N -bundle over the product Jacd2 x X (d1 −d2 ) x Jacd0 −d1 . Here, X (m) is the m-th symmetric product of the curve X, m > 0 (see [151], §3). In particular, R χ is a smooth projective variety of dimension d0 − d1 − d2 + 3g − 2. Remark 2.5.6.26. This example shows that the region R(2, 1, 1, d0, d1 , d2 ) really spreads out to infinity. By Exercise 1.5.1.30 in Chapter 1, the only stability parameters for which there exist semistable representations with dimension vector (2, 1, 1) in the category of '(X)-vector spaces are χλ = λ · (−1, 1), λ ∈ 1≥0 . According to the proposition in Exercise 2.5.6.15, the region R(2, 1, 1, d0, d1 , d2 ) may stretch to infinity only along lines of the form χ0 + λ · (−1, 1), λ ∈ 1≥0 . Our example nicely illustrates these findings. For the application to the topology of the moduli space of Higgs bundles, one has B to study the moduli space R χ to the parameter χB = (2g − 2, 4g − 4). One can show that this moduli space is birationally equivalent to R χ ([2], Proposition 6.9). Moreover, one can easily describe the changes which the moduli space undergoes when crossing a wall. Now, we have good control over the topology of R χ and we understand how χ χB the topology changes, if we pass from R to R . Finally, we will be able to determine B the topology of R χ . This application will be made precise in a forthcoming paper ´ of the author with L. Alvarez-C´ onsul, O. Garc´ıa-Prada, J. Heinloth, and M. Logares. The argument which we have just sketched is, of course, an adaptation of Thaddeus’s method as outlined in Section 2.3.7. All the material in this section has been taken from the paper [2]. That paper contains a state-of-the-art account on the moduli spaces of holomorphic chains.
2.6 Principal Bundles as Tumps n this section, we extend the formalism from Section 2.4 in order to handle arbitrary—even non-connected—reductive groups. Again, we build our theory on a faithful representation κ: G −→ GL(W) of the structure group G, such that κ(G) ⊆ SL(W). Such a representation clearly exists: By Chapter * Then, 1, Theorem 1.1.3.3, we may find a faithful representation * κ: G −→ GL(W). −1 * - ', is a faithful representation with κ := * κ - (det ◦* κ): G −→ GL(W), W := W image in SL(W). Since we have already constructed the moduli spaces for semistable principal G-bundles with connected reductive structure group in Section 2.4.8, we omit this application of the techniques of this section. The reader may consult [80] for the full account.
"
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258
2.6.1 Principal Bundles and Associated Tuples of Vector Bundles Suppose the G-module W decomposes as a direct sum W1 - · · · - Wt of G-modules. Then, ^ 4 κ(G) ⊆ GL(W1 ) x · · · x GL(Wt ) ∩ SL(W). Assume further that the radical R(G) of G maps to the center of GL(W1 ) x · · · x GL(Wt ). This may be achieved as follows: The radical R(G) is a torus ([30], 11.21, Proposition, p. 158), so that κ|R(G) may be diagonalized, by Chapter 1, Example 1.1.2.5. Therefore, W = W1 - · · · - Wt as R(G)-module. The Wi are non-trivial eigenspaces for different characters of R(G). Say Wi is the eigenspace to the character χi , i = 1, . . . , t. Let i ∈ { 1, . . . , t }, v ∈ Wi , and g ∈ G. For any element t ∈ R(G), we have t · g · v = g · t · v = g · (χi (t) · v) = χi (t) · (g · v), because R(G) lies in the center of G ([30], 11.21). This equality shows that g · v lies in Wi , too. The above decomposition of W is therefore indeed a decomposition of W as a G-module. Remark 2.6.1.1. We split the representation, so that we may construct principal Gbundles from principal (GL(W1 ) x · · · x GL(Wt ))-bundles, i.e., tuples of vector bundles, by reducing the structure group. This means that we use the representation κB : G −→ GL(W1 ) x · · · x GL(Wt ) rather than κ. The reason behind this is that κB maps the radical of G to the center of GL(W1 ) x · · · x GL(Wt ). In the sequel, we will use the following abbreviations: W := (W1 , . . . , Wt ), and E = (E1 , . . . , Et ) stands for a tuple of vector bundles, such that rk(Ei ) = dim(Wi ), i = 1, . . . , t, and det(E) ! OX , E := E1 - · · · - Et , H (E, W) I (E, W)
:= :=
H om(W1 * OX , E1 ) x · · · x H om(Wt * OX , Et ) X X ^ 4 . ∨ S pec S ym (W1 * E1 - · · · - Wt * Et∨ )
:=
I som(W1 * OX , E1 ) x · · · x I som(Wt * OX , Et ). X
X
The constructions and arguments of Section 2.4.1 can be easily adapted and extended to our new setting. So, a principal G-bundle P yields a tuple E = (E1 , . . . , Et ) of vector bundles with rk(Ei ) = dim(Wi ), i = 1, . . . , t, and det(E) ! OX together with a section σ: X −→ I (E, W)/G. Conversely, given a pair (E, σ) which consists of a tuple of vector bundles as above and a section σ: X −→ I (E, W)/G, we construct the principal G-bundle P(E, σ) from the cartesian diagram 7 I (W, E) P(E, σ) 3 X Lemma 2.4.1.4 generalizes to:
G- bundle
σ
3 7 I (W, E)/G.
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259
Proposition 2.6.1.2. The groupoid of principal G-bundles with isomorphisms is equivalent to the groupoid of tuples (E, σ: X −→ I (W, E)/G) with isomorphisms (defined in the obvious way). By a slight abuse of language, we will refer to a pair (E, σ: X −→ I (W, E)/G) as a principal G-bundle.
2.6.2 The Relevant GIT-Quotients In Section 2.4, we had to study some GIT problems over the field of complex numbers (see Section 2.4.2), i.e., the case where the base variety is a point rather than a curve, in order to solve the moduli problem of principal bundles. Here, the analogous analysis is necessary. Let us write I(W) H(W)
:= :=
Isom(W, 'r1 ) x · · · x Isom(W, 'rt ) and Hom(W, 'r1 ) - · · · - Hom(W, 'rt ), ri := dim(Wi ), i = 1, . . . , t.
We find the commutative diagram I(W) 7
7 H(W)
3 I(W)/G 7
3 7 H(W)//G.
(2.54)
Indeed, on H(W), we have the universal homomorphism
$r , H: W * OH(W) −→ OH(W)
r = r1 + · · · + rt .
Upon the choice of bases for the Wi , i = 1, . . . , t, the determinant of H is a G-invariant function on H(W). (Here, we use κ(G) ⊆ SL(W).) It descends to a function D on H(W)//G. Then, F L I(W)/G = h ∈ H(W)//G | D(h) % 0 . Let G be a reductive linear algebraic group. We fix a representation *: G −→ GL(W) on the finite dimensional k-vector space W with *(G) ⊆ (GL(W1 ) x · · · x GL(Wt )) ∩ SL(W). Set ri := dim(Wi ), *i : G −→ GL(Wi ), i = 1, . . . , t. To use previous notation, we set I := { 1, . . . , t } and r = (ri := dim(Wi ), i ∈ I). There is the canonical representation ^ 4 R: GL(I, r) x G −→ GL H(W) , GL(I, r) := X GLri ('). i∈I
The representation R provides actions of GL(I, r) x G on H(W) and !(H(W)∨ ) and induces GL(I, r)-actions on the categorical quotients
/(W) := H(W)//G
and
^
4
^
4
/(W) := ! H(W)∨ //G Ex. 1.5.3.3 = /(W) \ {0} //'. .
260 S 2.6: P B
S 2.6: P B 260
Here, the '. -action on /(W) is induced by R and the embedding '. '→ GL(I, r), z 1−→ (z · $r1 , . . . , z · $rt ). The coordinate algebra of the affine variety /(W) is Sym. (W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ )G .25 For s > 0, we set
,s :=
%0 , s
i=1
0i :=
i
!
&
4G Z∨ ^ , Symi W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨
i ≥ 0.
s If s is so large that i=0 Symi (W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ )G contains a set of generators for the algebra Sym. (W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ )G , there is the GL(I, r)-equivariant surjection of algebras
Sym.
^
,∨s
4
4G ^ −→ Sym. W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ ,
and, thus, the GL(I, r)-equivariant embedding ι s : /(W) '→ , s . Set SL(I, r) := GL(I, r) ∩ SLr ('), r := r1 + · · · + rt , and -(W) := I(W)/G. Lemma 2.6.2.1. i) Every point ι s (i), i ∈ -(W), is SL(I, r)-polystable. ii) A point ι s (h), h ∈ /(W) \ -(W), is not SL(I, r)-semistable. Proof. Fix a basis for Wi , i = 1, . . . , t. We then get the (SL(I, r) x G)-invariant function H(W) ( f1 , . . . , ft )
−→
'
1−→ det( f1 - · · · - ft ).
From here on, the proof runs exactly like the proof of Lemma 2.4.2.3 and is, therefore, left to the reader. " An important consequence is the following: Proposition 2.6.2.2. Let i ∈ I(W)/G be the class of a tuple of isomorphisms ϕi : Wi −→ 'ri , i = 1, . . . , t, t
ψ: X GL(Wi ) −→ GL(I, r) i=1
the induced isomorphism of algebraic groups, and ^t 4 ϕ: X GL(Wi ) /G −→ I(W)/G = -(W). i=1
the resulting ψ-equivariant isomorphism with ϕ([e]) = i. Then, for x := ι s (i) and a one parameter subgroup λ: '. −→ SL(I, r), the following conditions are equivalent: i) µκs (x, λ) = 0, κ s being the representation of SL(I, r) on , s . ii) There is a one parameter subgroup λB : '. −→ g · G · g−1 with 4 ^ 4 ^ Wi• (λi ), γ• (λi ) = Wi• (λBi ), γ• (λBi ) , i = 1, . . . , t. 25 We
have marked the dual in order to indicate the GL(I, r)-module structure.
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261
(Here, λ = (λ1 , . . . , λt ), λB = (λB1 , . . . , λBt ), λi , λBi : '. −→ GL(Wi ), Wi := 1, . . . , t.)
'r , i i
=
Proof. The proof is a straightforward generalization of the proof for the case t = 1, i.e., Proposition 2.4.2.4. " Next, we look at the categorical quotient & ^ 4 Z /(W) = Proj Sym. W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ G . For any positive integer d, we define 4G ^ Sym(d) W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ := ∞ ^ 4G Symid W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ . :=
% i=0
Then,
& ^ 4G Z Proj Sym. W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ ! & 4G Z ^ ! Proj Sym(d) W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ .
We can choose s, such that a) Sym. (W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ )G is generated by elements in degree ≤ s. b) Sym(s!) (W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨ )G is generated by elements in degree 1, i.e., by the elements in the vector space , s := Syms! (W1 *. ('r1 )∨ - · · · Wt *. ('rt )∨ )G . There is the natural surjection
^ 4G Sym. (, s ) −→ Sym(s!) W1 *. ('r1 )∨ - · · · - Wt *. ('rt )∨
which leads to the closed and GL(I, r)-equivariant embedding ιs : /(W) '→ !(, s ). We also define
^ 4 O"(W) (s!) := ι.s O(($s ) (1) .
Note that
^ 4 O"(W) (s + 1)! = O"(W) (s!)#(s+1) .
(2.55)
Lemma 2.6.2.3. For a positive integer s, such that a) and b) as above are satisfied, a G-semistable point f ∈ H(W), and a one parameter subgroup λ: '. −→ SL(I, r), one has µκs (h, λ) > (= / (= / 0 , and ε ∈ 3>0 . Set κ = (κi := 1, i = 1, . . . , t).26 Recall that we found in Section 2.6.3 integers a > 0, b > 0, and c ≥ 0, such that we can associate to every pseudo G-bundle (E, τ) a *a,b,c -tump (E, ϕτ ) with ϕτ : (E #a )$b * det(E)#−c −→ OX . A decorated pseudo G-bundle (E, τ, L, ϕ) is (χ, δ, ε)-(semi)stable, if Mκ,χ (E• , α• ) + δ · µ(E• , α• , ϕ) + ε · µ(E• , α• , ϕτ )(≥)0 holds for every weighted filtration (E• , α• ) of the I-split vector bundle (Ei , i ∈ I). The Moduli Functors Fix the integer l ∈ (, and choose a Poincar´e bundle L on Jacl x X. A family of pseudo G-bundles with a * *-decoration of type (d, l) parameterized by S is a tuple (E S , τS , κS , NS , ϕS ) where (E S , τS ) is a family of pseudo G-bundles on S x X and (E S , κS , NS , ϕS ) is a family of * *-tumps of type (d, l). The families (E 1S , τ1S , κ1S , NS1 , ϕ1S ) and 2 2 2 2 2 (E S , τS , κS , NS , ϕS ) are isomorphic, if κ1S = κ2S =: κS and there are isomorphisms 1 2 ψi,S : Ei,S −→ Ei,S , i = 1, . . . , t, and χS : NS1 −→ NS2 , such that the ψi,S , i = 1, . . . , t, yield an isomorphism between the families (E 1S , τ1S ) and (E 2S , τ2S ) of pseudo G-bundles and the ψi,S , i = 1, . . . , t, together with χS yield an isomorphism between the families (E 1S , κ1S , NS1 , ϕ1S ) and (E 2S , κ2S , NS2 , ϕ2S ) of * *-tumps. We derive the moduli functors *, d, l): Sch. M(χ,δ,ε)-(s)s (* S
−→ Set Isomorphy classes of families of (χ, δ, ε)-(semi)stable pseudo G-bundles 1−→ with a * *-decoration of type (d, l) parameterized by S
.
Associated Tumps Fix positive integers a1 and a2 . Recall that we have already fixed the homogeneous representations 4 ^ * *: GL(W) −→ GL(V) and *a,b,c : GL(W) −→ GL V(κ, r)a,b,c . Then, we may form the homogeneous representation % * := * *#a1
*
a2 % *# a,b,c : GL(W) −→ GL(V),
% V := V #a1
*
a2 V(κ, r)# a,b,c .
26 In the setting of tumps, the parameter κ reflected the choice of a faithful representation of GLr1 (") x · · · x GLrt ("). Here, we have the faithful representation κ as the parameter, so that it is reasonable to define κ as we did it.
272 S 2.7: D P B S 2.7: D P B 272 Applying this construction, we assign to a decorated pseudo G-bundle (E, τ, L, ϕ) the % *-tump (E, L#a1 , % ϕ) with #a1 * ^(E #a )$b * det(E)#−c 4#a2 −→ L#a1 . a2 % ϕ := ϕ#a1 * ϕ# * = E* τ : E% * Remark 2.7.2.1. Let (E, τ, L, ϕ) be a decorated pseudo G-bundle. The associated decorations ϕ: E** −→ L and ϕτ : Ea,b,c −→ OX , Ea,b,c := (E #a )$b * det(E)#−c define, by restriction to the generic point of X, points σ1 ∈ !(,** )
and σ2 ∈ !(,a,b,c ).
As usual, the use of the letter “,” indicates the restriction of the respective vector bundle to the generic point of X. We use the a1 th and the a2 th Veronese embedding and the Segre embedding in order to define the morphism
#a2 ) '→ !(,#a1 va1 ,a2 : !(,** ) x !(,a,b,c ) '→ !(,**#a1 ) x !(,a,b,c * *
*
#a ). ,a,b,c 2
The point va1 ,a2 (σ1 , σ2 ) is just the point defined by the restriction of the associated decoration % ϕ: E%* −→ L#a1 to the generic point of X. This observation has two consequences: ϕ) the associProposition 2.7.2.2. Let (E, τ, L, ϕ) be a pseudo G-bundle and (E, L#a1 , % ated % *-tump. Then, µ(E• , α• , % ϕ) = a1 · µ(E• , α• , ϕ) + a2 · µ(E• , α• , ϕτ ) for every weighted filtration (E• , α• ) of the I-split vector bundle (Ei , i ∈ I). Proof. This is immediate from the definitions and Remark 2.7.2.1.
"
*Now, suppose we are interested in (χ, δ, ε)-semistable pseudo G-bundles with a * decoration of type (d, l) for given stability parameters χ ∈ 3$t , δ, ε ∈ 3>0 . Then, we may clearly find % δ ∈ 3>0 and a1 , a2 ∈ (>0 , satisfying δ % δ= a1
and a2 =
ε ε · a1 = . % δ δ
(2.60)
Corollary 2.7.2.3. Assuming Relation 2.60, a pseudo G-bundle (E, τ, L, ϕ) with a * *decoration of type (d, l) is (χ, δ, ε)-(semi)stable, if and only if the associated % *-tump ϕ) of type (d, l) is (κ, χ, % δ)-(semi)stable. (E, L#a1 , % Thus, there is a natural transformation *, d, l) −→ M(κ,χ,δ)-(s)s (% *, d, l), NT: M(χ,δ,ε)-(s)s (* %
such that NT(Spec(')) is an injection. Proof. The assertion about semistability and the existence of the natural transformation is a direct consequence of Proposition 2.7.2.2. The assertion on NT(Spec(')) results from Proposition 2.6.3.2. "
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273
S-Equivalence and the Main Technical Result Fix the stability parameters χ and δ, ε ∈ 3>0 . We say that the pseudo G-bundles *-decoration of type (d, l) are S-equivalent, if (E 1 , τ1 , L1 , ϕ1 ) and (E 2 , τ2 , L2 , ϕ2 ) with a * a1 a1 , % ϕ ) and (E 2 , L# ϕ2 ) are S-equivalent. the associated % *-tumps (E 1 , L# 1 1 2 ,%
Theorem 2.7.2.4. Fix the topological data d ∈ ($t , l ∈ (, as well as the stability *, d, l) for the parameters χ ∈ 3$t , and δ, ε ∈ 3>0 . Then, the moduli space M (χ,δ,ε)-(s)s (* (χ,δ,ε)-(s)s (* *, d, l) exists as a projective scheme. functors M Asymptotic Behavior
Theorem 2.7.2.5. Fix the type (d, l) and the stability parameters χ and δ. Then, there exists a rational numbers ε∞ , such that for every rational number ε > ε∞ and every *-decoration of type (d, l) the following conditions pseudo G-bundle (E, τ, L, ϕ) with a * are equivalent: 1. (E, τ, L, ϕ) is (χ, δ, ε)-(semi)stable. 2.
(a) µ(E• , α• , ϕτ ) ≥ 0 holds for every weighted filtration (E• , α• ) of the I-split vector bundle (Ei , i ∈ I). (b)
Mκ,χ (E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0 holds for every weighted filtration (E• , α• ) with µ(E• , α• , ϕτ ) = 0.
Proof. To show “1.=⇒2.”, we proceed as in the proof of Theorem 2.5.5.2, i.e., assuming that (a) fails, we construct a weighted filtration (E• , α• ) with µ(E• , α• , ϕτ ) < 0. Again Chapter 1, Remark 1.7.2.3, shows that the tuple (rk(E ij ), j = 1, . . . , si , i ∈ I, α• ) belongs to a finite set, depending only on r, a, b, and c. This means that we can find a constant D, also depending only on r, a, b, and c, such that µ(E• , α• , ϕ) ≤ D. Thus, the same proof as for Theorem 2.5.5.2 works. The converse direction is proved exactly in the same manner as Corollary 2.5.5.4. " Recall from Exercise 2.6.3.3 that Condition (a) above means that P = (E, τ) is a principal G-bundle. Let (E• , α• ) be a weighted filtration of the I-split vector bundle (Ei , i ∈ I). This weighted filtration also provides the filtration E•total : {0} ! E1total ! · · · ! E total ! E. s We know from Proposition 2.6.2.2 that the following conditions are equivalent: 1. µ(E• , α• , ϕτ ) = 0.
274 S 2.7: D P B S 2.7: D P B 274 2. There are a one parameter subgroup λ: '. −→ G and a reduction β: X −→ P/QG (λ), such that ^ 4 (E•total , α• ) = E• (β), α• (β) . . (G) be a rational character of G. Set T G := R(G). The restriction of Next, let χ ∈ X% . . characters from G to T G yields the isomorphism X% (G) ! X% (T G ). By our construction, T G is a closed subgroup of F L T := (z1 · idW1 , . . . , zt · idWt ) | zi ∈ '. , i = 1, . . . , t ∩ SL(W).
A rational character x of T is specified by a tuple (x1 , . . . , xt ) of rational numbers with x1 + · · · + xt = 0. Now, the rational character χ of T G is induced by a rational character x of T . This character, in turn, is induced by a character of GL(I, r). In our former notation, the latter character is xi χ = (χi , i ∈ I) with χi = , i = 1, . . . , t. ri Applying (2.43) and Remark 2.5.3.5, i), we see that, for a weighted filtration (E• , α• ) of the I-split vector bundle (Ei , i ∈ I) with µ(E• , α• , ϕτ ) = 0, ^ 4 (2.61) Mκ,χ (E• , α• ) = M E• (β), α• (β) + Fλ, χ4. Setting
κ
* *
*B : G −→ GL(W) −→ GL(V), we may summarize our efforts as follows: Corollary 2.7.2.6. Fix the type (d, l) and the stability parameters χ and δ. Then, there exists a rational numbers ε∞ , such that for every rational number ε > ε∞ and every *-decoration of type (d, l) the following conditions pseudo G-bundle (E, τ, L, ϕ) with a * are equivalent: 1. (E, τ, L, ϕ) is a (χ, δ, ε)-(semi)stable decorated pseudo G-bundle of type (d, l). 2. (E, τ, L, ϕ) is a (χ, δ)-(semi)stable *B -bump of type (ϑ, l), where ϑ corresponds to one of the topological principal G-bundles whose associated tuple of topological vector bundles features the degrees d1 ,. . . ,dt . Exercise 2.7.2.7. Infer Theorem 2.7.1.4. More on Asymptotic Behavior Another important application of Proposition 2.7.2.2 concerns the asymptotic behavior of semistability when we make both δ and ε very large. This, in turn, has an important application to δ-semistability of *-bumps when δ becomes very large. Here is our set-up: We fix a1 and a2 with a2 /a1 > max{ η∞ , η.∞ } (see Chapter 1, Proposition 1.7.3.1 and Exercise 1.7.3.2). The stability parameters χB and χBB ∈ 3$t are fixed, and we look at the stability parameters χΔ := χB + Δ · χBB
and (δΔ , εΔ ) := Δ · (a1 , a2 ),
Δ ∈ 3>0 .
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275
Proposition 2.7.2.8. Fix the topological data d ∈ ($t and l ∈ ( and the stability parameter χ ∈ 3$t . In the above situation, there is a value Δ∞ , such that for every *-decoration of type (d, l), the Δ > Δ∞ and every pseudo G-bundle (E, τ, L, ϕ) with a * following conditions are equivalent: 1. (E, τ, L, ϕ) is (χΔ , δΔ , εΔ )-(semi)stable. 2. The following conditions are verified: (a) The inequality µ(E• , α• , ϕτ ) ≥ 0 is satisfied for every weighted filtration (E• , α• ) of E; (b) The inequality µ(E• , α• , ϕ) − rk(E) ·
s T
αi ·
t ^T j=1
i=1
4 j χBBj · rk(Ei ) ≥ 0
holds for every weighted filtration (E• , α• ) of E with µ(E• , α• , ϕτ ) = 0; (c) One has
Mκ,χB (E• , α• )(≥)0
for every weighted filtration (E• , α• ) of E with µ(E• , α• , ϕτ ) = 0 = µ(E• , α• , ϕ) − rk(E) ·
s T
αi ·
i=1
t ^T j=1
4 χBBj · rk(Eij ) .
Note that (a) in the second statement implies that (E, τ) is a principal G-bundle (Ex*-bump. The cunning reader will already recognize ercise 2.6.3.3), i.e., (E, τ, L, ϕ) is a * that (a) - (c) is just the version of asymptotic (semi)stability which we would expect for * *-bumps. We will spell this out below (Proposition 2.7.3.4). Proof. We use Corollary 2.7.2.3. This means that (E, τ, L, ϕ) is (χΔ , δΔ , εΔ )-(semi)stable, if and only if the associated % *-tump (E, L#a1 , % ϕ) is (κ, χΔ , Δ)-(semi)stable. Next, we apply Theorem 2.5.5.2. The condition that µ(E• , α• , % ϕ) ≥ 0 for every weighted filtration (E• , α• ) of E is equivalent to (a) and (b), by Proposition 1.7.3.1 in Chapter 1, Remark 2.7.2.1, and Proposition 2.7.2.2. Finally, assuming (a) and (b), the condition that µ(E• , α• , % ϕ) = 0 is equivalent to the property that µ(E• , α• , ϕτ ) = 0 = µ(E• , α• , ϕ) − rk(E) ·
s T i=1
αi ·
t ^T j=1
4 j χBBj · rk(Ei ) ,
by Chapter 1, Exercise 1.7.3.2. Thus, our assertion is, in fact, only a restatement of Theorem 2.5.5.2. "
276 S 2.7: D P B S 2.7: D P B 276 Proof of Theorem 2.7.2.4 Recall that we fix the type (d, l) and the stability parameters χ, δ, and ε. We also choose positive integers a1 and a2 and a positive rational number % δ, such that Equation 2.60 is verified. Having a1 and a2 at hand, we can associate to any pseudo G-bundle (E, τ, L, ϕ) ϕ) of type (d, l). Corollary 2.7.2.3 *-tump (E, L#a1 , % with a * *-decoration of type (d, l) the % and 2.5.4.3 now imply the boundedness of the family of (χ, δ, ε)-semistable pseudo Gbundles with a * *-decoration of type (d, l). Therefore, we may find an n = 0, such that for every (χ, δ, ε)-semistable pseudo G-bundle with a * *-decoration of type (d, l) and for every index i ∈ { 1, . . . , t }, it is true that • Ei (n) is globally generated and • H 1 (Ei (n)) = {0}. Furthermore, we can choose n so large that all the computations and proofs in Section 2.5.4 go through with this n. Let Yi be a complex vector space of dimension di + ri (n + 1 − g), and let Q0i be the quasi projective quot scheme that classifies quotients q: Yi * OX (−n) −→ Ei where Ei is a vector bundle of rank ri and degree di and H 0 (q(n)) is an isomorphism (Theorem 2.2.3.5), i = 1, . . . , t. As before, we define t
QB := X Q0i . i=1
Now, we leave it as an exercise to the reader to mimic the construction in Section 2.4.5 and find a scheme πB : Y −→ QB , a universal family (E Y , τY ) of pseudo G-bundles parameterized by Y, and a group action 4 ^t Γ B : X GL(Yi ) x Y −→ Y, i=1
B
such that π is equivariant and the following properties are verified: Proposition 2.7.2.9. For every pseudo G-bundle (E, τ) occurring in a (χ, δ, ε)-semistable decorated pseudo G-bundle is a point y ∈ Y with ^ 4 (E, τ) ! E Y|{y} x X , τY|{y} x X . Proposition 2.7.2.10. Two points y1 and y2 in Y belong to the same (Xti=1 GL(Yi ))orbit, if and only if the pseudo G-bundles (E Y|{y1 } x X , τY|{y1 } x X ) and (E Y|{y2 } x X , τY|{y2 } x X ) are isomorphic. By Exercise 2.5.4.4, there are also a scheme πBB : T −→ QB ,
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277
*-tumps of type (d, l) parameterized by T, and a a universal family (E T , κT , NT , ϕT ) of * group action 4 ^t Γ BB : X GL(Yi ) x T −→ T. i=1
BB
The morphism π is
(Xti=1
GL(Yi ))-equivariant, and one knows the following:
Proposition 2.7.2.11. Given a (χ, δ, ε)-semistable decorated pseudo G-bundle (E, τ, L, ϕ), there is a point u ∈ T, such that the * *-tump (E, L, ϕ) is isomorphic to the restriction of the universal family to {u} x X. Proposition 2.7.2.12. Let u1 and u2 be two points of the scheme T. Then, the restriction of the universal family to the fiber {u1 } x X will be isomorphic to the restriction of the universal family to the fiber {u2 } x X, if and only if there is a g ∈ Xti=1 GL(Yi ) with u1 = Γ BB (g, u2 ). We form the scheme
π: B := Y xB T −→ QB . Q
By construction, • there is a family (E B , τB , κB , NB , ϕB ) of pseudo G-bundles with a * *-decoration of type (d, l) parameterized by B, the universal family of pseudo G-bundles with a* *-decoration of type (d, l), such that the analog to Proposition 2.7.2.9 and 2.7.2.11 holds, • there is a group action &t Z Γ: X GL(Yi ) x B −→ B, i=1
such that π is equivariant and the analog to Proposition 2.7.2.10 and 2.7.2.12 is satisfied. Given these facts, it remains to prove that • the subsets B(s)s parameterizing the (χ, δ, ε)-(semi)stable decorated pseudo Gbundles are open and (Xti=1 GL(Yi ))-invariant, • the categorical quotient Bss //(Xti=1 GL(Yi )) exists as a projective scheme and contains the quotient Bs //(Xti=1 GL(Yi )) as an open subscheme, • the quotient Bs //(Xti=1 GL(Yi )) is also an orbit space. These properties will be established by a familiar method. First, we introduce the surjection ^t 4 ('. )t x X SL(Yi ) −→ i=1
(z1 , . . . , zt ; m1 , . . . , mt )
t
X GL(Yi )
i=1
1−→ (z1 · m1 , . . . ., zt · mt ).
278 S 2.7: D P B S 2.7: D P B 278 The quotient B := B//('. )t exists and inherits a projective morphism π: B −→ QB . The group Xti=1 SL(Yi ) still acts on B, and π is equivariant. On the other hand, we have also a parameter scheme % π: % T −→ QB for % *-bumps of type (d, l). We may assign to the universal family (E B , τB , κB , NB , ϕB ) of decorated pseudo G-bundles a family (E B ,% κB , NN ϕB ) of % *-tumps of type (d, l) B, % parameterized by B (Corollary 2.7.2.3). This morphism gives rise to a (Xti=1 GL(Yi ))equivariant morphism f : B −→ % T. The morphism f factorizes over B, so that we obtain the morphism f : B −→ % T. This morphism fits into the commutative diagram f
7% B 11 $T 11 π $ % π $$ 11 11 $$ $ 2 $$ QB . Since π is a projective morphism, f must be a projective morphism, too ([96], Corollary III.4.8, (e)). By Corollary 2.7.2.3, f is also injective, so that it is a finite morphism ([96], Exercise III.11.2). From the proofs of Section 2.5.4, we know that there are open and (Xti=1 SL(Yi ))invariant subsets % T(s)s which parameterize the (κ, χ, % δ)-(semi)stable % *-tumps, such that • % Tss //(Xti=1 SL(Yi )) exists as a projective scheme, • % Ts //(Xti=1 SL(Yi )) exists as an open subscheme of % Tss //(Xti=1 SL(Yi )) and is an orbit space. Using Chapter 1, Exercise 1.4.3.11, we see that the quotients f
−1 ^ (s)s 4 %
T
4 ^t // X SL(Yi ) i=1
−1 ss T )//(Xti=1 SL(Yi )) is projective and contains the scheme exist, that the scheme f (% −1 f (% Ts )//(Xt SL(Yi )) as an open subscheme. The latter quotient is also an orbit space, i=1
because f is injective. To conclude, let p: B −→ B be the quotient morphism. Corollary 2.7.2.3 implies ^ −1 4 B(s)s = p−1 f (% T(s)s ) .
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279
Therefore, according to Chapter 1, Exercise 1.5.3.3, we also have the quotients ^t 4 B(s)s // X GL(Yi ) i=1
4 ^ t = B(s)s // ('. )t x X SL(Yi ) i=1 ^ 4 t = B(s)s //('. )t // X SL(Yi ) =
^
i=1
−1
%(s)s
f (T
4 t ) // X SL(Yi ). i=1
Since p: B −→ B is a geometric quotient, Bs //(Xti=1 GL(Yi )) is also an orbit space. We now set 4 ^t M (χ,δ,ε)-(s)s (* *, d, l) := B(s)s // X GL(Yi ) . i=1
As in our former constructions, the universal properties of the parameter spaces and the categorical quotients imply the universal properties of the moduli space. "
2.7.3 Asymptotic Semistability Let κ: G −→ GL(W) be a faithful representation of G and *: G −→ GL(V) a homogeneous representation. We fix rational characters χB and χBB of G. A *-bump (P, L, ϕ) is said to be (χB , χBB )-asymptotically (semi)stable, if a) µ(β, ϕ) + F λ, χBB 4 ≥ 0 holds for every one parameter subgroup λ: '. −→ G and every reduction β: X −→ P/QG (λ) of P to λ, and ^ 4 b) M E• (β), α• (β) + F λ, χB 4(≥)0 is satisfied for every one parameter subgroup λ: '. −→ G and every reduction β: X −→ P/QG (λ) of P to λ with µ(β, ϕ) + F λ, χBB 4 = 0. Remark 2.7.3.1. For simplicity, we will again assume that χBB is a character of G, that is, a homomorphism χBB : G −→ '. of algebraic groups. Let (P, L, ϕ) be a *-bump. Then, the bundle G −→ X of local automorphisms of the principal G-bundle P is a reductive group scheme over X, and there is the action αX : G x !(P* ) −→ !(P* ). X
We restrict everything to the generic point η of X. Set * := '(X), Gη := G|{η} , $ := E|{η} , and ,* := P*|{η} . Then, Gη is a reductive group over the non-algebraically closed field *. (If G is a non-connected group, e.g., a non-trivial finite group, then Gη need not be isomorphic to G xSpec(.) Spec(*).) Furthermore, there is an action of Gη '→ SL($) on !(,* ). This action is canonically linearized in O(($* ) (1). We modify this linearization by the character χBB and will work with the resulting linearization in the sequel. Let ϕη ∈ !(,* ) be the point defined by the restriction of ϕ to η. We claim that Condition a) is equivalent to the fact that ϕη is Gη -semistable. Recall from
280 S 2.7: D P B S 2.7: D P B 280 Theorem 1.7.1.1 of Chapter 1 that we may pass to the algebraic closure * of write η := Spec(*), Gη := Gη x {η} = G x{η}, Spec(0)
$ := $
*0
*, and ,* := ,*
*0
*.
We
X
*. The point ϕη provides us with a point ϕη ∈ !(,* ).
The fact that ϕη is Gη -semistable is then equivalent to the fact that ϕη is Gη -semistable. Now, assume that ϕη is Gη -semistable. Then, ϕη is Gη -semistable, and the assertion that µ(β, ϕ) + F λ, χBB 4 ≥ 0 holds for all one parameter subgroups λ and all reductions β to λ is a consequence of the definition of the number µ(β, ϕ) (see Exercise 2.7.1.2). Conversely, suppose that ϕη fails to be Gη -semistable, so that there is a one paλ0 ) < 0, by Kempf’s theory of the rameter subgroup * λ0 : 4m (*) −→ Gη with µ(ϕη , * instability flag (Chapter 1, Section 1.7.2). Set G0 := G xSpec(.) Spec(*). In Exercise 2.7.1.2, we have already introduced an isomorphism ι: Gη ! G0 which is defined up to an inner automorphism of Gη . Since any two maximal tori inside the group G0 are conjugate, we may find a one parameter subgroup λ: '. −→ G, such λ0 : 4(*) −→ Gη is the one parameter that * λ0 and λ0 are conjugate inside G0 . Here, * * subgroup obtained by base change from λ0 , and λ0 : 4(*) −→ Gη is the one parameter subgroup constructed by base change and application of ι−1 from λ. Next, we write Parλ := P/QG (λ). We need some general result on this variety. To formulate it, let us recall some notions from Algebraic Geometry: If Y is a scheme, then a geometric point of Y is a morphism ξ: Spec(K) −→ Y where K is an algebraically closed field extension of the ground field '. A group scheme over Y is a tuple (G , e, µ, inv) which consist of a scheme G −→ Y, a section e: Y −→ G , and morphisms µ: G xY G −→ G and inv: G −→ G over Y, such that the axioms of a group are satisfied (expressed in diagrams as in [30], Section 1.5, or [45], Chapter III, §2). Of course, we will write just G instead of the whole quadruple. We say that G is of type G, if every point y ∈ Y possesses an e´ tale neighborhood U, such that G xY U and G xSpec(.) U are isomorphic as group schemes over U. Recall that we have associated to any principal G-bundle P over Y a group scheme of type G over Y, namely the scheme of local G-bundle automorphisms of P. Let G be a group scheme over Y. A parabolic subgroup of G is a subscheme Q ⊂ G , such that Q xY Spec(K) is a parabolic subgroup of G xY Spec(K) for every geometric point ξ: Spec(K) −→ Y. Let P be a principal G-bundle over X which yields the group scheme G . As before, we have, for every geometric point ξ: Spec(K) −→ X, an isomorphism ιξ : G x Spec(K) −→ G K := G x Spec(K) X
Spec(.)
of algebraic groups over the field K. If Y is a scheme over X, we set GY := G xX Y. This is the bundle of local automorphisms of the principal G-bundle PY := P xX Y over Y. Let λ: '. −→ G be a one parameter subgroup. We say that a parabolic subgroup Q ⊂ GY is of type λ, if Q xY Spec(K) is, for every geometric point ξ: Spec(K) −→ Y,
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281
conjugate to QGSpec(K) (λK ). In order to make sense out of the last statement, we define the one parameter subgroup λK as the “preimage” of λ x idSpec(K) , using the isomorphism ιξB associated to the geometric point ξ
ξB : Spec(K) −→ Y −→ X. We may now introduce the functor Parλ : Schemes over X −→ Sets F L Y 1−→ Parabolic subgroups of type λ of GY . Although the formalism is quite heavy, the following is easy to check, using local trivializations and cocycles. The doubtful reader may consult [49], p. 443ff, for more details. Exercise 2.7.3.2. The scheme Parλ −→ X represents the functor Parλ . In this construction, one assigns to a section β: Y −→ PY /QG (λ) its stabilizer in GY . Applying this exercise, the parabolic subgroup QGη (λ) of Gη supplies the point βη ∈ Parλ xX Spec(*). Since P/QG (λ) −→ X is a projective morphism, βη extends to a section β: X −→ P/QG (λ). It is a straightforward task to check from the definitions that we have µ(β, ϕ) = µ(ϕη , * λ0 ), whence also
µ(β, ϕ) + F λ, χBB 4 = µ(ϕη , * λ0 , χBB 4. λ0 ) + F *
The latter quantity is negative, so that a) is indeed violated. Theorem 2.7.3.3. Fix the topological data ϑ ∈ Π(G) and l ∈ . parameters χB , χBB ∈ X% (G). Set χδ := χB + δ · χBB ,
( and the stability
δ ∈ 3>0 .
Then, there is a positive rational number δ∞ ∈ 3>0 , such that a *-bump (P, L, ϕ) of type (ϑ, l) is (χδ , δ)-(semi)stable for some stability parameter δ > δ∞ , if and only if it is (χB , χBB )-asymptotically (semi)stable. Proof. Let (P, L, ϕ) be a *-bump of type (ϑ, l) which is not (χB , χBB )-asymptotically (semi)stable. If Condition a) in the definition of asymptotic (semi)stability is verified but Condition b) fails, it is clear that (P, L, ϕ) is (χδ , δ)-unstable for all δ ∈ 3>0 . Hence, we may assume that Condition a) does fail. Then, the construction from Remark 2.7.3.1 provides us with a one parameter subgroup λ: '. −→ G and a reduction β: X −→ P/QG (λ), such that µ(β, ϕ) + F λ, χBB 4 < 0. As in the proof of Theorem 2.5.5.2, we will be done, if we can find a constant C which depends only on ϑ, l, and χ, such that ^ 4 M E• (β), α• (β) + F λ, χB 4 ≤ C. This is, in principle, achieved by the same argument as in Theorem 2.5.5.2 with the sole difference that we may not assume that the principal QG (λ)-bundle is trivial in
282 S 2.7: D P B S 2.7: D P B 282 the Zariski topology. There is a finite extension K of the function field '(X), such that the parabolic subgroups Qη xSpec(.(X)) Spec(K) and ι−1 ξ (QGK (λK )) are conjugate within the group GK , ξ: Spec(K) −→ X. Of course, the notation is defined in analogy to the notation which was used above for geometric points. Let Y be the normalization of X in the field K. Then, Y is a smooth projective curve with function field K which comes with a finite morphism c: Y −→ X. We pull back P and β to Y and obtain the principal G-bundle PY together with the reduction βY : Y −→ PY /QG (λ) to the one parameter subgroup λ. If EY stands for the pullback of E to Y, then (EY• (βY ), α• (βY ) := α• (β)) features the filtration ^ 4 EY• (βY ) = c. E• (β) : {0} ! c. (E1 ) ! · · · ! c. (E s ) ! c. (E) = EY . This implies
^ 4 ^ 4 M EY• (βY ), α• (βY ) = deg(c) · M E• (β), α• (β) .
Pulling back the principal QG (λ)-bundle PY −→ PY /QG (λ) to Y via βY gives a principal QG (λ)-bundle Q(βY ) which yields PY when extending the structure group via QG (λ) ⊂ G. The restriction of Q(βY ) to the generic point of Y is trivial, by our construction of Y. Therefore, we may find a non-empty open subset U of Y and a trivialization ψ: Q(βY )|U ! U x QG (λ). Let
ϕY : PY,* ! c. (P* ) −→ LY := c. (L)
be the pullback of ϕ. After shrinking U, we may assume that ϕY|U is surjective and thus yields the morphism ϕY|U
“ψ”
σ: U −→ !(P*|U ) !
!(V) x U −→ !(V).
Our construction implies that λ is the instability one parameter subgroup of σ(y) for some point y ∈ U. The finiteness result stated in Remark 1.7.2.3 of Chapter 1 allows us to find a constant C B with F λ, χB 4 ≤ C B . Moreover, Chapter 1, Proposition 1.7.2.5, yields a certain character χ. of QG (λ). Let Lχ. be the line bundle associated to PY via χ. . It is clear that we may adapt all the arguments from the proof of Theorem 2.5.5.2, once we know that ^ 4 deg(Lχ. ) = M EY• (βY ), α• (βY ) . To see this, we observe that we may assume that there is a representation * *: GL(W) −→ GL(V), such that * is a direct summand of * * ◦ κ. Since χ. extends to a character of QGL(W) (λ), the same computations as in the proof Theorem 2.5.5.2 remain valid. We are now done. " Proposition 2.7.3.4. As before, we fix ϑ, l, and χ. The set of isomorphy classes of (χB , χBB )-asymptotically semistable *-bumps of type (ϑ, l) is bounded.
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283
Proof. Let d = (d1 , . . . , dt ) be the tuple of integers with the property that, for any principal G-bundle P of topological type ϑ, the tuple E = (E1 , . . . , Et ) associated to P by means of the fixed representation * κ: G −→ (Xti=1 GL(Wi )) ∩ SL(W) satisfies deg(Ei ) = di , i = 1, . . . , t. Recall from Section 2.7.2, p. 273ff, that the characters χB and χBB can be encoded as tuples χB and χBB ∈ 3$t . We must only show that the set of isomorphy classes of (χB , χBB )-asymptotically semistable *-bumps of type (ϑ, l) is a subset of the set of isomorphy classes of pseudo G-bundles with a * *-decoration of type (d, l) that satisfy Condition (a) - (c) in Proposition 2.7.2.8. Let (E, τ, L, ϕ) be such a decorated pseudo G-bundle. As we have already remarked after Proposition 2.7.2.8, Condition (a) is equivalent to the fact that (E, τ) is a principal G-bundle. Invoking Proposition 2.6.3.4, Condition (b) translates into the condition that µ(E• (β), α• (β)) + F λ, χBB 4 ≥ 0 holds for every one parameter subgroup λ of G and every reduction β of P to λ, i.e., into Property a) of χ-asymptotic semistability. Finally, Equation 2.61 tells us that Condition (c) is the same as Property b) of (χB , χBB )-asymptotic semistability. This shows that (E, τ, L, ϕ) is actually a (χB , χBB )asymptotically semistable * *-bump. It is also clear that any (χB , χBB )-asymptotically semistable * *-bump of type (ϑ, l) can be obtained in this way. Since the *-bumps form a subfamily of the * *-bumps, this settles our claim. " Remark 2.7.3.5. i) In our applications, we will only need the existence of the moduli spaces of (χB , χBB )-asymptotically (semi)stable *-bumps of fixed type. As the arguments in the proof of Proposition 2.7.3.4 show, this result is already contained in Theorem 2.7.2.4. ii) Theorem 2.7.3.3 is not an immediate consequence of Proposition 2.7.2.8, because we cannot grant that εΔ is large enough, so that the conclusion of Theorem 2.7.2.5 holds for εΔ and χΔ . Corollary 2.7.3.6. In the situation of Theorem 2.7.3.3 and Proposition 2.7.3.4, there is a rational number δB∞ , such that, for any δ > δB∞ , a *-bump (P, L, ϕ) of type (ϑ, l) is (χδ , δ)-(semi)stable, if and only if it is (χB , χBB )-asymptotically (semi)stable.
2.7.4 Hitchin Pairs In this section, we will give an application of the results obtained so far to the case of the adjoint representation. Again, the strategy which we shall use is a model for the general strategy which will be employed in Section 2.8 to obtain the main result. We assume that the group G is connected. So, let Ad: G −→ GL(g) be the adjoint representation of G on its Lie algebra g. If P is a principal G-bundle, we write Ad(P) rather than PAd . We fix a line bundle N on X. A Hitchin pair (of type (ϑ, N)) is a triple (P, ϕ, ε) which consists of a principal Gbundle P of topological type ϑ ∈ Π(G), a section ϕ: OX −→ Ad(P) * N, and a complex number ε. We say that the Hitchin pair (P1 , ϕ1 , ε1 ) is isomorphic to the Hitchin pair (P2 , ϕ2 , ε2 ), if there exist a number z ∈ '. and an isomorphism ψ: P1 −→ P2 , such that ^ 4 ε2 = z · ε1 and ϕ2 = z · (Ad(ψ) * idN ) ◦ ϕ1 .
284 S 2.7: D P B S 2.7: D P B 284 Of course, Ad(ψ): Ad(P1 ) −→ Ad(P2 ) is the isomorphism which is induced by ψ. Remark 2.7.4.1. Let Z be the center of G, z its Lie algebra, and D := [G, G] the derived group with Lie algebra gB . Note that Z and D are normal subgroups of G and that the surjective homomorphism Z x D −→ G, (z, h) 1−→ z · h, has finite kernel ([30], Chapter IV, 14.2, Lemma). Therefore, we have g = z - gB and this is also a decomposition as G-module. Note that z is a trivial G-module and thus isomorphic to the G-module z∨ . The adjoint representation of the semisimple Lie algebra gB is the homomorphism ad: gB −→ End(gB ), x 1−→ (ad(x): y 1−→ [x, y]), of Lie algebras. Then, the Killing form is the bilinear map gB x gB −→ ', (x, y) 1−→ Trace(ad(x) ◦ ad(y)). It is known to be nondegenerate (see [69], §14.2, p. 207). It is obviously invariant under the G-action by the adjoint representation. Therefore, gB and gB ∨ are isomorphic G-modules, by Chapter 1, Exercise 1.1.2.2, ii). The same goes for g and g∨ . This implies that Ad(P) and Ad(P)∨ are isomorphic vector bundles. In particular, the section ϕ appearing in the definition of a Hitchin pair corresponds to a homomorphism ϕt : Ad(P) −→ N. Let (P, ϕ, ε) be a Hitchin pair and Q = R " L a parabolic subgroup of G with unipotent radical R and Levi subgroup L. The Lie algebra q ⊂ g of Q therefore decomposes as r - l with r := Lie(R) and l := Lie(L). Since R is a normal subgroup of Q, the adjoint representation Ad|Q : Q −→ GL(g) leaves q and r invariant. Now, suppose that β: X −→ P/Q is a section. This gives a principal Q-bundle Q(β) ⊂ P, such that P is obtained from Q(β) by extending the structure group via Q ⊂ G. Hence, we have the subbundle Ad(Q(β)) ⊂ Ad(P). Furthermore, the Q-action on r also gives the subbundle R(Ad(Q(β))) ⊂ Ad(Q(β)). Let P be a principal G-bundle. The section ϕ: OX −→ Ad(P) * N is called nilpotent, if there exist a parabolic subgroup Q of G and a reduction β: X −→ P/Q, such that 4 ^ ϕ ∈ H 0 X, R(Ad(Q(β))) * N . A Higgs reduction of (P, ϕ: OX −→ Ad(P) * N) is a reduction β: X −→ P/Q to a parabolic subgroup Q of G with ^ 4 ϕ ∈ H 0 X, Ad(Q(β)) * N . A Hitchin pair (P, ϕ, ε) is called (semi)stable, if 1. either ε % 0, or ϕ is not nilpotent, and 2. the inequality
^ 4 deg L (Q, χ, β) (≥)0
is satisfied for every parabolic subgroup Q of G, every anti-dominant character χ of Q, and every Higgs reduction β: X −→ P/Q of (P, ϕ) to Q. Next, we proceed as follows: Fix a line bundle L together with embeddings σ0 : N −→ L and σ1 : OX −→ L. Define * := Ad - #: G −→ GL(g - '). Then, we associate to every Hitchin pair (P, ϕ, ε) the projective *-bump (P, L, ψ) with ψ := (σ0 ◦ ϕt ) - (ε · σ1 ): Ad(P) - OX −→ L.
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285
Lemma 2.7.4.2. Assume that G is semisimple. Then, a Hitchin pair (P, L, ϕ) is (semi)stable, if and only if the associated projective *-bump (P, L, ψ) is asymptotically (semi)stable.27 Proof. Let λ: '. −→ G be a one parameter subgroup and QG (λ) = Ru (QG (λ)) " LG (λ) the associated parabolic subgroup. Note that ^ 4 F L Lie R(QG (λ)) = X ∈ g | µ(X, λ) < 0 ^ 4 F L Lie QG (λ) = X ∈ g | µ(X, λ) ≤ 0 . The result is now a straightforward consequence of Proposition 2.4.9.1 and the definition of asymptotic (semi)stability. " We leave it to the reader to formulate the moduli problem for (semi)stable Hitchin pairs and infer the existence of a projective moduli space. The reader should also think of the Hitchin map. The geometry of the Hitchin map on the moduli space of Higgs bundles with structure group G is discussed in the papers [63], [192], and [161]. Remark 2.7.4.3. We have seen in Chapter 1, Exercise 1.5.1.39, that the invariant ring ⊂ G is a maximal torus, t is its Lie algebra, and W(T ) = N (T )/T is its Weyl group.
'[g]G agrees with '[t]W(T ) . Here, T
If G is not semisimple, then Lemma 2.7.4.2 need not to hold. Indeed, if λ: '. −→ G is a one parameter subgroup of the radical of G, then QG (λ) = G, and we have only the reduction β = idX : X −→ X = P/G. Of course, µ(β, ϕ) = 0, but the condition ^ 4 M E• (λ), β• (λ) (≥)0
will impose restrictions on the topology and prevent the existence of stable objects. As an illustration, let us consider the example G = GLr1 (') x GLr2 (') with the faithful representation κ: G −→ SLr1 +r2 +1 (') defined by (2.40). Let P be a principal G-bundle with associated vector bundle E. Note that E decomposes as E1 - E2 - det(E1 - E2 )∨ . Define the one parameter subgroups λ1 , λ2 : '. −→ G 4 ^ λ1 (z) := z · $r1 , 0, z−r1 4 ^ λ2 (z) := z−1 · $r1 , 0, zr1 . The resulting weighted filtrations of E are ^ 4 ^ 4 E• (λ1 ), α• (λ1 ) = {0} ! det(E1 - E2 )∨ ! det(E1 - E2 )∨ - E2 ! E, (r1 , 1) ^ 4 ^ 4 E• (λ2 ), α• (λ2 ) = {0} ! E1 ! E1 - E2 ! E, (1, r1 ) and give the conditions (r1 + 1) · deg(E1 ) + r1 · deg(E2 ) (≥) 0 −(r1 + 1) · deg(E1 ) − r1 · deg(E2 ) (≥) 0, 27 Since G
is assumed to be semisimple, there are no non-trivial characters and we have a canonical notion of asymptotic semistability.
286 S 2.7: D P B S 2.7: D P B 286 i.e.,
(r1 + 1) · deg(E1 ) + r1 · deg(E2 ) = 0.
(2.62)
We already see that this violates stability. In the same vein, we derive r2 · deg(E1 ) + (r2 + 1) · deg(E2 ) = 0.
(2.63)
Of course, (2.62) and (2.63) hold, if and only if deg(E1 ) = deg(E2 ) = 0. This is the unnecessary topological constraint which we meant.
2.7.5 Fine Tuning of the Theory In this section, we will sketch how we come by the deficiency that we have just sketched. The idea is, as in the Ramanathan–G´omez–Sols construction of Section 2.4.8, to use non-faithful representations. Let G be a reductive group and *: G −→ GL(V) a representation. The radical of * is defined as ^ 40 Rad(*) := ker(*|Rad(G) ) . By its construction, Rad(*) is a torus which is contained in the center of G, so that it is a normal subgroup and we may form GB := G/Rad(*). Then, * factorizes over a representation *B : GB −→ GL(V). We also choose a faithful representation κ: GB −→ GL(W). For a given principal G-bundle P, we let P B be the principal GB -bundle which it defines by extension of the structure group via G −→ GB . Note that, for a principal G-bundle P, we have a canonical isomorphism P* ! P*BB . In particular, a projective *-bump (P, L, ϕ) gives rise to the projective *B -bump (P B , L, ϕ). The stability parameters are a rational character χ of GB (which is the same as a rational character of G which is trivial on Rad(*)) and a positive rational number δ. We say that a projective *-bump (P, L, ϕ) is (χ, δ)-(semi)stable, if the associated projective *B -bump (P B , L, ϕ) is so. With these concepts, we also have the moduli functors M(χ,δ)-(s)s (*, ϑ, l): Sch. S
−→ Set Isomorphy classes of families of (χ, δ)-(semi)stable *-bumps 1−→ of type (ϑ, l) parameterized by S
.
A little care has to be spent to the definition of S-equivalence. Let T := Rad(G) be the radical of G and T B := Rad(*). Since T is a torus and T B a subtorus, we may find
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287
another subtorus T BB ⊂ T , such that the multiplication map T B x T BB −→ T is an isomorphism. Recall that the image of the multiplication map T x D(G) −→ G contains the connected component G0 of the neutral element, so that any one parameter subgroup of G factorizes over this image. We let G* ⊂ G be the image of the multiplication map T BB x D(G) −→ G. As an exercise, the reader may verify the following assertion. Lemma 2.7.5.1. A projective *-bump (P, L, ϕ) is (χ, δ)-(semi)stable, if and only if the inequality ^ 4 M E• (β), α• (β) + F λ, χ 4 + δ · µ(β, ϕ)(≥)0 holds for every one parameter subgroup λ: '. −→ G* and every reduction β: X −→ P/QG (λ) of P to λ. Now, let (P, L, ϕ) be a (χ, δ)-semistable projective *-bump. To any one parameter subgroup λ: '. −→ G* and any reduction β: X −→ P/QG (λ) with ^ 4 M E• (β), α• (β) + F λ, χ 4 + δ · µ(β, ϕ) = 0, we associate the admissible deformation df β (P, L, ϕ) as in Section 2.7.1. Hence, we may define S-equivalence in the usual way and state our result: Theorem 2.7.5.2. Fix ϑ ∈ Π(G), l ∈ (, χ ∈ X% (GB ), and δ ∈ 3>0 . If *: G −→ GL(V) is a homogeneous representation of G, the projective moduli space M (χ,δ)-ss (*, ϑ, l) for the functors M(χ,δ)-(s)s (*, ϑ, l) does exist. Proof. First, let S be a scheme and (PSB , κS , NS , ϕS ) a family of *B -bumps parameterized by S . Let α: G −→ GB be the quotient homomorphism. Recall that the functor Γ(PSB , α) assigns to a scheme f : T −→ S over S the set of isomorphy classes of pairs (PT , βT ) which consist of a principal G-bundle PT on T x S and an isomorphism βT : α. (PT ) −→ PTB , that two such pairs (PT,1 , βT,1 ) and (PT,2 , βT,2 ) are isomorphic, if there is an isomorphism ϕ: PT,1 −→ PT,2 with βT,1 = βT,2 ◦ α. (ϕT ), and that the * B , α) is the sheafification of Γ(P B , α). Let f : R −→ S be the scheme over functor Γ(P S S * B , α) (see Theorem 2.4.8.7). S that represents Γ(P S Now, assume that S is the parameter space Bss for (χ, δ)-(semi)stable projective B * -bumps that was constructed after Proposition 2.7.2.12 (observing Corollary 2.7.2.6) and that (PSB , κS , NS , ϕS ) is the universal family. Recall that there are a reductive group H, an action A: H x S −→ S , and an isomorphism C: π.S x X (PSB ) −→ (A x idX ). (PSB ) of principal GB -bundles on H x S x X with the following property: For any two points s1 , s2 ∈ S with κS (s1 ) = κS (s2 ) and any isomorphism ψ: P sB1 −→ P sB2 , such that there exists a number z ∈ '. with ϕS |{s1 } x X = z · (ϕS |{s2 } x X ◦ ψ), there is an element h ∈ H, such that h · s1 = s2 and ψ agrees with the restriction of C to (h, s1 ). As in Proposition 2.4.8.9, this implies that the action A lifts to R. By the usual reasoning, the quotient R//H exists as a projective scheme and is our moduli space. " Exercise 2.7.5.3 (Hitchin pairs). Verify that Lemma 2.7.4.2 holds for connected reductive groups and our new definition of semistability for projective bumps.
288
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Remark 2.7.5.4 (About connectedness). The formalism of Section 2.7.1 works for arbitrary reductive structure groups—connected or not. Assume that G is non-connected, and let G0 be the connected component of the neutral element. Fix a representation *: G −→ GL(V) and define *0 as its restriction to G0 . Then, we get the subgroup Rad(*0 ) ⊂ G0 and the quotient group GB := G/Rad(*0 ) with an induced representation *B : GB −→ GL(V). We choose a faithful representation κ: GB −→ GL(W) and assume that *B is homogeneous. Hence, we may apply the theory of projective *B bumps. Now, D := G/D(G) is a not necessarily connected diagonalizable linear algebraic group and G −→ GB x D is a central isogeny. Using the techniques of Section 2.6, we can construct a universal space for principal D-bundles of a given topological type together with an action of a general linear algebraic group GL(Y), such that the quotient exists as a projective scheme and is the moduli space for principal D-bundles of the given topological type (without stability condition). Hence, we can still apply the method of Section 2.4.8.
2.8 Decorated Principal Bundles: Affine Fibers ow, we are prepared to prove the main result of this monograph, namely the existence of the moduli space of semistable twisted affine *-bumps. By Remark 2.1.2.5, this result generalizes and unifies all the existence results of moduli spaces of decorated vector and principal bundles on a smooth projective algebraic curve.
+
2.8.1 The Moduli Functors Let us recall the classification problem which we would like to solve: We fix a reductive linear algebraic group G and a representation *: G −→ GL(V). We decompose the Gmodule V into its irreducible components: V = V1 - · · · - Vu . Hence, we obtain induced representations *i : G −→ GL(Vi ), i = 1, . . . , u, such that * = *1 - · · · - *u . In addition, we fix a tuple of line bundles L = (L1 , . . . , Lu ). Recall that a twisted affine *-bump of type (ϑ, L) is a tuple (P, ϕ) in which P is a principal Gbundle of topological type ϑ ∈ Π(G) and ϕ = (ϕ1 , . . . , ϕu ) is a tuple of homomorphisms ϕi : P*i −→ Li , i = 1, . . . , u. Note that we allow the case that the homomorphisms ϕi , i = 1, . . . , u, are all trivial. Also recall that the twisted affine *-bump (P1 , ϕ ) is 1 isomorphic to the twisted affine *-bump (P2 , ϕ ), if there is an isomorphism ψ: P1 −→ 2
P2 , inducing isomorphisms ψi : P1,*i −→ P2,*i , with ϕ1,i = ϕ2,i ◦ ψi ,
i = 1, . . . , u.
Families We can easily extend these definitions to families: Let S be a parameter scheme. A family of twisted affine *-bumps parameterized by S is a tuple (PS , ϕ ) with S
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• a principal G-bundle PS on S x X, • a tuple ϕ = (ϕS ,1 , . . . , ϕS ,u ) of homomorphisms ϕS ,i : PS ,*i −→ π.X (Li ), i = S 1, . . . , u. Two such families (PS ,1 , ϕ ) and (PS ,2 , ϕ ) are isomorphic, if there is an isomorS ,2 S ,1 phism ψS : PS ,1 −→ PS ,2 , such that one has ϕS ,1,i = ϕS ,2,i ◦ ψS ,i for the induced isomorphisms ψS ,i : PS ,1,*i −→ PS ,2,*i , i = 1, . . . , u. Semistability Our notions of semistability and stability for twisted affine *-bumps will depend on a rational character χ ∈ X% (G) := X(G) *# 3 of G. For a twisted affine *-bump (P, ϕ) and a reduction β: X −→ P/QG (λ), λ: '. −→ G a one parameter subgroup, we define the number µ(β, ϕ) as follows: We have the pull-back Q(β) of the principal QG (λ)-bundle P −→ P/QG (λ) via β and the weighted flag (W• (λ), γ• (λ)) associated to λ: '. −→ GL(W) (see Example 2.5.2.1). Note that P* equals the vector bundle associated to the principal QG (λ)-bundle Q(β) via *|QG (λ) : QG (λ) −→ GL(W). Since the action of QG (λ) on W preserves the flag W• (λ), there is the induced filtration {0} ! F1 ! · · · ! Fv ! Fv+1 := P* . Writing π j : P* −→ P* j for the projection, j = 1, . . . , u, we finally set L F − min γi | ∃ j ∈ { 1, . . . , u } : (ϕ j ◦ π j )|Fi & 0 , µ(β, ϕ) := 0,
if ϕ & 0 . if ϕ ≡ 0
A twisted affine *-bump (P, ϕ) is said to be χ-(semi)stable, if ^ 4 M E• (β), α• (β) + F λ, χ 4(≥)0 is verified for every one parameter subgroup λ: '. −→ G and every reduction β of P to λ, such that µ(β, ϕ) ≤ 0. Example 2.8.1.1. Let χ be trivial. Then, a twisted affine *-bump (P, ϕ) with ϕi ≡ 0, i = 1, . . . , u, is (semi)stable, if and only if P is a (semi)stable principal G-bundle. S-Equivalence The construction of admissible deformations and the definition of S-equivalence is analogous to the case of projective bumps (see Section 2.7.2). So, let (ϑ, L) be the type of the twisted affine *-bumps under consideration and χ the stability parameter which
290 S 2.8: D P B S 2.8: D P B 290 we are using. Let (P, ϕ) be a χ-semistable twisted affine *-bump, λ: '. −→ G a one parameter subgroup, and β: X −→ P/QG (λ) a reduction to λ, such that µ(β, ϕ) ≤ 0 and ^ 4 M E• (β), α• (β) + F λ, χ 4 = 0. As before, there is a principal QG (λ)-bundle Q(β), such that its extension of the structure group via QG (λ) ⊂ G gives P and its extension of the structure group via " 7 7 LG (λ) ! 7G QG (λ) is declared to be Pgr . As in the setting of projective *-bumps, there is a filtration {0} =: F0 ! F1 ! · · · ! Fv ! Fv+1 := P* with Pgr,* !
% F /F v+1 j=1
j
j−1 .
More precisely, there are filtrations i {0} = F0i ⊆ F1i ⊆ · · · ⊆ Fvi ⊆ Fv+1 = P*i ,
such that Pgr,*i ! and Fj =
% F /F v+1 j=1
i j
%F , u
i=1
i j
i j−1 ,
i = 1, . . . , u,
j = 1, . . . , v + 1.
If ϕi is trivial for i = 1, . . . , u or µ(β, ϕ) < 0, we define ϕgr,i ≡ 0, i = 1, . . . , u. Otherwise, let j0 ∈ { 1, . . . , v + 1 } be the index with γ j0 = 0, i.e., F L j0 := min j = 1, . . . , v + 1 | ∃i ∈ { 1, . . . , u } : ϕi|F ji & 0 . For each index i = 1, . . . , u, there is thus an induced homomorphism ϕ0i : F ji0 /F ji0 −1 −→ Li . We then define the homomorphism ϕgr,i : Pgr,*i −→ Li that is ϕ0i on F ji0 /F ji0 −1 and zero on the other summands. The twisted affine *-bump df β (P, ϕ) := (Pgr , ϕ = (ϕgr,1 , . . . , ϕgr,u )) is called the admissible deformation of gr (P, ϕ) associated to β. With admissible deformations, we may also introduce the notion of a χ-polystable twisted affine *-bump and of S-equivalence.
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The Main Result We now introduce the moduli functors Mχ-(s)s (*, ϑ, L): Sch. S
−→ Set Isomorphy classes of families of χ-(semi)stable twisted affine *-bumps 1−→ of type (ϑ, L) parameterized by S
.
The main result of this book reads as follows: Theorem 2.8.1.2. Fix ϑ ∈ Π(G), the tuple L = (L1 , . . . , Lu ) of line bundles, and χ ∈ X% (G). Then, the moduli space M χ-ss (*, ϑ, L) for the functors Mχ-(s)s (*, ϑ, L) exists as a quasi-projective scheme. As in the case of quiver representations (see Theorem 2.5.6.13), the moduli space will be projective over a certain affine space which is related to the invariant theory of the G-action on V via a Hitchin map. This, we will define next. Hitchin Maps Let
V = V1 - · · · - Vu
be the G-module with which we are working. For every positive integer e, we have Syme (V ∨ ) =
% $
(e1 ,...,eu )∈( >0 )x u : e1 +···+eu =e
Syme1 (V1∨ ) * · · · * Symeu (Vu∨ ).
(2.64)
Let us abbreviate
^ 4G S (e1 , . . . , eu ) := Syme1 (V1∨ ) * · · · * Symeu (Vu∨ ) ,
(e1 , . . . , eu ) ∈ ((>0 )x u .
Since the splitting in (2.64) is a splitting of G-modules, we have the decomposition Syme (V ∨ )G =
% $
(e1 ,...,eu )∈( >0 )x u : e1 +···+eu =e
S (e1 , . . . , eu ).
We may therefore find generators f1 , . . . , fq of the invariant ring Sym. (V ∨ )G , such that there exist tuples (ei1 , . . . , eiu ) ∈ ((>0 )x u with fi ∈ S (ei1 , . . . , eiu ),
i = 1, . . . , q.
Next, we introduce ^
/i := H 0 X, L#1 e
i 1
*
#eiu 4, i = 1, . . . , q, and
· · · * Lu
/ :=
%/ . q
i=1
i
(2.65)
292 S 2.8: D P B S 2.8: D P B 292 Exercise 2.8.1.3. Let S be a scheme and (PS , ϕ ) a family of twisted affine *-bumps S of type (ϑ, L). Mimic the construction of the generalized Hitchin map in Section 2.5.6, p. 246ff, in order to associate to the family a morphism hPS ,ϕ : S −→ /. S
Proposition 2.8.1.4. In the situation of Theorem 2.8.1.2, there is a projective morphism H: M χ-ss (*, ϑ, L) −→ /, called the (generalized) Hitchin map. Corollary 2.8.1.5. If '[V]G = ', then the moduli space M χ-ss (*, ϑ, L) is projective. Remark 2.8.1.6. In the special case L = (OX , . . . , OX ) and '[V]G = ', the existence of a semistability concept for twisted affine *-bumps, such that a projective moduli space for the semistable objects exists, have already been announced in [167].
2.8.2 Comparison with Projective *-Bumps We now introduce a little technical device which allows us to reduce the construction of the moduli spaces for semistable twisted affine bumps to the respective construction for (asymptotically) semistable projective bumps. In the transition from affine bumps to projective bumps, we have to alter the representation which is what we will do now. Associated Projective Bumps Let κ: G −→ GL(W) be the fixed faithful representation with image in SL(W). According to Chapter 1, Theorem 1.1.6.1, we may extend the irreducible representation *i to an irreducible whence homogeneous representation *i of GL(W) of degree, say, αi , i = 1, . . . , u. We assert that we can assume 0 < r := dim. (W) < α1 < · · · < αu . Indeed, since * takes values in SL(W), the determinant representation of GL(W) restricts to the trivial representation of G on '. Thus, we may twist *i by a suitable power of the determinant representation of GL(W), i = 1, . . . , u, in order to realize the assumption on the degrees of homogeneity. Let *0 : G −→ GL1 (') be the trivial representation and *0 : GL(W) −→ GL1 (') the determinant representation. Then, *0 - *1 - · · · - *u extends the representation *0 - * from G to GL(W). We let α be the least common multiple of r, α1 , . . . , αu , and form * :=
% $
x(u+1) (a0 ,...,au )∈ : ≥0 a0 r+a1 α1 +···+au αu =α
a0 *# 0
*
a1 *# 1
*
au · · · * *# u
and * * := * ◦ κ.
Next, we fix a sufficiently ample line bundle L on X and injective homomorphisms ι0 : OX −→ L#r , ιi : Li −→ L#αi , i = 1, . . . , u.
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To continue, let (P, ϕ) be a twisted affine *-bump of type (ϑ, L). Recall that we have direct sum decompositions % *i := (*i ◦ κ) = *i - *Bi
and P%*i = P*i
-
P*Bi , i = 1, . . . , u.
There are also the canonical projection operators πi : P%*i −→ P*i ,
i = 1, . . . , u.
These enable us to introduce the homomorphisms ϕi : P%*i −−−−−→ P*i −−−−−→ Li −−−−−→ L#αi , ϕi
πi
ιi
i = 1, . . . , u.
We also set ϕ0 := ι0 : P%*0 = OX −→ L#r . Altogether, we may define * ϕ :=
% $
(u+1) : (a0 ,...,au )∈ ≥0 a0 r+a1 α1 +···+au αu =α x
on
P** =
a0 ϕ# 0
% $
x(u+1)
*
a1 ϕ# 1
P%*#a0 0
*
*
au #α · · · * ϕ# * −→ L u : P*
P%*#a1 1
*
· · · * P%*#au . u
(a0 ,...,au )∈ : ≥0 a0 r+a1 α1 +···+au αu =α
So, we have associated to the twisted affine *-bump (P, ϕ) of type (ϑ, L) the projective ϕ) of type (ϑ, l), l := deg(L#α ). It is clear that isomorphic twisted * *-bump (P, L#α , * affine *-bumps yield isomorphic projective * *-bumps. Proposition 2.8.2.1. The assignment 7 0 7 0 Isomorphy classes of Isomorphy classes of −→ twisted affine *-bumps projective * *-bumps is finite-to-one. Proof. We have to investigate the following situation: Let P be a principal G-bundle and ϕ and ϕB two tuples of homomorphisms which lead by means of the above construction to the same homomorphism * ϕ. As in the proof of Proposition 2.4.3.2, we see, by restriction to the generic point, that there are αth roots of unity ζ1 , . . . , ζu with ϕBi = ζi · ϕi ,
i = 1, . . . , u.
From this, the assertion is immediate.
"
Note that we can also extend the assignment (P, ϕ) 1−→ (P, L#α , * ϕ) to tuples (P, ϕ, ε) where (P, ϕ) is as before and ε is a complex number: Simply set ϕ0 := ε · ι0 . ϕ) need not be finite-to-one on Note that the map (P, ϕ) 1−→ (P, ϕ, 0) 1−→ (P, L#α , * isomorphy classes. We need to carry out the latter construction also in families: Let S be a parameter scheme and (PS , ϕ , εS ) a tuple in which (PS , ϕ ) is a family of affine *-bumps paS S rameterized by S and εS : OS x X −→ OS x X is a homomorphism. As before, we cook up the homomorphisms ϕS ,i : PS ,%*i −−−−−→ PS ,*i −−−−−→ π.X (Li ) −−−−−→ π.X (L#αi ), ϕS ,i
π. (ιi )
i = 1, . . . , u,
294 S 2.8: D P B S 2.8: D P B 294 and
ϕS ,0 := εS
*
π.X (ι0 ): P%*0 −→ π.X (L#r ).
From these data, we construct the homomorphism * ϕS : PS ,** −→ π.X (L#α ) as before. In order to formulate the moduli problem for projective * *-bumps, we had to choose a Poincar´e line bundle L on Jacl x X. Let κS : S −→ Jacl , s 1−→ [L#α ], be the constant morphism. We then find a line bundle NS on S , such that π.X (L#α ) ! L [κS ] * π.S (NS ). Now, (PS , κS , NS , * ϕS ) is the associated family of projective * *-bumps. Behavior of Semistability The beautiful feature of the above assignment is that it is compatible with semistability notions on both sides: Proposition 2.8.2.2. Let χ ∈ X% (G) be a rational character of G. For every twisted *-bump (P, L, * ϕ), the following conaffine *-bump (P, ϕ) with associated projective * ditions are equivalent: i) (P, ϕ) is χ-(semi)stable. ii) (P, L#α , * ϕ) is (χ, 0)-asymptotically (semi)stable. Proof. By its construction, the representation * * contains the trivial representation. In (α/r) the decoration * ϕ, the homomorphism for that trivial summand is ϕ# . This homo0 morphism is non-trivial. From this, one sees that µ(β, * ϕ) ≥ 0 holds for every one parameter subgroup λ: '. −→ G and every reduction β: X −→ P/QG (λ) of P to λ. As in the proof of Lemma 2.4.2.7, one readily checks that µ(β, * ϕ) = 0
⇐⇒
µ(β, ϕ) ≤ 0
for every one parameter subgroup λ: '. −→ G and every reduction β: X −→ P/QG (λ) of P to λ. The assertion is now clear from the definitions. " It is obvious that this proposition is the key to the construction of the moduli spaces. Exercise 2.8.2.3. Show that the assignment which associates to a χ-semistable twisted *-bump of affine *-bump of type (ϑ, L) a (χ, 0)-asymptotically semistable projective * type (ϑ, l) commutes with the formation of admissible deformations and therefore induces a map between the sets of S-equivalence classes of the respective objects.
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Remark 2.8.2.4. The construction which associates to a family of semistable twisted affine *-bumps a family of semistable projective * *-bumps gives a natural transformation *, ϑ, l) Mχ-(s)s (*, ϑ, L) −→ M(χ,δ)-(s)s (* of functors. Here, δ > δB∞ as in Corollary 2.7.3.6. Example 2.8.2.5 (Higgs bundles). Let G be a semisimple group, g := Lie(G), and ϕ := Ad: G −→ GL(g) the adjoint representation. Then, a twisted affine *-bump of type (ϑ, L) is the same as a principal G-bundle P of topological type ϑ together with a Higgs field, i.e., a section ϕ: OX −→ Ad(P)* L. Such an object is called a Higgs bundle in the literature (see, e.g., [51], [63]). Since we assume G to be semisimple, we have X% (G) = {0}, so that we may omit the character in the definition of semistability. In Section 2.7.4, we introduced the notion of a Higgs reduction. It follows from Proposition 2.8.2.2 and Lemma 2.7.4.2 that a Higgs bundle (P, ϕ: OX −→ Ad(P) * L) is (semi)stable in the sense of the definition in this section, if and only if the inequality ^ 4 deg L (Q, χ, β) (≥)0 is satisfied for every parabolic subgroup Q of G, every anti-dominant character χ of Q, and every Higgs reduction β: X −→ P/Q of (P, ϕ) to Q. This is the usual notion of semistability for Higgs bundles (see, e.g., [51]). Remark 2.8.2.6. In the situation of the above example, we also have the adjoint representation GL(g) −→ GL(End(g)). Let A: G −→ GL(End(g)) be the restriction of that representation to G. The differential ad: g −→ End(g) of the adjoint representation makes g into a G-submodule of End(g). Hence, if P is a principal G-bundle with adjoint vector bundle E := Ad(P), we have the inclusion adP : E −→ End(E), so that any Higgs field ϕ: OX −→ E * L on P gives rise to the section (adP * idL ) ◦ ϕ: OX −→ End(E) * L and thus to the twisted endomorphism ad(ϕ): E −→ E * L. All in all, we may associate to a principal G-Higgs bundle (P, ϕ: OX −→ Ad(P) * L) a Higgs vector bundle (E, ψ: E −→ E * L). Then, (P, ϕ) is semistable as in the example, if and only if (E, ψ) is semistable (see Example 2.5.6.7), i.e., µ(F) ≤ µ(E) for every vector bundle {0} ! F ! E with ψ(F) ⊂ F * L. See [51], Proposition 12, for a proof. This result parallels our observation in Corollary 2.4.4.6.
2.8.3 Construction of the Moduli Spaces We recall from Section 2.6.3 that we may describe a principal G-bundle as as tuple (E, τ). Proposition 2.8.2.2 and 2.7.3.4 imply the following boundedness statement: Proposition 2.8.3.1. Let ϑ ∈ Π(G) and r = (r1 , . . . , rt ), d = (d1 , . . . , dt ) be the integers, such that a principal G-bundle P = (E = (E1 , . . . , Et ), τ) of topological type ϑ verifies rk(Ei ) = ri and deg(Ei ) = di , i = 1, . . . , t. Fix the twisting line bundles L and the rational character χ. For every index i0 ∈ { 1, . . . , t }, the family of isomorphy classes of vector bundles E of rank ri0 and degree di0 for which there exists a χ-semistable twisted affine *-bump (P = ((E1 , . . . , Et ), τ), ϕ) of type (ϑ, L) with Ei0 ! E is bounded. As usual, we fix an integer n0 , such that one has for every n ≥ n0 and every χsemistable twisted affine *-bump (P = ((E1 , . . . , Et ), τ), ϕ) of type (ϑ, l):
296 S 2.8: D P B S 2.8: D P B 296 • Ei (n) is globally generated, i = 1, . . . , t, • H 1 (Ei (n)) = {0}, i = 1, . . . , t. We choose n ≥ n0 in such a way that the construction of the moduli space of (χ, 0)asymptotically semistable projective * *-bumps of type (ϑ, l) can be performed with respect to this n. Now, set pi := di + ri (1 − g + n) and fix complex vector spaces Yi of dimension pi , i = 1, . . . , t. As before, we have the quot schemes Qi of quotients qi : Yi *OX (−n) −→ Ei in which Ei is locally free and H 0 (qi (n)) is an isomorphism, i = 1, . . . , t. We form the product t
QB := X Qi . i=1
Recall that there is the natural group action ^
4
t
B B X GL(Yi ) x Q −→ Q .
i=1
Next, there is the scheme
πB : Y −→ QB
which parameterizes tuples ((qi : Yi * OX (−n) −→ Ei , i = 1, . . . , t), τ) in which the part (E = (E1 , . . . , Et ), τ) is a principal G-bundle. On Y x X, there is the universal principal G-bundle ^ 4 E Y = (E1,Y , . . . , Et,Y ), τY (see Proposition 2.4.5.1 and Corollary 2.4.4.6). The (Xti=1 GL(Yi ))-action lifts to Y, and πB is equivariant. In the proof of Theorem 2.7.2.4 in Section 2.7.2, we introduced a parameter scheme B for projective * *-bumps of type (ϑ, l). It comes with a projective morphism π: B −→ Y, and the (Xti=1 GL(Yi ))-action lifts to B in such a way that π is equivariant. Furthermore, there is the universal family 4 ^ ϕB E B = (E1,B , . . . , Et,B ), τB , κB , NB , * of projective * *-bumps of type (ϑ, l) on B x X. We return to Y and Y x X. Let PY be the principal G-bundle defined by (E Y , τY ). There are the associated vector bundles PY,*0 ! OY x X
and PY,*i ,
i = 1, . . . , u..
For m = 0, we set H0 := OY and & ^ 4Z ^ 4 Hi := Hom πY. PY,*i * π.X (OX (m)) , πY. π.X (Li (m)) , Define the vector bundles
H := H1 x · · · x Hu Y
Y
i = 1, . . . , u.
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297
Hex := H0 x H1 x · · · x Hu ! ' x H1 x · · · x Hu Y
Y
Y
Y
Y
over Y. The group Xti=1 GL(Yi ) acts in a natural way on the schemes H and Hex , and the projections to Y are equivariant.
Remark 2.8.3.2. There is the '. -action on Hex given through multiplication by z−r on H0 and by z−αi on Hi , i = 1, . . . , u, z ∈ '. . The quotient ^ 4 πBB : Hex := Hex \ { zero section } /'. −→ Y is a weighted projective bundle over Y. In particular, πBB is a projective morphism. Set H.ex := '. x H1 x · · · x Hu . Y
Y
Then, H.ex /'. exists as an open subset of Hex . The '. -stabilizer of the complex number 1 is the group µr of the r-th roots of unity. The induced map ^ 4 {1} x H1 x · · · x Hu /µr −→ H.ex /'. Y
Y
is an isomorphism, so that the morphism {1} x H1 x · · · x Hu −→ H.ex /'. Y
Y
is finite. The '. -action commutes with the (Xti=1 GL(Yi ))-action. Using Proposition 2.3.5.1 once more, we construct a closed subscheme A '→ H, such that we have, on A x X, the pullbacks ^ 4 qA,i : Yi * π.X OX (−n) −→ EA,i , i = 1, . . . , t, of the quotients on Y x X, the pullback ^ 4 PA = E A = (EA,1 , . . . , EA,t ), τA of the universal principal G-bundle on Y x X, and, in addition, universal homomorphisms ϕA, j : PA,* j −→ π.X (L j ), j = 1, . . . , u. The family (PA , ϕ = (ϕA,1 , . . . , ϕA,u )) is the universal family of twisted affine *-bumps A of type (ϑ, L). The action of Xti=1 GL(Yi ) on H leaves A invariant and thus yields an action ^t 4 Γ: X GL(Yi ) x A −→ A. i=1
The usual properties hold for this group action, i.e.: Proposition 2.8.3.3. Given a χ-semistable twisted affine *-bump (P, ϕ) of type (ϑ, L), there is a point a ∈ A, such that (P, ϕ) is isomorphic to the restriction of the universal family to {a} x X.
298 S 2.8: D P B S 2.8: D P B 298 Proposition 2.8.3.4. For two points a1 and a2 in A, the restriction of the universal family to the fiber {a1 } x X is isomorphic to its restriction to the fiber {a2 } x X, if and only if there is a group element g ∈ Xti=1 GL(Yi ) with a1 = g · a2 . There is also the closed subscheme Aex := ' x A '→ Hex together with the projection Aex −→ A. Then, we may pullback PA to Aex x X in order to get the principal G-bundle PAex and the homomorphisms ϕA, j in order to get the homomorphisms ϕAex , j : PAex ,* j −→ π.X (L j ), j = 1, . . . , u.
The projection morphism Aex x X −→ ' provides us with εAex : OAex x X −→ OAex x X . With ϕ = (ϕAex ,1 , . . . ϕAex ,u ), we obtain the tuple (PAex , ϕ , εAex ). By our previous Aex Aex ϕAex ) of projecconstruction, this tuple leads to the associated family (PAex , κAex , NAex , * tive * *-bumps. Together with the pulled back quotients ^ 4 qAex ,i : Yi * π.X OX (−n) −→ EAex ,i , i = 1, . . . , t,
and the pulled back homomorphism τAex for which we have ^ 4 PAex = E Aex = (EAex ,1 , . . . , EAex ,t ), τAex , this family defines a morphism f : Aex ----.
Y
7B + + ++ ++ + +,
which is equivariant for the (Xti=1 GL(Yi ))-actions. The '. -action on Hex leaves the closed subscheme Aex invariant, so that Aex = (Aex \ {zero section})/'. exists a closed subscheme of Hex . In particular, it is projective over Y. By definition of the '. -action, the morphism f is '. -invariant and therefore factorizes over Aex : f : Aex ----.
Y.
7B + + ++ ++ + +,
Again, we see that f must be a projective morphism. As we pointed out in Remark 2.8.3.2, the '. -action on Aex commutes with the (Xti=1 GL(Yi ))-action, so that there is an induced (Xti=1 GL(Yi ))-action on Aex . Of course, f is equivariant. It is, however, not everywhere finite-to-one.
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299
More Hitchin Maps In order to detect a reasonable open subscheme over which it is finite-to-one, we have to go to the theory of projective * *-bumps: There is the morphism 6
κB : B −→ Jacl _ (qi : Yi * OX (−n) −→ Ei , i = 1, . . . , t), τ, * ϕ: E** −→ N 1−→ [N].
By construction, the image of f lands in B0 which we define as the fiber of κB over *-bump to B0 x X gives the homo[L#α ]. The restriction of the universal projective * morphism * ϕB0 : PB0 ,** −→ π.S (NB0 ) * π.X (L#α ),
PB0 := PB|B0 x X .
* as We may write the G-module V * ='-V V and view
* −→ ' η0 : V
* For sufficiently large as a G-invariant homogeneous function of degree one on V. degree b, we may find elements *∨ )G , τ0 , τ1 , . . . , τ M ∈ Symb (V
τ0 := ηb0 ,
such that the rational map
^ 4 *∨ )//G = Proj Sym. (V *∨ )G ι: !(V [w]
is a closed embedding. We define
* := !
&^
" " "7 / 7
H 0 (X, L#(b·α) )$ M+1
!M
6 _ τ0 (w), . . . , τ M (w) 4∨ Z .
(b·α/r) Let ι0 : OX −→ L#r be the inclusion which we fixed above. Set σ0 := ι# . We 0 complete σ0 to a basis σ0 , . . . , σD for H 0 (X, L#(b·α) )$(M+1) , use the induced coordinates on *, and set F L * := [s0 , . . . , sD ] ∈ * | s0 % 0 .
By the GIT construction of the moduli space of semistable projective * *-bumps in Section 2.7, there are (Xti=1 GL(Yi ))-invariant open subsets B(χ,0)-(s)s ⊂ B whose closed points are exactly those representing (χ, 0)-asymptotically (semi)stable projective * *-bumps, and we know that the quotient 4 ^t B(χ,0)-ss // X GL(Yi ) i=1
300 S 2.8: D P B S 2.8: D P B 300 exists as a projective scheme and that ^t 4 B(χ,0)-s // X GL(Yi ) i=1
exists as an open subscheme of B(χ,0)-ss //(Xti=1 GL(Yi )) and is an orbit space. Since B0 is a (Xti=1 GL(Yi ))-invariant closed subscheme, the quotient ^
4 ^t 4 B0 ∩ B(χ,0)-ss // X GL(Yi ) i=1
exists as a closed subscheme of B(χ,0)-ss //(Xti=1 GL(Yi )). Exactly as in the setting of quiver representations, the restriction of the universal homomorphism * ϕB0 to (B(χ,0)-ss ∩ B0 ) x X gives a morphism k: B0 −→ * with
^ 4 4 ^ k. O0 (1) ! π.S NB#0 b
|(B(χ,0)-ss ∩B0 ) x X
.
The morphism k is obviously invariant under the action of Xti=1 GL(Yi ). It therefore descends to a projective morphism 4 ^ 4 ^t K: B0 ∩ B(χ,0)-ss // X GL(Yi ) −→ *. i=1
Set
B. := K −1 (*)
and look at the composition f
a1−→(1,a)
A −−−−−−→ A.ex := Aex ∩ H.ex −−−−−→ B. By Proposition 2.8.2.2, the preimage of B. under this morphism is exactly the open subset Aχ-ss of those points that correspond to χ-semistable bumps, and we also have ^ 4 −1 f −1 (B0 ) = '. x Aχ-ss and f (B0 ) = '. x Aχ-ss /'. . Since the induced morphism −1
f (B0 ) =
^
4
'. x Aχ-ss /'. −→ B.
is proper, we see that the quotient ^ 4 ^t 4 ^ 4 ^ 4 t M 1 := ('. x Aχ-ss )/'. // X GL(Yi ) = '. x Aχ-ss // '. x X GL(Yi ) i=1
i=1
exists as a quasi-projective scheme and comes with a projective morphism 4 ^t M 1 −→ B. // X GL(Yi ) . i=1
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301
Since the quotient morphism
'. x Aχ-ss −→ M 1 is affine and (Xti=1 GL(Yi ))-invariant, it follows from Chapter 1, Exercise 1.4.3.11, i), that the quotient 4 ^t 4 ^ M1 := '. x Aχ-ss // X GL(Yi ) i=1
also exists. Now, A is embedded by the map a 1−→ (1, a) into '. x Aχ-ss as a closed, t (Xi=1 GL(Yi ))-invariant subscheme, so that the quotient χ-ss
^t 4 M χ-ss (*, ϑ, L) := Aχ-ss // X GL(Yi ) i=1
χ-s
exists, too. In the same vein, we see that the set A of the χ-stable bumps is also open and (Xti=1 GL(Yi ))-invariant, and that the categorical quotient ^t 4 M χ-s (*, ϑ, L) := Aχ-s // X GL(Yi ) i=1
exists as an open subset of M χ-ss (*, ϑ, L). The universal property of the categorical quotient (Chapter 1, Lemma 1.4.1.1), Proposition 2.8.3.3, and Proposition 2.8.3.4 show that M χ-(s)s (*, ϑ, L) corepresents the functor Mχ-(s)s (*, ϑ, L). In the final step, we have to understand the closed points of the moduli space M χ-(s)s (*, ϑ, L). By Remark 2.8.3.2, the morphism A −→ ('. x A)/'. −→ B. is finite and (Xti=1 GL(Yi ))-equivariant. It follows that the orbit of a point a is closed if and only if the orbit of its image in B. is closed. This shows already that M χ-s (*, ϑ, L) is an orbit space, so that it parameterizes the isomorphy classes of χ-stable twisted affine *-bumps of type (ϑ, L). Using Exercise 2.8.2.3, we also see that the closed points of M χ-ss (*, ϑ, L) naturally correspond to the S-equivalence classes of χ-semistable twisted affine *-bumps of type (ϑ, L). " Proof of Proposition 2.8.1.4 We use the notation and observations from the construction of the Hitchin map. Recall that we have the decomposition V = V1 - · · · - Vu . We also look at the G-module V B = V0 - V1 - · · · - Vu ,
V0 ! ' being the trivial G-module.
There is the isomorphism Sym. (V B ∨ ) ! Sym. (V0∨ ) * · · · * Sym. (Vu∨ ).
302 S 2.8: D P B S 2.8: D P B 302 We may introduce a new grading on Sym. (V B ∨ ), by assigning the weight αi to the elements in Vi∨ , i = 1, . . . , u, and r to the elements in V0∨ . The G-action on V respects this grading, so that it induces a grading on Sym. (V B ∨ )G ! Sym. (V0∨ )G * · · · * Sym. (Vu∨ )G . The generators f1 ,. . . , fq for Sym. (V ∨ )G which we found above are homogeneous of ' degree ε j := ui=1 αi · eij , j = 1, . . . , q. Let f0 be the generator of Sym. (V0∨ )G . It is homogeneous of degree r. It is clear that f0 , f1 , . . . , fq generate Sym. (V B ∨ )G . Recall that we have introduced the G-module
% $
* := V
'#a
0
x(u+1)
*
V1#a1
*
· · · * Vu#au .
(a0 ,...,au )∈ : ≥0 a0 r+a1 α1 +···+au αu =α
Next, we introduce the G-equivariant morphism * −→ V
p: V0 - · · · - Vu
T
v = (v0 , . . . , vu ) 1−→
$
x(u+1) : (a0 ,...,au )∈ ≥0 a0 r+a1 α1 +···+au αu =α
This morphism induces
a0 v# 0
*
a1 v# 1
*
au · · · * v# u .
* /G. p: V B //G −→ V/
* pull back to the invariThe G-invariant functions τ0 ,. . . ,τ M on the vector space V ant functions p. (τ1 ),. . . ,p. (τ M ) on V. Hence, there are polynomials h0 , . . . , h M ∈ '[x0 , . . . , xq ] with p. (τi ) = hi ( f0 , . . . , fq ), i = 0, . . . , M. The invariant function p. (τi ) is clearly homogeneous of degree b · α, i = 0, . . . , m (in ν the algebra Sym. (V B ∨ )G ). Hence, we may assume that any monomial xν00 · · · · · xqq appearing with non-zero coefficient in one of the hi ’s actually satisfies ν0 r + ν1 ε1 + · · · + νq εq = b · α.
(2.66)
Recall that the Hitchin space is
/ :=
%/ q
j=1
^
/ j := H 0 X, L#1 e
with
j
There are linear maps coming from
i=1
i
eij
*
4 euj , · · · * L# u
j = 1, . . . , q.
l j : / j −→ H 0 (L#ε j )
) ι# : ) L# u
j 1
u
i=1
i
eij
−→
) L# u
(αi ·eij )
= L#ε j ,
j = 1, . . . , q.
i=1
For any two positive integers k, l, we have the multiplication map H 0 (X, L#k ) * H 0 (X, L#l ) −→ H 0 (X, L#(k+l) ).
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303
Given sections si ∈ /i , i = 1, . . . , q, and the polynomials hi , i = 0, . . . , m, from above, these multiplication maps and Formula 2.66 enable us to define the section ^ 4 ^ 4 hi ι0 , l1 (s1 ), . . . , lq (sq ) ∈ H 0 X, L#(b·α) , i = 0, . . . , m. In this way, we obtain a morphism l: / −→ H 0 (X, L#(b·α) )$(M+1) .
(b·α/r) Note that h0 (ι0 , l1 (s1 ), . . . , lq (sq )) = ι# % 0, so that l leads to a morphism 0 &^ 4∨ Z l: / −→ * ⊂ ! H 0 (X, L#(b·α) )$(M+1) .
Finally, the universal family (PA|(Aχ-ss x X) , ϕ χ-ss ) on Aχ-ss x X gives the morA|(A x X) phism hPA|(Aχ-ss x X) ,ϕ χ-ss : Aχ-ss −→ /. A|(A
x X)
This morphism is certainly invariant under the action of the group Xti=1 GL(Yi ), and so descends to a morphism H: M χ-ss (*, ϑ, L) −→ /. Our constructions were made in such a way that the diagram ^ 4 7 B. // Xt GL(Yi ) i=1
M χ-ss (*, ϑ, L) H
3
/
l
7
3
K
*
commutes. The morphisms M χ-ss (*, ϑ, L) −→ B. //(Xti=1 GL(Yi )) and K are both proper, and so is their composition. By [96], Corollary II.4.8, again, the morphism H must be proper, too. "
2.8.4 Further Properties and Examples We conclude this section by the discussion of several properties and illustrations of the semistability concept for twisted affine *-bumps. First, we will show that we might base our theory also on faithful representations κ: G −→ GL(W) which do not factorize over the special linear group. Then, we apply this observation to check that Bradlow pairs fit into the new frame. As a new application, we discuss Higgs bundles with real reductive structure group. Next, we write down the notion of semistability for projective *-bumps for a not necessarily homogeneous representation *. Finally, we treat the case that the representation has a non-trivial radical. Embeddings into Special Linear Groups vs. Embeddings into General Linear Groups In order to state our condition of semistability and construct the moduli spaces for semistable twisted affine bumps, we always had to fix a faithful representation κ: G −→
304 S 2.8: D P B S 2.8: D P B 304 SL(V). The fact that the target is SL(V) was rather important for the success of our arguments. Still, it might be reassuring and aesthetically more appealing that one can build the whole theory also on a faithful representation in GL(V). In this way, our theory becomes totally parallel to the gauge theoretic setting in [156]. Let E be a vector bundle of rank r. A GL-weighted filtration of E is a pair (E• , γ• ) which consists of a filtration E• : {0} =: E0 ! E1 ! · · · ! E s ! E s+1 := E of E by subbundles and a vector γ• = (γ1 , . . . , γ s+1 ),
γ1 < · · · < γ s+1 ,
of integers.
Remark 2.8.4.1. Let W := 'r and λ: '. −→ GL(W) a one parameter subgroup whose weighted flag (W• , γ• ) (in the sense of Example 2.5.2.1) features W• : {0} =: W0 ! W1 ! · · · ! W s ! W s+1 := W with
dim. (Wi ) = rk(Ei ),
i = 0, . . . , s + 1.
Then, the datum of the filtration E• is equivalent to the datum of a section β: X −→ I som(W * OX , E)/QGL(W) (λ). Let det: GLr (') −→ '. be the determinant character. We can now compute F λ, det 4 =
s+1 T
s+1 ^ 4 T ^ 4 γi · dim. (Wi ) − dim. (Wi−1 ) = γi · rk(Ei ) − rk(Ei−1 ) .
i=1
i=1
Next, we will use the representation κ: GL(W) m
−→ GL(W - ') X J m 0 1−→ . 0 det(m)−1
Let E be a rank r vector bundle. The vector bundle associated to E by means of κ is * := E - det(E)∨ . Suppose (E• , γ• ) is a GL-weighted filtration of E. Let λ: '. −→ E GL(W) be a one parameter subgroup as in Remark 2.8.4.1. We form λ κ * * λ: '. −→ GL(W) −→ GL(W),
* := W - '. W
As we have remarked above, the filtration E• corresponds to a section β: X −→ I som(W * OX , E)/QGL(W) (λ), and the section β
* * OX , E)/Q * * * β: X −→ I som(W * OX , E)/QGL(W) (λ) '→ I som(W * (λ) GL(W)
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305
*• , * * We will now discuss this new weighted yields a weighted filtration (E α• ) of E. filtration. The weights of * λ are γ1 , . . . , γ s+1 and, by Remark 2.8.4.1, −γ = −
s+1 T
^ 4 γi · rk(Ei ) − rk(Ei−1 ) .
(2.67)
i=1
We order these weights: γ1 < · · · < γ j < −γ ≤ γ j+1 < · · · < γ s+1 . We assume that −γ < γ j+1 . The reader will see that the arguments easily transfer to the case γ = γ j+1 . We have *• : {0} ! E *1 ! · · · ! E *s+1 ! E * E with *i = E
7
Ei , Ei−1 - det(E)∨ ,
The tuple * α• = (* α1 , . . . , * α s+1 ) features (γi+1 − γi )/(r + 1), (−γ − γ j )/(r + 1), * αi = (γ j+1 + γ)/(r + 1), (γi − γi−1 )/(r + 1),
i≤ j . i≥ j+1
i≤ i= i= i>
j−1 j . j+1 j+1
We compute: *• , * M(E α• )
=
s+1 T
^ 4 * · rk(E *i ) − deg(E *i ) · (r + 1) * αi · deg(E)
i=1 * deg(E)=0
=
−
s+1 T
*i ) · (r + 1) * αi · deg(E
i=1
=
=
j−1 T − (γi+1 − γi ) · deg(Ei ) + (γ + γ j ) · deg(E j ) − i=1 ^ 4 −(γ j+1 + γ) · deg(E j ) − deg(E) − s T ^ 4 − (γi+1 − γi ) · deg(Ei ) − deg(E) −
i= j+1 s T
(γi+1 − γi ) · deg(Ei ) + (γ s+1 + γ) · deg(E)
i=1
=
s T i=1
γi · deg(Ei ) −
s T i=1
(2.68)
γi+1 · deg(Ei ) + γ s+1 · deg(E s+1 ) + γ · deg(E)
306 S 2.8: D P B S 2.8: D P B 306 E0 ={0}
=
s+1 T
γi · deg(Ei ) −
γi · deg(Ei−1 ) + γ · deg(E)
i=1
i=1
=
s+1 T
s+1 T
s+1 ^ 4 ^ 4 T γi · rk(Ei ) − rk(Ei−1 ) · deg(E) γi · deg(Ei ) − deg(Ei−1 ) + i=1
i=1
=:
* • , γ• ). M(E
We can relate the latter quantity also to a more familiar one. To this end, set α• (γ• ) := (α1 , . . . , α s )
with
αi :=
γi+1 − γi , r
i = 1, . . . , s.
Lemma 2.8.4.2. Define the rational character * χ := (µ(E) + 1) · det of GL(W). In the above setting, we have ^ 4 * • , γ• ) = M E• , α• (γ• ) + F λ, * M(E χ 4. Proof. Using Equation 2.68 and Remark 2.8.4.1, we compute * • , γ• ) − F λ, * M(E χ4 =
−
s T
αi · rk(E) · deg(Ei ) +
i=1
+γ s+1 · deg(E) −
s+1 T 4 γi ^ · rk(Ei ) − rk(Ei−1 ) · deg(E). r i=1
We manipulate the last line as follows: s+1 T 4 γi ^ · rk(Ei ) − rk(Ei−1 ) · deg(E) r i=1 s+1 s+1 T T γi γi deg(E) · γ s+1 − · rk(Ei ) + · rk(Ei−1 ) r r i=1 i=1 s s T T γi γi+1 · rk(Ei ) + · rk(Ei ) deg(E) · − r r i=1 i=1 s s T T γi+1 − γi · rk(Ei ) = deg(E) · αi · deg(E) · rk(Ei ). r i=1 i=1
γ s+1 · deg(E) −
= rk(E0 )=0
=
= Altogether, we see
* • , γ• ) − F λ, * M(E χ4 =
s T
^ 4 ^ 4 αi · deg(E) · rk(Ei ) − deg(Ei ) · rk(E) = M E• , α• (γ• ) ,
i=1
and that finishes our argument.
"
Now, let G be a reductive group and κ: G −→ GL(W) a faithful representation. Fix also a representation *: G −→ GL(V) and a rational character χ of G. For a principal G-bundle P and a reduction β: X −→ P/QG (λ) to a one parameter subgroup
S 2.8: D P B
307
λ: '. −→ G, we obtain a GL-weighted filtration (E• (β), γ• (β)) of E as follows: We take the weight vector γ• (β) := γ• (λ) from the weighted flag (W• (λ), γ• (λ)) associated to λ: '. −→ GL(W) (see Example 2.5.2.1) and let E• (β) be the filtration that comes from the section β
X −→ P/QG (λ) '→ I som(W * OX , E)/QGL(W) (λ). We say that a twisted affine *-bump (P, ϕ) is χ-(semi)stable, if ^ 4 M E• (β), α• (γ• (β)) + F λ, χ 4(≥)0 holds for every reduction β: X −→ P/QG (λ) of P to a one parameter subgroup λ: '. −→ G with µ(β, ϕ) ≤ 0. Lemma 2.8.4.2 and the discussion before it clearly show that we have a theory of moduli spaces for these semistable objects, because the notion of (semi)stability is equivalent to the one that we obtain from the faithful representation m1−→(m,det(m)−1 )
G −−−−−→ GL(W) −−−−−−−−−−−−→ SL(W - ') κ
and the rational character χ − * χ. Bradlow Pairs We start with the irreducible representation * := id: GLr (') −→ GL(V), V := 'r . We fix a line bundle L on X. Then, a twisted affine *-bump is a vector bundle E together with a homomorphism ϕ: E −→ L. This is a Bradlow pair. We have already considered these objects in Section 2.3.3. It is quite useful to see how our new formalism, indeed, reproduces the previous results. According to our discussion in the foregoing section, we may choose κ = * as our faithful representation. The stability parameter is an element of X% (GLr (')) ! 3. We make the following normalization: A rational number δ stands for the character δ·
det . r
Our first contention is that δ must be non-negative. To see this, we look at the one parameter subgroup λ: '. −→ GLr ('), z 1−→ z · $r . Since QGLr (.) (λ) = GLr ('), the only choice of a reduction to λ is β := id X : X −→ X = I som(V * OX , E)/QGLr (.) (λ). The GL-weighted filtration (E• , γ• ) is ({0} ! E, (1)). For this reduction, we have 7 −1, ϕ & 0 µ(β, ϕ) = . 0, ϕ ≡ 0 So, this reduction gives a condition. Since M(E• (β), α• (γ• (β))) = 0, this condition is δ = F λ, δ 4 ≥ 0.
308 S 2.8: D P B S 2.8: D P B 308 Next, let us check how semistability looks for δ = 0. In this case, (E, 0) is (semi)stable, if and only if E is a (semi)stable vector bundle (see Remark 2.8.1.1). On the other hand, if ϕ isn’t trivial, we have just constructed a reduction β to a one parameter subgroup λ with µ(β, λ) < 0 and ^ 4 M E• (β), α• (γ• (β)) + F λ, δ 4 = 0. The associated admissible deformation is (E, 0). Clearly, the resulting moduli space for δ-semistable Bradlow pairs will be the moduli space of semistable vector bundles. Finally, we have to study the case δ > 0. Suppose (E, ϕ) is a δ-(semi)stable Bradlow pair. We claim that ϕ must be non-trivial. If ϕ is trivial, then we work with the one parameter subgroup λ: '. −→ GLr ('), z 1−→ z−1 · $r . As before, we have a unique reduction β to this one parameter subgroup. Since ϕ is trivial, we have µ(β, ϕ) = 0. Then, ^ 4 M E• (β), α• (γ• (β)) + F λ, δ 4 = −δ < 0 is a contradiction to δ-semistability. Now, let us work out the exact notion of δ-semistability. To this end, let λ: '. −→ GLr (') be a one parameter subgroup. We look at det ◦λ: '. −→ '. . This homomorphism is of the shape z 1−→ zη , z ∈ '. , for some integer η. Note that, for a positive integer s, we have QGLr (.) (λ) = QGLr (.) (λ s ). A reduction β: X −→ I som(V * OX , E)/QGLr (.) (λ) to λ can therefore also be seen as a reduction to λ s . As such, we denote it by β s . It is then straightforward to check that µ(β s , ϕ) = s · µ(β, ϕ) and
^ 4 ^ ^ 4 4 M E• (β s ), α• (γ• (β s )) + F λ s , δ 4 = s · M E• (β), α• (γ• (β)) + F λ, δ 4 .
For this reason, we may assume without loss of generality that the integer η is divisible by r. Then, in additive notation, λ = λB + λBB ,
λB (z) = zη/r , z ∈ '. , and λBB = λ − λB .
Here, λBB is a one parameter subgroup of SLr ('). Any reduction β of the principal GL(V)-bundle I som(V * OX , E) to λ is also one to λBB . If we use this interpretation, we will write βBB . Then, one has µ(β, ϕ) = µ(βBB , ϕ) − k, and
k :=
η , r
^ 4 ^ 4 M E• (β), α• (γ• (β)) + F λ, δ 4 = M E• (βBB ), α• (γ• (βBB )) + δ · k.
Assume that µ(β, ϕ) ≤ 0 holds true. This means that k ≤ −µ(βBB , ϕ). Hence, ^ 4 ^ 4 M E• (β), α• (γ• (β)) + F λ, δ 4 ≥ M E• (βBB ), α• (γ• (βBB )) + δ · µ(βBB , ϕ). Without loss of generality, we may, therefore, assume that k = −µ(β, ϕ) holds true. We write down our observations:
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Proposition 2.8.4.3. A Bradlow pair (E, ϕ) is δ-(semi)stable, if and only if the condition ^ 4 M E• (β), α• (γ• (β)) + δ · µ(β, ϕ)(≥)0 is verified for every one parameter subgroup λ of the special linear group SL(V) and every reduction β of I som(V * OX , E) to λ. This looks already very familiar. To further simplify the notion of semistability, we look at the action of SLr (') on Hom('r , '). Let λ: '. −→ SLr (') be a one parameter subgroup. Then, there are a basis v = (v1 , . . . , vr ) of 'r and a weight vector γ = (γ1 , . . . , γr ) with γ1 ≤ · · · ≤ γr and γ1 + · · · + γr = 0, such that vi is an eigenvector for the action of λ with respect to the character z 1−→ zγi , z ∈ '. , i = 1, . . . , r. For a non-zero homomorphism l ∈ Hom('r , '), we compute F L µ(l, λ) = − min γi | l(vi ) % 0 . (2.69) Recall from Chapter 1, (1.11), that we have λ=
r−1 T i=1
αi · λ(v, γ(i) ), r
αi :=
γi+1 − γi , i = 1, . . . , r − 1. r
With Formula 2.69, it is clear that µ(l, λ) =
r−1 T
4 ^ αi · µ l, λ(v, γ(i) ) . r
i=1
This equation shows that it suffices to test the condition of δ-(semi)stability for those one parameter subgroups λ that are associated to a basis of V and a basic weight vector. These reductions correspond to the subbundles of E. Let {0} ! F ! E be a subbundle of rank s. Choose a basis v of V, set λ := λ(v, γr(s) ), and let β: X −→ I som(V * OX , E)/QGLr (.) (λ) be the reduction to λ that gives the weighted filtration (E• (β), α• (γ• (β))) = ({0} ! F ! E, (1)). With Formula 2.69, one sees directly 7 − rk(F), F ⊂ ker(ϕ) µ(β, ϕ) = . rk(E) − rk(F), F " ker(ϕ) Remembering M({0} ! F ! E, (1)) = deg(E) · rk(F) − deg(F) · rk(E), we have shown: Proposition 2.8.4.4. A Bradlow pair (E, ϕ) is δ-(semi)stable, if every subbundle {0} ! F ! E satisfies the condition δ , rk(E) δ δ µ(F) − (≤)µ(E) − , rk(F) rk(E) µ(F)(≤)µ(E) −
if
F ⊂ ker(ϕ)
if
F " ker(ϕ).
Thus, we have recovered Bradlow’s definition of semistability (Section 2.3.3).
310 S 2.8: D P B S 2.8: D P B 310 Γ-Higgs Bundles Here, we would like to give a first application of our general theorem to a moduli problem which might be of immediate interest. The set-up for this application is as follows: Suppose that Γ is a connected real reductive group. Then, there are a maximal compact subgroup K ⊂ Γ and a Cartan decomposition (see [31], I.4.8, p. 17) Lie(Γ) = Lie(H) - m. This is a decomposition of Lie(Γ) as an H-module. We introduce G := H
x
Spec(/)
Spec(').
This is a connected linear algebraic group. It is reductive, because K is compact (see Chapter 1, Section 1.1.7). We define *: G −→ GL(V) with V := (m */ ')∨ as the representation which is contragredient to the complexification of the representation of H on m. We choose all twisting line bundles to be equal to ωX . Then, a *-bump is a pair (P, σ) where P is a principal G-bundle and σ: X −→ P(V) * ωX is a section. Thus, these *-bumps are exactly the Γ-Higgs bundles studied in the literature. From the gauge theoretic side, one obtains a notion of (semi)stability for Γ-Higgs bundles and a moduli space for (semi)stable Γ-Higgs bundles as a real analytic variety. As such, it is isomorphic to the moduli space of representations of the fundamental group π1 (X) with target Γ. (The points of that moduli space parameterize the so-called reductive representations.) For more information on these topics, we refer the reader to the paper [40] and the lecture notes [85]. Our notion of semistability agrees with the gauge theoretically defined one. Hence, our main theorem gives the moduli space for semistable Γ-Higgs bundles also as a quasi-projective algebraic variety. In this generality, this is a new result. In general, the notion of semistability of Γ-Higgs bundles depends on many parameters (although only some specific ones appear in the application to representations of the fundamental group). It seems likely that one may simplify the notion of semistability to a more user friendly one, but this has been done only in a few examples. For the group U(p, q), these examples are the holomorphic chains at which we had a glance in Section 2.5.6. A more recent addition is the paper [72] in which the authors give a simplified version of semistability for Higgs bundles associated to the group Γ = Sp2r (1). Let us discuss the simplification procedure here. For Γ = Sp2r (1), the maximal subgroup H can be chosen as U(r), so that G = GLr ('). The representation is *: GLr (') −→ GL(Sym2 (W ∨ ) - Sym2 (W)), W := 'r . Hence, a Γ-Higgs bundle consists of a vector bundle E of rank r and homomorphisms β: OX −→ Sym2 (E) * ωX and ϕ: Sym2 (E) −→ ωX . We start by investigating the SLr (')-action on Sym2 (W)∨ in the spirit of Chapter 1, Example 1.5.1.19, and Section 2.3.3, p. 307ff, in this chapter. We choose a basis w = (w1 , . . . , wr ) for W. Then, the elements wi · w j with 1 ≤ i ≤ j ≤ r form a basis for Sym2 (W). We set F L I = (i, j) ∈ { 1, . . . , r }x 2 | i ≤ j . We introduce a partial ordering “C” on I via (i1 , j1 ) C (i2 , j2 )
:⇐⇒
i1 ≤ i2 and j1 ≤ j2 .
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311
L F K = (γ1 , . . . , γr ) ∈ 1r | γ1 ≤ · · · ≤ γr , γ1 + · · · + γr = 0 .
We know that this cone is spanned by the integral weight vectors γr(i) = ( i − r, . . . , i − r, i, . . . , i ), D!!!!!!!!!!!WB!!!!!!!!!!!\ D!WB!\ ix
i = 1, . . . , r − 1.
(r−i) x
Next, suppose we are given a linear map f : Sym2 (W) −→ from previous terminology, we let F L ST(w, f ) = (i, j) ∈ I | f (wi · w j ) % 0
'.
In slight deviation
be the set of states of f . Set F L M(w, f ) = (i, j) ∈ ST(w, f ) | (i, j) is minimal with respect to “C” . A proper decomposition of K takes only place, if #M(w, f ) > 1. We introduce a new partial ordering “!” on M(w, f ): (i1 , j1 ) ! (i2 , j2 )
:⇐⇒
i1 < i2 and j1 > j2 .
Remark 2.8.4.5. The relation “!” is an ordering on M(w, f ): Let (i1 , j1 ) and (i2 , j2 ) be two distinct pairs in M(w, f ). Since they are incompatible with respect to “C”, we have i1 % i2 . We consider the case i1 < i2 . It is impossible that j1 ≤ j2 , because this implies (i1 , j1 ) C (i2 , j2 ). Hence, (i1 , j1 ) ! (i2 , j2 ). This discussion also shows that (i1 , j1 ) ! (i2 , j2 ) if and only if i1 < i2 . First, we look at the cones ^ 4 F L C ± (i1 , j1 ), (i2 , j2 ) = (γ1 , . . . , γr ) ∈ K | ± (γi1 + γ j1 ) ≤ ±(γi2 + γ j2 ) for two given elements (i1 , j1 ) and (i2 , j2 ) in M(w, f ) with (i1 , j1 ) ! (i2 , j2 ). We also introduce W((i1 , j1 ), (i2 , j2 )) as the hyperplane in 1r that is spanned by the intersection C + ((i1 , j1 ), (i2 , j2 )) ∩ C − ((i1 , j1 ), (i2 , j2 )). Remark 2.8.4.6. Note that F L ^ 4 Γ0 := γr(k) | 1 ≤ k < i1 ∨ j1 ≤ k < r ∨ i2 ≤ k < j2 ⊂ W (i1 , j1 ), (i2 , j2 ) (2.70) L ^ 4 ^ 4 F (2.71) Γ− := γr(k) | j2 ≤ k < j1 ⊂ C − (i1 , j1 ), (i2 , j2 ) \ W (i1 , j1 ), (i2 , j2 ) L ^ 4 ^ 4 F + (k) (2.72) Γ+ := γr | i1 ≤ k < i2 ⊂ C (i1 , j1 ), (i2 , j2 ) \ W (i1 , j1 ), (i2 , j2 ) . We have to determine the intersection of W := W((i1 , j1 ), (i2 , j2 )) with the 2-dimensional faces of K. There are three possibilities: Either W contains the face, or it intersects the face in an edge of K, or in its relative interior. In the first case, the face is spanned by elements of Γ0 . In the second case, the edge is generated by an element of Γ0 , and, in the third case, the face is spanned by a vector γ− ∈ Γ− and a vector γ+ ∈ Γ+ . In that case, one readily checks that the intersection is spanned by γ− + γ+ . Altogether, we see that C + ((i1 , j1 ), (i2 , j2 )) is spanned by the set L F Γ0 ∪ Γ+ ∪ γ− + γ+ | γ± ∈ Γ± ,
312 S 2.8: D P B S 2.8: D P B 312 and C − ((i1 , j1 ), (i2 , j2 )) by L F Γ0 ∪ Γ− ∪ γ− + γ+ | γ± ∈ Γ± . The fan decomposition of K into subcones associated to the set of states of f is formed by the cones ^ 4 ^ 4 C − (i−k− , j−k− ), (i0 , j0 ) ∩ · · · ∩ C − (i−1 , j−1 ), (i0 , j0 ) ∩ ^ 4 ^ 4 (2.73) ∩C + (i0 , j0 ), (i+1 , j+1 ) ∩ · · · ∩ C + (i0 , j0 ), (i+k+ , j+k+ ) . Here, the index tuples which appear satisfy (i−k− , j−k− ) ! · · · ! (i−1 , j−1 ) ! (i0 , j0 ) ! (i+1 , j+1 ) ! · · · ! (i+k+ , j+k+ ). Claim 2.8.4.7. Set i±0 := i0 and j±0 := j0 . The cone in (2.73) is generated by the edges in Γ ∪ Γ B with L F Γ = γr(l) | 1 ≤ l < i−k− ∨ j0 ≤ l < r ∨ i0 ≤ l < j+k+ , F − L + ΓB = γr(l ) + γr(l ) | (l− , l+ ) ∈ L , − X kR −1F LJ L F − − − L := ik− −m , . . . , ik− −m−1 − 1 x j0 , . . . , jk− −m − 1 J m=0
J
+ X kR −1F
m=0
LJ L F i0 , .., i+m+1 − 1 x j+m+1 , . . . , j+m − 1 .
Proof. The proof is by induction on k := k− + k+ . We have already described the case k = 1. Now, let us explain the conclusion k −→ k + 1. We suppose that we add the cone C + ((i0 , j0 ), (i+k+ +1 , j+k+ +1 )) to the cone C in (2.73), the other case being similar. We first have to describe the intersection of the cone C with the hyperplane W = W((i0 , j0 ), (i+k+ +1 , j+k+ +1 )). As before, we determine the intersection of W with the 2dimensional faces of C. It is easy to check that these intersections are spanned by the vectors γr(l) with 1 ≤ l < i−k− , j0 ≤ l < r, or i+k+ +1 ≤ l < j+k+ +1 , by the vectors in Γ B , or by the vectors of the form γr(l) + γr(m) with i0 ≤ l < i+k+ +1 and j+k+ +1 ≤ m < j+k+ . It is clear that C ∩ C + ((i0 , j0 ), (i+k+ +1 , j+k+ +1 )) is generated by the edges we have just described and the edges (γ1 , . . . , γr ) of C that satisfy −(γi0 + γ j0 ) > −(γi+k+ +1 , γ j+k+ +1 ), i.e., by the edges γr(l) with i0 ≤ l < i+k+ +1 . This finishes the induction step.
"
Remark 2.8.4.8. Let f : Sym2 (W) −→ ' be as above. By Chapter 1, Example 1.5.1.19, we have to study the Hilbert–Mumford criterion only for one parameter subgroups λ(w, γ) where w is a basis for W and γ ∈ K ∩ ($r is a weight vector of the form γr(i) or
γr(l) + γr(m) , if #M(w, f ) > 1 and γr(l) + γr(m) is a generator of a cone C as in (2.73) (where the defining “index chain” is the maximal one (cf. Remark 2.8.4.5)). i) For γr(l) + γr(m) as just described, let W1 := F v1 , . . . , vl 4 and W2 := F v1 , . . . , vm 4. If one looks carefully at the construction, one sees that f is trivial on W1 · W2 (the image of W1 * W2 under the quotient map W * W −→ Sym2 (W)). Writing Wi⊥ :=
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313
ker(W ∨ −→ Wi∨ ), i = 1, 2, and using the isomorphism Sym2 (W)∨ ! Sym2 (W ∨ ), the latter observation translates into f ∈ Sym2 (W1⊥ ) + W2⊥ · W ∨ . Note also
^ 4 ^ 4 µ f, λ(w, γr(l) + γr(m) ) = 2 · (r − l − m) = 2 · r − dim(W1 ) − dim(W2 ) .
ii) Given the weight vector γr(i) , set W B := F v1 , . . . , vi 4. In order to compute the number µ( f, λ(w, γr(i) )), we distinguish three cases: a) The homomorphism f is non-trivial on W B · W B . Then, µ( f, λ(w, γr(i) )) = 2 · (r − dim(W B )). Setting W1 := {0} and W2 := W B , we see f ∈ Sym2 (W1⊥ ) + W2⊥ · W ∨ (= Sym2 (W ∨ )) and µ( f, λ(w, γr(i) )) = 2 · (r − dim(W1 ) − dim(W2 )). b) The homomorphism f is trivial on W B · W B but non-trivial on W B · W. Then, µ( f, λ(w, γr(i) )) = r − 2 · dim(W B ). This time, we set W1 := W2 := W B . We find f ∈ Sym2 (W1⊥ ) + W2⊥ · W ∨ (= W B ⊥ · W ∨ ) and ^ 4 µ f, λ(w, γr(i) ) = r − dim(W1 ) − dim(W2 ). c) The homomorphism f is trivial on W B · W but non-trivial. In this case, we observe µ( f, λ(w, γr(i) )) = −2 · dim(W B ). We define W1 := W B and W2 := W. Again, f ∈ Sym2 (W1⊥ ) + W2⊥ · W ∨ (= Sym2 (W B ⊥ )) and µ( f, λ(w, γr(i) )) = 2 · (r − dim(W1 ) − dim(W2 )). Exercise 2.8.4.9. Let us look at the representation *BB : GLr (') −→ GL(Sym2 (W)) and fix a twisting line bundle L. A *BB -bump is a pair (E, ϕ) which consists of a vector bundle E of rank r and a homomorphism ϕ: Sym2 (E) −→ L. (For r = 3, we studied this example in Section 2.3.3, p. 143f.) In this exercise, you should extend the arguments from the example of Bradlow pairs (p. 307f) to arrive at a simplified version of (semi)stability for *BB -bumps. This time, we let the rational number δ stand for the rational character 2 · det δ· r of GLr ('). i) Suppose that (E, ϕ) is a δ-semistable *BB -bump. Prove that δ ≥ 0. ii) Assume δ = 0. Show that a 0-semistable *BB -bump (E, ϕ) is S-equivalent to the BB * -bump (E, 0) and that (E, 0) is 0-(semi)stable, if and only if E is a (semi)stable vector bundle. Conclude that, for fixed degree d ∈ (, M 0-(s)s (*BB , d, L) is isomorphic to the moduli space M (s)s (r, d) of (semi)stable vector bundles of rank r and degree d. iii) Next, let δ > 0. Show that ϕ & 0 in a δ-semistable *BB -bump (E, ϕ). Conversely, let (E, ϕ) be a *BB -bump in which ϕ is non-trivial. Verify that (E, ϕ) is δ-(semi)stable, if and only if M(E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0 holds for every weighted filtration (E• , α• ) of E. iv) For a subbundle {0} ! F ! E, we set 2, if ϕ|F·F & 0 1, if ϕ|F·F ≡ 0 and ϕF·E & 0 . cϕ (F) := 0, if ϕ ≡0 |F·E
314 S 2.8: D P B S 2.8: D P B 314 A filtration E• : {0} ! E1 ! E2 ! E of E is called critical, if ϕ is trivial on E1 · E2 .28 Prove that (E, ϕ) is δ-(semi)stable, if and only if the following holds: • For every subbundle {0} ! F ! E, µ(F) − δ ·
cϕ (F) 2 (≤)µ(E) − δ · . rk(F) rk(E)
• For every critical filtration E• : {0} ! E1 ! E2 ! E, ^ 4 ^ 4 deg(E) · rk(E1 ) + rk(E2 ) − deg(E1 ) + deg(E2 ) · rk(E)+ ^ 4 +δ · rk(E) − rk(E1 ) − rk(E2 ) (≥)0. v) Set α := (−δ + deg(E))/ rk(E). Use Remark 2.8.4.8, ii), to show that (E, ϕ) is δ-(semi)stable, if and only ^ 4 deg(E) − deg(E1 ) − deg(E2 )(≥)α · rk(E) − rk(E1 ) − rk(E2 ) (2.74) holds for every filtration {0} ⊆ E1 ⊆ E2 ⊆ E of E by subbundles in which either E1 or E2 is a non-trivial proper subbundle and 4 ^ ϕ ∈ H 0 X, Sym2 (E1⊥ ) * L + (E2⊥ · E) * L . Remark 2.8.4.10. The condition δ ≥ 0 translates into the condition deg(E) ≥ α · rk(E). Formally, this is Condition 2.74 for semistability and the filtration with E1 = E2 = {0}.
A similar analysis applies to the SLr (')-action on Sym2 (W). We leave this to the ' reader and introduce only the following piece of notation: For a point b = 1≤i≤ j≤n ai j · wi · w j ∈ Sym2 (W), we define F L ST(w, b) = (i, j) ∈ I | ai j % 0 as the set of states of b and F L M(w, b) = (i, j) ∈ ST(w, b) | (i, j) is maximal with respect to “C” .
Exercise 2.8.4.11. Here, we work with *B : GLr (') −→ GL(Sym2 (W ∨ )) and a twisting line bundle L. A *B -bump is a pair (E, β) in which E is a vector bundle of rank r and β: OX −→ Sym2 (E) * L is a homomorphism. Again, we let the rational number δ stand for the rational character 2 · det δ· r of GLr ('). i) Let (E, β) be a δ-semistable *B -bump. Show that δ ≤ 0(!). ii) Demonstrate that a 0-semistable *B -bump (E, β) is S-equivalent to (E, 0) and that (E, 0) is 0-(semi)stable, if and only if E is a (semi)stable vector bundle. Infer that, for 28 Note
that, for r = 3, this condition is weaker than the one stated in Section 2.3.3, p. 143f.
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315
fixed degree d ∈ (, M 0-(s)s (*B , d, L) is isomorphic to the moduli space M (s)s (r, d) of (semi)stable vector bundles of rank r and degree d. iii) For δ < 0, let α := (−δ + deg(E))/ rk(E). Prove that (E, ϕ) is δ-(semi)stable, if and only ^ 4 deg(E) − deg(E1 ) − deg(E2 )(≥)α · rk(E) − rk(E1 ) − rk(E2 ) (2.75) holds for every filtration {0} ⊆ E1 ⊆ E2 ⊆ E of E by subbundles in which either E1 or E2 is a non-trivial proper subbundle and 4 ^ ϕ ∈ H 0 X, Sym2 (E2 ) * L + (E1 · E) * L . Remark 2.8.4.12. The condition δ ≤ 0 translates into the condition deg(E) ≤ α · rk(E). The latter is Condition 2.75 (with “≤”) for the filtration with E1 = E2 = E.
Finally, we have to investigate the GLr (')-action on Sym2 (W) - Sym2 (W)∨ . We fix a point (b, f ) ∈ Sym2 (W) - Sym2 (W)∨ of which we assume that both components are non-zero. We will have to describe all one parameter subgroups λ: '. −→ GLr (') with the property ^ 4 F L µ (b, f ), λ = max µ(b, λ), µ( f, λ) ≤ 0. A necessary condition for this is that µ(b, λ) + µ( f, λ) ≤ 0.
(2.76)
Formally, we can write any one parameter subgroup λ: '. −→ GLr (') in the form λ = λB + λBB with a one parameter subgroup λBB of SLr (') and the rational number λB = Fλ, det4/r. We then note that µ(b, λ)+µ( f, λ) depends only on λBB . For this reason, we will evaluate (2.76) for one parameter subgroups of SLr ('). We rewrite (2.76) as µ( f, λ) ≤ −µ(b, λ). Let λ = λ(w, γ): '. −→ SLr (') be a one parameter subgroup. We compute µ( f, λ) with the methods outlined before. This means that we find an ordering (i−k− , j−k− ) ! · · · ! (i−1 , j−1 ) ! (i0 , j0 ) ! (i+1 , j+1 ) ! · · · ! (i+k+ , j+k+ )
(2.77)
of the index pairs in M(w, f ), such that ^ 4 ^ 4 γ ∈ C := C − (i−k− , j−k− ), (i0 , j0 ) ∩ · · · ∩ C − (i−1 , j−1 ), (i0 , j0 ) ∩ ^ 4 4 ^ ∩C + (i0 , j0 ), (i+1 , j+1 ) ∩ · · · ∩ C + (i0 , j0 ), (i+k+ , j+k+ ) . The fact −µ(b, λ) ≥ µ( f, λ) means that we find an index pair (ˆı0 , ˆ0 ) ∈ M(w, b), such that F L γ ∈ C ∩ C B , C B = (γ1 , . . . , γr ) ∈ K | − (γi0 + γ j0 ) ≤ −(γıˆ0 + γ ˆ0 ) .
316 S 2.8: D P B S 2.8: D P B 316 To describe the intersection C ∩ C B , we distinguish three cases: a) If (ˆı0 , ˆ0 ) C (i0 , j0 ), then C ∩ C B = C. b) If (ˆı0 , ˆ0 ) is greater than or equal to one of the index pairs appearing in (2.77) with respect to the ordering “C”, then C ∩ C B is a (possibly empty) face of C. c) In the remaining cases, the index pairs in (2.77) which are not greater than or equal to (ˆı0 , ˆ0 ) form a chain (ˆı−kˆ − , ˆ−kˆ − ) ! · · · ! (ˆı−1 , ˆ−1 ) ! (ˆı0 , ˆ0 ) ! (ˆı+1 , ˆ+1 ) ! · · · ! (ˆı+kˆ + , ˆ+kˆ + ), such that
^ 4 ^ 4 C ∩ C B = C − (ˆı−kˆ − , ˆ−kˆ − ), (ˆı0 , ˆ0 ) ∩ · · · ∩ C − (ˆı−1 , ˆ−1 ), (ˆı0 , ˆ0 ) ∩ 4 ^ 4 ^ ∩C + (ˆı0 , ˆ0 ), (ˆı+1 , ˆ+1 ) ∩ · · · ∩ C + (ˆı0 , ˆ0 ), (ˆı+kˆ + , ˆ+kˆ + ) .
So, we may use Claim 2.8.4.7 to determine the generators for this cone. We find some of the form γr(i) and some of the form γr(l) + γr(m) with l < m. For a generator of the form γ := γr(l) + γr(m) , we define W1 := Fw1 , . . . , wl 4 and W2 := Fw1 , . . . , wm 4. Recall that f vanishes on W1 · W2 . The condition µ( f, λ(w, γ)) ≤ −µ(b, λ(w, γ)) translates into b ∈ Sym2 (W2 ) + W1 · W. For a generator of the form γr(i) , we define W B := F w1 , . . . , wi 4 and make the following conventions (compare Remark 2.8.4.8, ii): W1 := {0} and W2 := W B , W1 := W B and W2 := W B , W1 := W B and W2 := W,
if f is non-trivial on W1 · W1 if f is trivial on W1 · W1 and non-trivial on W1 · W if f is trivial on W1 · W.
We then have f ∈ Sym2 (W1⊥ ) + W2⊥ · W and the condition µ( f, λ(w, γ)) ≤ −µ(b, λ(w, γ)) reads b ∈ Sym2 (W2 ) + W1 · W. After these preparations, we are ready to prove the main result: Theorem 2.8.4.13 (Garc´ıa-Prada/Gothen/Mundet i Riera). Let (E, (β, ϕ)) be an ωX twisted affine *-bump, β: OX −→ Sym2 (E) * ωX and ϕ: Sym2 (E) −→ ωX . Fix the stability parameter δ ∈ 3 (representing the rational character 2 · δ · det /r) and define α := −δ + r · deg(E). Then: i) The *-bump (E, (β, ϕ)) is δ-semistable, if and only if ^ 4 deg(E) − deg(E1 ) − deg(E2 ) ≥ α · rk(E) − rk(E1 ) − rk(E2 ) holds for every filtration {0} ⊆ E1 ⊆ E2 ⊆ E of E, such that ϕ ∈ Sym2 (E1⊥ ) + E2⊥ · E
and β ∈ Sym2 (E2 ) + E1 · E.
ii) The *-bump (E, (β, ϕ)) is δ-stable, if and only if ^ 4 deg(E) − deg(E1 ) − deg(E2 ) > α · rk(E) − rk(E1 ) − rk(E2 )
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317
holds for every filtration {0} ⊆ E1 ⊆ E2 ⊆ E of E, such that ϕ ∈ Sym2 (E1⊥ ) + E2⊥ · E
and β ∈ Sym2 (E2 ) + E1 · E
and {0} ! E1 ! E or {0} ! E2 ! E. Proof. First, we explain the semistability conditions for the filtrations with E1 = E2 = {0} and E1 = E2 = E. The semistability condition in the first case implies β ≡ 0 and gives deg(E) ≥ α · r, i.e., δ ≥ 0. In the second case, we find ϕ ≡ 0 and deg(E) ≤ α · r which is equivalent to δ ≤ 0. So, these two filtrations produce the necessary conditions which we have found in Exercise 2.8.4.9, i), and 2.8.4.11, i). Note also that the filtration with E1 = {0} and E2 = E gives no condition. Next, we note that we have already proved the theorem in the cases β ≡ 0 (Exercise 2.8.4.9) and ϕ ≡ 0 (Exercise 2.8.4.11). Hence, we may assume that both β & 0 and ϕ & 0. A generalization of the arguments used in the example of Bradlow pairs and Exercise 2.8.4.9 and 2.8.4.11 shows the following: Claim 2.8.4.14. Let (E, (β, ϕ)) be a *-bump in which both β and ϕ are non-trivial. i) If δ ≥ 0, then (E, (β, ϕ)) is δ-(semi)stable, if and only if M(E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0 holds for every weighted filtration (E• , α• ) of E with µ(E• , α• , β) + µ(E• , α• , ϕ) ≤ 0. ii) If δ ≤ 0, then (E, (β, ϕ)) is δ-(semi)stable, if and only if M(E• , α• ) + δ · µ(E• , α• , β)(≥)0 holds for every weighted filtration (E• , α• ) of E with µ(E• , α• , β) + µ(E• , α• , ϕ) ≤ 0. The discussion preceding the statement of the theorem shows that we have to look only at weighted filtrations of the form ({0} ! E1 ! E, (1)) or ({0} ! E1 ! E2 ! E, (1, 1)) where ϕ ∈ Sym2 (E1⊥ ) + E2⊥ · E
and β ∈ Sym2 (E2 ) + E1 · E.
This observation and the claim easily translate into the assertion of the theorem (compare Exercise 2.8.4.9 and 2.8.4.11). " Fine Tuning Let G be a connected29 reductive group and *: G −→ GL(V) a representation. Recall that the radical of * is defined as ^ 40 Rad(*) := ker(*|Rad(G) ) . 29 The
non-connected case can be dealt with as sketched in Remark 2.7.5.4.
318 S 2.8: D P B S 2.8: D P B 318 Set
GB := G/Rad(*).
Obviously, * factorizes over a representation *B : GB −→ GL(V). Fix a faithful representation κ: GB −→ GL(W) and twisting line bundles L. For a principal G-bundle P, we define P B as the principal GB -bundle that is obtained from P by extension of the structure group via G −→ GB . With these conventions, there is the canonical isomorphism P* ! P*BB . Consequently, a twisted affine *-bump (P, ϕ) induces a twisted affine *B -bump (P B , ϕ). This time, the stability parameter is a rational character χ of GB . We say that a twisted affine *-bump (P, ϕ) is χ-(semi)stable, if the associated twisted affine *B -bump (P B , ϕ) is so. Let T := Rad(G) be the radical of G and T B := Rad(*). As before, there exists a subtorus T BB ⊂ T , such that the multiplication map T B x T BB −→ T is an isomorphism. We let G* ⊂ G be the image of the multiplication map T BB x D(G) −→ G. The new notion of semistability can be rephrased as follows: Lemma 2.8.4.15. A twisted affine *-bump (P, ϕ) is χ-(semi)stable, if and only if the inequality ^ 4 M E• (β), α• (β) + F λ, χ 4(≥)0 holds for every one parameter subgroup λ: '. −→ G* and every reduction β: X −→ P/QG (λ) of P to λ, such that µ(β, ϕ) ≤ 0. We leave it to the reader to define the notion of S-equivalence and the moduli functors and to infer the result on the existence of moduli spaces. Projective *-Bumps This time, we look at a reductive group G and a representation *: G −→ GL(V). For any principal G-bundle P, we have the associated vector bundle P* with fiber V. A projective *-bump is a triple (P, L, ϕ) which consists of a principal G-bundle P on X, a line bundle L on X, and a non-trivial homomorphism ϕ: P* −→ L. Two such projective *-bumps (P1 , L1 , ϕ1 ) and (P2 , L2 , ϕ2 ) are isomorphic, if there are isomorphisms ψ: P1 −→ P2 and χ: L1 −→ L2 , such that ϕ2 = χ ◦ ϕ1 ◦ ψ−1 * . In Section 2.7, we studied and solved the resulting moduli problem in the case that * is homogeneous. Now, we want to explain that we have solved it indeed for arbitrary representations. We stress again that, even for the structure group GLr ('), the inhomogeneous case has not been investigated, so far. In order to apply the theory of affine bumps, we define H := G x '. and η: H −→ Hom(', V). The decomposition Hom(', V) = V = V1 - · · · - Vu
S 2.8: D P B
319
of V into irreducible G-modules is also the decomposition into irreducible H-modules. We choose L = (OX , . . . , OX ). Using these input data, the resulting twisted affine ηbumps are triples (P, L, ϕ) where P is a principal G-bundle on X, L a line bundle, and ϕ: P* −→ L a homomorphism. The isomorphy relation is exactly as for projective *-bumps. The sole difference at the moment is that ϕ is allowed to be trivial for twisted affine η-bumps. The stability parameter for twisted affine η-bumps is a rational character of H. We write such a rational character as a pair (χ, δ) in which χ is a rational character of G and δ is a rational number, representing the character δ · id.. . Likewise, we write a one parameter subgroup of H as (λ, γ) where λ is a one parameter subgroup of G and γ is an integer which stands for the one parameter subgroup z 1−→ zγ of '. . In order to have a semistability concept for the twisted affine η-bumps, we need a faithful representation of H. To this end, we fix a faithful representation κ: G −→ GL(W) and use * κ: H
−→
(g, z) 1−→
* * := W - ', GL(W), W X J g 0 0 z
as the faithful representation of H. As before, we view a principal H-bundle as a pair (P, L) which consists of a principal G-bundle P and a line bundle L. Set E := Pκ . Then, the vector bundle with fiber * that is associated to (P, L) by means of * W κ is E - L. Note that for any one parameter subgroup (λ, γ) of H, we have QH (λ, γ) = QG (λ) x '. . Hence, H/QH (λ, γ) = G/QG (λ), and there is the natural embedding (P, L)/QH (λ, γ) = P/QG (λ) '→ I som(W * OX , E)/QGL(W) (λ).
(2.78)
(We apologize for the obvious abuse of notation.) Hence, a reduction β of the principal H-bundle (P, L) gives a GL-weighted filtration (E• , γ• ) of E. On the other hand, we *• ,* also obtain a GL-weighted filtration (E γ• ) of the vector bundle E - L. Let us clarify how the two objects are related. Suppose γ• = (γ1 , . . . , γ s+1 ). Let j ∈ { 0, . . . , s + 1 } be the index that is determined by the condition γ j < γ ≤ γ j+1 . We assume again that γ < γ j+1 . Then, * γ• = (* γ1 , . . . ,* γ s+2 ) = (γ1 , . . . ., γ j , γ, γ j+1 . . . , γ s+1 ), and the GL*• ,* weighted filtration (E γ• ) features 7 Ei , i ≤ j * . Ei = Ei−1 - L, i ≥ j + 1 We compute: *E *• ,* M( γ• ) =
s+2 T
s+2 ^ 4 T ^ 4 *i ) − deg(E *i−1 ) + *i ) − rk(E *i−1 ) · deg(E) * * γi · deg(E * γi · rk(E
i=1
=
s+1 T
i=1 s+1 ^ 4 T ^ 4 γi · deg(Ei ) − deg(Ei−1 ) + γi · rk(Ei ) − rk(Ei−1 ) · deg(E) +
i=1
* + +γ · deg(L) + γ · deg(E)
i=1 s+1 T i=1
^ 4 γi · rk(Ei ) − rk(Ei−1 ) · deg(L).
320 S 2.8: D P B S 2.8: D P B 320 * = 2 · deg(L) + deg(E). Then, our Set χ0 := deg(L) · det and γ0 := deg(L) + deg(E) computation shows 1 P * E *• ,* * • , γ• ) + (λ, γ), (χ0 , δ0 ) . M( γ• ) = M(E Fix a rational character χ of G and a rational number δ. Then, our calculations and Lemma 2.8.4.2 demonstrate that we may sensibly define a twisted affine η-bump (P, L, ϕ) to be (χ, δ)-(semi)stable, if ^ 4 M E• (β), α• (γ• (β)) + Fλ, χ4 + δ · γ(≥)0 holds for every reduction β of (P, L) to a one parameter subgroup (λ, γ) of H, such that µ(β, ϕ) ≤ 0. We would like to formulate the (semi)stability condition for triples (P, L, ϕ) in terms of reductions of the principal G-bundle P to one parameter subgroups of G. We have already explained that a reduction of the principal H-bundle (P, L) to the one parameter subgroup (λ, γ) of H is the same as a reduction of the principal G-bundle P to λ. Let λ: '. −→ G be a one parameter subgroup of G and β: X −→ P/QG (λ) a reduction of the principal G-bundle P to λ. For any integer γ, we may interpret β as a reduction of the principal H-bundle (P, L) to (λ, γ). If we use this interpretation, we write βγ . Let (P, L, ϕ) be a twisted affine η-bump. For a reduction β: X −→ P/QG (λ) of P to the one parameter λ of G, we set µ(β, ϕ) := µ(β0 , ϕ). For any integer γ, we check that µ(βγ , ϕ) = µ(β, ϕ) + γ. Now, we may proceed as in the example of Bradlow pairs. We first note that δ must be non-negative. If δ = 0, the S-equivalence classes of (χ, 0)-semistable twisted affine η-bumps equal the S-equivalence classes of principal H-bundles, so that we may neglect this case. If δ > 0, then ϕ & 0. So, a (χ, δ)-(semi)stable twisted affine η-bump is indeed a projective *-bump. Hence, we switch to the terminology of projective bumps. At the end, we find that a projective *-bump (P, L, ϕ) is (χ, δ)-(semi)stable, if and only if ^ 4 M E• (β), α• (γ• (β)) + Fλ, χ4 + δ · µ(β, ϕ)(≥)0 holds for every reduction of P to a one parameter subgroup λ: '. −→ G of G. Here, χ is a rational character of G and δ is a positive rational number.
Example 2.8.4.16. We look at the structure group GLr (') and use the identity representation κ: GLr (') −→ GL('r ) for the definitions. Assume that *: GLr (') −→ GL(V) is a homogeneous representation of degree α. Again, we use vector bundles rather than principal GLr (')-bundles. Let (E, L, ϕ) be a *-swamp (=projective *-bump in this situation). By evaluating the semistability condition for the one parameter subgroups λ± : '. −→ GLr ('), z 1−→ z±1 · $r , we see that there are no semistable objects except for δ χ = · α. r
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321
If we make this choice, then we see that every one parameter subgroup λ of the center of GLr (') gives the value zero for the term in the semistability condition. Furthermore, the associated admissible deformation is (E, L, ϕ) itself. Thus, we may exclude one parameter subgroups of the center. In this way, we recover the definition that a *swamp is δ-(semi)stable, if and only if the inequality M(E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0 is satisfied for every weighted filtration (E• , α• ) of E. This closes the circle of our arguments.
2.9 More Generalizations he restriction to a smooth projective curve over the field of complex numbers (or over an algebraically closed field of characteristic theory) was made in order to obtain the results in their strongest and most elegant form. As soon as one relaxes one of the above conditions, one faces serious restrictions and arrives at (even) more technical results. Still, some of these generalizations might be of interest. Hence, we give a brief overview how one may extend the techniques of this book to more general settings and which results one may hope for.
*
2.9.1 Positive Characteristic The first obvious question is whether our results are also true for algebraically closed fields of positive characteristic. To begin with, the theory of vector bundles over curves is independent of the characteristic of the base field. The reader will check that the proofs in Section 2.2 (with the exception of the topological ones) work for all algebraically closed fields. The theory of moduli for the other objects which we have considered such as decorated vector bundles and principal bundles requires additional assumptions in positive characteristic. The reason for the technical difficulties is the fact that reductive affine algebraic groups over fields of positive characteristic are, in general, not linearly reductive. Some useful tools such as the Reynolds operator are not available and several results such as the Hilbert–Mumford criterion over nonalgebraically closed fields (see Chapter 1, Section 1.7) are false, if we work in characteristic p > 0. We will now survey the known results. In the sequel, k denotes an algebraically closed field of positive characteristic p. Decorated Vector Bundles and Tuples of Vector Bundles Let a, b, and c be non-negative integers and *a,b,c : GLr (k) −→ GL(Wa,b,c ) the usual representation on r 4#−c 4$b ^8 ^ kr * . Wa,b,c := (kr )#a The reader may check that all the definitions and constructions in Section 2.3.1-2.3.5 work for the representation *a,b,c in positive characteristic, too. There is one little exception: The tensor product of two semistable vector bundles need not be semistable.
322
S 2.9: G 322
This was used in the proof of Theorem 2.3.4.3. However, by work of A. Langer [131], one can effectively control the defect from being semistable of the tensor product of semistable vector bundles. With a little care, almost the same argument gives then the proof of Theorem 2.3.4.3. The reader may consult [80] for the details. Even most results on asymptotic semistability and the consequences described in Section 2.3.6 and 2.3.7 remain valid, although some additional arguments are required. (The characterization of Remark 2.3.6.4 is however not valid in positive characteristic due to the problems with the instability flag.) These arguments may be found again in [80]. The same can, therefore, be said about representations *: GLr (k) −→ GL(V) where V is a quotient of a module of the form Wa,b,c . Unfortunately, Chapter 1, Corollary 1.1.5.4, is no longer valid in positive characteristic, so that we do not capture all homogeneous representations in that way. Example 2.9.1.1. Let u be a positive integer. The uth divided power is the representation (Symu (id∨GLr (k) ))∨ , i.e., the representation of GLr (k) on ^ 4∨ Du (W) := Symu (W ∨ ) ,
W := kr .
If the characteristic p of the ground field is less than or equal to u, the GLr (k)-module Du (W) is not isomorphic to Symu (W) and cannot be written as the quotient of a module of the form Wa,b,c . Since we are on a curve, we can fix the situation. Let u, v, and w be positive integers and define the GLr (k)-modules ^ 4 Du1 (W) * · · · * Duv (W) , &u,v (W) := (2.79)
%
(u1 ,...,uv ): ' ui ≥0, v ui =u i=1
&u,v,w (W)
:=
&u,v (W)
*
r ^8
W
4#−w
.
Lemma 2.9.1.2. Let *: GLr (k) −→ GL(V) be a homogeneous representation. Then, there exist non-negative integers u, v, and w, such that V is a quotient of the GL(V)module &u,v,w (W). Proof. We refer to [103], Lemma 5.1.5.
"
One can then redo the whole business of Section 2.3.1, 2.3.2, 2.3.4, and 2.3.5 for representations of the form &u,v,w (W) and consequently get the main results on the existence of moduli spaces for semistable *-swamps with respect to an arbitrary homogeneous representation *: GLr (k) −→ GL(V). This is carried out in detail in [103] in the more complicated setting of decorated parabolic vector bundles (compare Section 2.9.3). Even the proof of Theorem 2.3.4.3 can be adapted (see [79]). This brings the results on asymptotic semistability to positive characteristic. (One can, however, not deduce the semistability of the point defined by the decoration at the generic point of X as, e.g., in Remark 2.3.6.4.) So, the important Theorem 2.3.7.1 also holds true. The issues that we have discussed in this section apply to decorated tuples of vector bundles, too. We do not go into more detail here.
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323
Pseudo G-Bundles and Principal G-Bundles The reference for the material in this section are the papers [79] and [80]. The theory of semistable principal G-bundles is a delicate subject in positive characteristic. Fortunately, the fundamental result on the existence of moduli spaces of δ-semistable pseudo G-bundles is valid in any characteristic. Let us have a closer look at the specific ingredients of Section 2.4. First, we note that the set-up described in Section 2.4.1 makes perfect sense in positive characteristic as well. As we will see (Theorem 2.9.1.4 and 2.9.1.7), a slightly weaker form of Theorem 2.4.1.8 is valid, if the characteristic of the ground field is positive. The GIT background outlined in Section 2.4.2 works also over ground fields of positive characteristic. Let us just make the following comments: • In the proof of Lemma 2.4.2.3, one has to observe that, in positive characteristic, the formation of the GIT quotient does not commute with closed embeddings. However, if V is an affine variety on which the reductive group G acts and if W '→ V is a G-invariant closed subvariety, then the induced morphism W//G −→ V//G is injective and proper. This is sufficient for our purposes. • At the end of Section 2.4.1, one has to replace the GL(W)-modules Wa,b,c from Chapter 1, Corollary 1.1.5.4, by the GL(W)-modules &u,v,w (W) from (2.79). Our exposition of the theory of decorated vector bundles in positive characteristic readily implies that the results on pseudo G-bundles contained in Section 2.4.3 do carry over to positive characteristic. The place where things start to become problematic is Section 2.4.3. The problems are the same as those in the theory of the instability flag (Chapter 1, Section 1.7). We first point out that Proposition 2.4.4.5 is also true in positive characteristic. Next, the family of semistable principal G-bundles of fixed topological type is bounded (see [131]). With this one gets the following statement: Proposition 2.9.1.3. There is a rational number δ∞ , such that for every δ > δ∞ , a principal G-bundle (E, τ) is semistable, if and only if (E, τ) is δ-semistable as a pseudo G-bundle. By Lemma 2.4.3.1, given a family of pseudo G-bundles on X parameterized by the scheme S , the locus S B parameterizing principal G-bundles is open. It is also straightforward to check with Proposition 2.4.4.5 that the set of isomorphy classes of semistable principal G-bundles is closed under S-equivalence within the family of isomorphy classes of δ-semistable pseudo G-bundles, assuming δ > δ∞ as in Proposition 2.9.1.3. Going to the GIT construction in Section 2.4.5, this means that the locus YB ⊂ Yδ-ss parameterizing the semistable principal G-bundles is actually a saturated GL(Y)-invariant open subset. This signifies that, given a point y ∈ YB , the orbit closure GL(Y) · y '→ Yδ-ss is entirely contained in YB . The saturatedness property, in turn, implies that the categorical quotient YB // GL(Y) exists as an open subscheme of the categorical quotient Yδ-ss // GL(Y). We have thus shown:
324 S 2.9: G
S 2.9: G 324
Theorem 2.9.1.4. There exists a quasi-projective moduli space M ss (ϑ) for the moduli functors M(s)s (ϑ). The remaining question is to find sufficient conditions for the properness of the moduli spaces. One method is to use an analog of Theorem 2.4.4.1. We may offer the following: Theorem 2.9.1.5 (Semistable reduction). Assume that κ: G −→ GL(W) is of low separable index or that κ is the adjoint representation and is of low height. Then, there is a rational number δB∞ , such that, for every δ > δB∞ , a δ-semistable pseudo G-bundle (E, τ) is a principal G-bundle. Remark 2.9.1.6. i) The low separable index is defined in [11], Definition 6. ii) For the notion of low height representations, we refer the reader to [148] and [196]. The low height assumption for the adjoint representation amounts to the following restrictions on the characteristic of the base field: • Char(k) > 2n, if G contains a simple factor of type An . • Char(k) > 4n − 2, if G contains a simple factor of type Bn , or Cn . • Char(k) > 4n − 6, if G contains a simple factor of type Dn . • Char(k) ≥ 11, if G contains a simple factor of type G2 . • Char(k) ≥ 23, if G contains a simple factor of type F4 or E6 . • Char(k) ≥ 37, if G contains a simple factor of type E7 . • Char(k) ≥ 59, if G contains a simple factor of type E8 . iii) The lifting method of Ramanathan-G´omez-Sols works also in positive characteristic (see [80]). Using this and a few extra arguments (see [79]), one can show that one has a projective moduli space for semistable principal G-bundles with connected reductive structure group G of fixed topological type ϑ, and k of arbitrary characteristic, if the simple factors are all of type A, k of characteristic at least 3, if the simple factors are all of type A, B, C, D, and Char(k) satisfying the bounds in ii), if the respective exceptional groups occur as simple factors. The other approach is to give an independent proof of the semistable reduction theorem as stated in Remark 2.4.4.2, ii). This approach was pursued in [11], the proof of the semistable reduction theorem being based on Bruhat-Tits theory. A new and very efficient approach based on the theory of affine Graßmannians has been recently devised by Heinloth ([101] and [102]). It gives the following result: Theorem 2.9.1.7. The moduli space M ss (ϑ) is proper, if the following conditions are met: • Char(k) ≥ 3, if G contains a simple factor of type G2 . • Char(k) ≥ 11, if G contains a simple factor of type F4 or E6 . • Char(k) ≥ 17, if G contains a simple factor of type E7 . • Char(k) ≥ 37, if G contains a simple factor of type E8 .
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Decorated Pseudo G-Bundles The results on decorated vector bundles and decorated tuples of vector bundles in positive characteristic permit to develop the theory of decorated pseudo G-bundles, as introduced in Section 2.7.2, up to (and including) Theorem 2.7.2.4 in positive characteristic as well. To get the results on moduli spaces of decorated principal G-bundles in Section 2.7 and 2.8 in characteristic zero, one has to carry out a subtle analysis of the asymptotic behavior of the semistability concept in various directions. This analysis is, as it stands, based on the results on the instability flag presented in Chapter 1, Section 1.7. These results do not work over non-perfect fields of positive characteristic (such as the function field of the curve X). At the moment, we do not have an elegant statement how the conditions look like so that, e.g., the main result of Section 2.8 does hold. We will therefore not give any additional theorems in that direction. It is our hope that in important special cases, such as Higgs bundles, one may more easily come by explicit and low bounds on the characteristic of the ground field k which grant the existence of moduli spaces.
2.9.2 Higher Dimensional Base Varieties Here, we will discuss to what extent the results of Chapter 2 generalize to base varieties of dimension two or higher. We assume that the ground field is ' and make some comments on positive characteristic from time to time. Torsion Free Sheaves Let X be a connected smooth projective variety over the field of complex numbers. We denote by '(X) its function field and the associated constant sheaf. Using the latter interpretation, there is the injective homomorphism OX −→ '(X) of OX -modules. For a coherent sheaf F on X, we have therefore a homomorphism lF : F −→ F *OX '(X). The kernel Tors(F ) of lF is called the torsion subsheaf of F . We say that F is torsion free, if Tors(F ) = {0}. If F = Tors(F ), then F is said to be a torsion sheaf. Remark 2.9.2.1. i) For any point x ∈ X, the stalk at x of the torsion subsheaf of F is computed to be F L Tors(F ) x = f ∈ F x | ∃ h ∈ OX,x \ {0} : h · f = 0 . ii) The stalk
lF ,x : F x −→ F x *OX,x
'(X)
at a point x ∈ X is the localization at the generic point. iii) One checks that F is a torsion sheaf, if and only if the support of F is a proper closed subset of X. iv) A sheaf F is torsion free, if and only if the support of every non-trivial subsheaf of F is the whole of X. v) Let F be a coherent OX -module of rank r. There is the maximal open subset UF ⊂ X where F is locally free. If F is torsion free, then UF is a big open subset, i.e., the complement X \ UF has codimension at least two.
326 S 2.9: G
326 S 2.9: G
Unlike the situation on curves, we have to fix an additional structure on X in order to introduce a notion of semistability. This extra structure is an ample line bundle OX (1) on X. Having fixed OX (1), we associate to every coherent OX -module F its Hilbert polynomial P(F ). It is the polynomial of degree at most dim(X) with the property ^ 4 ∀n ∈ ( : P(F )(n) = χ F (n) . Furthermore, the degree of F is the integer ^ 4dim(X)−1 ∈ H 2·dim(X) (X, () ! (. deg(F ) := c1 (F ).c1 OX (1) The latter isomorphism comes from the natural orientation of X as a topological manifold. Remark 2.9.2.2. i) By the theorem of Riemann–Roch ([96], Appendix A, Theorem 4.1), we have P(F )(n) =
^ 4 ndim(X) ^ 4 1 ndim(X)−1 r · deg OX (1) · + deg(E ) − r deg(KX ) · + dim(X)! 2 (dim(X) − 1)! + lower order terms, n ∈ (.
ii) For a non-trivial torsion sheaf F , we have rk(F ) = 0, P(F ) : 0, and deg(F ) ≥ 0 (with “>” holding, if and only if the support of F contains a divisor). A coherent OX -module E is said to be slope (semi)stable, if one has deg(F ) · rk(E )(≤) deg(E ) · rk(F ) for every subsheaf {0} ! F ! E . If the torsion subsheaf F := Tors(E ) of E has support on a divisor, the deg(F ) > 0 and F will violate this condition, so that a slope semistable OX -module is automatically “torsion free in codimension one”. For our purposes, we will admit only torsion free sheaves. For a non-zero torsion free OX -module, we may rewrite the condition of slope (semi)stability as the condition µ(F ) :=
deg(F ) (≤)µ(E ) rk(F )
for every subsheaf {0} ! F ! E . The rational number µ(E ) is referred to as the slope of E . A coherent OX -module E is said to be (semi)stable, if one has P(F ) · rk(E )(C)P(E ) · rk(F ) for every subsheaf {0} ! F ! E . The symbols “≺” and “C” refer to the lexicographic ordering of polynomials (see [216], §33). Remark 2.9.2.3. i) In view of Remark 2.9.2.2, i), we have the implications E slope stable =⇒ E stable =⇒ E semistable =⇒ E slope semistable.
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ii) Remark 2.9.2.2, ii), implies that a semistable sheaf is torsion free: Just plug in the torsion subsheaf into the semistability condition. iii) Slope (semi)stability is also referred to as Mumford–Takemoto-(semi)stability and (semi)stability as Gieseker–Maruyama-(semi)stability. Let E be a torsion free sheaf. A subsheaf F ⊂ E is called saturated, if E /F is a torsion free. The saturation of a subsheaf F ⊂ E is ^ 4 Fsat := ker E −→ (E /F )/Tors(E /F ) . Obviously, Fsat is a saturated subsheaf of E which coincides with F at the generic point. By Remark 2.9.2.2, we have rk(Fsat ) = rk(F ),
deg(Fsat ) ≥ deg(F ),
and
P(Fsat ) 8 P(F ).
It therefore suffices to check the condition of (slope) (semi)stability for saturated subsheaves. Example 2.9.2.4. If X is a curve, E a vector bundle on X, and F ⊂ E a subsheaf, then Fsat is the subbundle generated by F . Let S be a scheme. A family of torsion free sheaves parameterized by S is a coherent sheaf ES on S x X which is flat over S , such that ES |{s} x X is torsion free for every point s ∈ S . Remark 2.9.2.5. Let ES be a family of torsion free sheaves on S x X. Then, there is the maximal open subset UES ⊂ S x X where ES is locally free. For every point s ∈ S , we have ^ 4 UES ∩ {s} x X = UES |{s} x X , i.e., UES ∩ ({s} x X) is the maximal (big) open subset of X where the torsion free sheaf ES |{s} x X is locally free. The proof is given in [116], Lemma 2.1.7. There are the moduli functors M(s)s (P): Sch. S
−→ Sets Isomorphy classes of families of (semi)stable torsion free sheaves ES on S x X 1−→ with P(E ) = P for all s ∈ S S |{s} x X
.
The notion of S-equivalence from Section 2.2.4 readily generalizes to the setting of torsion free sheaves (see [116], Definition 1.5.3). Theorem 2.9.2.6. Let (X, OX (1)) be a connected smooth projective variety endowed with an ample line bundle and fix a polynomial P. Then, there exists a projective moduli scheme M ss (P) for the functors M(s)s (P). Proof. The theorem was originally proved over surfaces by Gieseker [75]. His proof was extended to higher dimensional base varieties by Maruyama [143], [144]. A new proof was later given by Simpson [204]. Simpson’s techniques work more generally on a polarized projective scheme. Recently, Langer proved in [129] and [130] the missing
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boundedness results and estimates on the number of global sections which permit to establish the result also over algebraically closed fields of positive characteristic. Simpson’s construction over a base field of characteristic zero is exposed in [116], Section 4. " Remark 2.9.2.7. The theory of slope semistability does, in general, not lead to good moduli functors and one does not have moduli spaces. There is a certain substitute for the moduli space of slope semistable sheaves over surfaces, due to Li (see [116], Section 8.2). It played a major rˆole in the application of Algebraic Geometry to Donaldson theory. Decorated Sheaves First, we fix non-negative integers a, b, and c, and the representation *a,b,c : GL(W) −→ GL(Wa,b,c ), W := 'r , r 4#−c ^8 W . Wa,b,c := (W #a )$b * For a torsion free sheaf E of rank r, we then define Ea,b,c := (E #a )$b * det(E )#−c . Next, let u ∈ NS(X) be a class in the N´eron–Severi group of X. There is a moduli space Picu (X) for the line bundles L on X whose algebraic equivalence class is u. We also fix a polynomial ^ 4 ^ 4 tdim(X) 1 tdim(X)−1 + d − · r · deg(KX ) · + ···. P = r · deg OX (1) · dim(X)! 2 (dim(X) − 1)! A *a,b,c -swamp of type (P, u) is a triple (E , L, ϕ) which consists of a torsion free OX module E with Hilbert polynomial P(E ) = P (this implies rk(E ) = r, by Remark 2.9.2.2, i), a line bundle L whose algebraic equivalence class is u, and a non-zero homomorphism ϕ: Ea,b,c −→ L. Let E1 and E2 be two torsion free OX -modules and ψ: E1 −→ E2 an isomorphism. There is the induced isomorphism ψa,b,c : E1,a,b,c −→ E2,a,b,c. We say that the *a,b,c -swamp (E1 , L1 , ϕ1 ) is isomorphic to the *a,b,c -swamp (E2 , L2 , ϕ2 ), if there are isomorphisms ψ: E1 −→ E2 and χ: L1 −→ L2 , such that ϕ2 = χ ◦ ϕ1 ◦ ψ−1 a.b,c . These concepts easily generalize to families. For this, we have to fix a Poincar´e line bundle Lu on Picu (X) x X. If S is a scheme and κS : S −→ Picu (X) is a morphism, we define L [κS ] as the pullback of Lu to S x X via κS x idX . A family of *a,b,c -swamps of type (P, u) parameterized by the scheme S is a quadruple (ES , κS , NS , ϕS ) in which ES is an S -flat family of torsion free sheaves with Hilbert polynomial P parameterized by the scheme S , κS : S −→ Picu (X) is a morphism, NS is a line bundle on S , and ϕS : ES ,a,b,c −→ L [κS ] * π.S (NS ) is a homomorphism whose restriction to {s} x X is non-trivial for every point s ∈ S .
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Remark 2.9.2.8. In general, we may not expect that the sheaf ES ,a,b,c will be again flat over S . The fibers over S of this sheaf, i.e., its restrictions to subschemes of the form {s} x X, s ∈ S , will usually acquire torsion, if ES |{s} x X is not locally free. We say that the families (ES ,1 , κS ,1 , NS ,1 , ϕS ,1 ) and (ES ,2 , κS ,2 , NS ,2 , ϕS ,2 ) are isomorphic, if κS ,1 = κS ,2 =: κS and there are isomorphisms ψS : ES ,1 −→ ES ,2 and χS : NS ,1 −→ NS ,2 , such that ^ 4 ϕS ,2 = id L [κS ] * π.S (χS ) ◦ ϕS ,1 ◦ ψ−1 S ,a,b,c . Next, we have to define (semi)stability. First, let E be a torsion free sheaf. A weighted filtration of E is a pair (E• , α• ) which consists of a filtration E• : {0} ! E1 ! · · · ! E s ! E =: E s+1 of E by saturated subsheaves and a tuple α• = (α1 , . . . , α s ) of positive rational numbers. For such a weighted filtration (E• , α• ), we define the polynomial s T ^ 4 M(E• , α• ) := αi · P(E ) · rk(Ei ) − P(Ei ) · rk(E ) . i=1
By Remark 2.9.2.2, i), the degree of this polynomial is at most dim(X) − 1. To a weighted filtration (E• , α• ), we associate as in (2.6) the weight vector 4 ^ γ1 , . . . , γ1 , γ2 , . . . , γ2 , . . . , γ s+1 , . . . , γ s+1 . D!!!!!!!!!!WB!!!!!!!!!!\ D!!!!!WB!!!!!\ D!!!!!WB!!!!!\ rk(E1 ) x
(rk(E2 )−rk(E1 )) x
(rk(Es+1 )−rk(Es )) x
If ϕ: Ea,b,c −→ L is a non-zero homomorphism, we may define the rational number A ! µ(E• , α• , ϕ) := − min γi1 + · · · + γia !! (i1 , . . . , ia ) ∈ { 1, . . . , s + 1 }x a : ϕ is non-trivial > on (Ei1 * · · · * Eia )$b * det(E )#−c . This time, the stability parameter is a polynomial δ ∈ 3[t] which is positive (in the lexicographic ordering of polynomials) and has degree at most dim(X) − 1. Fix such a polynomial δ. A *a,b,c -swamp (E , L, ϕ) is said to be δ-(semi)stable, if the inequality M(E• , α• ) + δ · µ(E• , α• , ϕ)(8)0 between polynomials in the lexicographic ordering is satisfied for every weighted filtration (E• , α• ) of E . For fixed data P, u, and δ, we have the moduli functors Mδ-(s)s (*a,b,c , P, u): Sch. −→ Sets Isomorphy classes of families of δ-(semi)stable * -swamps S 1−→ . a,b,c of type (P, u) parameterized by S There are notions of polystable *a,b,c -swamps and of S-equivalence which naturally generalize the corresponding concepts from Section 2.3.2 (see [80]).
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Theorem 2.9.2.9. There exists a projective moduli scheme M δ-(s)s (*a,b,c , P, u) for the functors Mδ-(s)s (*a,b,c , P, u).
Proof. For the ground field ', this is Theorem 1.8 in [82]. Thanks to the work of A. Langer ([129], [130]), the construction extends to algebraically closed ground fields of positive characteristic as well (see [80]). "
Remark 2.9.2.10. As in Section 2.3.6, one can study the behavior of (semi)stability when the stability parameter δ becomes very large. This is done in [189] for the ground field ' and in [80] for algebraically closed ground fields of arbitrary characteristic. It is quite remarkable that the qualitative result is the same regardless of the characteristic of the ground field. Finally, we may look at a homogeneous representation *: GLr (') −→ GL(V). The problem for more general representations is that we do not have a good definition of E* , if E is a non locally free torsion free sheaf. Thus, we restrict to the open set UE ⊂ X over which E is locally free and define E* := (E|UE )* . A *-swamp is a triple (E , L, ϕ) which consists of a torsion free OX -module E , a line bundle L, and a non-zero homomorphism ϕ: E* −→ L|UE . The type of (E , L, ϕ) is the pair (P, u) in which P is the Hilbert polynomial of E and u ∈ NS(X) is the algebraic equivalence class of L. The *-swamp (E1 , L1 , ϕ1 ) is isomorphic to the *-swamp (E2 , L2 , ϕ2 ), if there are isomorphisms ψ: E1 −→ E2 and χ: L1 −→ L2 , such that ϕ2 = χ ◦ ϕ1 ◦ ψ−1 * ,
ψ* : E1,* −→ E2,* the induced isomorphism.
Remark 2.9.2.11. The first thing which we have to explain is that we get the same objects as before, if we start with the representation *a,b,c . First, suppose we are given (E , L, ϕ) with ϕ: Ea,b,c −→ L. Obviously, we may restrict ϕ to the open subset UE in order to get * ϕ: E*a,b,c = (Ea,b,c)|UE −→ L|UE .
(2.80)
Next, suppose we are given (E , L, * ϕ) with * ϕ as in (2.80). Writing ι: UE −→ X for the inclusion, we define ^ 4 ι. (* ϕ) ϕ: Ea,b,c −→ ι. ι. (Ea,b,c) = ι. (E*a,b,c ) −→ ι. (L|UE ) = L. The last equality follows, because UE is big and X is a normal variety. Finally, we have to show that the two constructions we have just outlined are inverse to each other. It suffices to verify that, for a coherent OX -module F , and a torsion free sheaf G of OX -modules, two homomorphisms ϕ1,2 : F −→ G are equal, if and only if they agree at the generic point of X. Look at F / ker(ϕ1 − ϕ2 ). If ϕ1 and ϕ2 agree at the generic point, then this is a torsion sheaf (see Remark 2.9.2.1, iii). At the same time, it is a subsheaf of the torsion free OX -module G . Hence, it is trivial and ker(ϕ1 −ϕ2 ) = F .
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Example 2.9.2.12. i) Set W := 'r , H := Hom(W, W * W), and let *: GL(W) −→ GL(H) be the natural representation. One easily sees that a *-swamp is the same as a triple (E , L, ϕ) with E a torsion free sheaf, L a line bundle, and ϕ: E
*
E −→ E ∨∨ * L
a non-zero homomorphism. One may impose extra (closed) conditions, e.g., L ! OX and that ϕ satisfies the Jacobi identity. The resulting moduli problem plays a central rˆole in the work [83] of G´omez and Sols on principal G-bundles on higher dimensional varieties. ii) In some applications, the set-up which we have introduced above might be too general. E.g., let W := 'r , H := Hom(W, W), and *: GL(W) −→ GL(H) the natural representation. Then, a *-swamp is a triple (E , L, ϕ) with E a torsion free sheaf, L a line bundle, and ϕ: E −→ E ∨∨ * L a non-zero homomorphism. However, one would rather look at triples (E , L, * ϕ) with E a torsion free sheaf, L a line bundle, and * ϕ: E −→ E
*
L
a non-zero homomorphism. We will briefly explain in Remark 2.9.2.16 how we may integrate the solution of the corresponding moduli problem into our framework. Note also that, for a torsion free sheaf E , the restriction map H 0 (X, E nd(E ) * L) −→ H 0 (UE , E nd(E|UE ) * L|UE ) is always injective but might fail to be surjective. This might lead to some extra components in the boundary of the moduli spaces which we are going to construct. Fix a polynomial P, u ∈ NS(X), and a Poincar´e line bundle Lu on Picu (X) x X. Let S be a scheme. A family of *-swamps of type (P, u) parameterized by S is a quadruple (ES , κS , NS , ϕS ) in which ES is an S -flat family of torsion free sheaves with Hilbert polynomial P parameterized by the scheme S , κS : S −→ Picu (X) is a morphism, NS is a line bundle on S , and ^ 4 ϕS : ES ,* := (ES |UES )* −→ L [κS ] * π.S (NS ) |UES
is a homomorphism whose restriction to ({s} x X) ∩ UES is non-trivial for every point s ∈ S . The families (ES ,1 , κS ,1 , NS ,1 , ϕS ,1 ) and (ES ,2 , κS ,2 , NS ,2 , ϕS ,2 ) are isomorphic, if κS ,1 = κS ,2 =: κS and there are isomorphisms ψS : ES ,1 −→ ES ,2 and χS : NS ,1 −→ NS ,2 , such that 4 ^ ϕS ,2 = id L [κS ] * π.S (χS ) ◦ ϕS ,1 ◦ ψ−1 S ,* . Again, ψS ,* : ES ,1,* −→ ES ,2,* is the isomorphism induced by ψS . Remark 2.9.2.13. i) Remark 2.9.2.5 demonstrates that the above notion of a family makes sense and that we have the operation of pullback of families via a base change morphism T −→ S . ii) The content of Remark 2.9.2.11 remains true for families. This is a consequence of the following result due to Maruyama ([143], p. 111f, see also [186], Proposition 2.1, and [82], Lemma 0.9):
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Theorem 2.9.2.14. Let S be a scheme and U ⊂ S x X an open subset with the property that the codimension of U ∩ ({s} x X) in X is at least two for every point s ∈ S . Writing ι: U −→ S x X for the inclusion, the equality ES = ι. (ES |U ) holds for every locally free sheaf ES on S x X. Suppose (E , L, ϕ) is a *-swamp and that (E• , α• ) is a weighted filtration of E . We may use the same technique of restricting to the generic point as in Section 2.3.2 in order to define µ(E• , α• , ϕ). Let δ be a positive rational polynomial of degree at most dim(X) − 1. We say that a *-swamp (E , L, ϕ) is δ-(semi)stable, if M(E• , α• ) + δ · µ(E• , α• , ϕ)(8)0 holds for every weighted filtration of E . We skip again the definition of polystability and S-equivalence. Fix the data P, u, and δ. Then, we may define the moduli functors Mδ-(s)s (*, P, u): Sch. S
−→ Sets Isomorphy classes of families of δ-(semi)stable *-swamps 1−→ of type (P, u) parameterized by S
.
Theorem 2.9.2.15. There is a projective moduli scheme M δ-(s)s (*, P, u) for the functors Mδ-(s)s (*, P, u). Proof. According to Chapter 1, Corollary 1.1.5.4, we may find non-negative integers a, b, and c, such that *a,b,c = * - *. δ-(s)s It suffices to show that M (*, P, u) is a closed subfunctor of Mδ-(s)s (* , P, u). a,b,c
We first have to define a natural transformation between those functors. So let S be a scheme and (ES , κS , NS , * ϕS ) a family of δ-(semi)stable *-swamps of type (P, u) parameterized by S . Let ι: UES −→ S x X be the inclusion. We have the direct sum decomposition ES ,*a,b,c = ES ,* - ES ,* . First, we define 4 * ϕS ^ ϕBS : ES ,*a,b,c −→ ES ,* −→ L [κS ] * π.S (NS )
|UES
and then
ι. (ϕBS )
ϕS : ES ,a,b,c −→ ι. (ES ,*a,b,c ) −→ L [κS ] * π.S (NS ).
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333
Here, we have used Theorem 2.9.2.14 which implies that ^ 4 L [κS ] * π.S (NS ) = ι. (L [κS ] * π.S (NS ))|UES . As in the case of curves, it is straightforward to check that this assignment is compatible with (semi)stability, so that (ES , κS , NS , ϕS ) is a family of δ-(semi)stable *a,b,c -swamps of type (P, u) parameterized by S . Next, let (ES , κS , NS , ϕS ) be a family of δ-(semi)stable *a,b,c -swamps of type (P, u) parameterized by S . There is the restriction homomorphism ^ 4 res: ES ,a,b,c −→ ι. ι. (ES ,a,b,c) = ι. (ES ,*a,b,c ). Set
^ 4 E*S ,* := res−1 ι. (ES ,* ) .
Since S x X is a noetherian scheme and ES ,a,b,c is a coherent OS x X -module, the OX module E*S ,* is coherent, too. By Exercise 2.3.5.3, the locus where the homomorphism E*S ,* −→ L [κS ] * π.S (NS ) induced by ϕS vanishes carries the structure of a closed subscheme S of S . The subscheme S is characterized by the condition that ϕS |UES factorizes over the projection ES ,*a,b,c −→ ES ,* . " Remark 2.9.2.16. i) If k is an algebraically closed ground field of positive characteristic, then the above proof only works for homogeneous representations which are quotients of representations of the form *a,b,c . Recall from Example 2.9.1.1 that these are not all representations. On higher dimensional base varieties over fields of positive characteristics, we have no result for divided powers. ii) We return to the set-up W := 'r , H := Hom(W, W), and the canonical representation *: GL(W) −→ GL(H). We would like to classify triples (E , L, * ϕ) with E a torsion free sheaf, L a line bundle, and * ϕ: E −→ E * L a non-zero homomorphism. We leave it to the reader to define the moduli functors for these objects. Given such a triple (E , L, * ϕ), we may restrict * ϕ to the open subset UE and define a *-swamp (E , L, ϕ). The same construction works in families and leads to a natural transformation from the moduli functor for triples (E , L, * ϕ) into the moduli functor for *-swamps. We may construct a parameter space for triples (E , L, * ϕ) which is projective over some quot scheme. Likewise, we have a parameter scheme for *-swamps which is projective over the same quot scheme. The natural transformation leads to an injective morphism relative to the quot scheme from the parameter space of the triples to the parameter space of *-swamps. The usual arguments then show that the existence of a projective moduli scheme for *-swamps implies the existence of a projective moduli scheme for triples (E , L, * ϕ: E −→ E * L). Bradlow Pairs Let us look at the representation *: GLr (') −→ GL(Hom('r , ' s )). We take all twisting sheaves to be equal to a fixed line bundle L. Then, a *-bump is a pair (E , ϕ) which consists of a torsion free sheaf E and a homomorphism ϕ: OU$ s −→ (E * L)|U .
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Exercise 2.9.2.17. We fix a positive rational polynomial δ of degree at most dim(X)−1. Show that a *-bump (E , ϕ) is (−δ)-(semi)stable, if and only if ϕ & 0 and for every saturated subsheaf {0} ! F ! E : δ P(E ) δ P(F ) + (C) + , if Im(ϕ|U ) ⊂ (F * L)|U , rk(F ) rk(F ) rk(E ) rk(E ) P(F ) P(E ) δ (C) + , otherwise. rk(F ) rk(E ) rk(E ) Let us fix a torsion free reference sheaf E0 . A Bradlow pair of type (E0 , P) is a pair (E , ϕ) with a torsion free sheaf E with Hilbert polynomial P and a homomorphism ϕ: E0 −→ E . Two pairs (E , ϕ) and (E B , ϕB ) are isomorphic, if there is an isomorphism ψ: E −→ E B with ϕB = ψ ◦ ϕ. Let δ be a positive rational polynomial of degree at most dim(X) − 1. We call a Bradlow pair (E , ϕ) δ-(semi)stable, if and only if ϕ & 0 and for every saturated subsheaf {0} ! F ! E : δ P(E ) δ P(F ) + (C) + , if Im(ϕ) ⊂ F , rk(F ) rk(F ) rk(E ) rk(E ) P(E ) δ P(F ) (C) + , otherwise. rk(F ) rk(E ) rk(E ) We choose a line bundle L and a surjection τ: L∨ $ s −→ E0 . Let (E , ϕ) be a Bradlow pair. Then, ϕ ◦ τ: L∨ $ s −→ E gives rise to a homomorphism * ϕ: OX$ s −→ E
*
L.
The assignment (E , ϕ) −→ (E , * ϕ) is injective and compatible with isomorphisms. We have to check that is also compatible with the notions of semistability. For a saturated subsheaf F of E , the fact that Im(ϕ) ⊂ F is equivalent to the property Im(* ϕ) ⊂ F * L. We have to verify that the latter condition is equivalent to Im(* ϕ|U ) ⊂ (F * L)|U , U the maximal open subset where E is locally free. Note that Im(* ϕ) ⊂ F * L holds precisely when the homomorphism ϕ: OX$ s
* ϕ
7E
*
L
7 (E /F ) * L
is trivial. Now, OX$ s and (E /F ) * L are torsion free sheaves. So, ϕ is zero, if and only if ϕ|U is zero. The latter condition is equivalent to the condition Im(* ϕ|U ) ⊂ (F * L)|U . Finally, let (E , ϕ) be a *-bump where * ϕ is defined on all of X. We obtain * ϕt : L∨
$s −→ E .
Then, (E , * ϕ) will come from a Bradlow pair, if and only if * ϕt vanishes on ker(τ). δ-(s)s (E0 , P) for δ-(semi)stable Bradlow pairs. TakWe define the moduli functors M ing Remark 2.9.2.16, ii), into account, we see: Theorem 2.9.2.18. For given type (E0 , P) and stability parameter δ, there exists a projective moduli scheme M δ-ss (E0 , P) for the functors Mδ-(s)s (E0 , P). I was informed by R. Thomas that the corresponding moduli problem for sheaves with proper support rather than torsion free sheaves is interesting in connection with the work [168].
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Decorated Tuples of Vector Bundles There is also a generalization of the results in Section 2.5 to higher dimensional base varieties. It is explained in the paper [190]. We give here a very brief summary. We fix the following parameters: • a finite index set V, • a tuple r = (rv , v ∈ V) of positive integers, • a tuple κ = (κv , v ∈ V) of positive integers, • a tuple η = (ηv , v ∈ V) of rational numbers subject to the condition 0, and
'
v∈V
ηv · rv =
• a positive polynomial δ ∈ 3[x] of degree at most dim(X) − 1. We also define
χ = (χv , v ∈ V)
with χv := δ · ηv , v ∈ V.
We may clearly speak of the category of V-split sheaves on X (compare Section 2.5.3). The (κ, χ)-Hilbert polynomial of the V-split sheaf (Fv , v ∈ V) is Pκ,χ (Fv , v ∈ V) :=
T^
4 κv · P(Fv ) − χv · rk(Fv ) .
v∈V
The κ-rank of the V-split sheaf (Fv , v ∈ V) is the integer T κv · rk(Fv ). rkκ (Fv , v ∈ V) := v∈V
A torsion free V-split sheaf is a V-split sheaf (Ev , v ∈ V) in which Ev is a torsion free OX -module, v ∈ V. Let (Ev , v ∈ V) be a torsion free V-split sheaf. A weighted filtration of (Ev , v ∈ V) is a pair (E• , α• ) which consists of a filtration E• : {0} ! (E1v , v ∈ V) ! · · · ! (E sv , v ∈ V) ! (Ev , v ∈ V) of the V-split sheaf (Ev , v ∈ V) in which Eiv is a saturated subsheaf of Ev , v ∈ V, i = 1, . . . , s, and a tuple α• = (α1 , . . . , α s ) of positive rational numbers. For such a weighted filtration, we introduce the polynomial Mκ,χ (E• , α• ) =
s T i=1
4 ^ αi Pκ,χ (Ev , v ∈ V)rkκ (Eiv , v ∈ V) − Pκ,χ (Eiv , v ∈ V)rkκ (Ev , v ∈ V) .
This polynomial has degree at most dim(X) − 1. As in Section 2.5, we look at the structure group GL(V, r) := Xv∈V GLrv (') and ' define W(κ, r) := ' v∈V κv ·rv as well as the GL(V, r)-modules '
W(κ, r)a,b,c
κv ·rv Z#−c 8 ^ 4$b & v∈V := W(κ, r)#a * , W(κ, r)
a, b, c ∈ (≥0 .
336 S 2.9: G We write
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4 ^ *a,b,c : GL(V, r) −→ GL W(κ, r)a,b,c ,
a, b, c ∈ (≥0 ,
for the corresponding representations. For the following, we fix the non-negative integers a, b, and c and a class u ∈ NS(X). If (Fv , v ∈ V) is a V-split sheaf, we set Ftotal := and
% F$ v∈V
v
κv
^ #a 4$b #−c . Fa,b,c := Ftotal * det(F )
A *a,b,c -tump of type (P, u) is a tuple (Ev , v ∈ V, L, ϕ) where (Ev , v ∈ V) is a V-split vector bundle, L is a line bundle whose algebraic equivalence class is u, ϕ: Ea,b,c −→ L is a non-trivial homomorphism, and P = (P(Ev ), v ∈ V). The notion of isomorphy is as for tumps over curves. For a *a,b,c -tump (Ev , v ∈ V, L, ϕ) and a weighted filtration (E• , α• ) of (Ev , v ∈ V), we set F L µ(E• , α• , ϕ) := − min γi1 + · · · + γia | ϕ|(Eitotal #···#Eitotal )"b is non-trivial . (i1 ,...,ia )∈ { 1,...,s+1 }x a
Here,
and, with r =
1
Eitotal := '
v∈V
%E $ v∈V
v κv , i
a
i = 1, . . . , s,
κv rv and ri := rk Eitotal , i = 1, . . . , s,
(γ1 , . . . , γ s+1 ) =
s T i=1
αi · (ri − r, . . . , ri − r, ri , . . . , ri ). D!!!!!!!!!!!!!WB!!!!!!!!!!!!!\ D!!!WB!!!\ ri x
(r−ri ) x
A *a,b,c -tump (Ev , v ∈ V, L, ϕ) is said to be (κ, η, δ)-(semi)stable, if Mκ,χ (E• , α• ) + δ · µ(E• , α• , ϕ)(8)0 is verified for every weighted filtration (E• , α• ) of (Ev , v ∈ V). We leave it to the reader to explain the moduli functors M(κ,η,δ)-(s)s (*a,b,c , P, u). Theorem 2.9.2.19. There is a projective moduli space M (κ,η,δ)-ss (*a,b,c , P, u) for the functors M(κ,η,δ)-(s)s (*a,b,c , P, u). Remark 2.9.2.20. i) In this form, the result is also true over an algebraically closed ground field k of positive characteristic. However, we may not replace the tensor powers by divided powers. ii) As in the foregoing section on decorated sheaves, one may develop a theory for an arbitrary homogeneous representation * of the structure group GL(V, r). We leave it to the reader to spell this out.
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337
An important ingredient in the theory of decorated tuples of vector bundles on curves is the asymptotic behavior of the semistability concept. Let us generalize this to higher dimensional base varieties. We fix two tuples ηB = (ηBv , v ∈ V) and ηBB = (ηBBv , v ∈ ' ' V) of rational numbers with v∈V ηBv = 0 = v∈V ηBBv and a positive rational polynomial δ of degree exactly dim(X) − 1. We define χB = (χBv := δ · ηBv , v ∈ V), & Z 1 ηΔ = ηΔv := · ηBv + ηBBv , v ∈ V , Δ and
^ 4 χΔ = χΔv := Δ · δ · ηΔv , v ∈ V ,
Δ ∈ 3>0 .
We say that the *-tump (Ev , v ∈ V, L, ϕ) is (κ, χB , ηBB )-asymptotically (semi)stable, if a) µ(E• , α• , ϕ) − rk
%E$
^
v
v∈V
κv
s Z &T 4 T αi · · ηBBv · rk Eiv ≥ 0
(2.81)
v∈V
i=1
holds for every weighted filtration (E• , α• ) of (Ev , v ∈ V) and b)
Mκ,χB (E• , α• )(8)0
holds for every weighted filtration (E• , α• ) for which there is equality in (2.81). Theorem 2.9.2.21. Fix the input data a, b, and c, the type (P, l), the tuples κ, ηB , ηBB , and the polynomial δ. Then, there exists a positive rational number Δ∞ , such that, for any stability parameter (κ, ηΔ , Δ · δ), such that Δ ≥ Δ∞ , a *a,b,c -tump (Ev , v ∈ V, ϕ) of type (P, l) is (κ, ηΔ , Δ · δ)-(semi)stable, if and only if it is (κ, χB , ηBB )-asymptotically (semi)stable. Proof. Let us explain the (little) modifications in the proof of Theorem 2.5.5.2 (which corresponds to the implication “=⇒” in the above theorem). First of all, we replace χBB by ηBB . We then perform the same operations at the generic point of X as before. Next, we let U be a big open subset where all the sheaves Ev , v ∈ V, are locally free. To simplify notation, we write Ev for the restriction of Ev to U, v ∈ V. Then, there are morphisms Spec(K) −→ G r(Ev , ri,v ), i = 1, . . . , s, v ∈ V. * ⊂ U, such that all these morphisms extend to U. * We may find a big open subset U * = U. These extended morphisms proWithout loss of generality, we may suppose U vide a weighted filtration (E• , α• ) of the V-split vector bundle (Ev , v ∈ V) on U. It extends to a weighted filtration (E• , α• ) of (Ev , v ∈ V). The construction implies µ(E• , α• , ϕ) − rk
%E$
^
v∈V
v
κv
s &T Z 4 T · αi · ηBBv · rk Eiv < 0. i=1
v∈V
We go back to the open subset U and may then continue as in the proof of Theorem 2.5.5.2, shrinking U from time to time (but keeping it big!).
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At the end, we find a line bundle Lχ. on the big open subset U. We write δ :=
and
δ · tdim(X)−1 + lower order terms, (dim(X) − 1)! ^ 4 χΔ = χΔv := Δ · δ · ηΔv , v ∈ V ,
degκ,χΔ (Fv , v ∈ V) :=
T^ 4 κv · deg(Fv ) − χΔv · rk(Fv ) v∈V
for a V-split sheaf (Fv , v ∈ V) on X or a big open subset of X. Given a torsion free Vsplit sheaf (Fv , v ∈ V) on X or a big open subset of X and a weighted filtration (F• , α• ) of it, we define ^ 's αi · degκ,χΔ (Fv , v ∈ V) · rkκ (Fiv , v ∈ V)− Lκ,χΔ (F• , α• ) = i=1 4 − degκ,χΔ (Fiv , v ∈ V) · rkκ (Fv , v ∈ V) . We compute
deg(Lχ. ) = Lκ,0 (E• , α• ) = Lκ,0 (E• , α• ).
The semistability condition yields 0 C Mκ,χΔ (E• , α• ) + Δ · δ · µ(E• , α• , ϕ). This inequality implies 0 ≤ Lκ,χΔ (E• , α• ) + Δ · δ · µ(E• , α• , ϕ). There is an estimate BB
Lκ,χΔ (E• , α• ) + Δ · δ · µ(E• , α• , ϕ) ≤ lχ + C − Δ · δ. BB
We see that the conclusion is true for Δ∞ := (lχ + C)/δ.
"
Example 2.9.2.22 (Quiver representations). One may apply Theorem 2.9.2.19 and 2.9.2.21 in order to get a solution to the moduli problem of quiver representations. We let Q = (V, A, t, h) be a quiver. We fix a tuple M := (Ma , a ∈ A) of vector bundles on X. An M-twisted representation (Ev , v ∈ V, ϕa , a ∈ A) of the quiver Q consists of torsion free sheaves Ev , v ∈ V, on X and homomorphisms ϕa : Et(a) −→ Eh(a) * Ma , a ∈ A. We refer to P := (Pv := P(Ev ), v ∈ V) as the type of the representation. The concepts of homomorphisms, subrepresentations, and quotient representations are straightforward generalizations of the respective notions on curves, keeping in mind that subbundles have to be replaced by saturated subsheaves. A quiver representation (Ev , v ∈ V, ϕa , a ∈ A) is (κ, χ)-(semi)stable, if Pκ,χ (Fv , v ∈ V) rkκ (Fv , v ∈ V)
(C)
Pκ,χ (Ev , v ∈ V) rkκ (Ev , v ∈ V)
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339
holds for every non-trivial proper subrepresentation (Fv , v ∈ V, ϕa|Ft(a) , a ∈ A). One can then construct a moduli space R (κ,χ)-(s)s (Q, M, P) for (κ, χ)-semistable M-twisted representations of Q of type P, if δ has degree exactly dim(X) − 1. By means of a generalized Hitchin map, this moduli space is projective over an affine variety. We refer the reader to [190] for more details on the subject. Pseudo G-Bundles Fix a faithful representation κ: G −→ SL(W). Recall that a pseudo G-bundle on a curve X is a pair (E, τ) in which E is a vector bundle of rank r = dim(W) and τ: S ym. (W * E ∨ )G −→ OX is a non-trivial homomorphism of OX -algebras. Of course, the same definition makes sense on any projective manifold X. However, we know from the case of vector bundles that we also have to admit certain non-locally free torsion free sheaves. Since dualizing is not a well-behaved operation for torsion free sheaves, we slightly modify the definition. A pseudo G-bundle is a pair (A , τ) in which A is a torsion free OX -module and τ: S ym. (W * A )G −→ OX is a non-trivial homomorphism of OX -algebras. We skip the obvious definition of isomorphic pseudo G-bundles. Let S be a scheme. A family of pseudo G-bundles parameterized by S is a pair (AS , τS ) which consists of a family AS of torsion free sheaves on S x X and a homomorphism τS : S ym. (W * AS )G −→ OS x X of OS x X -algebras whose restriction to every fiber {s} x X, s ∈ S , is non-trivial. Again, the definition of isomorphic families is left to the reader. Remark 2.9.2.23. In characteristic zero, the GIT quotients are universal categorical quotients. This implies that, given a scheme S , a family AS of torsion free sheaves on S x X, and a morphism f : T −→ S , 4 ^ 4G ^ ( f x id X ). S ym. (W * AS )G = S ym. W * ( f x id X ). (AS ) . This shows that the above definition of families is a sensible one to make and that there is the natural operation of pulling back families via f : T −→ S . In positive characteristic, one has to be a little careful with the pullback operation. In this case, one has a natural base change homomorphism 4G 4 ^ ^ ( f x id X ). S ym. (W * AS )G −→ S ym. W * ( f x id X ). (AS ) which is an isomorphism on the maximal open UT ⊂ T x X subset where AT := ( f x idX ). (AS ) is locally free. By [116], Lemma 2.1.7., UT = ( f x idX )−1 (US ). Here, US ⊂ S x X is the maximal open subset where AS is locally free. Hence, given a family
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(AS , τS ) of pseudo G-bundles on S x X, we may pull it back to the family (AT , τT ) on T x X with AT = ( f x idX ). (AS ) and & ^ ^ 4G 4G Z = τT : S ym. W * ( f x id X ). (AS ) −→ ιT . ι.T S ym. W * ( f x id X ). (AS ) Z & 4 ιT . ι.T (( f x id X ). (τT )) ^ 7 ιT . ι. (OUT )=OT x X , = ιT . ι.T ( f x id X ). S ym. (W * AS )G T using the inclusion ιT : UT −→ T x X. The last equality results from Theorem 2.9.2.14. For the following, we fix a positive integer s, such that Sym. (W * 'r )G is generated by elements of degree s or less. Then, any pseudo G-bundle (A , τ) on X gives rise to and is determined by the non-zero homomorphism & s Z τs : S ymi (W * A )G −→ OX
% i=1
of OX -modules. For any coherent OX -module F , we set & ^ 4 ^ 4Z S ymd1 (W *. F )G * · · · * S ymds S yms (W *. F )G . ,s(F ) :=
%
(d1 ,...,d s ),di ≥0,i=1,...,s: d1 +2·d2 +···+s·d s =s!
If (A , τ) is a pseudo G-bundle, then τ s yields a non-zero homomorphism ϕ s : , s (A ) −→ OX of OX -modules. Remark 2.9.2.24. For the manifold {pt}, we obtain the GLr (')-module , s ('r ). If A is a torsion free sheaf of rank r on X, then , s (A )|UA is the vector bundle that is associated to the vector bundle A|U∨A and the GLr (')-module , s ('r ), or, alternatively to A|UA and the GLr (')-module that is obtained from , s ('r ) by precomposing with the involution g − 1 → (g−1 )t of GLr ('). By this remark, we may use the method of restricting to the generic point from Section 2.3.2 in order to define µ(A• , α• , ϕ s ) for every pseudo G-bundle (A , τ) and every weighted filtration (A• , α• ) of A . Next, we fix a positive rational polynomial δ of degree at most dim(X) − 1. We say that a pseudo G-bundle (A , τ) is δ-(semi)stable, if M(A• , α• ) + δ · µ(A• , α• , ϕ s )(8)0 holds for every weighted filtration (A• , α• ) of A . Given a Hilbert polynomial P and a stability parameter δ, we are now ready to introduce the moduli functors Mδ-(s)s (P, κ): Sch. −→ Sets Isomorphy classes of families (AS , τS ) of δ-(semi)stable pseudo G-bundles, such that S 1−→ . P(A ) = P, for all s ∈ S S |{s} x X
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341
Theorem 2.9.2.25. Fix P and δ as before. Then, there exists a projective moduli space M δ-ss (P, κ) for the functors Mδ-(s)s (P, κ). Proof. It is quite interesting that this theorem holds in characteristic zero as well as in positive characteristic. Let us first explain the proof over the complex numbers. Note that for a torsion free sheaf A , the OX -module , s (A ) is a quotient of & ^ 4 ^ 4Z S ymd1 (W *. A ) * · · · * S ymds S yms (W *. A )
%
(d1 ,...,d s ),di ≥0,i=1,...,s: d1 +2·d2 +···+s·d s =s!
by means of the Reynolds operator (Chapter 1, Remark 1.4.2.1). That module is clearly a quotient of Aa,b,c for appropriate non-negative integers a, b, and c (indeed, a = s!). Thus, there is a natural transformation from the moduli functor of δ-(semi)stable pseudo G-bundles into the moduli functor of δ-(semi)stable *a,b,c -swamps. With our standard arguments, we can then infer the existence of the moduli space M δ-(s)s (P, κ) δ-(s)s from the existence of the moduli space Ma,b,c (P, 0). The details are given in [186]. In positive characteristic, we do not know whether we can represent , s (kr ) as 5 the quotient of a GLr (k)-module of the shape ((kr )#a )$b * ( r kr )#−c . Hence, we mimic the arguments in the construction of the moduli scheme for *a,b,c -swamps in order to directly construct a moduli scheme for pairs (A , ϕ s ), consisting of a torsion free OX -module A of rank r and a non-zero homomorphism ϕ: , s (A ) −→ OX . This construction is explained in [79]. " There is also the variant of pseudo G-bundles adapted to deal with non-semisimple groups. So, assume the G-module W decomposes as a direct sum W1 - · · · - Wt of G-modules and the radical R(G) of G maps to the center of GL(W1 ) x · · · x GL(Wt ). Under these assumptions, κ(G) ⊆ (GL(W1 ) x · · · x GL(Wt )) ∩ SL(W). We will use the following abbreviations: W := (W1 , . . . , Wt ), and A = (A1 , . . . , At ) stands for a tuple of torsion free sheaves, such that rk(Ai ) = dim(Wi ), i = 1, . . . , t, and det(A ) ! OX , A := A1 - · · · - At . We define UA as the maximal open subset where A1 ,. . . ,At are all locally free and introduce H (A , W) I (A , W)
:=
H om(W1 * OX , A1∨ ) x · · · x H om(Wt * OX , At ∨ ) X X ^ 4 . S pec S ym (W1 * A1 - · · · - Wt * At )
:=
∨ ∨ I som(W1 * OUA , A1|U ) x · · · x I som(Wt * OUA , At|U ). A A
:=
X
X
A pseudo G-bundle is a pair (A , τ) which consists of a tuple A of torsion free sheaves and a homomorphism τ: S ym. (W1 * A1 - · · · - Wt * At )G −→ OX of OX -algebras which is non-trivial in the sense that it is not just the projection onto the component of degree zero. Note that a pseudo G-bundle is the same as a pair (A , σ: X −→ H (W, A )//G), σ not being the zero section. The notions of semistability and the moduli spaces for these pseudo G-bundles are discussed in [80].
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Singular Principal G-Bundles Let (A , τ) be a pseudo G-bundle on X. The restriction of the homomorphism τ to UA corresponds to a section σ: UA −→ H om(W * O|UA , A|U∨A )//G. Lemma 2.9.2.26. In the above situation, we either have σ(UA ) ⊂ I som(W * O|UA , A|U∨A )/G or
4 ^ 4 ^ σ(UA ) ⊂ H om(W * O|UA , A|U∨A )//G \ I som(W * O|UA , A|U∨A )/G .
Proof. The arguments of the proof of Lemma 2.4.3.1 show that σ gives rise to a ho5 momorphism OUA ! ( r W) * OUA −→ OUA ! det(A|U∨A ). Since UA is a big open subset of X, this homomorphism extends to a homomorphism OX −→ OX which is constant. " If the first alternative in the lemma holds, we say (A , τ) is a singular principal G-bundle. Note that we obtain the principal G-bundle 7 I som(W * O|UA , A|U∨ ) A
P(A , τ) 3 UA
σ
3 7 I som(W * O|UA , A|U∨ )/G A
on UA . Remark 2.9.2.27. Ramanan and Ramanathan call a pair (U, P) which is composed of a big open subset U of X and a principal G-bundle P on U a rational principal Gbundle (see [173]). Singular principal G-bundles are global objects on X which give rise to rational principal G-bundles. This property turns them into analogs of torsion free sheaves on X. Observe, however, that the concept of a singular principal G-bundle depends on the choice of a faithful representation κ: G −→ SL(W). Let (A , τ) be a singular principal G-bundle and λ: '. −→ G a one parameter subgroup of G. A reduction of (A , τ) to λ is a pair (U, β) in which U ⊂ UA is a big open subset in X and β: U −→ P(A , τ)|U /QG (λ) is a section. Let (W• (λ), α• (λ)) be the weighted flag of λ in W. Suppose α• (λ) = (αB1 , . . . , αBs ). Then, we set α• (β) := (α1 , . . . , α s ) := (αBs , . . . , αB1 ). The section β
U −→ P(A , τ)|U /QG (λ) '→ I som(W * OU , A|U∨ )/QGL(W) (λ) endows A|U∨ with a filtration A•B : {0} ! A1B ! · · · ! A sB ! A|U∨
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343
by subbundles. We dualize this filtration in order to get A•BB : {0} ! A1BB := A sB ∨ ! · · · ! A sBB := A1B ∨ ! A|U . The filtration A•BB extends to a filtration A• (β) : {0} ! A1 ! · · · ! A s ! A of A by saturated subsheaves. Altogether, (U, β) gives rise to the weighted filtration (A• (β), α• (β)) of A . We say that the singular principal G-bundle (A , τ) is (semi)stable, if ^ 4 M A• (β), α• (β) (8)0 holds for every one parameter subgroup λ: '. −→ G and every reduction (U, β) of (A , τ) to λ. Remark 2.9.2.28. For a weighted filtration (A• , α• ) of A , we define the rational number s T ^ 4 L(A• , α• ) := αi · deg(A ) · rk(Ai ) − deg(Ai ) · rk(A ) . i=1
A singular principal G-bundle is defined to be slope (semi)stable, if ^ 4 L A• (β), α• (β) (≥)0 is verified for every one parameter subgroup λ: '. −→ G and every reduction (U, β) of (A , τ) to λ. Then, using the arguments sketched in Section 2.4.9, one sees that the notion of slope (semi)stability coincides with the notion of (semi)stability that Ramanan and Ramanathan defined for the rational principal G-bundle (UA , P(A , τ)) in [173]. We have the moduli functors M(s)s (P, κ): Sch.
−→
S
1−→
Sets Isomorphy classes of families (AS , τS ) of (semi)stable singular principal G-bundles, such that P(A ) = P, for all s ∈ S S |{s} x X
.
Theorem 2.9.2.29. Fix P. There is a polynomial δ∞ , such that for every rational polynomial δ : δ∞ , the moduli functors Mδ-(s)s (P, κ) and M(s)s (P, κ) are identical. Hence, for such a polynomial δ, M ss (P, κ) := M δ-ss (P, κ) is the projective moduli space for the functors M(s)s (P, κ). Proof. One just needs the analogs of Theorem 2.4.4.1 and Proposition 2.4.4.5. For this, the reader may consult [188]. " Remark 2.9.2.30. i) The moduli space M ss (P, κ) will depend in general on the representation κ. ii) If X is an algebraic surface, there is again a kind of moduli space for slope semistable singular principal G-bundles [6]. The topological space underlying that moduli space does not depend on the representation κ.
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iii) In positive characteristic, there are certain sufficient conditions on the representation κ, so that the above theorem holds. These conditions require the characteristic of the base field to be very large [79]. iv) Over an algebraic ground field k of arbitrary characteristic, the following theorem holds true: Theorem 2.9.2.31. Fix P. There is a polynomial δ∞ , such that for every rational polynomial δ : δ∞ , the moduli space M ss (P, κ) for the functors M(s)s (P, κ) exists as an open subscheme of M δ-ss (P, κ). Proof. This is the main theorem of [79].
"
Finally, if we begin with a representation κ: G −→ GL(W), such that κ(G) ⊆ (GL(W1 ) x · · · x GL(Wt )) ∩ SL(W) and the radical R(G) of G maps to the center of GL(W1 ) x · · · x GL(Wt ), a singular principal G-bundle is a pseudo G-bundle (A , τ), such that the section σ: X −→ H (A , W)//G that corresponds to τ maps the maximal open subset UA where A1 ,. . . ,At are locally free to I (A , W)/G. Such a singular principal G-bundle gives rise to the rational principal G-bundle (UA , P(A , τ)). The principal G-bundle P(A , τ) is defined via the cartesian diagram: 7 I (A , W)
P(A , τ) 3 UA
σ|UA
3 7 I (A , W)/G.
The theory of semistable singular principal G-bundles is developed in [80]. Note that, on higher dimensional base varieties, this approach is more interesting and more important, because the moduli spaces will depend on the representation κ and be different from those constructed by G´omez and Sols [83]. The Ramanathan–G´omez–Sols Method The lifting technique originally due to Ramanathan (see Section 2.4.8) was generalized by G´omez and Sols to higher dimensional base varieties [83]. We will sketch the result. Let α: G −→ GB be a central isogeny. We assume that we are given a scheme S , an open subscheme US ⊂ S x X, such that US |{s} x X is big in X = {s} x X, for every closed point s ∈ S , and a principal GB -bundle PSB on US . This time, we investigate the functor Γ(PSB , α) that associates to a morphism f : T −→ S the set of isomorphy classes of pairs (PT , ξT ) which consist of a principal G-bundle PT on UT := ( f x idX )−1 (US ) and an isomorphism ξT : α. (PT ) −→ ( f x idX ).|UT (PSB ). We consider two such pairs (PT,1 , ξT,1 ) and (PT,2 , ξT,2 ) to be isomorphic, if there is * B , α) be the an isomorphism ψT : PT,1 −→ PT,2 with ξT,1 = ξT,2 ◦ α. (ξT ). Let Γ(P S sheafification of Γ(PSB , α) in the e´ tale topology. * B , α) can be represented by a Proposition 2.9.2.32 (G´omez/Sols). The functor Γ(P S scheme f : T −→ S , such that f is finite and e´ tale over its image.
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345
Proof. This is Proposition 3.6 in [83].
"
Let κ: G −→ GB be a homomorphism, such that the connected component Rad(κ) of the neutral element of the kernel is contained in the radical of G. Fix a maximal torus T which contains Rad(κ) and a subtorus T B , such that the multiplication map Rad(κ) x T B −→ T is an isomorphism. Then, T := G/(T B · [G, G]) is a torus. We fix an isomorphism T ! ('. )x s . Suppose that U ⊂ X is a big open subset and that P is a principal G-bundle on U. By extension of the structure group, it provides a principal T -bundle on U. Using the above isomorphism, that principal T -bundle can be seen as a tuple L˚ = (L˚ 1 , . . . , L˚ s ) of line bundles on U. Since U is a big open subset of X, this tuple uniquely extends to a tuple L = (L1 , . . . , L s ) of line bundles on X. Let P be a Hilbert polynomial and u = (u1 , . . . , u s ) a tuple of classes in NS(X). A principal κ-sheaf of type (P, u) is a quadruple (A , τ, P, ξ) in which (A , τ) is a singular principal GB -bundle with P(A ) = P, P is a principal G-bundle on UA such that ui is the algebraic equivalence class of Li in the tuple L of line bundles on X associated to P, i = 1, . . . , s, and ξ: α. (P) −→ P(A , τ) is an isomorphism. The principal κ-sheaf (A1 , τ1 , P1 , ξ1 ) is isomorphic to the principal κ-sheaf (A2 , τ2 , P2 , ξ2 ), if there are an isomorphism ψ: A1 −→ A2 between the singular principal GB -bundles (A1 , τ1 ) *: P1 −→ P2 , such that the induced isomorphism and (A2 , τ2 ) and an isomorphism ψ ψU : P(A1 , τ1 ) −→ P(A2 , τ2 ) gives the commutative diagram α. (P1 )
*) α. (ψ
ξ1
3 P(A1 , τ1 )
ψU
7 α. (P2 ) 3
ξ2
7 P(A2 , τ2 ).
The principal κ-sheaf (A , τ, P, ξ) is (semi)stable, if the singular principal GB bundle (A , τ) is so. We obtain the moduli functors M(s)s (P, u, κ) of (semi)stable principal G-sheaves of type (P, u) Theorem 2.9.2.33. Fix the type (P, u). Then, there exists a projective moduli space M ss (P, u, κ) for the functors M(s)s (P, u, κ). Proof. The result goes back to [83], Theorem 0.8. It was refined to the above form in [80]. Note that the result is known only in characteristic zero and over curves. " Decorated Pseudo G-Bundles We fix a faithful representation κ: G −→ SL(W), as well as non-negative integers a, b, and c, giving the representation *a,b,c : GL(W) −→ GL(Wa,b,c ). A pseudo G-bundle with a decoration of type (a, b, c, u) is a tuple (A , τ, L, ϕ) in which (A , τ) is a pseudo G-bundle, L is a line bundle whose algebraic equivalence class is u ∈ NS(X), and ϕ: Aa,b,c −→ L is a non-trivial homomorphism. The notion of isomorphy for decorated pseudo G-bundles is the obvious one. In order to define families, we choose a Poincar´e line bundle Lu on Picu (X) x X. A family of pseudo G-bundles with a decoration of type (a, b, c, u) parameterized by
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the scheme S is a tuple (A S , τS , κS , NS , ϕS ) where (A S , τS ) is a family of pseudo Gbundles parameterized by S , κS : S −→ Picu (X) is a morphism, NS is a line bundle on S , and ϕS : AS ,a,b,c −→ L [κS ] * NS is a homomorphism whose restriction to every fiber {s} x X, s ∈ S , is non-trivial. We leave it to the reader to figure out the notion of isomorphic families. To define (semi)stability, we fix a positive rational polynomial δ of degree at most dim(X) − 1, a positive rational number ε, and a tuple η = (ηi , i = 1, . . . , t) of rational numbers. We set κ = (1, . . . , 1),
χ = (χi := δ · ηi , i = 1, . . . , t).
A pseudo G-bundle (A , τ, L, ϕ) with a decoration of type (a, b, c, u) is defined to be (η, δ, ε)-(semi)stable, if Mκ,χ (A• , α• ) + δ · µ(A• , α• , ϕ) + δ · ε · µ(A• , α• , ϕτ )(8)0 holds for every weighted filtration (A• , α• ) of A . We fix the Hilbert polynomials P = (P1 , . . . , Pt ), u ∈ NS(X), a, b, c, and the stability parameters δ, ε, and η. There are the moduli functors M(η,δ,ε)-(s)s (*a,b,c , P, u): Sch.
S
−→ Sets Isomorphy classes of families (A S , τS , κS , NS , ϕS ) of (η, δ, ε) 1−→ (semi)stable pseudo G-bundles with a decoration of type (a, b, c, u), such that P(A ) = P , i = 1, . . . , t i,S |{s} x X
i
.
Next, we choose positive integers a1 and a2 , such that a2 /a1 = ε and define % δ := δ/a1 . Using our previous construction, any pseudo G-bundle (A , τ) gives rise to a *d,e, f -tump a1 #a2 * := *# (A , ϕτ ). We set % a,b,c * *d,e, f and associate to a pseudo G-bundle (A , τ, L, ϕ) with ϕ) with a decoration of type (a, b, c, u) the % *-tump (A , L#a1 , % % ϕ := ϕ#a1
*
a2 ϕ# τ .
By our choice of % δ, a1 , and a2 , (A , τ, L, ϕ) is an (η, δ, ε)-(semi)stable pseudo G-bundle ϕ) is a (κ, η, % δ)-(semi)stable with a decoration of type (a, b, c, u), if and only if (A , L#a1 , % % *-tump, and we find a natural transformation *, P, u). NT: M(η,δ,ε)-(s)s (*a,b,c , P, u) −→ M(κ,η,δ)-(s)s (% %
The standard argument proves: Theorem 2.9.2.34. Fix the tuple P of Hilbert polynomials, u ∈ NS(X), a, b, c, and the stability parameters η, δ, and ε. Then, the moduli space M (η,δ,ε)-(s)s (*a,b,c , P, u) for the functors M(η,δ,ε)-(s)s (*a,b,c , P, u) exists as a projective scheme.
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347
We explained in Section 2.7 and 2.8 that Theorem 2.9.2.34 together with Theorem 2.9.2.21 is the technical basis for the construction of the moduli space of twisted affine *-bumps. Hence, we shorten our discussion on the theory on higher dimensional base manifolds and proceed directly to twisted affine *-bumps. Let *: G −→ GL(V) be a representation of G and V = V1 - · · · - Vu , * = *1 - · · · - *u with *i : G −→ GL(Vi ), i = 1, . . . , u, its decomposition into irreducible components. Furthermore, we fix a faithful representation κ: G −→ GL(W) and a tuple L = (L1 , . . . , Lu ) of line bundles on X. Let (A , τ) be a singular principal G-bundle. This gives the principal G-bundle P(A , τ) and the associated vector bundles P(A , τ)*i with fiber Vi , i = 1, . . . , u, on UA . A twisted affine *-bump of type (P, L) is a tuple (A , τ, ϕ) in which (A , τ) is a singular principal G-bundle, such that P = (P(A1 ), . . . ., P(At )), and ϕ = (ϕ1 , . . . , ϕu ) is a tuple of homomorphisms ϕi : P(A , τ)*i −→ Li|UA , i = 1, . . . , u. The reader may conceive on his or her own how the notions of isomorphism, family, and isomorphism of families have to look like. To define (semi)stability, we fix a positive rational polynomial δ of degree exactly dim(X) − 1 and a rational character η of G. Given a twisted affine *-bump (A , τ, ϕ) and a reduction β: U −→ P(A , τ)|U /QG (λ) to a one parameter subgroup λ: '. −→ G, the number µ(β, ϕ) is defined as on curves: The reduction β induces the filtration {0} ! F1 ! · · · ! Fv ! Fv+1 := P(A , τ)*|U . Using the projections π j : P(A , τ)*|U −→ P(A , τ)* j |U , j = 1, . . . , u, we finally set L F − min γi | ∃ j ∈ { 1, . . . , u } : (ϕ j ◦ π j )|Fi & 0 , if ϕ & 0 . µ(β, ϕ) := 0, if ϕ ≡ 0 A twisted affine *-bump (A , τ, ϕ) is said to be (δ, η)-(semi)stable, if ^ 4 M E• (β), α• (β) + δ · F λ, η 4(8)0 holds for every one parameter subgroup λ: '. −→ G and every reduction β of (A , τ) to λ, such that µ(β, ϕ) ≤ 0. We have the moduli functors M(δ,η)-(s)s (*, P, L): Sch. −→ Set Isomorphy classes of families of (δ, η)-(semi)stable twisted affine *-bumps S 1−→ . of type (P, L) parameterized by S Using the generalization of tumps to higher dimensional varieties, the construction on curves can be transferred to higher dimensional base varieties and gives the following result: Theorem 2.9.2.35. Fix the tuple P = (P1 , . . . , Pu ) of Hilbert polynomials and the tuple L = (L1 , . . . , Lu ) of line bundles, the polynomial δ and the character η. Then, the moduli space M (δ,η)-(s)s (*, P, L) for the functors M(δ,η)-(s)s (*, P, L) exists as a quasi-projective scheme.
348 S 2.9: G
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It is also possible to extend the construction of the Hitchin map to the setting of higher dimensional base varieties. Given a family (A S , τS , ϕS ) of (δ, χ)-semistable twisted affine *-bumps of type (P, L) parameterized by the scheme S , the constructions presented in the setting of curves work on the open subset UA S ⊂ S x X where all the sheaves AS ,i , i = 1, . . . , t, are locally free. With the help of Maruyama’s theorem 2.9.2.14, one produces sections of the respective line bundles on S . We omit the details here. Higgs Sheaves As a nice application, one can define moduli spaces of Higgs bundles on higher dimensional base varieties. If G is semisimple, we may directly apply our theory to the adjoint representation. In general, we assume that G is a connected reductive group and look at the adjoint representation Ad: G −→ GL(g) of G on its Lie algebra g. Let GB be the image of Ad. We first apply our results to the semisimple group GB with the induced faithful representation κ: GB −→ GL(g). A singular GB -pre-Higgs bundle is a triple (A , τ, ϕ) in which (A , τ) is a singular principal GB -bundle and ϕ: A −→ Ω1X is a homomorphism. Remark 2.9.2.36. We tacitly choose a sufficiently ample line bundle L and an injective homomorphism ι: Ω1X −→ L$k . In this way, we can interpret a singular GB -pre-Higgs bundle as a twisted affine *-bump for the representation * := Ad$k . As on curves (see Section 2.7.4), the general theory of (semi)stability for twisted affine bumps gives the following notion of semistability for singular GB -pre-Higgs bundles: A Higgs reduction of (A , τ, ϕ) is a reduction β: U −→ P(A , τ)|U /QG (λ) to a one parameter subgroup λ of G over a big open subset U ⊂ UA with ^ 4 ϑ|U ∈ H 0 U, Ad(Q(β)) * Ω1U . To define ϑ, note that A|U∨A identifies with the vector bundle P(A , τ)κ . Thus, ϕ|UA may be viewed as a homomorphism ϑ: OUA −→ P(A , τ)κ * Ω1UA . Moreover, Ad(Q(β)) is the vector bundle with fiber Lie(QG (λ)) that is associated to the principal Q-bundle Q(β) on U defined by β and the adjoint representation of QG (λ). A singular GB -pre-Higgs bundle (A , τ, ϕ) is called (semi)stable, if ^ 4 M A• (β), α• (β) (8)0 is satisfied for every one parameter subgroup λ: '. −→ G of G and every Higgs reduction of (A , τ) to λ. The moduli space of semistable singular GB -pre-Higgs bundles can now be realized as a closed subscheme of M δ-(s)s (*, P, L = (L, . . . , L)) (observe Remark 2.9.2.36) for a sufficiently large polynomial δ (note that η is obsolete, because GB is semisimple). Next, let (A , τ, ϕ) be a singular GB -pre-Higgs bundle. Set P B := P(A , τ) and
#
ϑ ϑ
ϑ ∧ ϑ: OUA −→ PκB * PκB * Ω1UA
*
Ω1UA −→ PκB * Ω2UA .
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349
The last homomorphism is the tensor product of the Lie bracket PκB * PκB −→ PκB and the wedge product Ω1UA * Ω1UA −→ Ω2UA . A singular GB -Higgs bundle is a singular GB -pre-Higgs bundle (A , τ, ϕ), such that ϑ ∧ ϑ = 0. Let us explain why the moduli functor of (semi)stable singular GB -Higgs bundles is a closed subfunctor of the moduli functor of (semi)stable singular GB -pre Higgs bundles. Let S be a scheme and (AS , τS , ϕS ) a family of singular GB -pre-Higgs bundles parameterized by S . The above construction works in families, too. Hence, we get a homomorphism ϑS ∧ ϑS : OUA −→ AS∨|UA * π.X (Ω1X )|UAS . Then, ϑS ∧ ϑS S
corresponds to a homomorphism * τS : AS |UAS −→ π.X (Ω2X )|UAS . This extends to ι. ι. (* τS )
τS : AS −→ ι. ι. (AS |UAS ) −→ ι. ι. (π.X (Ω2X )|UAS )
Thm. 2.9.2.14
=
π.X (Ω2X ).
According to Exercise 2.3.5.3, the locus S B ⊂ S where τS vanishes carries a natural structure of a closed subscheme of S . This proves our contention. Hence, we obtain the moduli space for (semi)stable singular GB -Higgs bundles of fixed type (P, u). Finally, a principal G-Higgs sheaf of type (P, u) is a tuple (A , τ, P, ξ, ϕ) in which (A , τ, P, ξ) is a principal κ-sheaf of type (P, u) and (A , τ, ϕ) is a singular GB -Higgs bundle. We call (A , τ, P, ξ, ϕ) (semi)stable, if (A , τ, ϕ) is a (semi)stable singular GB -Higgs bundle. Using the Ramanathan–G´omez–Sols lifting method, we obtain the moduli space H (P, u)ss for (semi)stable principal G-Higgs sheaves of type (P, u). Remark 2.9.2.37. A similar notion of Higgs sheaves and a construction of moduli spaces was previously obtained by G´omez and Sols in [84], building on the approach in [83]. ADHM-Sheaves In this section, we would like to present an example of a more complicated moduli problem on higher dimensional base varieties. It was studied on !n by Jardim [117] and on arbitrary projective base manifolds by Diaconescu [52]. Let the quiver Q = ({ •, ◦ }, { A1 , A2 , a, b }) be given by Figure 2.3. We will also A1
1•# A2
'
b
4
◦
c
Figure 2.3: The ADHM-quiver. impose the relation
r := A2 A1 − A1 A2 − bc.
(2.82)
We fix a dimension vector n = (n• , n◦ ), vector spaces N1 , N2 , and define W as Hom(N1 * 'n• , 'n• ) - Hom(N2 * 'n• , 'n• ) - Hom(N1 * N2 * 'n• , 'n◦ ) - Hom('n◦ , 'n• ).
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There is a natural representation *B : G −→ GL(W) of the group G := GLn• ('). The relation (2.82) defines the G-invariant closed subvariety A ⊂ W whose points are the tuples ^ 4 Fi : Ni * 'n• −→ 'n• , i = 1, 2, f : N1 * N2 * 'n• −→ 'n◦ , g: 'n◦ −→ 'n• ∈ W, such that
F2 ◦ (idN2
*
F1 ) − F1 ◦ (idN1
*
F2 ) − g ◦ f = 0.
The following assertion is readily verified: Lemma 2.9.2.38. Let w = (Fi , i = 1, 2, f, g) ∈ W be a point and λ: '. −→ G a one parameter subgroup with associated weighted filtration (W• , γ• = (γ1 , . . . , γ s+1 )). Then, µ(w, λ) ≤ 0 occurs if and only if Fi (Ni * W j ) ⊂ W j , i = 1, 2, j = 1, . . . , s, N1 * N2 * W j− ⊂ ker( f ), and W j0 ⊃ Im(g), j− := max{ 1, . . . , s + 1 | γ j < 0 }, j0 := max{ 1, . . . , s + 1 | γ j ≤ 0 }. Let us first study the formal application of our general results to the representation * := *B ∨ : G −→ GL(V), V := W ∨ . We fix line bundles L1 , L2 , L3 , and L4 and associate L1 to all irreducible representations in Hom(N1 * 'n• , 'n• ), L2 to all irreducible representations in Hom(N2 * 'n• , 'n• ), L3 to those appearing in Hom(N1 * N2 * 'n• , 'n◦ ) and L∨4 to those in Hom('n◦ , 'n• ). Then, a *-bump is a tuple (E , Φ1 , Φ2 , ϕ, ψ) in which n◦ E is a torsion free sheaf, Φi : Ni * E|U −→ E|U * Li|U , i = 1, 2, ϕ: N1 * N2 * E|U −→ L$ 3|U , n◦ and ψ: L$ 4|U −→ E|U are homomorphisms. Here, U is the maximal open subset where E is locally free. Let '. = '. · $n• be the center of GLn• ('). The weights for the '. -action on W are 0 (on Hom(N1 * 'n• , 'n• ) - Hom(N2 * 'n• , 'n• )), −1 (on Hom(N1 * N2 * 'n• , 'n◦ )), and 1 (on Hom('n◦ , 'n• )). It will be our first task to understand the notion of δ-semistability. We will only deal with the case δ : 0, the case δ ≺ 0 being “dual”. We leave it as an exercise to the reader to verify the following result, by applying the arguments used in the proof of Proposition 2.8.4.3 and Claim 2.8.4.14, taking into account Lemma 2.9.2.38. Proposition 2.9.2.39. A *-bump (E , Φ1 , Φ2 , ϕ, ψ) is δ-(semi)stable, if and only if ϕ & 0 and the inequality M(E• , α• ) + δ · µ(E• , α• , ϕ)(8)0 holds for every weighted filtration (E• , α• ) of E with Φi (Ni * E j|U ) ⊂ E j|U * Li|U , i = 1, 2, j = 1, . . . , s, and µ(E• , α• , ϕ) + µ(E• , α• , ψ) ≤ 0. Now, let (E , Φ1 , Φ2 , ϕ, ψ) be a *-bump. We fix a saturated subsheaf {0} ! F ! E and take the weighted filtration (E• , α• ) = ({0} ! F ! E , (1)). Then, the condition µ(E• , α• , ϕ) + µ(E• , α• , ψ) ≤ 0 is equivalent to
N1 * N2 * F ⊂ ker(ϕ)
or F ⊃ Im(ψ).
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Corollary 2.9.2.40. Assume δ : 0 is a polynomial of degree at most dim(X) − 1 and that (E , Φ1 , Φ2 , ϕ, ψ) is δ-semistable, then we have P(F ) P(E ) ≺ rk(F ) rk(E ) for every saturated subsheaf {0} ! F ! E with N1 * N2 * F|U ⊂ ker(ϕ) and Φi (Ni * F|U ) ⊂ F|U , i = 1, 2. Proof. For the weighted filtration (E• , α• ) = ({0} ! F ! E , (1)), we compute µ(E• , α• , ϕ) = − rk(F ), so that δ-semistability requires P(E ) − δ P(F ) C . rk(F ) rk(E ) This gives the assertion.
"
As in the case of Higgs bundles or Hitchin pairs (see [163], [183], and also Example 2.8.2.5), this condition suffices to grant boundedness: Proposition 2.9.2.41. Fix the Hilbert polynomial P. Then, the set of isomorphy classes of torsion free sheaves E with Hilbert polynomial P for which there exists a *-bump (E , Φ1 , Φ2 , ϕ, ψ) which satisfies the condition in Corollary 2.9.2.40 and P(E ) = P is bounded. Proof. This is an easy modification of Nitsure’s argument ([163]; Example 2.8.2.5). We look at the slope Harder–Narasimhan filtration ([116], Theorem 1.6.7) {0} =: E0 ! E1 ! · · · ! E s ! E s+1 := E of E . If µ(E1 ) ≤ deg(L3 ), we are done. Otherwise, the restriction of ϕ to N1 * N2 * E1|U is trivial and we let j0 ∈ { 1, . . . ., s } be the maximal index, such that ϕ restricts to the zero map on N1 * N2 * E j|U . (Note that j0 ≤ s, because ϕ & 0.) On the other hand (compare Exercise 2.2.4.5, i), µ(E j ) > µ(E ),
j = 1, . . . , j0 .
By the condition stated in Corollary 2.9.2.40, Φ1 (N1 * E j|U ) " E j|U * L1|U or Φ2 (N2 * E j|U ) " E j|U * L2|U . We infer F L µ(E j /E j−1 ) ≤ µ(E j+1 /E j ) + l, l := max deg(L1 ), deg(L2 ) , j = 1, . . . , j0 . For j0 , the implies µ(E j0 /E j0 −1 ) ≤ µ(E j0 +1 /E j0 ) + l ≤ deg(L3 ) + l. Altogether, we deduce that µ(E1 ) ≤ deg(L3 ) + j0 · l ≤ deg(L3 ) + (rk(E ) − 1) · l. This bounds µmax (E ) and yields the assertion.
"
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Corollary 2.9.2.42. Given the Hilbert polynomial P, there is a polynomial δ∞ , such that, for any polynomial δ : δ∞ , a *-bump (E , Φ1 , Φ2 , ϕ, ψ) with P(E ) = P is δ(semi)stable, if and only if ϕ is non-trivial and there is no saturated subsheaf {0} ! F ! E with Φi : Ni * F|U ⊂ F|U * Li|U , i = 1, 2, and N1 * N2 * F ⊂ ker(ϕ). Proof. First, observe that a *-bump (E , Φ1 , Φ2 , ϕ, ψ) with P(E ) = P which possesses a saturated subsheaf {0} ! F ! E with Φi : Ni * F|U ⊂ F|U * Li|U , i = 1, 2, and N1 * N2 * F ⊂ ker(ϕ) is δ-unstable for all δ :: 0, by the computations in the proof of Corollary 2.9.2.40. On the other hand, the boundedness established in Proposition 2.9.2.41 implies that there is a polynomial δ∞ , such that the condition of δ1 -(semi)stability and δ2 (semi)stability are equivalent for all δ1 , δ2 : δ∞ . This shows that the stated condition is necessary (see Corollary 2.9.2.40). For the converse, we observe that the hypothesis implies µ(E• , α• , ϕ) > 0 for every weighted filtration (E• , α• ) of E . As in the discussion of asymptotic semistability, p. 164ff, this suffices to conclude. " Now, we can discuss ADHM-sheaves. We fix vector bundles M1 and M2 , a torsion free coherent OX -module E∞ , and a polynomial P as the background data. Then, an ADHM-sheaf of type (M1 , M2 , E∞ , P) is a tuple (E , Φ1 , Φ2 , ϕ, ψ) in which E is a torsion free coherent OX -module, Φi : Mi * E −→ E , i = 1, 2, ϕ: M1 * M2 * E −→ E∞ , and ψ: E∞ −→ E are homomorphisms. Moreover, the ADHM-sheaf (E , Φ1 , Φ2 , ϕ, ψ) is isomorphic to the ADHM-sheaf (E B , ΦB1 , ΦB2 , ϕB , ψB ), if there is an isomorphism Ψ : E −→ E B with ΦBi = Ψ ◦Φi ◦(id Mi *Ψ )−1 , i = 1, 2, ϕB = ϕ◦(id M1 * id M2 *Ψ )−1 , and ψB = Ψ ◦ψ. An ADHM-sheaf (E , Φ1 , Φ2 , ϕ, ψ) is co-stable, if ϕ is non-trivial and there is no saturated subsheaf {0} ! F ! E with Φi : Mi * F ⊂ F , i = 1, 2, and M1 * M2 * F ⊂ ker(ϕ). After fixing the Hilbert polynomial P, we may define a moduli functor -s (M , M , E , P) for co-stable ADHM-sheaves of type (M , M , E , P). McADHM 1 2 ∞ 1 2 ∞ Now, choose vector spaces Ni , i = 1, 2, line bundles Li , i = 1, 2, 3, 4, and n◦ > 0 as well as surjections σi : Ni * L∨i −→ Mi , i = 1, 2, an injective homomorphism n◦ n◦ ι: L1 * L2 * E∞ −→ L# and a surjection τ: L$ −→ E∞ . 3 4 Let (E , Φ1 , Φ2 , ϕ, ψ) be an ADHM-sheaf of type (M1 , M2 , E∞ , P). We get Ni * L∨i * E and thus
# 7 Mi * E
σi idE
*i : Ni * E −→ E Φ
*
Li ,
Φi
7E
i = 1, 2.
Furthermore, we have N1 * N2 * L∨1 * L∨2 * E which gives
# #
σ1 σ2 idE
7 M1 * M2 * E
N1 * N2 * E −→ L1 * L2 * E∞ .
Combining with ι, we arrive at a homomorphism n◦ * ϕ: N1 * N2 * E −→ L$ 3 .
ϕ
7 E∞
S 2.9: G Finally, there is
353
*: L$n◦ ψ 4
The assignment
τ
7 E∞
ψ
7 E.
*1 , Φ *2 , * *) (E , Φ1 , Φ2 , ϕ, ψ) 1−→ (E , Φ ϕ, ψ
is clearly injective and compatible with isomorphisms. *1 , Φ *2 , * *) will come from an ADHM-sheaf, if and only if the map A tuple (E , Φ ϕ, ψ *ti : Ni * L∨i * E −→ E Φ defined by Φi vanishes on the kernel of σi * idE , i = 1, 2, the map N1 * N2 * E is zero, the map
* ψ
7 L$n◦ 3
7 L$n◦ /(E∞ * L1 * L2 ) 3
* ϕt : N1 * N2 * L∨1 * L∨2 * E −→ E∞
* vanishes on ker(τ). In induced by * ϕ vanishes on the kernel of σ1 * σ2 * idE , and ψ families, these will be closed conditions. It remains to check that the above assignment is also compatible with the notions of semistability: Lemma 2.9.2.43. An ADHM-sheaf (E , Φ1 , Φ2 , ϕ, ψ) is co-stable, if and only if the *1 , Φ *2 , * *) satisfies the condition of Corollary 2.9.2.42. associated *-bump (E , Φ ϕ, ψ Proof. Let (E , Φ1 , Φ2 , ϕ, ψ) be an ADHM-sheaf and F a saturated subsheaf. Note that Φi (Mi * F ) ⊂ F holds if and only if the induced homomorphism Φi : Mi * F −→ E /F is zero, i = 1, 2. Since F and E /F are torsion free, this is equivalent to the fact that Φi|U is trivial, i = 1, 2, U being the maximal open subset where E is locally free. Likewise, M1 * M2 * F is contained in the kernel of ϕ, if and only if M1|U * M2|U * F|U is a subsheaf of ker(ϕ|U ). These observations immediately yield the claim. " Observing Remark 2.9.2.16, ii), our constructions also imply: Theorem 2.9.2.44. Fix the type (M1 , M2 , E∞ , P). Then, there is a quasi-projective -s (M , M , E , P). c-s moduli space MADHM (M1 , M2 , E∞ , P) for the functors McADHM 1 2 ∞ An ADHM-sheaf (E , Φ1 , Φ2 , ϕ, ψ) is stable, if ψ is non-trivial and there is no saturated subsheaf {0} ! F ! E with Φi : Mi * F ⊂ F , i = 1, 2, and F ⊃ Im(ϕ). There is the moduli functor MsADHM (M1 , M2 , E∞ , P) for stable ADHM-sheaves of type (M1 , M2 , E∞ , P). If we perform the discussion of semistability for parameters δ ≺≺ 0, we find: Theorem 2.9.2.45. Given the type (M1 , M2 , E∞ , P), there is a quasi-projective moduli s space MADHM (M1 , M2 , E∞ , P) for the functors MsADHM (M1 , M2 , E∞ , P). s The moduli scheme MADHM (M1 , M2 , E∞ , P) was originally constructed by Diaconescu [52] (as algebraic spaces) and, on projective spaces, by Jardim [117].
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2.9.3 Parabolic Structures To conclude the monograph, we discuss the framework of parabolic vector and principal bundles with decorations. We return to the setting of a smooth projective curve X. Again, we will work over the field of complex numbers and add some remarks concerning positive characteristics. Parabolic Vector Bundles The notion of a parabolic vector bundle was introduced by Mehta and Seshadri in [149] to study certain vector bundles on an open Riemann surface. More precisely, let (x1 , . . . , xb ) be a tuple of b distinct points on the curve X. We would like to understand ˚ −→ the vector bundles on X˚ := X\{ x1 , . . . , xb } that come from representations σ: π1 (X) ˚ ˚ ˚ Ur (') of the fundamental group of X. The vector bundle E on X associated to σ extends to a vector bundle E on X, and the monodromy of σ around the points x1 ,. . . ,xb equips the fibers of E at x1 ,. . . ,xb with flags and gives tuples of real numbers. We do not spell this out here and refer the reader to the paper [149] or the book [199] instead. A quasi-parabolic sheaf is a tuple (F , q) which consists of a coherent sheaf F on X and a tuple q = (qi j , j = 1, . . . , si , i = 1, . . . , b) of quotients qi j : F|{xi } −→ Qi j onto vector spaces Qi j , j = 1, . . . , si , i = 1, . . . , b, such that {0} ⊆ ker(qi1 ) ⊆ · · · ⊆ ker(qisi ) ⊆ F|{xi } ,
i = 1, . . . , b.
(2.83)
The tuple (r, d, r) with r = rk(F ), d = deg(F ), and r = (ri j := dim(Qi j ), j = 1, . . . , si , i = 1, . . . , b) is referred to as the type of (F , q). An isomorphism between quasi-parabolic sheaves (F1 , q ) and (F2 , q ) is an isomorphism ψ: F1 −→ F2 , such 1 2 that q1i j = q2i j ◦ ψ|{xi } , j = 1, . . . , si , i = 1, . . . , b. A family of quasi-parabolic sheaves parameterized by the scheme S is a tuple (FS , q ) S in which FS is a coherent sheaf on S x X which is flat over S and q is a tuple which S consists of quotients qS ,i j : FS |S x{xi } −→ QS ,i j , with
ker(qS ,i j ) ⊆ ker(qS ,i j+1 ),
Two families (FS1 , q1 ) and S ψS : FS1 −→ FS2 , such that
(FS2 , q2 ) S
j = 1, . . . , si , i = 1, . . . , b, j = 1, . . . , si − 1, i = 1, . . . , b.
are isomorphic, if there exists an isomorphism
q1S ,i j = q2S ,i j ◦ ψS |S x{xi } ,
j = 1, . . . , si , i = 1, . . . , b.
A parabolic sheaf is a tuple (F , q, a) which consists of a quasi-parabolic sheaf (F , q) and a tuple a = (ai j , j = 1, . . . , si , i = 1, . . . , b) of real numbers. The type of (F , q, a) is (r, d, r, a). The entries of a are the parabolic weights. Two parabolic
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sheaves (F1 , q , a1 ) and (F2 , q , a2 ) are isomorphic, if a1 = a2 and (F1 , q ) and 1 2 1 (F2 , q ) are isomorphic quasi-parabolic sheaves. 2 The parabolic degree of the parabolic sheaf (F , q, a) is α-deg(F ) := deg(F , q, a) := deg(F ) −
si b T T
^ 4 ai j · dim qi j (F ) .
i=1 j=1
Let (F , q, a) be a parabolic sheaf and G ⊆ F a subsheaf. Then, we set ^ 4 q = qG i j : G|{xi } −→ q(G|{xi } ), j = 1, . . . , si , i = 1, . . . , b . G
Thus, we obtain the parabolic sheaf (G , q , a). We call a parabolic sheaf (F , q, a) G (semi)stable, if we have deg(G , q , a) · rk(F )(≤) deg(F , q, a) · rk(G ), G
or, in short hand notation, α-deg(G ) · rk(F )(≤) α-deg(F ) · rk(G ), for every subsheaf {0} ! G ! F of F . We say that the parabolic weights a are admissible, if • ai j > 0, j = 1, . . . , si , i = 1, . . . , b, and 'i ai j < 1, i = 1, . . . , b. • sj=1 Exercise 2.9.3.1. Assume that the parabolic weights a are admissible. i) Let (F , q, a) be a parabolic sheaf in which F is a non-trivial torsion sheaf. Verify α-deg(F ) = deg(F , q, a) > 0. ii) Let (F , q, a) be a semistable parabolic sheaf. Show that F is torsion free. iii) Let (E, q, a) be a parabolic vector bundle (i.e., (E, q, a) is a parabolic sheaf and E is locally free). Show that (E, q, a) is (semi)stable, if and only if it satisfies the (semi)stability condition for all subbundles {0} ! F ! E. iv) Give the definition of a Harder–Narasimhan filtration of a parabolic vector bundle (E, q, a) and prove its existence. It is an easy task to find the appropriate notions of Jordan–H¨older filtrations, S-equivalence, and of polystable parabolic vector bundles. Therefore, we also get the moduli functors M(s)s (r, d, r, a) for (semi)stable parabolic vector bundles of type (r, d, r, a). Theorem 2.9.3.2 (Mehta/Seshadri). Fix the type (r, d, r, a), such that the parabolic weights a are admissible. Then, there exists a projective moduli space M ss (r, d, r, a) for the functors Mss (r, d, r, a). Note that the parabolic weights are allowed to be real numbers. In fact, as for holomorphic chains (see Remark 2.5.6.21), one may define a certain chamber structure on the set of parabolic weights in which every chamber contains rational weights. This reduces the construction of the moduli spaces to the case of rational parabolic weights. We refer the reader to the papers [27] and [111] for some information on the geometry of the moduli space Mss (r, d, r, a).
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Decorated Parabolic Bundles Let *: GLr (') −→ GL(V) be a homogeneous representation. A parabolic *-swamp of type (r, d, r, a, l) is a tuple (E, q, L, ϕ) which consists of a parabolic vector bundle (E, q, a, l) of type (r, d, r, a), a line bundle L of degree l, and a non-trivial homomorphism ϕ: E* −→ L. The notion of isomorphic parabolic *-swamps is evident in view of the background developed, so far. The same goes for the notion of families and their isomorphisms. Let (E, q, a) be a parabolic vector bundle and (E• , α• ) a weighted filtration of E. We define Pa (E• , α• ) :=
s T
^ 4 αi · α-deg(E) · rk(Ei ) − α-deg(Ei ) · rk(E) .
i=1
The quantity µ(E• , α• , ϕ) is the same as in Section 2.3.2. The stability parameter is again a positive rational number δ. A parabolic *-swamp (E, q, a, L, ϕ) is δ-(semi)stable, if the inequality Pa (E• , α• ) + δ · µ(E• , α• , ϕ)(≥)0 holds for every weighted filtration (E• , α• ) of E. We let Mδ-(s)s (r, d, r, a, l) be the moduli functors for δ-(semi)stable parabolic *swamps of type (r, d, r, a, l). Theorem 2.9.3.3 (Heinloth/Schmitt). For fixed type (r, d, r, a, l) and fixed stability parameter δ, there is a projective moduli scheme M δ-ss (r, d, r, a, l) for the functors Mδ-(s)s (r, d, r, a, l). This theorem holds again over algebraically closed ground fields of arbitrary characteristic. The proof is given in [103] in the case the line bundle L is fixed, i.e., does not vary over Jacl . This restriction can be easily removed. From here on, one may rewrite Section 2.1 until 2.8, replacing everywhere the ordinary degree by the parabolic degree. As an exercise, the reader may reformulate the theory of decorated tuples of vector bundles as a theory of decorated tuples of parabolic vector bundles. Parabolic Principal Bundles Let G be a connected reductive affine algebraic group. We keep the tuple (x1 , . . . , xb ) of b distinct points on X and fix a tuple Q = (Q1 , . . . , Qb ) of parabolic subgroups of G. A quasi-parabolic principal G-bundle of type (ϑ, Q) is a tuple (P, s) which consists of a principal G-bundle P of topological type ϑ ∈ Π(G) and a tuple s = (si , i = 1, . . . , b) of points si ∈ P|{xi } /Qi , i = 1, . . . , b. Two quasi-parabolic principal G-bundles (P1 , s1 ) and (P2 , s2 ) are isomorphic, if there is an isomorphism ψ: P1 −→ P2 , such that the induced isomorphisms ψi : P1|{xi } /Qi −→ P2|{xi } /Qi satisfy s2i = ψi (s1i ), i = 1, . . . , b. We have to explain what the replacements for the parabolic weights should be. Let Q ⊂ G be a parabolic subgroup, T ⊂ Q ∩ [G, G] a maximal torus, and X.,% (T ) :=
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357
X. (T ) *# 3 the 3-vector space of rational one parameter subgroups of T . To a nontrivial element a ∈ X.,% (T ), we may associate the parabolic subgroup QG (−a) of G.30 Then, F L + (T ) := a ∈ X.,% (T ) | QG (−a) = Q X., % is a rational polyhedral cone inside X.,% (T ). We fix a maximal torus T i ⊂ Qi ∩ [G, G], + (T i ), i = 1, . . . , b). We i = 1, . . . , b. Then, a parabolic weight is a tuple a = (ai ∈ X., % say that a is admissible, if |Fα, ai 4| ≤ 1/2 holds for every root α of T i and every index i ∈ { 1, . . . , b }. Remark 2.9.3.4. Let κ: G −→ GL(W) be a representation of G. Let a be a rational one parameter subgroup of G. We associate to it the weighted flag (W• (a), α• (a)). Suppose that a = (a1 , . . . , ab ) is a parabolic weight. We define aκ = (aκi j , j = 1, . . . , si , i = 1, . . . ., b) with (aκi1 , . . . ., aκisi ) := dim(W)·α• (ai ), i = 1, . . . , b. We say that a is κ-admissible, if si T j=1
aκi j < 1,
i = 1, . . . , b.
Lemma 2.9.3.5. A parabolic weight a is admissible, if and only if it is Ad-admissible. For the proof, we refer to the paper [103]. We fix a maximal torus T ⊂ [G, G] and a parabolic weight a = (ai ∈ X.,% (T ), i = 1, . . . , b). Then, ai gives the parabolic subgroup Qi := QG (−ai ) which contains T , and + (T ) in the above conventions, i = 1, . . . , b. A parabolic principal G-bundle of ai ∈ X., % type (ϑ, a) is a tuple (P, s, a) in which P is a principal G-bundle of topological type ϑ ∈ Π(G), s = (si , i = 1, . . . , b) is a tuple of points si ∈ P|{xi } /Qi , i = 1, . . . , b, and a is the fixed parabolic weight. A parabolic principal G-bundle (P, s, a) of type (ϑ, a) gives rise to the quasi-parabolic principal G-bundle (P, s) of type (ϑ, Q = (Q1 , . . . , Qb )), and we call two parabolic principal G-bundles isomorphic, if the corresponding quasiparabolic principal G-bundles are so. Let us a assume that our group G is semisimple. We fix a faithful representation κ: G −→ GL(W) and prescribe T and a as before. The rational one parameter subgroup ai yields the weighted flag (W• (−ai ), α• (−ai )) inside W with W• (−ai ) : {0} ! Wi1 ! · · · ! Wisi ! W,
i = 1, . . . , b.
We set r := dim(W), d = 0, r := (ri j := r − dim(Wi j ), j = 1, . . . , si , i = 1, . . . , b). A parabolic principal G-bundle (P, s, a) gives rise to a parabolic vector bundle (E, q, aκ ) of type (r, d, r, aκ ) as follows: The tuple aκ was defined in Remark 2.9.3.4, E is the vector bundle with fiber W that is associated to P by means of κ, the points si ∈ P|{xi } /QG (−ai ) '→ I som(W * OX , E)/QGL(W) (−ai ) give flags E•i : {0} ! Ei1 ! · · · ! Eisi ! E|{xi } ,
i = 1, . . . , b,
30 We exclude one parameter subgroups of the center, because we would like to have proper parabolic subgroups.
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and we define q = (qi j , j = 1, . . . , si , i = 1, . . . , b) via qi j : E|{xi} −→ Qi j := E|{xi } /Ei j ,
j = 1, . . . , si , i = 1, . . . , b.
The parabolic principal G-bundle is now defined to be (semi)stable, if ^ 4 Paκ E• (β), α• (β) (≥)0 holds for every reduction β: X −→ P/QG (λ) to a one parameter subgroup λ of G. As usual, there are moduli functors M(s)s (ϑ, a) for (semi)stable parabolic principal Gbundles of type (ϑ, a). Theorem 2.9.3.6. Fix ϑ ∈ Π(G) and the parabolic weight a. If a is κ-admissible, then there exists a projective moduli space M ss (ϑ, a) for the moduli functors M(s)s (ϑ, a). Proof. This theorem is demonstrated in [103]. In that paper, it is also shown that the moduli space M ss (ϑ, a) exists as a quasi-projective scheme, if k is an algebraically closed field of positive characteristic. " Remark 2.9.3.7. i) There is a definition of semistability which does not involve the representation κ. Let T be a maximal torus of [G, G], a ∈ X.,% (T ) a rational one parameter subgroup of T , Q a parabolic subgroup, and χ a character of Q. The intersection Q1 ∩ Q2 of any two parabolic subgroups Q1 and Q2 of G contains a maximal torus. Therefore, we might find an element g ∈ QG (−a), such that g · a · g−1 is a rational one parameter subgroup of Q. We set Fa, χ4 := Fg · a · g−1 , χ4. It is readily verified that this definition does not depend on the choice of the group element g (see [103]). A parabolic principal G-bundle (P, s, a) is (semi)stable, if and only if, for every parabolic subgroup Q ⊂ G, every anti-dominant character χ of Q, and every reduction β: X −→ P/Q of P to Q, the inequality b ^ 4 T deg L (Q, χ, β) + Fai , χ4(≥)0 i=1
is verified. (The line bundle L (Q, χ, β) was defined in Theorem 2.4.9.3.) ii) If k is an algebraically closed ground field of positive characteristic, then the moduli space M ss (ϑ, a) is projective under the following assumptions on the characteristic of k: 1. The simple factors of G are of type A and k is arbitrary. 2. The simple factors of G are of type A, B, C, D and Char(k) % 2. 3. The simple factors of G are of type A, B, C, D, G and Char(k) ≥ 11. 4. The simple factors of G are of type A, B, C, D, G, F, E6 and Char(k) ≥ 23.
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5. The simple factors of G are of type A, B, C, D, G, F, E6, E7 and Char(k) ≥ 37. 6. The simple factors of G are of type A, B, C, D, G, F, E6, E7 , E8 and Char(k) ≥ 59. iii) Over ', the moduli spaces of semistable principal G-bundles have already been constructed in [18] and [212]. iv) The notion of a parabolic vector bundle was generalized in a different direction to the setting of principal G-bundles in [8]. We use the definition of Remark 2.9.3.7, i), to define (semi)stability for parabolic principal G-bundles with arbitrary connected reductive structure group G. We get the moduli functors M(s)s (ϑ, a) for (semi)stable parabolic principal G-bundles of type (ϑ, a). Theorem 2.9.3.8. Fix ϑ ∈ Π(G) and the parabolic weight a. If a is admissible, then there exists a projective moduli space M ss (ϑ, a) for the moduli functors M(s)s (ϑ, a). Proof. We look at the adjoint representation Ad: G −→ GL(g). Let GB be the image of Ad and κ: GB −→ GL(g) the induced faithful representation. Write ϑB ∈ Π(GB ) for the topological type induced by ϑ via extension of the structure group. We have fixed a maximal torus T ⊂ [G, G] and define T B ⊂ GB to be its image in G. The parabolic weight a gives a parabolic weight aB = (aBi ∈ X.,% (T B ), i = 1, . . . , b), and the parabolic subgroups QGB (−aBi ) are the images of the parabolic subgroups QG (−ai ) under the projection G −→ GB . Let P be a principal G-bundle and P B the principal GB -bundle obtained from it by extending the structure group. There are canoniB cal isomorphisms P|{xi } /QG (−ai ) −→ P|{x /QGB (−aBi ), i = 1, . . . , b. Using these isoi} morphisms, any parabolic principal G-bundle (P, s, a) gives a parabolic principal GB bundle (P B , s, aB ). The definition of semistability in the form of Remark 2.9.3.7, i), is such that (P, s, a) is (semi)stable, if and only if (P B , s, aB ) is so. Taking into account Lemma 2.9.3.5, Theorem 2.9.3.8 gives the moduli space M ss (ϑB , aB ). The straightforward generalization of the Ramanathan–G´omez–Sols lifting method finally yields the moduli space M ss (ϑ, a). " The moduli stack of parabolic principal G-bundles plays a prominent rˆole in the ramified global Langlands program. The details are exposed in the article [66]. Decorated Parabolic Principal Bundles We start with a reductive linear algebraic group G and a representation *: G −→ GL(V). The G-module V decomposes into its irreducible components V = V1 - · · · - Vu , and we write *i : G −→ GL(Vi ), i = 1, . . . , u, for the induced representations, such that * = *1 - · · · - *u . In addition, let L = (L1 , . . . , Lu ) be a tuple of line bundles. Then, a twisted affine parabolic *-bump of type (ϑ, a, L) is a tuple (P, s, a, ϕ) in which (P, s, a) is a parabolic principal G-bundle of type (ϑ, a) and ϕ = (ϕ1 , . . . , ϕu ) is a tuple of homomorphisms ϕi : P*i −→ Li , i = 1, . . . , u. The twisted affine parabolic *-bump (P1 , s1 , a1 , ϕ ) is isomorphic to the twisted affine parabolic *-bump (P2 , s2 , a2 , ϕ ), 1 2 if there is an isomorphism ψ: P1 −→ P2 which gives an isomorphism between the parabolic principal G-bundles (P1 , s1 , a1 ) and (P2 , s2 , a2 ) and for which the induced isomorphisms ψ*k : P1,*k −→ P2,*k satisfy ϕ1,k = ϕ2,k ◦ ψ*k , k = 1, . . . , u.
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Let S be a parameter scheme. A family of twisted affine parabolic *-bumps parameterized by S is a tuple (PS , sS , a, ϕ ) with S
• a principal G-bundle PS on S x X, • a parabolic weight a = (a1 , . . . , ab ), • sections sS ,i : S x{xi } −→ P|S x{xi } /QG (−ai ), i = 1, . . . , b, • a tuple ϕ = (ϕS ,1 , . . . , ϕS ,u ) of homomorphisms ϕS ,i : PS ,*i −→ π.X (Li ), i = S 1, . . . , u. Two such families (PS ,1 , s1S , a1 , ϕ ) and (PS ,2 , s2S , a2 , ϕ ) are isomorphic, if a1 = a2 S ,1 S ,2 and there is an isomorphism ψS : PS ,1 −→ PS ,2 , such that one has ϕS ,1,k = ϕS ,2,k ◦ ψS ,*k for the induced isomorphisms ψS ,*k : PS ,1,*k −→ PS ,2,*k , k = 1, . . . , u, and s2S ,i = ψS ,i ◦ s1S ,i for the induced isomorphisms ψS ,i : PS ,1|S x{xi } /QG (−ai ) −→ PS ,2|S x{xi } /QG (−ai ), i = 1, . . . , b. The stability parameter is a rational character χ of G. For a twisted affine parabolic *-bump (P, s, a, ϕ) and a reduction β: X −→ P/QG (λ), λ: '. −→ G a one parameter subgroup, we define the number µ(β, ϕ) as in Section 2.8. A twisted affine parabolic *-bump (P, s, a, ϕ) is said to be χ-(semi)stable, if ^ 4 Pa E• (β), α• (β) + F λ, χ 4(≥)0 is verified for every one parameter subgroup λ: '. −→ G and every reduction β of P to λ, such that µ(β, ϕ) ≤ 0. We introduce the moduli functors Mχ-(s)s (*, ϑ, a, L): Sch. −→ Set Isomorphy classes of families of χ-(semi)stable twisted affine parabolic S 1−→ . L) *-bumps of type (ϑ, a, parameterized by S The techniques developed in Chapter 2 give the following result: Theorem 2.9.3.9. Fix ϑ ∈ Π(G), the parabolic weight a, the tuple L = (L1 , . . . , Lu ) of line bundles, and χ ∈ X% (G). Then, there is a quasi-projective moduli space M χ-ss (*, ϑ, a, L) for the functors Mχ-(s)s (*, ϑ, a, L). As in the case of twisted affine *-bumps (Proposition 2.8.1.4), there is a projective Hitchin map Hpar : M χ-ss (*, ϑ, a, L) −→ / to the affine variety / defined in (2.65).
Example 2.9.3.10 (Parabolic Higgs bundles). An interesting application arises again for the adjoint representation. Set D = x1 + · · · + xb . The twisting line bundles are all set to be ωX (D). (This means that we allow simple poles in the points xi , i = 1, . . . , b.) We also choose a tuple Q = (Q1 , . . . , Qb ) of parabolic subgroups of G, maximal tori
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+ T i ⊂ Qi , i = 1, . . . , b, and admissible weights a = (ai ∈ X., (T i ), i = 1, . . . , b). % A parabolic pre-Higgs bundle is a tuple (P, s, a, ϕ) where (P, s, a) is a parabolic principal G-bundle and ϕ: OX −→ Ad(P) * ωX (D) is a section. A parabolic pre-Higgs bundle (P, s, a, ϕ) is (semi)stable, if, for every parabolic subgroup Q ⊂ G, every antidominant character χ of Q, and every Higgs reduction β: X −→ P/Q of P to Q, the inequality b ^ 4 T Fai , χ4(≥)0 deg L (Q, χ, β) + i=1
is verified. If we fix the topological type ϑ of P, then Theorem 2.9.3.9 gives the moduli space for semistable parabolic pre-Higgs bundles (P, s, a, ϕ) where (P, s, a) is of type (ϑ, a). (Note that we use the formalism for adjoint groups together with the Ramanathan–G´omez–Sols method, so that we may work with any admissible parabolic weight, by Lemma 2.9.3.5.) In the literature, some closed subvarieties of this moduli space are considered. These are moduli spaces for semistable parabolic pre-Higgs bundles (P, s, a, ϕ) with certain conditions on the residues at the points xi , i = 1, . . . , b. There are the following variants: • A parabolic ωX (D)-pair is a pre-Higgs bundle (P, s, a, ϕ), such that Im(ϕ|{xi } ) ⊂ Lie(Qi ) *. (ωX (D)|{xi} ), i = 1, . . . , b. (For G = GLr ('), the vector bundle E of rank r corresponding to P, and the Higgs field ϕ: E −→ E * ωX , this means that ϕ|{xi } preserves the flag in E|{xi } which is determined by the quasi-parabolic structure s, i = 1, . . . , b.) • A parabolic Higgs bundle is a pre-Higgs bundle (P, s, a, ϕ) with Im(ϕ|{xi } ) ⊂ Lie(Ru (Qi )) *. (ωX (D)|{xi } ), i = 1, . . . , b. (For G = GLr ('), the vector bundle E of rank r corresponding to P, and the Higgs field ϕ: E −→ E * ωX , this condition amounts to the fact that ϕ|{xi } is nilpotent with respect to the flag in E|{xi } corresponding to si in the quasi-parabolic structure s, i = 1, . . . , b.) The notions of (semi)stability for parabolic ωX (D)-pairs and parabolic Higgs bundles are the same as for parabolic pre-Higgs bundles. From the above definitions, it is straightforward to conclude that the moduli functors for (semi)stable ωX (D)-pairs and parabolic Higgs bundles are closed subfunctors of the moduli functor for (semi)stable pre-Higgs bundles. Thus, we also have moduli spaces for (semi)stable ωX (D)-pairs and parabolic Higgs bundles (of fixed type (ϑ, a)). To end the discussion, let us give a few references to the literature: For the structure group G = GLr ('), parabolic Higgs bundles were probably considered first by Simp˚ −→ son [203]. He establishes a correspondence between certain representations π1 (X) GLr (') and stable parabolic Higgs bundles. A GIT-construction of the moduli space of semistable parabolic ωX (D)-pairs with structure group GLr (') was given by Yokogawa in [221] (see also [222]). A construction of the moduli space of stable parabolic Higgs bundles as a hyperk¨ahler quotient is contained in [122]. The Betti numbers of moduli spaces for stable parabolic Higgs bundles with structure group GLr (') for parabolic weights of the coprime type (so that stability is equivalent to semistability) have been determined for r = 2 in [28] and [159] and r = 3 in [73]. Information on parabolic
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Higgs bundles with other structure groups will appear in the forthcoming paper [25]. Finally, parabolic Higgs bundles for some real reductive groups (compare the section on Γ-Higgs bundles, p. 310ff) were examined in [136] and [74].
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[196] J.-P. Serre, The notion of complete reducibility in group theory, Part II of the Moursund lectures 1998, 32 pp, available at http://math.uoregon.edu/resources/serre/. [197] C.S. Seshadri, Quotient spaces modulo reductive algebraic groups, Ann. of Math. 95 (1972), 511-56. [198] C.S. Seshadri, Geometric reductivity over arbitrary base, Adv. Math. 26 (1977), 225-74. [199] C.S. Seshadri, Fibr´es vectoriels sur les courbes alg´ebriques, notes by J.-M. Drezet, Ast´erisque 96 (1982), 209 pp. [200] C.S. Seshadri, Geometric reductivity (Mumford’s conjecture) — revisited in Commutative Algebra and Algebraic Geometry, edited by S. Ghorpade, H. Srinivasan, and J. Verma, 137-45, Contemp. Math., 390, Amer. Math. Soc., Providence, RI, 2005. [201] K.S. Sibirski˘ı, Unitary and orthogonal invariants of matrices, Russian, Dokl. Akad. Nauk SSSR 172 (1967), 40-3. [202] K.S. Sibirski˘ı, Algebraic invariants of a system of matrices, Russian, Sibirsk. ˘ 9 (1968), 152-64. Mat. Z. [203] C.T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), 713-70. [204] C.T. Simpson, Moduli of representations of the fundamental group of a ´ smooth projective variety. I, Inst. Hautes Etudes Sci. Publ. Math. 79 (1994), 47-129. [205] C.T. Simpson, Moduli of representations of the fundamental group of a ´ smooth projective variety. II, Inst. Hautes Etudes Sci. Publ. Math. 80 (1994), 5-79. [206] Ch. Sorger, Lectures on moduli of principal G-bundles over algebraic curves, School on Algebraic Geometry (Trieste, 1999), 1-57, ICTP Lect. Notes, 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000. [207] T.A. Springer, Linear algebraic groups, second edition, Progress in Mathematics, 9, Birkh¨auser Boston, Inc., Boston, MA, 1998, xiv+334 pp. [208] N. Steenrod, The topology of fibre bundles, reprint of the 1957 edition, Princeton Landmarks in Mathematics, Princeton Paperbacks, Princeton University Press, Princeton, NJ, 1999, viii+229 pp. [209] R. Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes ´ Etudes Sci. Publ. Math. 25 (1965), 49-80. [210] B. Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993, vi+197 pp.
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[211] B. Sury, An elementary proof of the Hilbert–Mumford criterion, Electron. J. Linear Algebra 7 (2000), 174-7. [212] C. Teleman, C. Woodward, Parabolic bundles, products of conjugacy classes and Gromov–Witten invariants, Ann. Inst. Fourier (Grenoble) 53 (2003), 713-48. [213] M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), 317-53. [214] M. Thaddeus, Geometric Invariant Theory and flips, J. Amer. Math. Soc. 9 (1996), 691-723. [215] E. Viehweg, Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 30, Springer-Verlag, Berlin, 1995, viii+320 pp. [216] B.L. van der Waerden, Algebra I, based in part on lectures by E. Artin and E. Noether, translated from the seventh German edition by Fred Blum and John R. Schulenberger, Springer-Verlag, New York, 1991, xiv+265 pp. [217] A. Weil, L’int´egration dans les groupes topologiques et ses applications, Actualit´es Scientifiques et Industrielles, 869, Hermann, Paris, 1940, 158 pp. ¨ [218] R. Weitzenb¨ock, Uber die Invarianten von linearen Gruppen, Acta Math. 58 (1932), 231-93 [219] H. Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939, xii+302 pp. [220] J. Winkelmann, Invariant rings and quasiaffine quotients, Math. Z. 244 (2003), 163-74. [221] K. Yokogawa, Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves, J. Math. Kyoto Univ. 33 (1993), 451-504. [222] K. Yokogawa, Moduli of parabolic Higgs sheaves in Moduli of vector bundles (Sanda, 1994; Kyoto, 1994), 287-96, Lecture Notes in Pure and Appl. Math., 179, Dekker, New York, 1996.
Index action, 19, 49 adjoint —, 84 closed —, 56 free —, 104 left —, 19 ADHM-quiver, 349 ADHM-sheaf, 352 co-stable —, 352 isomorphic —s, 352 stable —, 353 type of an —, 352 adjoint action, 84 group, 200 representation, 284 admissible deformation, 222, 269, 287, 290 of a bump, 269, 287, 290 of a principal bundle, 177 of a tump, 222 parabolic weight, 355, 357 affine algebraic group, 17 action of an — —, 19 character of an — —, 18 homomorphism of — —s, 17 radical of an — —, 106 reductive — —, 23 representation of an — —, 19 semisimple — —, 106, 174 subgroup of an — —, 17 bump, 117 moduli space of — —s, 291, 347 parabolic bump moduli space of — —s, 360 algebraic form, 34 group, 17 action of an — —, 19, 49 affine — —, 17
character of an — —, 18 homomorphism of — —s, 17 linear — —, 17 reductive — —, 23 representation of an — —, 19 special — —, 106 subgroup of an — —, 17 torus, 18 ´ Alvarez-C´ onsul, 131, 257 anti-dominant character, 208 arrow, 45 head of an —, 45 tail of an —, 45 associated admissible deformation, 265 family of projective bumps, 294 fiber space, 108 graded object, 128, 241 swamp, 186 asymptotic irreducibility, 163 asymptotically (semi)stable bump, 279 swamp, 164 tump, 230, 337 augmented quiver representation (semi)stable — —, 243 family of — —s, 243 moduli space of — —s, 244 polystable — —, 243 twisted representation of a quiver, 242 Balaji, 103, 179, 190 basic weight vector, 134, 154 Białynicki-Birula, 92 big open subset, 325 binary cubic form, 38 discriminant of a — —, 38
380 form, 34, 37 discriminant of a — —, 37 quartic form, 38 discriminant of a — —, 38 Biswas, 195 Bogomolov, 96 Borel subgroup, 81, 111 boundary, 201 bounded family criterion for a —, 123 of vector bundles, 121 boundedness criterion, 123 Bradlow, 143 pair, 143, 169, 307, 334 isomorphic — —s, 334 semistable — —, 143, 334 stable — —, 143, 334 Bruasse, 96 Brundu, 43 bump affine —, 117 asymptotically (semi)stable —, 279 family of —s, 268, 288 isomorphic —s, 116, 117, 318 moduli space of —s, 270, 287, 291, 347 polystable —, 269, 290 projective —, 116, 318 semistable —, 267, 286, 289, 307, 318, 320, 347 stable —, 267, 286, 289, 307, 318, 320, 347 twisted affine —, 117, 288, 347 type of a —, 117, 347 bundle Higgs —, 237, 241 semistable —, 241 stable —, 241 Burghelea, 202 Cartan decomposition, 68, 310 categorical quotient, 50 ˇ Cech-cohomology, 201 central isogeny, 201, 344 centralizer, 85 chain holomorphic —, 251 chamber, 254 one-dimensional —, 254 two-dimensional —, 254 zero-dimensional —, 254
I character, 18 anti-dominant —, 208 characteristic polynomial, 248 characterization of polystable points, 57 of semistable points, 57 of stable points, 57 Chevalley, 44, 85 closed action, 56 subgroup, 17 co-stable ADHM-sheaf, 352 coarse moduli space, 128, 129 cochain, 201 cocycle, 107, 201 compact real form, 29 completely reducible, 23, 27 representation, 23 complexification, 29 conic bundle, 143 conjugacy class, 84 constructible, 68 contragredient representation, 19 corepresents, 129 covariant, 39 covering, 103 degree of a —, 103 e´ tale —, 107 Galois —, 104, 108 unramified —, 103 critical filtration, 144, 314 cubic form, 34, 38 ternary —, 41 normal form of a —, 41 decorated pseudo G-bundle, 270, 345 isomorphic —s, 270 moduli space of —s, 273, 346 S-equivalent —s, 273 semistable —, 271, 346 stable —, 271, 346 universal family of —s, 277 deformation admissible — of a principal bundle, 177 of a swamp, 140 of a tump, 222 associated admissible —, 265 degree, 217, 239
I
381
of augmented twisted quiver representations, 243 isomorphic —s —, 243 of bumps, 268 isomorphic —s —, 268, 289 of decorated pseudo G-bundles, 270, 271, 345 isomorphic —s —, 271 of hypersurfaces, 137 of parabolic bumps isomorphic —s —, 360 of pseudo G-bundles, 264 of quasi-parabolic sheaves, 354 of swamps, 142, 328, 331 isomorphic —s —, 142, 329, 331 of torsion free sheaves, 327 of tumps, 222 isomorphic —s —, 223 of twisted affine bumps, 288 affine parabolic bumps, 360 quiver representations, 248 quiver representations, isomorphic — s —, 248 of vector bundles, 121 of rank r and degree d is unbounded, 122 universal —, 130 edge, 74 of decorated pseudo G-bundles, 277 elementary symmetric function, 44 of pseudo G-bundles, 193 equivalent of swamps, 149 families of vector bundles, 130 of tumps, 225 line bundles, 59 of twisted affine bumps, 297 representations, 19 fan decomposition, 73 equivariant map, 19 fiber space, 104 e´ tale associated —, 108 covering, 107 fibered system, 105 topology, 106, 344 filtration extension critical —, 144, 314 of the structure group, 111 Harder–Narasimhan —, 125, 145, 355 universal —, 171 Jordan–H¨older —, 128 exterior power, 19 weighted —, 138, 217, 304, 329, 335 fine moduli space, 128, 130, 195 face, 74 flag failure of the Hilbert–Mumford criterion in posuniversal —, 208 itive characteristic, 101 form faithful representation, 26 binary —, 34, 37 Faltings, 179, 190 discriminant of a —, 37 family binary cubic —, 38 criterion for boundedness of a —, 123 discriminant of a —, 38 of a coherent OX -module, 120, 326 of a homogeneous representation, 25 of a line bundle, 118 of a parabolic sheaf, 355 of a torsion sheaf, 119 of a twisted quiver representation, 239 of a vector bundle, 119 of an unramified covering, 103 parabolic —, 355 determinant of a vector bundle, 118 Diaconescu, 349 diagonalizable representation, 21 dimension vector, 46 direct complement, 23 discriminant of a binary cubic form, 38 form, 37 quartic form, 38 of a quadratic form, 36 divided power, 322, 333 divisor universal —, 171 Dolgachev, 92 Donaldson, 328 theory, 328 dual representation, 19
382 I binary quartic —, 38 discriminant of a —, 38 cubic —, 34 Killing —, 284 normal —, 35 for ternary cubics, 41 Jordan —, 43 quadratic —, 34, 36 discriminant of a —, 36 quartic —, 34 quaternary —, 34 ternary —, 34 Formanek, 45 fppf topology, 106 frame bundle, 109 free action, 104 functor of points, 129 fundamental group representation of the —, 251, 354 space of the —, 252 GAGA, 107 Galois covering, 104, 108 group, 104 Garc´ıa-Prada, 143, 257 general linear group, 17 generalized Hitchin map, 248 geometric point, 280 quotient, 50 geometrically reductive, 24, 32 Gieseker, 131, 132, 225, 327 map, 132, 151 is proper, 134 space, 132, 151, 225 Gieseker–Maruyama-(semi)stable sheaf, 327 gluing property, 150 G´omez, 103, 179, 201, 324, 331, 344, 349, 359, 361 good quotient, 49 Graßmann bundle, 231, 236 variety, 231 Graßmannian, 216, 235 Grothendieck, 106, 122, 124 ’s quot scheme, 122 ’s splitting theorem, 114, 124
I 382 group adjoint —, 200 object, 110 scheme, 110, 280 unitary —, 28 Weyl —, 84, 114, 285 Gurevich, 45 Haar, 27 Harder–Narasimhan filtration, 125, 145, 241, 355 of a twisted quiver representation, 240 head, 45, 211 Heinloth, 179, 204, 257, 324 Hesselink, 96 Higgs bundle, 167, 237, 241, 252, 295, 310, 348 moduli space of — —s, 167, 252 parabolic — —, 361 semistable — —, 241 stable — —, 241 field, 295 reduction, 284, 348 sheaf, 349 semistable — —, 349 stable — —, 349 Hilbert, 31, 32, 51 polynomial, 121, 218, 326, 335 Hilbert–Mumford criterion, 32 failure of the — for non-algebraically closed fields of positive characteristic, 101 Hilbert–Nagata theorem, 52 Hitchin, 248, 252 fibration, 248 generalized — map, 248 map, 248, 291, 292, 360 pair, 167, 283 isomorphic — —s, 167, 283 moduli space of — —s, 167 semistable — —, 167, 284 stable — —, 167, 284 space, 167 HM-semistable, 97 Hochster, 199 Hoffmann, 164, 195 holomorphic chain, 251 moduli space of — —s, 251 principal bundle, 107
I homogeneous representation, 25, 135, 211, 268, 322 degree of a —, 135, 211 homomorphism of G-modules, 19 of (affine/linear) algebraic groups, 17 of augmented quiver representations, 242 of principal bundles, 106 is an isomorphism, 106 of quiver representations, 237 of split sheaves, 217 tautological —, 148 universal —, 149 Hu, 92 Huybrechts, 152 hypersurface, 35 family of —s, 137 inner automorphism, 110 instability flag, 232 rationality of the — —, 101 one parameter subgroup, 96, 99, 231 invariant, 23 bilinear from, 20 subbundle, 167 subspace, 23 inversion, 17 irreducible, 23, 27 representation, 23 isogeny central —, 201 isomorphic ADHM-sheaves, 352 affine bumps, 117 parabolic bumps, 359 Bradlow pairs, 334 decorated pseudo G-bundles, 270 families of augmented twisted representations, 243 of bumps, 268, 289 of decorated pseudo G-bundles, 270, 271 of parabolic bumps, 360 of pseudo G-bundles, 264 of quasi-parabolic sheaves, 354 of quotients, 130 of swamps, 142, 329, 331
383 of tumps, 223 of twisted representations, 248 Hitchin pairs, 167, 283 parabolic principal bundles, 357 sheaves, 355 principal sheaves, 345 projective bumps, 116, 318 pseudo G-bundles, 185 quasi-parabolic principal bundles, 356 sheaves, 354 quotient families of swamps, 149 representations, 19 of a quiver, 46 swamps, 136, 328, 330 tumps, 219 twisted affine bumps, 117 isomorphism of orthogonal bundles, 113 of symplectic bundles, 113 isotropic subspace, 48 Jacobian (variety), 124 Jardim, 349 Jordan normal form, 43 quiver, 237 Jordan–H¨older filtration, 128 of a twisted quiver representation, 240 Kempf, 96, 101, 280 Killing form, 284 King, 47, 131 Kraft, 181 Kronecker, 46 Kuttler, 181 Langer, 146, 322, 327, 330 Langlands program, 359 left action, 19 module, 19 Lehn, 152 Levi subgroup, 82, 100, 141, 177 Li, 328 Lie group, 27 complex —, 27 real —, 27 reductive —, 27
384 I representation of a —, 27 linear algebraic group, 17 action of a —, 19 character of a —, 18 homomorphism of —s, 17 radical of a —, 106 reductive —, 23 representation of a —, 19 semisimple —, 106, 174 subgroup of a —, 17 linearization, 56, 108 of a principal bundle, 197 linearized principal bundle, 108 linearly reductive, 24 local sections, 112 universal property, 150 locally finite module, 51 Logar, 43 Logares, 257 Lopshitz, 45 Luna, 105, 182 Marinˇcuk, 45 Maruyama, 327, 331 Maschke, 23, 27 master space, 94 maximal destabilizing subbundle, 125 torus, 68, 97, 98 Mehta, 354 minimal destabilizing quotient, 125, 153 module, 19 left —, 19 locally finite —, 51 semisimple —, 23 simple —, 23 moduli space, 124 coarse —, 128, 129 fine —, 128, 130, 195 for semistable affine bumps, 291, 347 affine parabolic bumps, 360 augmented quiver representations, 244 decorated pseudo G-bundles, 273, 346 Higgs bundles, 167, 252 Hitchin pairs, 167 holomorphic chains, 251 parabolic principal bundles, 358, 359 principal bundles, 178
I 384 projective bumps, 270, 287 pseudo G-bundles, 188, 340 quiver representations, 249, 338 swamps, 142, 330, 332 torsion free sheaves, 327 tumps, 223, 336 vector bundles, 129 multiplication, 17 Mumford, 34, 92 Mumford–Takemoto-(semi)stable sheaf, 327 Narasimhan, 125 neutral element, 17 Newton function, 44 nilpotent, 284 Nitsure, 242, 351 non-separated quotient, 56 normal form, 35 for ternary cubics, 41 Jordan —, 43 normalizer, 84 nullform, 32 one parameter subgroup, 18 instability —, 96, 99 orbit, 30 oriented cycle, 47 orthogonal bundle, 113 isomorphism of — —s, 113 form, 113 group, 113 pair
Bradlow —, 143, 169, 307, 334 Hitchin —, 283 parabolic bump family of — —s, 360 isomorphic — —s, 359 moduli space of — —s, 360 semistable — —, 360 stable — —, 360 twisted affine — —, 359 type of a — —, 359 degree, 355 Higgs bundle, 361 pair, 361 pre-Higgs bundle, 361
I principal bundle, 357 isomorphic — —s, 357 moduli space of — —s, 358, 359 semistable — —, 358 stable — —, 358 type of a — —, 357 sheaf, 354 isomorphic — —s, 355 semistable — —, 355 stable — —, 355 type of a — —, 354 subgroup, 81, 141, 280 swamp, 356 semistable — —, 356 stable — —, 356 vector bundle, 355 weight, 354, 357 admissible — —, 355 path, 47 algebra twisted — —, 238 of length zero, 238 Pl¨ucker embedding, 235, 236 Poincar´e line bundle, 131, 142, 222 point geometric —, 280 polarization, 59 polyhedral cone rational —, 73 polynomial representation, 24 polystable, 56, 78, 91, 239, 243 augmented twisted quiver representation, 243 bump, 269, 290 point characterization of — —s, 57 principal bundle, 177 swamp, 142, 329 tump, 222 twisted quiver representation, 239 Popov, 52 principal bundle, 105, 107, 185 holomorphic — —, 107 homomorphism of — —s, 106 linearized — —, 108 moduli space of — —s, 178, 324 parabolic — —, 357 polystable — —, 177 pull-back of a — —, 106
385 quasi-parabolic — —, 356 s on the projective line, 114 S-equivalent — —s, 178 semistable — —, 176, 200 stable — —, 176, 200 structure group of a — —, 105 swamp associated to a — —, 186 trivial — —, 105 universal — —, 195 Higgs sheaf, 349 pre-Higgs bundle semistable — —, 361 stable — —, 361 sheaf, 345 isomorphic — —s, 345 semistable — —, 345 stable — —, 345 type of a — —, 345 Procesi, 45 projective bump, 116, 318 moduli space of — —s, 270, 287 type of a — —, 116 equivalence, 35, 137 space weighted — —, 60 projectivization of a vector bundle classical —, 137 pseudo G-bundle, 185, 263, 339, 341 decorated —, 270, 345 isomorphic —s, 270 semistable —, 271, 346 stable —, 271, 346 family of —s, 271 isomorphic —s, 185 moduli space of —s, 188, 340 universal family of —s, 193 pull-back of a principal bundle, 106 quadratic form, 34, 36 discriminant of a —, 36 quartic form, 34, 38 quasi-parabolic principal bundle, 356 isomorphic — —s, 356 type of a — —, 356 sheaf, 354 isomorphic — —s, 354 type of a — —, 354 quaternary form, 34
386 I quiver, 45, 211 arrow of a —, 45 augmented — representation, homomorphism of —s, 242 twisted — representation, 242 twisted — representation, type of an —, 242 Jordan —, 237 oriented cycle in a —, 47 path in a —, 47 representation degree of a — —, 239 equivalent — —s, 46 family of — —s, 248 Harder–Narasimhan filtration of a — —, 240 homomorphism of — —s, 237 Jordan–H¨older filtration of a — —, 240 moduli space of — —s, 249, 338 polystable — —, 239 rank of a — —, 239 semistable — —, 239, 338 slope of a — —, 239 stable — —, 239, 338 twisted representation of a —, 237, 338 type of a —, 237, 338 varieties, 131 vertex of a —, 45 quot scheme, 122 universal quotient on the —, 122, 148 quotient categorical —, 50 family, 198 of swamps, 149 of swamps, isomorphic —s —, 149 geometric —, 50 good —, 49 minimal destabilizing —, 125, 153 non-separated —, 56 representation of a quiver representation, 238 universal —, 72, 122, 148, 224 variety, 90 radical, 106, 258, 341, 344 of a representation, 286, 317 unipotent —, 82, 177 Ramanan, 101, 130, 190, 342, 343
I 386 Ramanathan, 101, 174, 177, 178, 190, 201, 324, 342–344, 349, 359, 361 rank, 217, 239, 335 of a coherent OX -module, 120 of a twisted quiver representation, 239 rational polyhedral cone, 73 principal bundle, 342, 344 representation, 19 on an algebra, 51 stably —, 164 rationality of the instability flag, 101 Razmyslov, 45 real form compact — —, 29 point, 29 structure, 29 reduction Higgs —, 284, 348 of a principal bundle to a one parameter subgroup, 176, 265 of the structure group, 111 semistable —, 189 reductive, 23 geometrically —, 24, 32 Lie group, 27 linearly —, 24 representable, 195 representation, 19 augmented — of a quiver, 242 completely reducible —, 23 contragredient —, 19 diagonalizable —, 21 dual —, 19 equivalent —s, 19 faithful —, 26 homogeneous —, 25, 135, 268, 322 degree of a —, 135 irreducible —, 23 isomorphic —s, 19 of !. , 20 of a Lie group, 27 of a quiver, 46, 237, 338 of a torus, 21 of the fundamental group, 251, 354 polynomial —, 24 rational —, 19 space, 237, 252 represents, 130
I residue, 361 resultant, 37, 40 Reynolds operator, 51, 52, 341 Richardson, 45 Riemann–Roch theorem of —, 120, 326 Roberts, 199 Rousseau, 96 S-equivalence, 127, 140, 222, 241, 243, 253, 255, 269, 273, 287, 290, 329 S-equivalent augmented semistable twisted representations, 243 bumps, 269, 290 decorated pseudo G-bundles, 273 principal bundles, 178 semistable twisted representations, 241 tumps, 222 vector bundles, 128 saturated, 68, 323, 327 saturation, 327 scheme group —, 280 semisimple, 23 algebraic group, 106, 174 element in an algebraic group, 84 module, 23 semistable, 32, 56, 78, 91, 96, 239, 243, 338 augmented twisted quiver representation, 243 Bradlow pair, 143, 334 bump, 267, 286, 289, 307, 318, 320, 347 decorated pseudo G-bundle, 271, 346 Higgs bundle, 241 sheaf, 349 Hitchin pair, 167, 284 parabolic bump, 360 principal bundle, 358 sheaf, 355 swamp, 356 point characterization of — —s, 57 principal bundle, 176, 200 sheaf, 345 principal pre-Higgs bundle, 361 reduction, 189
387 sheaf, 326 singular pre-Higgs bundle, 348 swamp, 140, 329, 332 tump, 220, 336 vector bundle, 125 Serman, 195 Serre, 106, 111, 119 Seshadri, 125, 179, 190, 354 set of states, 73, 98, 311, 314 sheaf parabolic —, 354 quasi-parabolic —, 354 split —, 217, 335 homomorphism of —s, 217 torsion —, 119 Sibirski˘ı, 45 simple module, 23 Simpson, 127, 152, 252, 327, 361 singular Higgs bundle, 348 pre-Higgs bundle, 348 semistable — —, 348 stable — —, 348 principal bundle, 342, 344 slope, 239 of a torsion free sheaf, 326 of a twisted quiver representation, 239 semistable sheaf, 326 stable sheaf, 326 Sols, 179, 201, 324, 331, 344, 349, 359, 361 Sorger, 103 special algebraic group, 106 classification of — —s, 106 linear group, 18 orthogonal group, 18, 113 spectral curve, 248 split sheaf, 217, 335 homomorphism of — —s, 217 weighted filtration of a — —, 217, 335 vector space, 212 splitting theorem, 124 stable, 32, 56, 78, 243 ADHM-sheaf, 353 augmented twisted quiver representation, 243 Bradlow pairs, 143, 334 bump, 267, 286, 289, 307, 318, 320, 347 decorated pseudo G-bundles, 271, 346
388 I Higgs bundle, 241 sheaf, 349 Hitchin pair, 167, 284 parabolic bump, 360 principal bundle, 358 sheaf, 355 swamp, 356 point characterization of — —s, 57 principal bundle, 176, 200 sheaf, 345 principal pre-Higgs bundle, 361 sheaf, 326 singular pre-Higgs bundle, 348 swamp, 140, 329, 332 tump, 220, 336 twisted quiver representation, 239, 338 vector bundle, 125 stably rational, 164 stack, 125 state set of —s, 73, 98, 311, 314 Steinberg, 111 strong topology, 68, 107, 201 structure group extension of the —, 111 reduction of the —, 111 subbundle generated by a subsheaf, 15 subgroup Borel —, 81, 111 Levi —, 82, 100, 141, 177 parabolic —, 81, 141, 280 subrepresentation of a quiver representation, 238 subspace invariant —, 23 supporting hyperplane, 74 swamp, 136, 328, 330 associated —, 186 asymptotically (semi)stable —, 164 family of —s, 142, 328, 331 isomorphic —s, 136, 328, 330 moduli space of —s, 142, 330, 332 parabolic —, 356 polystable —, 142, 329 semistable —, 140, 329, 332 stable —, 140, 329, 332 type of a —, 136, 330
I 388 universal family of —s, 149 symbolic method, 39 symmetric function elementary — —, 44 Newton function, 44 power, 19, 34 of a curve, 171 product, 257 symplectic bundle, 113 isomorphism of — —s, 113 form, 113 group, 18, 113 vector space, 113 tail, 45, 211 tautological homomorphism, 148 Teleman, 96 ternary cubic form, 41 normal form of a — —, 41 form, 34 Thaddeus, 143 theorem of Hilbert–Nagata, 52 of Riemann–Roch, 120, 326 Thomas, 334 topology e´ tale —, 106 fppf —, 106 strong —, 68, 107, 201 torsion free, 325, 335 free sheaf family of — —s, 327 moduli space of — —s, 327 sheaf, 119, 325 subsheaf, 325 torus, 18, 21 maximal —, 68, 97, 98 trivial principal bundle, 105 tump, 218, 336 asymptotically (semi)stable —, 230, 337 family of —s, 222 isomorphic —s, 219 moduli space of —s, 223, 336 polystable —, 222 S-equivalent —s, 222 semistable —, 220, 336
I stable —, 220, 336 twisted affine bump, 117, 288, 347 type of a — —, 117, 347 affine parabolic bump, 359 path algebra, 238 representation of a quiver, 237, 338 type of a Bradlow pair, 334 of a parabolic bump, 359 principal bundle, 357 sheaf, 354 of a principal Higgs sheaf, 349 sheaf, 345 of a projective bump, 116 of a quasi-parabolic principal bundle, 356 sheaf, 354 of a quiver representation, 237, 338 of a swamp, 136, 330 of a twisted affine bump, 117, 347 of an ADHM-sheaf, 352 of an augmented quiver representation, 242 unipotent radical, 82, 177 unitary group, 28 universal bundle, 170 divisor, 171 extension, 171 family, 130 of decorated pseudo G-bundles, 277 of pseudo G-bundles, 193 of swamps, 149 of tumps, 225 of twisted affine bumps, 297 flag, 208 homomorphism, 149 principal bundle, 195 property local — —, 150 quotient, 72, 122, 148, 224 unnecessary topological constraint, 286 unramified covering, 103 valuative criterion of properness, 134 vector bundle
389 bounded family of —s, 121 degree of a —, 119 determinant of a —, 118 family of —s, 121 moduli space of —s, 129 projectivization classical — of a —, 137 S-equivalent —s, 128 semistable —, 125 stable —, 125 weighted filtration of a —, 138 vector space split —, 212 Verlinde, 174 Verona, 202 weight admissible parabolic —, 357 parabolic —, 354, 357 vector basic — —, 134, 154 weighted filtration, 138, 217, 304, 329, 335 flag, 83, 212 projective space, 60 Weitzenb¨ock, 52 Weyl, 27 ’s unitarian trick, 27 chamber, 74 group, 84, 97, 98, 114, 285 Winkelmann, 52 Yokogawa, 361
