Advanced Courses in Mathematics CRM Barcelona
Michèle Audin Ana Cannas da Silva Eugene Lerman
Symplectic Geometry of Integrable Hamiltonian Systems
Birkhä user Verlag Basel . Boston Berlin
Authors' addresses:
Michéle Audin Institut de Recherche Mathematique Avancée Université Louis Pasteur et CNRS 7 rue René Descartes 67084 Strasbourg Cedex France
[email protected] Ana Cannas da Silva Departamento de Matemática Institute Superior Técnico Av. Rovisco Pais 1049-001 Lisboa Portugal
[email protected] Eugene Lerman
Department of Mathematics University of Illinois at Urbana-Champaign Urbana. IL 61820 USA
[email protected] 2000 Mathematical Subject Classification 14J32. 14M25, 14R05, 52B20, 53C22, 53010. 53012
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ISBN 3-7643-2167-9 Birkhauser Verlag, Basel — Boston — Berlin
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Contents Preface
ix
A Lagrangian Submanifolds Michèle Audin Introduction I
Lagrangian and special Lagrangian imxnersions in C'1 1.1 Symplectic form on C", symplectic vector spaces I.1.a Symplectic vector spaces I.1.h Symplectic bases 1.1 .c The symplectic form as a differential form Li .d The symplectic group 1.1 .e Orthogonality, isotropy 1.2 Lagrangian subspaces I.2.a Definition of Lagrangian subspaces 1.2.b The symplectic reduction 1.3 The Lagrangian Grassmannian 1.3.a The Grassmannian A,, as a homogeneous space L3.b The manifold A, 1.3.c The tautological vector bundle I.3.d The tangent bundle to A, I.3.e The case of oriented Lagrangian snbspaces I.3.f The determinant and the Maslov class 1.4 Lagrangian submanifolds in C" 1.4.a Lagrangian submanifolds described by functions I.4.b Wave fronts 14.c Other examples I.4.d The Gauss map
1 3 5
5
5
6 7 8 9 9 9 10 11
ii 11
13 14
15 15 16 16
20 25 28
Contents
vi
Special Lagrangian submanifolds in I.5.a Special Lagrangian subspaces I.5.b Special Lagrangian submanifolds Graphs of forms 1.5.c I.5.d Normal bundles of surfaces From integrable systems T.5.e I.5.f Special Lagrangian submanifolds invariant under SO(n) Appendices 1.6 I.6.a The topology of the symplectic group I.6.b Complex structures I.6.c Hamiltonian vector fields, integrable systems Exercises
29
1.5
H Lagrangian and special Lagrangian submanifolds in symplectic and Calabi-Yau manifolds 11.1 Symplectic manifolds 11.2 Lagrangian submanifolds and immersions II.2.a In cotangent bundles 11.3 Tubular neighborhoods of Lagrangian submanifolds I1.3.a Moser's method 113.h Tubular neighborhoods II.3.c "Moduli space" of Lagrangian submanifolds 11.4 Calabi-Yau manifolds II.4.a Definition of the Calabi-Yau manifolds I1.4.b Yau's theorem II.4.c Examples of Calabi-Yau manifolds II.4.d Special Lagrangian submanifolds 11.5 Special Lagrangians in real Calabi-Yau manifolds I1.5.a Real manifolds II.5.b Real Calabi-Yau manifolds II.5.c The example of elliptic curves II.5.d Special Lagrangians in real Calabi-Yau manifolds 11.6 Moduli space of special Lagrangian submanifolds 11.7 Towards mirror symmetry? 1L7.a Fibrations in special Lagrangian submanifolds I1.7.b Mirror symmetry
29 32 33 35 36 .
.
38
39 39 41 41 44
49 49 51 51
53 53 57 59 60 61
62 63 66 66 66 67 68
68
Exercises
70 73 74 75 76
Bibliography
81
Contents
vii
B Symplectic Toric Manifolds Ana Cannas da Silva
85
Foreword
8T
I
Symplectic Viewpoint Symplectic Toric Manifolds 1.1 Symplectic Manifolds 1.1.1 1.1.2 Hamiltonian Vector Fields 1.1.3 Integrable Systems Hamiltonian Actions 1.1.4 Hamiltonian Torus Actions 1.1.5 Symplectic Toric Manifolds 1.1.6 1.2 Classification Delzant's Theorem 1.2.1 Orbit Spaces 1.2.2 Symplectic Reduction 1.2.3 1.2.4 Extensions of Symplectic Reduction 1.2.5 Delzant's Construction 1.2.6 Idea Behind Del.zant's Construction Moment Polytopes 1.3 Equivariant Darboux Theorem 1.3.1 1.3.2 Morse Theory Homology of Symplectic Tone Manifolds 1.3.3 Symplectic Blow-Up 1.3.4 Blow-Up of Toric Manifolds 1.3.5 Symplectic Cutting 1.3.6
II Algebraic Viewpoint 11.1
11.2
Toric Varieties 11.1.1 Affine Varieties 11.1.2 Rational Maps on Affine Varieties 11.1.3 Projective Varieties 11.1.4 Rational Maps on Projective Varieties 11.1.5 Quasiprojective Varieties 11.1.6 Toric Varieties Classification 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.2.6
Spectra Tone Varieties Associated to Semigroups Classification of Affine Toric Varieties Fans Toric Varieties Associated to Fans Classification of Normal Toric Varieties
89 89 89 90 91
94 96 98 101
101 103
104 106 109 113 118 118 119 120 122 124 126
129 129 129 131 132
135 136 137 140 140 142 143 144 148 153
Contents
VII'
11.3 Moment Polytopes 11.3.1 Equivariantly Projective Tone Varieties 11.3.2 Weight Polytopes 11.3.3 Orbit Decomposition 11.3.4 Fans from Polytopes 11.3.5 Classes of Tone Varieties 11.3.6 Symplectic vs. Algebraic
154 154 155 156 158 160 161
Bibliography
165
Index
169
C
Geodesic Flows and Contact Toric Manifolds Eugene Lerman
Foreword I
175 177
From toric integrable geodesic flows to contact tone manifolds Introduction 1.1 1.2 Symplectic cones and contact manifolds
179 179 184
II Contact group actions and contact moment maps
193
III Proof of Theorem 1.38 111.1 Homogeneous vector bundles and slices 111.2 The 3-dimensional case 111.3 Uniqueness of sympleet,ic toric manifolds 111.3.1 Cech cohomology 111.4 Proof of Theorem 1.38, part three 111.4.1 Morse theory on orbifolds
197
Appendix
Hypersurfaces of contact type
Bibliography
198 202 205 211 213 216
221
223
Preface
This book contains an expanded version of the lectures delivered by the authors at the Euro Summer School Symplectic Geometry of Integrable Hamiltonian
Systems. The summer school took place at the Centre de Recerca Matemàtica (Barcelona) from the 10th to the 15th of July 2001. It consisted of three main courses of 7.5 hours each and of some complementary talks dealing with integrable Hamiltonian systems.
Thus the book has three parts. 'I'he first part by Michèle Audin is devoted to special Lagrangian submanifolds, the second part by Ana Cannas da Silva deals with syrnplec tic tomc manifolds, and the last part by Eugene Lerman centers on contact tone manifolds. Hamiltonian systems arose, as their name suggests, from the formulation (by Hamilton) of classical niechaiiics: a Hamiltonian system is a dynamical system that describes the motion of a mechanical system whose total energy is conserved. This is where symplectic geometry comes from. It has since
developed into an area of mathematics in its own right, but it remains a point of contact between physics and geometry. Among all the Hamiltonian systems, the integrable ones -- those which have many conserved quantities — have special
geometric properties; in particular, their solutions are very regular and quasiperiodic. Moreover, their study has been central in symplectic geometry, as it stands at a crossroads between dynamical systems, algebraic geometry, and group representation theory (to name a few areas of mathematics involved). The quasi-periodicity of the solutions of an integrable system is a result of the fact that the system is invariant under a (semi-global) torus action. It is thus natural to investigate the symplectic manifolds that can be endowed with a (global) torus action. This leads, in a natural way, to symplectic toric manifolds (part B of this book), which are examples of extremely symmetric Hamiltonian systems. Physics makes a surprising come-back in part A: to describe Mirror Symmetry, one looks for a special kind of Lagrangian submanifolds and integrable systems, the special Lagrangians. Furthermore, integrable Hamiltonian systems on punctured cotangent bundles are a starting point for the study of contact tone manifolds (part C of this book). Along the way, tools from many different areas of mathematics are brought to bear on the questions at hand. Thus actions of Lie groups in symplectic and contact manifolds, the Delzant theorem, Morse theory, sheaves and Cech cohomology, and aspects of Calabi-Yau manifolds make an appearance in this book.
We hope that this book can serve as an introduction to syinpiectic and contact geometry for graduate students and that it. can also be useful to research mathematicians interested in integrable systems.
x
Preface
Acknowledgments
This Euro Summer School was supported by the European Commission under contract number HPCF-CT-2000-001 10 of the Improving Human Research Potential Programme, by the Spanish Ministry of Science and Technology under contract number PGC2000-2266-E, and by the Department of Universities, Research and Information Society of the Catalan Government under contract number ARCSO1-152.
We would like to thank the Centre de Recerca Matemàtica and its director Manuel Castellet for sponsoring and hosting this Advanced Course, the CRM administrative staff Maria Julia and Consol Roca for smoothly working out innumerable details, and Caries Curràs and Eva Miranda for the mathematical organisation of the course and for making it such a pleasant experience. Thanks are also due to the participants of the course for their many comments on the material. It was indeed a very stimulating audience.
Part A
L agrangian S ubmanifolds Michèle Audin
Introduction This text is an introduction to Lagrangian and special Lagrangian submanifolds. Special Lagrangian submanifolds were invented twenty years ago by Harvey and Lawson [18]. They have become very fashionable recently, after the work of McLean [25], leading to the beautiful speculations of Strominger, Yau and Zaslow [32] and the remarkable papers of Hitchin [19, 20] and Donaldson [11]. My aim here is mainly to present as many examples as possible. I have taken some time to explain why we know so many Lagrangian and so few special Lagrangian submanifolds and immersions. There are mainly two reasons:
• To be Lagrangian is, eventually, a linear property. On the other hand, the property to be special Lagrangian is, in dimension 3 and morc, non linear. • The moduli space of Lagrangian submanifolds that are close to a given one is an infinite dimensional manifold, while the corresponding moduli space of special Lagrangian submanifolds is finite dimensional. This will be apparent in the number and nature of the examples I describe in these notes.
To prepare these lectures, in addition to the papers mentioned above, I have used standard textbooks on manifolds and vector fields as [22], on symplectic geometry as [4, 7, 24, 30] and on complex manifolds and Hodge theory as [8, 15].
I have used standard notation but, although this text pretends to be written in English, I have kept a preference for (transparent) French standards, for instance
P'1 (K) for the projective space of dimension n over the field K and tA for the transpose of a matrix A. I thank Etienne Mann, Edith Socié, Thomas Vogel and Jean-Yves Welschinger for their comments and their help during the preparation of these notes. Special thanks to MiMi Damian, Alicia Jurado and Sébastien Racanière.
Chapter I
Lagrangian and special Lagrangian immersions in C n S
S
To In this chapter, 1 define Lagrangian and special Lagrangian immersions in is the standard real vector space endowed with a begin with, I explain that non degenerate alternated bilinear form 1.1) and use this "symplectic structure" 1.2, 1.3 and 1.4). Later, I use to define Lagrangian subspaces and immersions the complex structure as well, to define special Lagrangian immersions 1.5).
1.1
I.1.a
symplectic vector spaces
Symplectic form on Symplectic vector spaces
with the Hermitian form
Consider the vector space
(Z,Z') = (note that it is anti-linear in the first entry and linear in the second). Decompose
it in real and imaginary parts:
(Z, Z') = (Z, Z') —
Z').
The real part is the standard scalar product (Euclidean structure) of C'1 = R'
(Z, Z') =
+
= X X' + Y . .
x
InCa
6
a symmetric non degenerate (real) bilinear form. The imaginary part defines a (real) bilinear form
that is alternated, this meaning that w(Z, Z) =
0
for all Z. Equivalently, w is
skew-symmetric, that is,
w(Z',Z) = —w(Z,Z'). To write these formulas, I have decomposed the complex vectors of
as
Z=X+iY, and I have used the scalar product X• Y of
The form w is non degenerate
too, as
X =0.
w(X, Y) =0 for all Y
More generally, on a real vector space E, a symplectic form is a non degenerate alternated bilinear form. A vector space endowed with a symplectic form is said to be a symplectic vector space.
I.1.b Symplectic bases Fix a complex unitary basis (ej,.
.
.
(e1,.. is
a basis of the real vector space
,e,,) of Ctm. Put .
.
=
—ie3,
so that
.
Compute w on the vectors of this basis:
= Im(e;,ej) =
1m513
=
0,
also
w(f2,f3) =
= Im(e2,e3) =
0
and eventually
=
—ie3) =
=
Inspired by these properties, we say that a basis (ej,. symplectic vector space is a symplectic basis if
=
and
=
. .
, e,,,
fi,.
.
.
, f,,) of a
0 for all i and j.
There are symplectic bases in all symplectic spaces, thanks to the following proposition.
Li. Symplectic form on C", symplectic vector spaces Proposition 1.1.1. Let to
be
a symplectic form on a finite dimensional vector space of E such that and =
E. There exists a basis (ei,... w(ej,ej) = = 0.
Proof As to is non degenerate, it is not identically zero so that one can find two vectors e1 and Ii such that w(ej, 11) = 1. One then checks that the restriction of to to the orthogonal complement (with respect to to) of the plane (ei, is non degenerate. One eventually concludes by induction on the dimension once noticed that an alternated bilinear form on a 1-dimensional vector space is zero. 0
In particular, the dimension of P is an even number and this is the only invariant of the isomorphism type of (E, to). If E has dimension 2n, then P with its symplectic form is isomorphic to C" with the form to. This result can be called a "linear Darboux theorem", in reference with the forthcoming (Darboux) theorem 11.3.6.
More generally, an alternated bilinear form has a rank, that is the dimension of the largest subspace on which it is non degenerate, and is an even number. Matrices
In a symplectic basis, the matrix of the symplectic form is
(
0
Id
Notice that the matrix J satisfies
J2 =
— Id.
As the matrix of an endomorphism, this is a complex structure. In the symplectic basis of C" associated with the canonical (complex) basis (ej,... e4, J is nothing other that the matrix of multiplication by i. ,
I.1.c
The symplectic form as a differential form
One can write to
as
a differential form to
A
dxi.
= This is an exact differential form (the differential of a degree 1-form): to
=
= d(Y• X).
The form A = Y dX is called Liouvifle form (see § 11.1 below). .
8
I.1.d The syinpiectic group This is the group of isometries of satisfies
A transformation g of
w(gZ,gZ') =w(Z,Z')
for
is symplectic if it
all Z,Z' E C".
Call Sp(2n) the symplectic group of the space C" of dimension 2n. Consider all the groups O(2n), GL(n; C), U(n) and Sp(2n) as subgroups of GL(2n; R).
Proposition 1.1.2. The foUov,ing equalities hold
Sp(2n) fl O(2n) = Sp(2n) fl GL(n; C) = O(2n) fl GL(n; C) = U(n). Proof. Let us characterize our subgroups of GL(2n; R):
(1) g e GL(n; C) if and only if g is C-linear, that is, if and only if
g(iZ) = ig(Z) for all Z. For a matrix A, this is to say that AJ = JA. (2) g E Sp(2n) if and only if g preserves w, that is, if and only if w(gZ, gZ') = w(Z, Z') for all Z and Z'. For a matrix A, this is
tAJA =
J.
(3) g E O(2n) if and only if(gZ,gZ') = (Z,Z'). For a matrix A, this is LAA = Id. One
then checks that two of these conditions imply the third:
• (2) and (3) imply that (gZ, gZ') = (Z, Z')
thus that g
U(n) C GL(n; C).
• (3) and (1) imply that
gZ') = w(gZ, —ig(iZ')) = (gZ, g(iZ'))
(Z, iZ') = w(Z, Z')
thus that g E Sp(2n). • in the same way, (1) and (2) imply (3). In matrix terms, the intersection Sp(2n) fl O(2n) is the set of matrices
;j')
GL(n; C) C GL(2n;
such that
I
= Id. This is exactly the condition that U + iV be a unitary matrix.
1.2.
9
Lagrangian subspaces
I.1.e
Orthogonality, isotropy and F° for
Write F' for the Euclidean orthogonal of the real subspace F of
its symplectic (that is, with respect to w) orthogonal. As w is non degenerate, one has dimF + dimF° = 2n = dimRC". and (F°)° = F
Notice however that a subspace and its orthogonal may have a non trivial intersection. The restriction of the non degenerate form w to a subspace is not always a non degenerate form, in contradiction with what happens in the Euclidean case (which is due to the positivity of the scalar product). In other words, all the subspaces of a symplectic space do not have the same behaviour with respect to the symplectic form. See Exercises 1.6 and 1.7.
One says that a subspace F is isotropic if F C F°, co-isotropic if F F°. For instance, a (real) line is always isotropic, as it lies in its orthogonal which is a (real, co-isotropic) hyperplane. Notice that F is isotropic if and only if F° is co-isotropic. Notice also that the dimension of an isotropic subspace is at most equal to n, half the dimension of 1.2
Lagrangian subspaces
I.2.a Definition of Lagrangian subspaces The isotropic subspaces of maximal dimension n are Lagrangian. For instance, C is a Lagrangian subspace. More generally, a subspace generated by "one half" of a symplectic basis is Lagrangian. Conversely, if F is an isotropic subspace of dimension k n, it is possible to complete any basis (e1 of F in a symplectic basis and thus to obtain Lagrangian subspaces containing F. Let us use now the complex multiplication in C" to state: Lemma 1.2.1. A real subspace P of C" is Lagrangian if and only if P' = iP. Proof. This is a straightforward computation:
w(Z, Z') =
0
Im(Z, Z') =
Re(Z,iZ') = (Z,iZ') = 0.
0 0
U
Lemma 1.2.2. Let P be a Lagrangian subspace of C" and let (x1,. be an orthonormal basis of this real subspace. Then n,,) is a complex unitary basis of C". Conversely, if (xi,. .. Zn) a unitary basis of C", the real subspaee it spans is Lagrangian. . .
,
,
InCa
10
an orthonormal basis of the Lagrangian P, the previous is an orthonormal basis of the . lemma says that the basis (x1,. ,
Proof. If (xi,... ,
is
,
- .
thus that (x1,...
real space
(XZ,X3)
is a complex basis of
= (x,,x1) —
=
Moreover, one has
—0,
[J
thus this is a unitary basis. The converse is even more obvious.
I.2.b The symplectic reduction This is a simple but useful operation, essentially contained in the next lemma. Lemma 1.2.3. Let P be a Lagrangian stthspace and F be a co-isotropic subspace of slAch that
Then the restriction of the projection
PnFcF is injective, the space F/F° is symplectic and the image of P fl F is a Lagrangian subspace.
Proof The symplectic form of clearly induces a non degenerate form on F/F°, as F° is the kernel of the restriction of w to F. The kernel of the composition
PnFcF is
P n F fl F° = P fl F° F being co-isotropic, F D F° = (P° + F)°, since (A + B)° = A° fl B°,
=(P+F)° asPisLagrangian,P=P° =(C'1)° =0
as w is non degenerate.
The map is thus injective. Eventually P fl F is isotropic and has dimension dim P fl F = dim P + dim F - dim(P + F) = dim F
- n,
half the dimension of the symplectic space F/F°, that is
dim F/F° = dimF
—
(2n
—
dim F)
2(dimF — n).
0 See more generally Exercise 1.9.
1.3.
The Lagrangian Grassmannian
11
The Lagrangian Grassmannian
1.3
We consider now the set A,, of all Lagrangian subspaces of
I.3.a The Grassinannian
as a homogeneous space
Look again at lemma 1.2.2. If P1 and P2 are two Lagrangian subspaces of choose an orthonormal basis for each. We thus have two unitary bases of C11. There exists a unitary transformation (an element of the unitary group U(n)) that maps the basis of P1 on that of P2... and thus a fortiori the Lagrangian P1 on the Lagrangian P.2.
In other words, the group U(n) acts transitively on the set of Lagrangian The stabilizer of the Lagrangian subspaces of is the group 0(n) of orthonormal basis changes in R'7. We have defined this way a bijection
U(n)/0(n)
A,,
with the help of which we identify the two sets. Notice that this provides A,, with a topology, namely that of U(n)/ 0(n), the quotient topology of the topology of the matrix group U(n). Example 1.3.1. As all lines are isotropic, the space A1 is the space of real lines in C = R2, namely the projective space P' (R). The unitary group U( 1) is a circle and the orthogonal group 0(1) is the group with two elements {±1 }. As
the unitary group U(n) is compact (being closed and bounded in the
space of matrices) and path-connected (exercise), the space A,, is a compact pathconnected topological space.
I.3.b
The manifold
Let us firstly describe a neighbourhood of P E
(4. = {Q
A,,
in A,,. Put
A,, jQn(iP) =CJ}.
This is an open subset: using a unitary matrix, one can assume that P =
but then is the image in A,, of the (saturated) open subset of U(n) consisting of all the unitary bases the real parts of whose vectors form a basis of This is, clearly, a neighbourhood of P.
Lemma 13.2. The open set Up is homeomorphic to the real vector space of all symmetric endomorphisms of P.
Proof. The subspaces Q that intersect iP only at 0 are the graphs of the linear P iP. It is more convenient to call icc the linear map, so that is a linear map from P to itself. Write now that Q is Lagrangian, namely that maps
Vx,y
P,
w(x + içc(x),y + iço(y)) =
0.
InCa
12
We have = — Im(x + iço(x), y + iw(y)) = w(x, y) + 'p(y)) +
w(x + iço(x), y +
= (w(x),y)
y) — (x,
—
P being Lagrangian. The subspace Q is Lagrangian if and only if the last expression vanishes for all x and y in P, namely if and only if is symmetric'. We have thus defined a bijection that maps 0 to P
End Sym(P)
>
Up
W'
0
a homeomorphism. Remark 1.3.3. Consider for instance the "vertical" Lagrangian C is a disjoint union = U where is the set of all Lagrangians that are not transversal to identified with the space of n x n real symmetric matrices.
We see
that
and
is
We intend to prove now that the open sets Up define the structure of a x R"
Notice firstly that any n-dimensional subspace Q of manifold on may be represented by a rank-n matrix Z
=
fx\
with 2n lines and n columns,
the column vectors of which form a basis of Q. Two matrices Z and Z' describe the same subspace if and only if there exists an n x n invertible matrix g E GL(n; R), such that Zg = Z'.
Lemma 1.3.4. The subspace Q is Lagrangian if and only if the two matrices X and
Y are such that tXY Proof. Let u, u' E
tYX.
and let z, z' be the corresponding vectors in Q:
(x\ Note that Xu, YtL,
,
fx\,
Xii and Yu' are vectors of
We compute:
z') = ,((Xu, Yu), (Xv!, Yu')) = (Xii) . (Yu') — (Yu) (Xu') (scalar product in Ri') = tUtXTuI — = (try —
(as
U• V = tUV)
0 'See also Exercise 1.8.
1.3.
13
The Lagrangian Grassmannian
it can be represented Remark 1.3.5. If Q is the graph of a linear map R'2 —' The relation in lemma 1.3.4 simply expresses the fact that by a matrix Z = the matrix A is symmetric. , n} and the Lagrangian subspace Consider more generally a subset J of { 1, by Uj (for simplicity). Denote Pj of R" x RTh spanned by Any element of Ui is described by a unique matrix Z such that, if we extract from .
. .
Z the matrix containing the lines j (for j J) and j + ii (for j J), we get the Moreover, as we have said open sets Uj clearly cover identity matrix. The
it, each of them can be identified with the subspace Sym(n; R) of n x n symmetric matrices. The Uj's, with their identification with Sym(n; R) are coordinate charts. Change of coordinates are given by Sym(n; R)
A'
U,j fl Uj'
'Zj(A)=Z.r(B)i
where Zj (A) is the matrix obtained from
by
Sym(n; R) B
mapping the first n lines on
the lines j (for j E J) and j + n (for j J). The matrix Zj'(B) is obtained by multiplying Zj (A) by the inverse matrix of the (invertible!) matrix of the lines corresponding to J' in Zj (A). The coordinate change A '—' B is clearly smooth (it is actually rational, thus analytic).
Proposition 1.3.6. The Grassmannian dimension
n(n+1)
is a compact and connected manifold of
0
2
I.3.c The tautological vector bundle Consider the space
E P}. Together with its projection on this is a rank-n vector bundle over The a reason why this fiber of E bundle is qualified as "tautological". The property expressed in Lemma 1.2.1, namely pi = iP, is translated, in terms of the bundle in the fact that ®R C, the complexified bundle, is trivial (has a canonical trivialization). The (global) trivialization is the isomorphism of complex vector bundles
(P,x®(a+ib)) I
' (P(a+ib)x).
InC"
14
I.3.d The tangent bundle to The canonical identification of the open subset Up with the space of synunetric with the bundle endomorphisms of P allows to identify the tangent bundle of It is also possible to describe this bundle from the tangent bundle End of U(n). The group U(n) is described as a submanifold of the space of all complex matrices by the equation LAA = Id, so that we have TA U(n) = {X E GL(n; C) I tAX + tXA = O}.
Call u(n) the vector space Tjd U(n) of skew-Hermitian matrices. There is an isomorphism u(n) TA U(n) >
identifying the tangent bundle T U(n) with the trivial bundle U(n) x u(n) — as image in any Lie group, U(n) is parallelizable. Consider the Lagrangian of the identity matrix Id. One can write
TRn(U(fl)/O(n)) = u(n)/o(n), this is the quotient of the vector space of anti-Hermitian matrices by that of skewsymmetric real matrices. We thus identify
= iSym(n;R), as the real part of a skew-Hermitian matrix is skew-symmetric and its imaginary part is symmetric. Let P be any Lagrangian subspace. Choose a unitary matrix A such that P=A. As we have identified the quotient u(n)/o(n) with the subspace i Sym(n; R) of u(n), we identify the quotient T(A1An with a subspace of TA U(n):
iSym(n;R)
T(AAfl
I
u(n)
We derive an isomorphism
iSym(n;R) —-f X Remark 1.3.7. This isomorphism depends on the choice of A, this is why it does not follow that is parallelizable (it is actually not, as soon as n 2).
1.3.
The Lagrangian Grassmannian
15
I.3.e The case of oriented Lagrangian subspaces of oriented Lagrangian subspaces. Replacing One can also consider the space "orthonormal basis" by "positive orthonormal basis" in what precedes, we get an identification of with U(n)/ SO(n).
I.3.f The determinant and the Maslov class The "determinant" mapping det : U(n)
St
descends to the quotient by SO(n) and, in the same way, its square
S'
det2 U(n)
to the quotient by 0(n). This allows to compute the fundamental groups of and
Proposition 1.3.8. The fundamental group of (resp. is isomorphic to Z. The covering shows as an index-2 subgroup in Proof. Recall first that the group SU(n) is simply connected. This can be proved by induction on n: SU(1) is a point and SU(n + 1) acts transitively on the unit + 1 of sphere with stabilizer SU (n), so that the exact sequence ir1 SU(n)
1r1 SU(n + 1)
gives the result. As the determinant mapping
S'
det : U(n)
is a fibration with fiber SU(n), it induces an isomorphism :
in(S1).
7r1 U(n)
The fiber of the determinant mapping simply connected, thus
S' is SU(n)/ SO(n), which is in1 S'
in1
is an isomorphism. What is left to prove is a consequence of the fact that the diagram
det
det2
SI is
commutative.
j
o
InC'2
16
"The" generator of class" the cohomology class
is called the Maslov class. One also calls "Maslov
that it defines by duality. Using the notation of Remark 1.3.3, it can be shown that (see [1, 12]). is the dual class to the integral homology class represented by
1.4
Lagrangian submanifolds in
We are going now to globalize the notion of Lagrangian subspace, considering submanifolds of C'2 whose tangent space at any point is Lagrangian. We will not really need actual submanifolds, but maps
f:V from some n-dimensional manifold to
C" the tangent mapping of which
is an injection for any point x of V, with image a Lagrangian subspace. It is then said that f is a Lagrangian immersion. For instance, any immersion of a curve (real manifold of dimension 1) in C is a Lagrangian immersion. Any product of Lagrangian immersions is a Lagrangian immersion (into the product target space), we thus obtain Lagrangian immersions of tori (products of circles). Our next aim is to describe examples of Lagrangian submanifolds and immersions in C" and to give a necessary (and sufficient) condition for a given manifold to have a Lagrangian immersion into C".
I.4.a Lagrangian submanifolds described by functions We consider firstly graphs.
Proposition 1.4.1. The graph of a map F: R" —' (i)R" is a Lagrangian submanifold if and only if F is the gradient of a function f : R" R. Proof. The tangent space to the graph at the point (x, F(x)) is the graph of the differential of F at the point x. This graph is a Lagrangian subspace if and only if is a symmetric endomorphism (see the proof of Lemma 1.3.2). The matrix is symmetric for all x if and only if the differential form over R'2 is closed or, equivalently, exact:
F = Vf.
1.4.
Lagrangiari submanifolds in C" See, more generally, Proposition 11.2.1. The Lagrangian submanifolds obtained as graphs have a very specific prop-
erty: the projection of the Lagrangian submanifold on R" is a diffeomorphism. We would like to consider more general Lagrangian immersions, for instance immersions of compact manifolds. Here is a way to construct Lagrangian immersions using the reduction process of § 1.2.h. We start from a Lagrangian submanifold2 We want to construct a Lagrangian immersion into C". To write C" LC
as F/F°, we choose the co-isotropic subspace F = C" Rk, the orthogonal of which is F° = 0 Rk. We suppose that the submanifold L is "transversal to F" in the sense that, for all x,
+F= The Lagrangian subspace L thus satisfies the assumption of the reduction lemma (Lemma 1.2.3). Hence the composition
is the injection of a Lagrangian subspace.
Consider now the intersection V of the submanifold L with F. With the transversality assumption we have made on L, V is an n-dimensional submanifold of F (a consequence of the inverse function theorem) whose tangent space V is the intersection of with F. Thus, the reduction lemma asserts, at the level of each tangent space, that, for all x in V = L fl F, we have the injection of a Lagrangian subspace
In other words, the composition
V=LnFCF is a Lagrangian immersion. Remark 1.4.2. Even if one starts from a Lagrangian submanifold, what we get in general is only an immersion.
Generating functions We generalize the "graph" construction, using the reduction process as explained. Let us start with a nice and useful example. Example 1.4.3 (The Whitney immersion). Consider the unit sphere in
R and the map
a Lagrangian immersion.
1x112
+ a2 =
f:S" (x,a)'
20r
f
'(l+2ia)x.
i}
18
The tangent space to the sphere is + aa =
x RI x
= {(e,a) E
O}
and the tangent mapping to f is T(x,a)f :
C'2
a) S'2
+ 2i(ae + ax).
I
is injective for all (x,a) e S'2: if = 0, then = 0 if x = 0, we have a = ±1 and the equality x + aa = 0 gives a = 0. Thus we have = 0 and a = 0, so that f is an immersion. Moreover, we have The map
and ax =
+
0;
+ dx)) =
+ ax),e' +
(at' + dx) —
+ ax))
so that the image of is an isotropic subspace of dimension n, a Lagrangian subspace. In conclusion, the map f is a Lagrangian immersion. It has a unique double point (North and South poles of the sphere are mapped to 0). In dimension 1, this is a "figure eight". Below (in § I.4.b) we will draw pictures in dimensions 1 and 2.
Obviously, the Whitney sphere is not the graph of a map from to R'2. Let us show that it can nevertheless be described from the graph of a. map defined on a larger space. We start from a function
As we have seen it above, the graph of Vf is a Lagraugian subspace of We reduce as in § I.2.b using the co-isotropic subspace F = C'2 Rc. Here we intersect the graph of Vf with F, namely we consider 0a1
j
Oak
The transversality assumption above is equivalent to the assumption that V is a submanifold of R'2 x in other words that the map
R'1 x Rc (x,a) I is
(Of '
Of Oak
a submersion along V. In terms of partial derivatives, this is to say that the
matrix
(82f\ I
1.4.
19
Lagrangian subrnanifolds in
has maximal rank k. In terms of tangent subspaces, this is to say that the Lagrangian subapaces that are tangent to the graph of Vf are transversal to the co-isotropic subspace F. The reduction lemma 1.2.3 says that the map V
(x,a) is
Of Ix'—
\
-,
I
Ox1
of —
a Lagrangian immersion.
Example 1.4.4 (The Whitney immersion, again). With k =
1
and f(x, a) =
a3
2
2ax gives the x R, and Of/Ox C equation which describes the sphere Whitney map. Example 1.4.5 (Unfolding). Unfoldings are deeply related with Lagrangian submanifolds (see (2]). I will not explain here the general theory but rather show an example. Let P E R.[X] be a degree-(n + 1) polynomial an
P(X) =
+
+
R. These coefficients are going to vary, this is the reason where Xj,. . , the polynomial corresponding to x = why they are named as variables. Call and consider the map (x1,. . .
.
(xj,.
. .
I
P1(a)
to which we apply the previous techniques. The manifold V is
V=
a)
a) E
O}
(zeroes of its derivative when x varies. The this is the set of critical points of condition that V actually be a submanifold is that the matrix of partial derivatives
(02f has
rank 1. But
=
= (ii +
+ (n —
1)xia"2 +... +
InCtm
20
so
that
is
identically 1. Thus V is indeed a submanifold. The La-
grangian immersion is V
(x,a) I
/ 1
8Px
—(a),
.
.
(a)
ax1
instance, starting from the family
For
X4+xiX2+x2X, we
get
V = {(xi,x2,a) E R3 4a3 + 2x1a + x2 = 0) and the Lagrangian immersion from V into R2 x R2 is the map I
(xl,x2,a) Figure
(x1,x2,a2,a).
1.1 shows V with its projection on the plane R2 of coefficients (x1, X2). The
cusp curve is the discriminant of the family of degree-3 polynomials, the set of points x such that has a multiple root. It is obtained here as the set of critical values of the projection V —, R2. Over such a point x in the space of coefficients are the (one or three) roots of the polynomial
Figure 1.1: The discriminant of degree-3 polynomials
I.4.b
Wave fronts
Exact Lagrangian iininersions
If /
V —, is a Lagrangian immersion, the 2-form f*w is zero, so that d(f*A) = 0 and f* A is a. closed 1-form on V. If, for some reason, for instance because Hj,R(V) = 0, this form is exact, there exists a function
F:V
1.4.
21
Lagrangian subrnanifolds in
such that f*A = dF. The immersion f is qualified as exact Lagrangian immersion. The mapping
Fxf:V
C"xR
has the property3
=0.
(1 xF)* Wave fronts
Instead of looking at the Lagrangian immersion f, consider the projection
fxF
R'1xR (X+iY,z)
(X,z).
We will assume here that, at a general point of the Lagrangian, the tangent space is transversal to the subspace of coordinates Y. The image of the Lagrangian imx R. This hypersurface is the wave front. mersion is then a hypersurface of
Of course, it will in general be singular. Precisely, at a point of V where V is X is singular. However, as the projection X + iY not a graph over at every point of the wave front, there is a y3dxj) = 0, (f x F)* (dz — tangent hyperplane, the hyperplane in the space
z
R of (x, z) coordinates
=
at the point image of (X, Y, z). Notice that, as the coefficient of z in this equation is non zero, the hyperplane is always transversal to the z-axis. Conversely, R has at every point a tangent hyperplane if a singular hypersurface of that is transversal to the z-axis, this hyperplane has a unique equation of the Y3x3 and it is possible to reconstruct a (maybe singular) Lagrangian form z = submanifold from the "slopes" Y3. We begin with an example of dimension 1, that of the Whitney immersion
again. Notice that this is indeed an exact Lagrangian immersion: the restriction of the Liouville form ydx to the curve is exact because f ydx 0 (the area surrounded by the curve is zero). A primitive of ydx is easily found. The curve is parametrized by t '—i (cost, sin 2t) and
ydx = —2sin2tcost = 3The manifold C" x R is a "contact manifold" and F x f is a "Legendrian immersion" lifting the Lagrangian immersion F.
InCa
22
Figure 1.2: Eye
Figure 1.3: Crossbow
A map to the (x, z) space is thus t'—' (x,z) = This is depicted on Figure 1.2, in an old-fashion "descriptive geometry" mood. It can be seen that the singular points of the (x, z) curve correspond to the tangents to the (x, y) curve that are vertical, and that the double point of the latter corresponds to the two tangents to the wavefront (the "eye") at points with the same x coordinate that are parallel. Figure 1.3 represents an example in which we start from the wave front (a "crossbow") to reconstruct the Lagrangian. From the wave front, it is seen that the Lagrangian curve has two double points and two "vertical" tangents.
Figure 1.4: Flying saucer
Figure 1.5: Cylinder
One could wonder what it is useful for to replace an immersed curve by a singular one. Notice that, in higher dimensions, the wave front is a hypersurface in x RTh. Even for R and it replaces a submanifold of the same dimension n in n = 2, this is very useful as this allows to represent exact Lagrangian surfaces of R4 by (singular) surfaces in a dimension-3 space. Here are some beautiful examples. Rotate the eye (Figure 1.2) about the z-a.xis to get the flying saucer depicted on
Figure 1.4. The corresponding Lagrangian surface in R2 x It2 is a Lagrangian
1.4.
Lagrangian submanifolds in
23
immersion of the dimension-2 sphere in C2 with a double point. In Exercise 1.14,
one checks that this is, indeed, the Whitney immersion... eventually drawn in dimension 2! Figurc 1.5 represents a cylinder constructed on the eye, namely a Lagrangian immersion of a cylinder, product of a figure eight with an interval, with two whole lines of singular points.
Singularities
Wave fronts are, as we have said it, singular hypersurfaces. We have seen, in dimension 1, cusps, in dimension 2, lines of cusps, but this can be more complicated, as Exercise 1.15 shows it.
Wave fronts of non exact Lagrangian linmersions
Wave fronts are so nice that it is a pity not to have them for all Lagrangian immersiolls. In dimension 1, the problem is to represent by wave fronts curves that do not surround a zero area. Consider for instance the standard (round) circle in C. As f ydx 0, it seems that nothing can be done. Look, however, at the parametrization (cost,sint).
t
It gives
ydx =
—
sin2
tdt =
d
(sin2t
—
z
Figure 1.6: Wave front of the circle
24
Nothing forbids us to represent the Lagrangian (non exact) immersion of the circle by a piece of the (non closed) wave front4 parametrized by
tt
I
sin2t
t
and depicted on Figure
Figure 1.7
If we rotate the (unbounded) wave front of Figure 1.6 around a line parallel to the z-axis that does not intersect the wave front, we get the wave front of a Lagrangian torus, the one depicted on Figure 1.7. One can then use the cylinder represented on Figure 1.5 to perform connected sums of wave fronts. This way, Figure 1.8 represents (the wave front of) a genus-2 Lagrangian surface. In the same way, one constructs Lagrangian immersions of all orientable surfaces in C2.
These figures are copied from Givental's paper [13], that contains many other examples.
Figure 1.8: A genus-2 surface
Remark 1.4.6. Except for the torus, all the surfaces depicted here have double points, that show up in the wave fronts as points having the same projection on the horizontal plane and parallel tangent planes. It is rather easy to prove that the torus is the only orientable surface that can be embedded as a Lagrangian submanifold in C2. As for non orientable surfaces, they can be embedded as Lagrangian surfaces when (and only when) their Euler characteristic is divisible by 4 (with the exception of the Klein bottle). See the pictures in [13]. As for the Klein bottle, it has long been unknown whether it had or had not a Lagrangian embedding. Mohnke [27J has recently proved that it has not. 4This is a place where one can really appreciate the difference between closed and exact
1-forms.
5Notice that wave fronts are defined only up to a "vertical" translation, the actual constant
C used in Figure 1.6 is (ir + 1)/4.
1.4.
Lagrangian submanifokis in C"
25
Exact Lagrangian embeddings
Notice that, in all the examples of exact Lagrangian immersions we have given, there are double points. This is obviously necessary in dimension 2 (n = 1), due to Jordan theorem: an embedded curve cannot surround a zero area. This is also true in higher dimensions, due to a (hard) theorem of Gromov [16]: there is no exact Lagrangian submanifold in C".
I.4.c
Other examples
Here are a few other examples.
Consider the map
Sym(n;C)
U(n) A
from the group U(n) to the complex vector space of symmetric matrices. tAA defines a Lagrangian immersion
Proposition 1.4.7. The map A
Sym(n;C).
Proof As tAA = Id when A 0(n), the map 4 is well defined. Call [A] the class of a unitary matrix A in We have seen in I.3.d that the tangent space to at the point [A] can be identified with = {AH H I
It is mapped into Sym(n; C) par AH
iSym(n;R)}.
as follows
'tA (AHLA + AtHtA) A.
The matrix AHtA+AtHtA has the form K-k for K = AWA =
in u(n) and this describes all the matrices in the vector space i Sym(n; R) when H varies in i Sym(n; R). The image of the tangent mapping TEAl 4' is, thus, the subspace p(A) . i Sym(n; R) where
p:U(n) is the representation of U(n) operating on complex symmetric matrices. This image is, indeed, a Lagrangian subspace, being the image of the real part of the complex vector space i Sym(n; C) by a unitary matrix. 0
In C" Tori,.intpgrabte systems
systems (mechanical systems with many conserved quantities) yield iii'any Lagrangian tori. We use here a few standard symplectic notions: Hamilt9nian vector fields, Poisson bracket, commuting functions. See if necessary Ap= is a pendix I 6.c. Recall for instance that an integrable system on —+ R" whose components ft,... , f,, are functionally independent map f : commuting functions.
which is locally free at the regular This defines a local R''-action on are points of the system (the points at which the derivatives of the functions actually independent). Call X1, . , X,., the Hamiltonian vector fields associated These vector fields commute: with the functions . .
=
=
0.
The R'2-action is given by integration: t x=
E R'' is close to 0 (in order and t = (t1,.. denotes the flow of be defined). This local action is indeed locally free on the open set of that give independent tangent vectors at regular points because the vector fields where
. ,
these points. are complete, namely that the Assume moreover that the vector fields flows are defined for all values of t. We then have a locally free action of R'' on the whole set of regular points. The vector fields being tangent to the common this action preserve the level sets. The connected components level sets of the
of the regular level sets of f are thus homogeneous spaces, quotients of R" by discrete subgroups. The discrete subgroups of R'' are the lattices Z' in the linear suhspaces of dimension k. The connected components of the regular level sets are thus diffeomorphic to Rn—k x Tk for some k such that 0 k n. In particular, the compact connected components are tori T" and these tori are Lagrangian6, they are called the Liouviile tori. The next proposition is the easiest part of the Arnold-Liouville theorem (see for instance [2, 6]). Proposition 1.4.8. Compact connected components of the regtdar common level sets of an integrable system are Lagrangian tori. 0
There are many examples of integrable systems and thus of Lagrangian tori, coming from mechanical systems (spinning top, pendulum. ..)7. The most classical example is that of the standard action of the torus
T''={(ti,...,tn)EC'2IItjI=1, i=1,...,n} 6Notice that on a compact connected component, the flows are complete. 7See for instance (6].
1.4.
on
Lagrangian submanifolds in by
(21,.... z,,) (tjzi,. . the orbits of which are the common level sets of the functions (t1,.
.
.
,
.
,
1
1
2
91 =
,9n =
'
2 IZnt
x 51 indeed, for the regular values of the gz's (namely every g2 non tori S1 x zero). We will come back to these examples in § I.5.e.
Normal bundles
be any immersion of a k-dimensional manifold into R". Let now I : V —+ Consider the total space of its normal bundle
Nf = {(s,v)
x R" Is
V
It is naturally mapped into
x
vE
V,
by
f: Nf
(f(x).v).
(x,v)
The manifold Nf has dimension k + n — k = n,
and I
is
clearly an immersion.
Moreover, it is Lagrangian. More precisely, we have:
Lemma 1.4.9. If A is the Liouville form on
x
one has f*A = 0.
Nf. Use the commutative diagram
Proof. Consider a vector X E
If
NI
irj
J.
V
I
f
to compute
(j*A) (xv)
(X) = =
(T(X.L)J(x)) 0
= V (T1f 0 T(XV)lr(X)) = 0 since v is orthogonal to
0 This method allows to construct many (non compact) examples and can be generalized by replacing x R't = T* by the cotangent bundle T* M of a manifold and V —+ by an immersion into M. See § 1I.2.a.
InCtm
28
I.4.d
The Gauss map
Let f : V —' C" be a Lagrangian immersion. Its tangent space at any point is a Lagrangian subapace of C". One can globalize the data consisting of all these tangent spaces to define the "Gauss map" V
,y(f)
xI
By definition of the tautological bundle
I.3.c), one has
= TV. In particular, the tangent bundle to V must have the same properties as Proposition 1.4.10. For a manifold to have a Lagrangian immersion into C's, it is necessary that the complexification of its tangent bundle be trivializable. 0 The converse is true, but less easy to prove. This is an application of Gromov's h-principle [17], see also [23]. (1) Spheres. We have seen examples of Lagrangian immersions of spheres in C" (in I.4.a). One deduces that TS" C is a trivial complex bundle. Notice however that it is not true that the tangent bundle TS" itself is trivial (except for n = 0, 1, 3 and 7).
Examples 1.4.11.
(2) Surfaces. All orientable surfaces and half the non orientable surfaces have Lagrangian immersions in C2 (as we have seen it in §I.4.b). This is not the case, neither for the real projective plane nor for the connected sums of an odd number of copies of this plane.
(3) Normal bundles. This is a case where the tangent bundle itself is trivial (before complexification):
=
e TZV,U
and this is canonically isomorphic to the ambient space
(4) Grassmannians. The Gauss map
of the Lagrangian immersion 4)
satisfies of course
= End
1.5.
29
Special Lagrangian subinanifolds in
The Maslov class
Every Lagrangian immersion has a Maslov class: use the Gauss map
to pull back
to a class
E
p(f) E H1(V;Z). One can also, with the notation of Remark 1.3.3, define see [24] for example. ogy class dual to
1.5
as
the cohomol-
Special Lagrangian submanifolds in
whose tangent space at each Lagrangian submanifolds are submanifolds of point is a Lagrangian subspace. They have a Gauss map into the Grassmannian namely into U(n)/ 0(n). We look now at the submanifolds whose Gauss map takes = SU(n)/ SO(n). These are the special Lagrangian submanifolds, values in invented by Harvey and Lawson [18}.
I.5.a Special Lagrangian subspaces is said to be special Lagrangian if it has a positive An oriented subspace P of orthonormal basis that is a special unitary basis of C'2. For instance, if n = 1, as C has a unique special unitary basis (the group SU(1) is the trivial group), there is only one special Lagrangian subspace in C, the line R c C... this will not be a very interesting notion in dimension 1. Fortunately, for n ? 2, this is more exciting. Identify the space C2 with the skew-field H of quaternions:
Z=(zi,z2) =X+iY =(x1
+iy1,x2+iy2)
= (x1 +iy1)+j(x2+iy2) = (Xi + jX2) + i(yi — jy2). The 2 x 2 matrices that are in SU(2) are the matrices of the form
(zi
with ziI2 +
)z212
= 1.
Thus the special Lagrangian planes are those who have an orthonormal basis (Z, Z') with Z and Z' of the form
JZ=(xj +iy1)+j(x2+iy2) —iy1).
30
Notice that
Z' = [(xi +iy1)+j(x2 = Zj. Thus a basis (Z, Z') of C2 is special unitary if and only if Z' = Zj. Now use multiplication by j to give H the structure of a complex vector space. One has: Proposition 7.5.1. The special Lagrangian subspaces of C2 are the complex lines with respect to the complex structure defined by the multiplication by j. The Grassmannian SA2 is a complex projective line. 0
Remark 1.5.2. Notice also that Sit2 = SU(2)/SO(2)
S3/S'. This is indeed a
dimension-2 sphere.
To distinguish the special Lagrangian subspaces among all the Lagrangian subspaces or the special unitary matrices among all the unitary matrices, one uses the (complex) determinant. To globalize the notion of special Lagrangian subspace and define special Lagrangian submanifolds, it will be practical (and natural) to describe the linear objects by differential forms. The form corresponding to the complex determinant is Il = dxi A ••. A Expressing the definition of the determinant, namely
= (dot A)e1A
(Aei)A
A
e GL(n; C), we have indeed (detA)1l. Hence
detA =
1
A*fl
In order to work with real subspaces, we need an additional notation: call the two degree n real forms:
and
a=Refl, For instance, in dimension 1, 11 = dx,
= dx and L9 =
dy. In dimension 2,
= dzj A dz2 (dx1 + idy1) A (dx2 + idy2) =dx1 Adx2 —dy1 Ady2+i(dyi Adx2 +dx1 Ady2), that is
f
dx1 A dx2 — dy1 A dy2
Adx2+dx1 Ady2.
Proposition 7.5.3. Let P be an oriented (real) vector subspace of dimension n in The number A ... A depends only on P and not on the positive orthonormal basis (x1,. , of P used to express it. .
.
1.5.
31
Special Lagrangian subinanifolds in
x,,, ix1.... , ix,,) and the linear mapping Proof. Consider the 2n vectors (x1 defined by the images of the vectors of the canonical basis: A C" —+ C" x3.
A(e3)
A(ie3) = ix)
(so that A is complex linear). Then
cl(xiA ... A x,,) = detc A.
If (gx1,... ,gx7,) is a positive orthonormal basis of P (that is, if g
SO(n)), one
gets cI(gx1
(since g
Agx,,) = detc(gA) = detcgdetcA = detRgdetcA = detc A = cl(x1A •••A x,,)
0
SO(n) C GL(n; R) C GL(n; C)).
We will thus denote 11(P) the number 11(x1 A- . Ax,,). Similarly, denote ci(P) its real and imaginary parts. and Remark 1.5.4. Notice that 11(P) is non zero if and only if the 2n vectors .
(x1,.. .
,ix,,)
form a basis of C" over R, that is, if and only if P fl iP = {O} or P does not contain any complex line. These subspaces are said to be totally real. This is in particular the case for Lagrangian subspaces.
Proposition 1.5.5. A real subspace P of C" has an orientation for which it is a special Lagrangian subspace tf and only if P is Lagrangian and j3(P) = 0. Proof Let P be a Lagrangian subspace. Choose (x1 an orthonormal basis which is the image of the canonical basis of C" by a unitary matrix A. Thus
S'.
11(P) = detc A
For P to have a positive basis that is special unitary, it is necessary and sufficient
that detc A be equal to ±1, that is, that 1m11(P) =
0.
0 Here
is a last elementary remark on linear suhspaces:
Proposition 1.5.6. Let Q C C" be an oriented isotropic linear subspace of dimension n — 1. There exists a unique special Lagrangian subspace that contains Q. Proof. Choose a positive orthonormal basis (x1,. . , x,, of Q. In the complex line that is the orthogonal, with respect to the Hermitian form, of the complex subspace spanned by the xi's, there is a unique vector such that the basis is a special unitary basis of C". 0 .
InCa
32
I.5.b Special Lagrangian submanifolds A Lagrangian immersion
f:V
is a special Lagrangian is special if of an oriented manifold into C subspace for every x. The Gauss map then takes values in (1) In dimension 1, the tangent space must be the unique special Lagrangian R C C for all x. If V is connected, f must thus be the immersion of an open subset of R by t '—* t + ia. We have already noticed that this dimension will not be very exciting.
Examples 1.5.7.
must be a j-complex line for all x, is thus the (2) In dimension 2, immersion of a j-complex curve into C2. This gives quite a lot of examples. is Remark 1.5.8. The Maslov class of a special Lagrangian immersion into zero. Of course, as the examples above show it, there are much more Lagrangian immersions with zero Maslov class than there are special Lagrangian immersions.
In terms of forms, to say that the immersion
f:V is special Lagrangian is to say that it satisfies
• firstly f*w =
0
(it is Lagrangian)
• secondly
=
0
(it is special).
Proposition 1.5.9. If f is a special Lagrangian immersion, f"T is a volume for,n
onV. Proof. The complex form has type (n, 0) and defines an n-form on V, which is real since its imaginary part vanishes on V. Let x be a point in V and let One has (f*fl)
=
because of Remark 1.5.4 and since V is Lagrangian. Thus
never vanishes. 0
In dimensions 1 and 2, the special Lagrangian submanifolds are non compact (in dimension 2, Liouville's theorem forbids complex curves in C2 to be compact). This is actually always the case, a straightforward application of Proposition 1.5.9: Corollary 1.5.10. There is no special Lagrangian immersion from a compact manifold into C't.
___ 1.5.
33
Special Lagrangian submanifolds in
C" is a special Lagrangian immersion, is an exact complex form:
Proof. If f: V V. But
Decompose z1 dz2 A
is a volume form on
into its real and imaginary parts to get
A
a. =
d Re(z 1dz2 A
and eventually
... A
= dil
= f*a. =
The manifold V thus has an exact volume form, and this prevents it of being
0
compact. Let us give now examples of special Lagrangian submanifolds in from the examples of Lagrangians constructed in section 1.4.
I.5.c
starting
Graphs of forms
Let us begin with Proposition 1.4.1. Let f: R" —i R be a function. We require the graph of Vf, a Lagrangian submanifold, to be a special Lagrangian submanifold. The n = 1 case is not interesting. For n = 2, the Lagrangian immersion associated with the function f is
F:(x,y)'-
(
ofof
and the form /3 is /3 =
dy1 A dx2
+ dx1 A dy2.
Then
= We thus have:
Proposition 1.5.11. Let U be an open subset of R2 and f : U R a function of class e2. The graph of Vf is a special Lagrangian submanifold of C2 if and only 0 if f is a harmonic function. Notice that the condition is linear. Starting from dimension 3, this is no more the case. The function f must satisfy a complicated non linear partial differential
34
equation, expressed in Proposition 1.5.12 below. Let us begin by a notation. Denote
by Hess(f) the Hessian matrix of f, namely the matrix = and by crk(Hess(f)) the k-th elementary symmetric functions of its eigenvalues. More generally, for an n x n real matrix A, write det (A —
X Id) =
example, 01 (Hess(f)) is the trace of the Hessian matrix, the Laplacian of f. R a function of Proposition 1.5.12. Let U be an open subset of if and only graph of Vf is a special Lagrangian submanifold of The class e2. if f satisfies the partial differential equation For
= 0. k>O
Examples 1.5.13. For n = 1, the differential equation is f"(t) = 0 or f'(t) constant, and this is precisely the differential equation of the special Lagrangian submanifolds. For ii = 2, again, only oi appears in the (linear) relation, which expresses the fact that the function f must be harmonic. For n = 3, the relation is
01(Hess(f)) = cr3(Hess(f)) or
= det(Hess(f)). Remark 1.5.14. The only order-i term (in dl) in this partial differential equation is so that the "linear part" of this equation is = 0. this should be compared with McLean's theorem (Theorem 11.6.1 below).
Proof of the proposition. The tangent space to the graph of Vf at the point This is the image of the plane RTh under the linear map Id (x, is a special Lagrangian subspace if and only if 0. Im (detc(Id We still must check that, for any real symmetric matrix A, one has
Im(detc(Id+iA)) = k>O
Since A is real symmetric, it is diagonalizable in an orthonormal basis. It is clear that the two sides of the relation to be proved are invariant under conjugation
by matrices in 0(n). One may thus assume that the matrix A is the diagonal (A1,. .. , The left hand side is then Imfl,(1 + iA3) and it clearly coincides with the right hand side.
0
Special Lagrangian submanifolds in
1.5.
35
I.5.d Normal bundles of surfaces Let f V —+ R" be an immersion of a dimension-k manifold into :
We
know (see I.4.c) that its normal bundle has a natural Lagrangian immersion into
x W'. Look now for the conditions under which this is a special Lagrangian immersion. For the sake of simplicity, suppose here that k = in R3). There is a more general discussion in [18].
2
and n =
3
(case of surfaces
Fix a point x0 in V, a unit normal vector field n = n(x) on a neighbourhood of x0. The restriction of the normal bundle
NJ = {(x,v) x E V,v E R3,v I to this neighborhood is isomorphic with V x R by (x, JL) '—* (x, jin(x)). We map
Nf to C3 by iin(x) + if(x) (notice that, this time, the immersion f appears in the second copy of R", that of purely imaginary vectors). Let us now choose an orthonormal basis V. Assume that this , e2) of basis is orthogonal with respect to the second fundamental form, that is, to the symmetric bilinear form defined on V by (x,
I1(X,Y) = We have
= where A1 and A2 are the two "principal curvatures" of V at x0. Consider now the tangent space to Nf at (x0, v) where v = R. n(xo). The tangent mapping to our immersion is —A1e1,
P0
=
=
—'
e
x
+
t
The images of the basis vectors are e1 e2
i
I
ni
(—IL)12e2.e2)
(n,O).
Thus
fZ(P0) = (dz1 A dz2 A dz3) (((i
= (i so
—
pA1)(i
-
1zAi)ci) A ((i
that P0 is a special Lagrangian if and only if p(A1 + A2) = is zero. In other words, we have shown:
the trace of
An)
—
—
0.
This is to say that
InC"
36
Proposition 1.5.15. The immersion of the normal bundle of
f:V
R3
into C3 is a special Lagrangian immersion if and only if / is a minimal immersion.
0
For more information on minimal surfaces, see, for example, the beautiful surveys in [29] and the references quoted there. Remark 1.5.16. It is true that we have already mentioned Riemannian metrics
in these notes, but up to now, they have had only an auxiliary role. The result presented here is a genuine Riemannian one.
I.5.e From integrable systems Being compact, Lagrangian tori obtained as "Liouville tori" cannot be special Lagrangian submanifolds in C". One can try to replace them by special Lagrangian
submanifolds with the help of the remark included in Proposition 1.5.6: the idea is to consider a (necessarily isotropic) subtorus in a Liouville torus 1"' and to add a direction to construct another Lagrangian submanifold, which will be special. Here is an example, coming from [18], of such a construction. Start from an orbit L of the standard action of T'2 on C" (see § I.4.c), namely a common level set of the functions
gi(z)=
1
1
2
2
9i = none of the as's being zero, so that L is a Lagrangian torus. Choose a subtorus ofT": say
Let V be an orbit of this subtorus, an isotropic torus of dimension n — 1. Consider the Hamiltonian vector fields Y1,. , Y,, associated to the functions gj: . .
J z = (zj,... , z,,) be a point of V. The tangent space to L at z is spanned by values of the 1' 's, the tangent space to V is the hyperplane consisting of the vectors >2 X,Y, satisfying >2 A, = 0. It is spanned by the values at z of the vector Let
the
fields
X1 = Yn,. . ,Xn_i = Yn_.i — that are the Hamiltonian vector fields of the functions .
11
=91
9n,...,fn—i
Yn,
—gn.
37
Special Lagrangian submanifolds in
1.5.
We are looking now for an n-th function f such that the subspace spanned by and X1 is a special Lagrangian at each point where the the vectors X1,. . . , is isotropic and has vectors are independent. The subspace F = (X1,.. , .
dimension ri —
1.
such that:
We look for X1 as a linear combination Xf =
• The vector field Xj is in the subspace orthogonal to (X1,. . the Hermitian form), that is, (Xf,Xk) = 0 for 1 k n — must have the form = 0. Thus )'k —
,
(for
1.
This gives
.
2 I
• The determinant iz1
0
0•..
0
... is real. This allows to determine the function p. from the last vector, this Subtracting the linear combination A1X1+• Zn. so that the determinant is t.\1 + vector becomes +
We are thus looking for functions f and p such that is real.
and
For any index j, we must have:
9f
=
--
Of
and z /.i(Z1,.. ,zn)zi
= )t3z3,
.
.
.
ER.
The functions
give a solution when take
E R, namely when n is even. When n is odd, we rather
Proposition 1.5.17. The functions fi,..
fj(zi,..
.
=
.
,
fn defined by
—
.
.
=
1z12)
and
— form an integrable system on x = which are special Lagrangian cylinders
un is odd all the regular common level sets of
x R.
InC'2
38
Proof. The only thing that is left to prove is that the regular levels are "cylinders" T'2' x R. As we are dealing with an integrable system, we know that the levels are endowed with an R'2-action. Here the n — 1 first vector fields are periodic and in particular complete; the last one is complete too, because the level is a closed submanifold of C". The action is thus an action of T'2' x R and this is a free 0 action, as the level, being special Lagrangian, cannot be compact. Exercise 1.19 describes essentially the same construction.
I.5.f Special Lagrangian subinanifolds invariant under SO(n) The next and sporadic examples also come from [18]. Start from a smooth curve
r
C =C x {O} cCx C'2' = C"
and "rotate" it with the help of the diagonal SO(n)-action, namely
g. (X +iY) = g• X +ig• Y forgE SO(n) and X,Y ER". If we assume the curve does not pass through 0, we get a submanifold of C":
V = {(x+ iy)u x+ iy E F,u
R",u = g(ei) for some g€ SO(n)}
(notice that u describes a sphere S'2' C R"). The tangent space to V at (x + iy)u with is spanned by the vectors (x+iy)U with U TUS'21 and the as is easily tangent to r at x + iy. The submanifold V is always checked:
w((x + iy)U, (x + iy)U') = xy(U
U' — U'
U) =
0,
a basis of TUS"1,
It is special Lagrangian if and only if, denoting (U1,. ..
or to + i?7)detc(Uj, . . But this determinant is equal to (x + (x + iy)'2_i + iii) detR(Ul,. .. , Un_i, u) since these vectors are in R" c C". The condition is thus that -
(x +
+ iii)
R for any tangent vector
+
to I'.
We get eventually:
Proposition 1.5.18. The Lagrangian submanifold of C"
V = {(x+iy)uI (x+iy)
r,u€
R"}
is special Lagrangian if and only if, on F, the function Im ((x + iy)'2) is constant.
0
1.6.
39
Appendices
iVL Figure 1.9
Remark 1.5.19. This method gives essentially one special Lagrangian submanifold in any dimension, which is not much! Remark 1.5.20. Any connected component of r is diffeomorphic to It, the special x R. Lagrangian submanifolds obtained are (unions of) copies of
To draw a picture of the special Lagrangian submanifold, one draws first the
curve F (in the (x,y) plane), then its wave front (in the (x, z) plane). One then notices that the Liouville form = Y• dX is, on V:
= (yu) ((dr)u + xdu) = ydx .
(since udu =
=
0)
{(xu,z) E
so that the wave front of V is x
(x,z) is a point of the wave front of F}.
For example, for n = 2, the curve F is a hyperbola xy constant, its wave front is the curve z = log x and the wave front of the special Lagrangian submanifold is the surface of revolution obtained by rotating the graph of the logarithm function about the z-a.xis (Figure 1.10).
1.6
Appendices
I.6.a The topology of the symplectic group Proposition 1.6.1. The manifold Sp(2n) is diffeomorphic to the Cartesian product of the group U(n) with a convex open cone of a vector space of dimension n(n +1). Corollary 1.6.2. The symplectic group Sp(2n) is path connected. The injection of an isomorphism U(n) in Sp(2n)
Z=
ir1
U(n)
ir1 Sp(2n).
In z
Figure 1.10
Proof of the proposition. Let A E Sp(2n). As any invertible transformation of A can be written in a unique way as a product
A=Sfl and fl is the orthogwhere S is the positive definite symmetric matrix S = onal matrix fl = S'A. As A is symplectic, the matrix S is also symplectic: tA and AtA are symplectic, the matrix AtA is symmetric, positive definite, thus it is diagonalizable in an orthonormal basis and S is the matrix that, in this basis, is the diagonal of the square roots of the eigenvalues of AtA, so that S is indeed symplectic as is AtA. One deduces that
= and thus that
S'A
Sp(2n) fl O(2n) = U(n)
is a unitary matrix. We have thus obtained a bijection Sp(2n)
)
U(m) x S
A' where S denotes the set of positive definite symmetric matrices that are symplectic. We still have to prove that this space is an open convex cone in a vector space of
dimension n(n + 1). Write the matrices as block matrices in a symplectic basis. Let S E 5, we have
s=
with A and C positive definite symmetric and tSJS = J.
The last condition, that expresses the fact that S is symplectic, is equivalent to BA is symmetric and C =
A'(Id +B2).
The mapping S
Si
Sym(n;R) x (BA,A)
1.6.
Appendices
41
(n; R) of all positive definite is the desired diffeomorphism. The open set symmetric real matrices is obviously an open convex cone in the vector space Sym(n; R) of all symmetric matrices, the product is an open convex cone of the product space, that has dimension
2n(n± 1)
Proof of the corollary. The convex cone Sym(n; R) x
R) is contractible.
Remark 1.6.3. There is another beautiful proof of this type of contractibility results, due to Sévennec, in
I.6.b Complex structures If E is a vector space endowed with a symplectic from w, it is said that an endomorphism J of E is a complex structure calibrated by w if J2 = — Id (J is a complex structure),
w(Jv,Jw)=w(v,w) (J is symplectic) and
g(v,w) =c,i(v,Jw) is a scalar product (namely a positive definite bilinear form) on E.
I.6.c Hanilltonian vector fields, integrable systems In this appendix, denote for simplicity symplectic manifold W (see § 11.1).
=
by W. It can be replaced by any
Hamiltonian vector fields
To any function H : W R, the symplectic form allows to associate a vector field, a kind of gradient, the Hamiltonian vector field (sometimes called the "symplectie gradient" H). This is the vector field defined by the relation for all YE
= or by
tXHL) = —dH.
In coordinates, one has
Xjj(xj
=
(OH
— \Oyi I
OH OH —,—-—— Oy,, Ox1
OH Ox,,
Notice that the vector field XH vanishes at x if and only if x is a critical point of the function H:
Xy(x) =0
(dH)1 =0.
InCtm
42
In particular, the singularities (or zeroes) of a Hamiltonian vector field are the critical points of a function. Notice also that the function H is constant along the trajectories, or integral is skew symmetric, we have (dH) (XH) = 0 curves, of the vector field XH: as
or XH H =0. The Poisson bracket
Assume now that f and g {f,g} by the formula
are
two functions on W. Define their "Poisson bracket"
{f,g} = X1 g = dg(Xj). In coordinates, one has
c9gaf Notice
that X1 •g = dg(X1) = w(X1,X9) = —w(X9,Xj) = —df(X9) = —X9 .f,
— {g, 1). This shows that the Poisson bracket is skew-symmetric in f and g. By definition, this is also a derivation (in both entries); in other words, the Poisson bracket satisfies the Leibniz identity
so that {f, g} =
{f,gh} = Using
{f,g}h+g{f,h}.
the general relation LXtY — wCx = t[X,y1
and Cartan formula
Lx =
+ Lxd,
we get
= Lx,tx9w — = dtX1LX9W +
—
— tx9tx1dw
—dtx1tx9w—d(w(X9,X1)) = —d{f,g}, in other words
[X1,X9J
X{f,9}.
We also have
[X1,X9].h={{f,g},h}. From this, we deduce that the Poisson bracket satisfies the Jacobi identity
{f,{g,h}}+
{g,{h,f}} + {h,{f,g}} =0
43
Appendices
1.6.
and thus defines a Lie algebra structure on C°°(W), e°°(W)
the
mapping
X(W)
f'
(W) (with the Poisson bracket) into the being a morphism of Lie algebras from Lie algebra of vector fields (with the Lie bracket of vector fields). Proof of the Jacobi identity. Apply the definition of the bracket of vector fields: [Xj,X91 h = X1• (X9 h) — X9 . (X1
.
and the equality above to get
{{f,g}
,h} =
[Xj,X9J
h
=Xf(X9.h)—X9'(Xfh) = X1 .{g,h} — X9 .{f,h} = {f,{g,h}} —{g.{f,h}}.
taking into account the skew-symmetry of the Poisson bracket, is equivalent to the Jacobi identity. U This,
Integrable systems As any vector field does it, the Hamiltonian vector field XH defines a differential system on W, namely, = XH(X(t)),
the Hamiltoniari system associated with H. The function H is constant along the trajectories of this system, in other words
XH•H=OordH(XH)=O. It is said that H is a first integral of the system. More generally, a function R that is constant along the integral curves of a vector field X is f :W called a first integral of X. In the case of a Hamiltonian vector field XH, the
f
equality XH = 0 is equivalent to {f, H) = 0, we say that the functions I and H commute. It is said that a Hamiltonian system is integrable if it has "as many commuting first integrals as possible". Let us explain this: .
• Let Ii,.
..
=
0
be commuting first integrals of the system XH, so that for all i and j. Each one is constant on the trajectories of
the Hamiltonian system associated to each other one.
InC"
44
• The expression "as many as possible": at any point x of W, the subspace of W spanned by the Hamiltonian vector fields of the functions is isotropic:
=
=
0.
Its dimension is thus at most n = dim W. It is required that, at least for x in an open dense subset of W, this subspace has maximal dimension n.
• Notice that the vectors forms
are independent at x if and only if the linear
are independent.
Definition 1.6.4. The function H or the Hamiltonian vector field XH on W is qualified as integrable if it has n independent commuting first integrals. Examples 1.6.5. Every function depending only of the coordinates
is integrable: the functions are independent commuting first integrals. Every Hamiltonian system on C is integrable. Similarly, a Hatniltonian system on C2 is integrable if and only if it has a "second first integral".
Exercises Exercise Li. Let V be a real vector space and V* be its dual. Check that the form by (w,13))
ct(w) — /3(v)
is a symplectic form
Exercise 1.2 (Relative linear Darboux theorem). Let F be a vector subspace of a symplectic vector space E. Assume that the restriction of the symplectic form to F has rank 2r. Show that there exists a symplectic basis (ei, . , er,, fl,... , of . .
Esuchthat (ej,...,ef,er+1,...,er+k,fl,...,fr)isabasisofF(kistheinteger defined by 2r + k = dim F).
Exercise 1.3. Show that the symplectic group of C is isomorphic with the special linear group SL(2;R).
Exercise 1.4. Prove directly that the symplectic group Sp(2) is diffeomorphic to the product of a circle by an open disk.
Exercise 1.5. Let A E Sp(2n). Check that the matrices tA and A' are similar8. Show that A is an eigenvalue of A if and only if A' is also an eigenvalue, and that both occur with the same multiplicity. 8Thus A and
A' are similar too.
Exercises
45
Exercise 1.6. Check that a non zero vector of a symplectic space can be mapped to any other non zero vector by a symplectic transformation (in other words, the symplectic groups acts transitively on the set of non zero vectors). Show that, for n> 1, the symplectic group does not act transitively on the set of real 2-dimensional subspaces of Exercise 1.7. Let n> 1 be an integer. Let P be a real plane (dimension-2 subspace)
Show that P is either isotropic or symplectic. What are the orbits of the action of the symplectic group on the set of planes in Exercise 1.8. Let V be a vector space and V* be its dual. Endow V V* with V* be a linear map. the symplectic form defined in Exercise 1.1. Let A V Prove that the graph of A is a Lagrangian subspace if and only if the bilinear form defined by A on V is symmetric. in
Exercise 1.9. Let E be a vector space endowed with a symplectic form
and let
F be (any) subspace of E. Prove that w induces a symplectic structure on the quotient F/F fl F°. Exercise 1.10. Let E be an even dimensional vector space and let w, w' be two symplectic forms on E. Prove that the symplectic groups Sp(E, w) and Sp(E, w') are conjugated subgroups of GL(E). Let be the space of all symplectic forms on the vector space E. Prove that the linear group of E acts on this space by (g . w)(X, Y)
= w(9X,9Y).
Deduce that is in one-to-one correspondence° with the homogeneous space GL(E)/Sp(E), where Sp(E) is the symplectic group Sp(E,wo) for a given form w0 onE. Exercise 1.11. Prove that, on any symplectic vector space, there are complex structures. Prove that a complex structure is an isometry and that it is skew-symmetric for the scalar product it defines. Exercise 1.12. Let V be a real vector space. Using a scalar product on V, construct a complex structure calibrated by the standard symplectic form on V and
such that
(J(v),w) =
v
w for all v,w E V.
Exercise 1.13. Assume that the wave front t
(x(t),z(t))
has an ordinary cusp for t = 0 with a tangent line transversal to the z-axis. Prove that this is the wave front of a Lagrangian immersion of j — ct[ into R2. 9This is actually a homeomorphism.
inCa
46
Exercise 1.14. Prove that the wave front of the Whitney immersion by image of the sphere the hypersurface in
/
(x,a)u
2
—'
is
a3
(using the notation of Example 1.4.3). Find the singular points of this wave front
and draw it in the cases n =
1
(this is the eye, Figure 1.2) and n =
2
(this is the
flying saucer, Figure 1.4).
Exercise 1.15 (The swallow tail). Determine.., and draw the wave front of the Lagrangian immersion described in § 1.4.5 and on Figure 1.1. Exercise 1.16. Prove that the Maslov class of the standard (Lagrangian) embedding
of the circle is ±2. What is that of the Whitney immersion? Of the immersion defined by the crossbow10? is endowed with its LiExercise 1.17 (Lagrangian cobordisms [3]). The space ouville form A and its symplectic form dA. it is said that a Lagrangia.n immersion is "cobordant to zer&' if there exists an oriented manifold V of dif: L mension n +1, with boundary, whose boundary is L, and a Lagrangian immersion
f:V transversal to the co-isotropic subspace F = C" ® iR C
such that
f'(FnV)_—OV=L and such that the cornposiUon L
F
is the immersion 1. 1.4.3) is cobordant to zero.
(1) Prove that the Whitney immersion S"
f*A? —, C is cobordant to zero. What can be said of (2) Assume that I : Prove that, if a Lagrangian immersion St —* C is cobordant to zero, it is
exact.
(3) Consider an exact Lagrangian immersion
f:S1
C
and its wave front in R2. Assume the singularities of the wave front are ordinary cusps. The tangent line to the front at any point is transversal '0Hint: orient the circle and notice that the unit tangent vector to the Whitney immersion does not take all the values in the circle. For the crossbow, notice that this immersion of the circle into C may be deformed, among immersion, into the standard embedding.
47
Exercises
to the z-axis. The circle S' is oriented. Count the cusps of type (a) with a + sign, those of type (b) with a — sign (Figure Lii) and get a number N(f) E Z. What is the value of N(f) for the Whitney immersion? For the crossbow (Figure 1.3)?
(b)
(a)
Figure 1.11
(4) The Lagrangian immersion f : S'
C has a Gauss map 'y(f), taking its
values in the Grassmannian A1 of oriented Lagrangians in C, that is a circle S'. Call a the closed 1-form "dO" on this circle. Prove that1'
N(f) (5) Consider the mapping
j:A, PtIt can
be shown (this is an additional question, use § I.3.f) 3*
: H1(A2)
is an isomorphism. Prove that if f
'
S'
that
H'(A,) C is cobordant to zero, then
N(f) = 0. Does there exist a Lagrangian immersion of a disk into C2 whose boundary is the crossbow?
"This is tn say that N(f)
the Maslov class of the immersion f.
48
Exercise 1.18 (&om (x,y) to
Writing
dz=dx+idy, one gets a couple of relations between the expressions of the vector fields in coorProve for instance that dinates (x,y) or
Of 0
0 i
(*91
— 2
*9
a;
Of
*9
a; 0z3
Exercise 1.19. Consider the vector field X given on C2 by
X(zi,z2) = (icxizi,icx2z2) (a1
and a2 being two real parameters).
(1) Check that
X(zj,z2) and show that the form
*9
8z1
._zzi___) +a2 *921
*9
*9
822
022
is holomorphic.
(2) Show that X preserves w and find a function H such that X = Xy. (3) Under which condition does the vector field X preserve Il? Assume now that this condition holds. Find two functions g and h from C2 to R such that
Consider fl h'(b). Show that, if a is a regular value of H, this is a special Lagrangian submanifold.
(4) Describe the special Lagrangian submanifolds H1 (a) i-curves, that is, by equations.
fl
(b) as complex
(5) Check that they are diffeomorphic to S' x R. Hint: they are conies.
Chapter II
Lagrangian and special Lagrangian submanifolds in symplectic and C alabi-Yau manifolds 11.1
Symplectic manifolds
we must understand how a In order to deform a Lagrangian submanifold in tubular neighbourhood looks like. We prove here that a Lagrangian submanifold has a neighbourhood which is diffeomorphic to a neighbourhood of the zero section in its cotangent bundle. To be precise and explicit, we need to define a symplectic structure on the cotangent bundles and more generally to say what a symplectic structure on a manifold is. A symplectic manifold is a manifold W endowed with a non degenerate 2form w, namely, a non degenerate alternated bilinear form on each tangent space T1W, which is required to be closed, (dw = 0). Notice that a symplectic manifold is even dimensional. (1) The first example is of course C" with the symptectic form we have used so far, considered as a differential form:
Examples 11.1.1.
w=
(where (x1 +
.
A
+ iyn) stands for the complex coordinates in C").
II. In symplectic and Calabi-Yau manifolds
50
One also has: w(Z, Z')
-
=
X Y'.
And this is an exact, hence closed, form: w = d(Eyjdx3). x = (2) The next example is that of cotangent bundles. Think that = T*Rfl and simply replace by any manifold V. On x then that W = T1'V, there is a canonical 1-form, the Liouville form A, defined by the "compact" formula:
= in which x denotes a point of V, an element of TV (namely a linear V of the form on the tangent space TXV), and ir the projection T*V cotangent bundle. If (x1,.. . are local coordinates on V and (yj,... the cotangent coordinates, then ,
,
A
=
The 2-form dA is both closed (!) and non degenerate.
(3) Surfaces. On a surface W, any 2-form is closed. Moreover, in dimension 2, to say that a 2-form is non degenerate means that it nowhere vanishes, in other words that this is a volume form: all the orientable surfaces may be considered as symplectic manifolds. (4) The sphere. Consider, in particular, the unit sphere S2 in R3, whose tangent space at a point v is the plane orthogonal to the unit vector v. Put
Y) = v• (X A Y) = det(v, X, Y). This is a non degenerate 2-form and thus a symplectic form.
(5) The projective space is a symplectic manifold. The nicest thing to do is to define its syrnplectic form starting from that of and using the symplectic reduction process. To define we factor out the unit sphere of by the S'-action (multiplication of coordinates): =
Lagrangian submanifo)ds and immersions
11.2.
51
At each point x of the sphere the tangent space is the Euclidean orthogonal of x and the kernel of the restriction of the symplectic from is the line generated by ix. This line is also the tangent space to the circle through x on the sphere. The symplectic form of defines a non degenerate alternated bilinear form w on Its pull-back on the sphere is closed, so that is closed. It is actually a (the standard) Kähler form on (6) Complex submanifolds of the projective space are symplectic. The compatibility of w with the complex structure gives that w(X, iX) > 0 for any vector X that is tangent to the submanifold, so that w is indeed non degenerate on this submanifold. (7) More generally, all Kähler manifolds are symplectic. We will come back to this remark.
Notice that, on cotangent bundles, as on the symplectic form is exact. This cannot be the case on a compact symplectic manifold. Proposition 11.1.2. On a compact manifold, there exists no 2-form that is both non degenerate and exact.
Proof Let w be a non degenerate 2-form on the 2n-dimensional manifold W. To is a volume form. But then, if
say that w is non degenerate is to say that w
is
dcv,
also exact, thus W cannot be compact.
Haniiltonian vector fields XH for functions H: W R are defined exactly as in Appendix I.6.c and so is the Poisson bracket of two functions on W. Exercise 11.3 explains why it is required that a symplectic form be dosed.
11.2
Lagrangian submanifolds and immersions
An immersion f: L —* W into a symplectic manifold is Lagrangian if and dimW = 2dimL.
II.2.a In cotangent bundles All what was done in
in § I.4.a works as well in a cotangent bundle.
=
0
II. In symplectic and Calabi-Yau manifolds
52
Grapbs Proposition 1.4.1 generalizes as: T* L be a section of a cotangent bundle. Its image L Proposition 11.2.1. Let is a Lagrangian submanifold if and only if the 1-form is closed.
Proof. The most elegant thing to do is to state first a property of the Liouville form (which explains why it is called the "canonical" 1-form): for any form has
one
= is considered as a section of the cotangent bundle in the left In this equality, hand side and as a form in the right hand side. One has indeed:
= = =
o
by definition of by definition of becanse is a section.
Eventually, = 0 if and only if d(a*A) = 0, thus the graph of Ck is a Lagrangian 0 submanifold if and only if is closed.
Remark 11.2.2. In particular, the zero section of L C T*L is a Lagrangian submanifold. What we plan to do next is to show that L C T* L is a model for all Lagrangian embeddings of L into a symplectic manifold (Theorem 11.3.7). Generating functions A function
allows to construct a Lagrangian submanifold (the graph of dF) into T* M x Ck and then, by reduction, a Lagrangian immersion into T*M. Wave fronts
Exact Lagrangian immersions into T*M define wave fronts in M x ft and conversely.
Conorinal bundles Let
f:V
be any immersion. The conorrnal bundle is the subbundle of the pull back bundle
f*T*M =
)
x
Tf*(X)M}
v
11.3.
Tubular neighborhoods of Lagrangian submanifolds
53
defined by
N*f =
= o} = 0}.
E f*T*M I
=
E
f*T*M
I
Map N*f into T*M by )(f(x),co).
F:(x,co)u This is an immersion, since
= It is Lagrangian, as we have F*A = 0. Indeed, calling ir the two projections T*M M et N*f V, we get
= =
0
= cp
=0 as
vanishes on the vectors that are tangent to V.
0
One should check that the proof given for the normal bundle in § I.4.c for is identical to the one given here, the orthogonality used the case where M = there being an ersatz of the duality used here.
11.3
Tubular neighborhoods of Lagrangian submanifolds
Let us now present a method, invented by Moser [28], which allows to describe a symplectic manifold in the neighbourhood of a point (they are all the same) or a neighbourhood of a Lagrangian submanifold in a symplectic manifold.
II.3.a Moser's method The next "lemma" contains all these results. Lemma 11.3.1. Let W be a 2n-dimensional manifold and let Q C W be a compact submanifold. Assume that wo and w1 are two closed 2-forms on W such that, at any point x of Q, w0 and w1 are equal and non degenerate on Then there exists open neighborhoods V0 and V1 of Q and a diffeomorphism
such that
= IdQ and
= w0.
11.3.
Thbu)ar neighborhoods of Lagrangian submanifolds
55
First proof. The vector bundle NQ retracts on its zero section. The inclusion As j*1wiI = V0 thus induces an isomorphism f —' Q
j
and the cohomology classes of that their difference is an exact form.
are equal in
Second proof. We explicitly construct a 1-form the dilatation of factor t in the fibers cOt'
0
that is a primitive of r. Consider
V0
VO
which means
tE1O 1
This is a diffeomorphism (onto its image) for t > 0 and we have cOo(Vo) = Q, Consider its = Idv0 and c°dQ = IdQ. The form i- = — w0 is a 2-form on restriction to Vo. It is identically zero along Q by assumption. We have
Consider now the (time depending) radial vector field dilatation) on Vo. This is the vector field defined by
Is=t'
=
It is defined only for t > 0, in the same way that
is a diffeomorphism only for
t > 0. In a very concrete way, the vector field is
+ v) = For all t, consider also the 1-form
defined by
= Notice that, if y is in Q, one has
cot(y) = y and
thus
(tangent to the
=
is zero along Q. For t> 0, one has
(Y) = (txer)x+tv
t > 0,
=
+
0
II. In symplectic and Calabi-Yau manifolds
56
and consequently
dat
+ tx,dr) = =
d
(tx,r) * r).
Eventually, we get
dat = for t >
0
and thus also for all t E T
= T —0 =
— WO*T
(0,
=
1]. Now
f
=
f
=
da
atdt. We has thus proved that, in a neighbourhood of Q, writing a = da is an exact form (with a identically zero on Q).
—
=
0
To finish the proof of Lemma 11.3.1, we use the actual method of Moser. We consider the path of symplectic forms
=
= w0 + tdcr.
w0 + t(w1 —
For t = 0, this is the non degenerate form Also, along Q, this is the very same form w0. Restricting again Vo if necessary (using compactness again) one can assume that is non degenerate on V0 for all t E [0,11. Let Yt be the vector
field defined by
= (the existence and uniqueness of degenerate). Let be its flow:
—a
are consequences of the fact that Wt is non
We have
=
(d(u) + +
=0 by definition of
Hence
=
=
wo
and eventually
=
0
11.3.
Tubular neighborhoods of Lagrangian subrnanifolds
57
Remark 11.3.5. In a general symplectic manifold W, the proof is identical to the one given here; what we need is the notion of a normal bundle, that is, of orthogonality in TW, and a way to replace the mapping (x, v) '—f x + v. One uses a Riemannian that replaces x + v is metric on W and its exponential mapping: the point (at time 0) with tangent the point reached at time 1 by a geodesic2 starting from x vector v. The most direct application of Lemma 11.3.1 is the Darboux theorem. This is a symplectic form on W and w0 is the is the case where Q is a point Xo, symplectic form induced on
Theorem 11.3.6 (Darboux theorem). Let x be a point of a manifold W endowed with a symplectic form w. There exists local coordinates
(xi,...,xn,y1 centered at x in which w =
yn)
A dx2.
W defines, using a diffeomorphism from on Proof. The form induced by a neighbourhood of 0 in Tro W onto a neighbourhood of xo in W, a symplectic on a neighbourhood of xo. Lemma 11.3.1 gives a diffeomorphism from a = wo. By definition neighbourhood of into itself, that fixes x0 and satisfies form
of w0, there exists local coordinates centered at x0 in which it can be written
o
II.3.b
Tubular neighborhoods
The next application is a theorem of Weinstein that describes the tubular neighborhoods of the Lagrangian submanifolds. Theorem 11.3.7 (Weinstein [34J). Let (W,w) be a symplectic manifold and let L C W be a compact Lagrangian submanifold. There exists a neighbourhood of the zero section in T* L, a neighbourhood V0 of L in W and a diffeomorphism Vo such that N0 —d,\
and PIL=Id
Proof. Let us check that we can apply Lemma 11.3.1. The submanifold Q is the is the restriction of w. The form w1 Lagraiigian submanifold L and the form is the symplectic form of T*L. We are going to compare them in T*L. As in the previous proof, let us assume firstly that W = C". Let be the composed mapping T*L
NL
2To extend the geodesics, we also need an assumption on the completeness of the metric, or on the manifold W.
II. In symplectic and Calabi-Yau manifolds
58
where
•
is the isomorphism between cotangent and tangent spaces given by restricted to L: the Eudlidean structure of '—p
a(u) = • J is the multiplication by i. Recall (see Lemma 1.2.1) that L is Lagrangian if and only if TL1- = JTL. a neighbourhood of the zero section in T*L, mapped onto a suitable Call is a diffeomorphism onto its image. We want to compare, so that To apply Lemma 11.3.1, we = --dA and w0 = C T*L, the two forms in have to check that they coincide along the zero section. Let (x, 0) e L c N0. We have T(xo)
Recall that
there is an exact
TL.
(T*L)
sequence a)IT
T(x,a) (T*L)
0 which splits along
the zero section s,
0
using
'T(so) (T*L), and that the kernel Ker T(X,Q)ir
is
For
v,
along
the
zero
section.
canonically identified with TL. Compute then /3 E TL, we have and w
((v,a),(w,/3)) = = =
+
(v +
+ —
= /3(v) — But
we have seen
(in
((v,
The forms the lemma.
Exercise 11.1) that
(w, /3))
=
dy3 A
dxi) ((v,
(w, /3)) =
—
/3(v).
and —dA coincide along the zero section, therefore we can apply 1i
In the general situation where W is a symplectic manifold, we need a Riemannian metric and an analogue of J. We use an "almost complex structure" J calibrated by w, namely an endomorphism J of the tangent bundle TW such that J2 = — Id and
(X,Y)u
w(X,JY)
is a Riemannian metric. Such structures exist and form a contractible set. See for instance [5, 24j. Notice that this notion is a globalization of the linear notion,
mentioned in §1.6.b.
11.3.
II.3.c
59
Thbular neighborhoods of Lagrangian submanifolds
"Moduli space" of Lagrangian submanifolds
We consider now, for a given manifold L, the space of Lagrangian immersions
f:L We call it a "space" because this set is actually a topological space, a fact which allows to consider immersions that are "close" to a given immersion. We use the Whitney e'-topology. The
Let V and W be two manifolds. The €1-topology is a topology on the space of e'TW) over V x W, the fiber maps from V to W. Consider the vector bundle The total space is usually called at (x, y) of which is the vector space £(TXV, J' (V, W) rather than L(TV, TW). Every map f e C1 (V, W) defines a mapping
j1f: V
J1(V,W)
x'
If U is an open subset of J1 (V, W), denote
V(u)=—{fEe1(v,w)Ij'feu}. The C1-topology is the topology for which the V(U) are a basis. It is said that a map f is "C'-close" to if it is close to fo for the C1-topology. Diffeomorphism group
f
The group of diffeomorphisms of L acts on this space by p I = o We want to consider Lagrangian immersions only up to this action: we do not want to take into account the way the manifold L is "parametrized". Moduli space
We consider the space of Lagrangian e'-ininiersions from L to W up to the action of the diffeomorphism group. The quotient space is called the "moduli space" of Lagrangian immersions from L to W and denoted 41(L). The next theorem describes the Lagrangian immersions that are close to a fixed Lagrangian embedding of L into W. Theorem 11.3.8. Let L be a compact and connected manifold. A neighbourhood of a Lagrangian embedding L
in the space 41(L) can be identified with a neighbourhood of 0 in the vector space of closed 1-forms of class C1 on L.
II. In symplectic and Calabi-Yau manifolds
60
W be a W be a Lagrangian embedding and I : L Proof. Let fo : L f is close to fo for the "e°Lagrangian immersion close to topology3", we can consider that everything lies in a neighbourhood of L. Thanks
to the tubular neighbourhood theorem (here Theorem 11.3.7) we can assume that everything takes place in a neighbourhood of the zero section in T*L. The map f is T*L (this is what ía has become e1-close to the inclusion of the zero section L when we have identified the neighbourhood of ía (L) in W with a neighbourhood of the zero section in T*L). Thus the composition of f with the projection of the cotangent is a e'-mapping L L, close to the identity. Recall the next lemma, which is a consequence of the inverse function theorem.
Lemma 11.3.9. Let L be a compact and connected manifold. Let I be a L —÷ L that is e' -close to the identity. Then f is a diffeomorphism.
-map D
According to this lemma, the composition is a diffeomorphism g of L. Com-
posing with g', we get an embedding
,T*L which is still
to the zero section... but now the composition L
T*L
is the identity. Thus is a section, that is, a 1-form on L, and is closed because the embedding is Lagrangian. Conversely, all the closed 1-forms that are close to the zero section define Lagrangian embeddings close to 0 Remark 11.3.10. One should have noticed that the section L —p T*L defined by a 1-form is a C1-mapping if and only if the form is a e1-form. The e'-topology thus defines the structure of a topological vector space on the space of 1-forms. In § 11.6 below, we will need a Banach space structure. Remark 11.3.11. The vector space we have obtained is infinite dimensional. It can be considered as a neighbourhood of ía in the "manifold" of deformations of fo, or as its tangent space at fo.
11.4
Calabi-Yau manifolds
We want now to describe, in a way analogous to what we have done in § 11.3.b, the moduli space of special Lagrangian submanifolds. In order to apply Theorem 11.3.7 (special Lagrangian submanifolds are, firstly, Lagrangian submanifolds) we need a compactness assumption on the Lagrangian submanifold. Unfortunately, as we have seen it in § I.5.b, the special Lagrangian submanifolds of C'2 are never compact. We thus need to consider more general manifolds, in which it is possible to define special Lagrangian submanifolds. These are the "Calabi-Yau" manifolds. 3The e°-topology, defined similarly to the e1-topology, is simply the compact open topology.
11.4.
61
Calabi-Yau manifolds
which The point is to define a structure that globalizes the structures on have allowed us to speak of special Lagrangian submanifolds. Recall that, in addition to the R-bilinear alternated form w, we have used the form Il = dz1 A• A of the complex determinant. We will use here the best adapted definition of a Calabi-Yau manifold, the
point is not to spend time on the Calabi-Yau manifold itself but rather on its special Lagrangian submanifolds. For more information on Calabi-Yau manifolds, see [33, 81 and the references they contain.
II.4.a Definition of the Calabi-Yau manifolds Our manifolds should be complex and endowed with a symplectic form w and a that is nowhere zero (this is sometimes called a type-(m, 0) holomorphic form holomorphic volume form). Consider thus a manifold M, on which are given
• a complex structure J (multiplication by i), • a closed non degenerate type (1, 1)-form w (the Kähler form)
• a Riemannian metric g(X, Y) =
iY),
• a Hermitian metric h(X, Y) = g(X, Y) — iw(X, Y), • a trivialization of the "canonical" bundle morphic form which is nowhere zero.
namely a type-(n, 0) holo-
We still need a relation between the forms w and Notice that both forms and A are of type (n, n) and both do not vanish on M, in particular, both arc volume forms. We thus have = A
f on M. The additional compatibility condition is that f should be constant. Let us look at the case of C". We have w"
Writing
dy =
—
and dx =
and noticing that
dyAdx =
—
(dz +d2) =
+
II. In symplectic and Calabi-Yau manifolds
62
we can also write
=
A d21
A
A
gives
A
We thus have
-
We will use the same normalization formula to define a Calabi-Yau manifold in general.
Definition 1L4.1. A complex manifold M is said to be a Calabi-Yau manifold if it is Kähler, has a trivialized canonical bundle, and if the Kàhler form w and the type-(n, 0) form trivializing the bundle AflT*M are related by 'n Remark
11.4.2. Recall that it is possible to express the fact that the form w is
Kähler by saying that the complex structure is "parallel" with respect to the LeviSimilarly, it is Civitá connection associated with the metric it defines with possible to express the compatibility condition for by saying that it is parallel with respect to the same connection. Remark 11.4.3. In general, it is required that the Kähler metric be complete, in other words that it is possible to extend geodesics. This is equivalent to requiring that the manifold be complete (in the sense of metric spaces).
II.4.b Yau's theorem Consider a (complex algebraic) projective smooth manifold M of complex dimension n. Assume that all the HP'°(M) are zero for 1 < p < n — 1 and that the canonical bundle = KM is trivialized by a type-(n, 0) form Notice that M is Kähler, call the Kähler form Rescaling w if necessary, we get
f
JM
•z
[
JM
A hard theorem of Yau [35] asserts that there exists a unique Kähler form on M such that e and which, together with f1, gives M the structure = of a Calabi-Yau manifold.
II.4.c
Examples of Calabi-Yau manifolds
Of course
is a Calabi-Yau manifold.
11.4.
63
Calabi-Yau manifolds
Affine quadrics
We have defined in § 11.1 a symplectic form on the unit sphere S2 C R3 by the formula
= det(x,X,X').
Similarly, the formula
Z, Z') defines a "holomorphic. symplectic" form of the complex quadric
+
+
Q=
=
i}
In "differential" terms,
=
z1dz2 A dz3 + z2dz3 A dz1 + z3dz1 A dz2.
0, z1 and z2 are coordinates and, using the
On the open subset of Q where z3 relation
z1dz1 + z2dz2 + z3dz3 = 0,
we can write
=
!dzi
A dz2,
Z3
so that A
=
A dz2 A 1z3
A d22.
I
Modifying the restriction to Q of the standard Kähler form of C3, let us construct a Kähler form w on Q such that wA W =
1
A
—
Call h the restriction to Q of the function z12. We look for w of the form
w= for some function f. A straightforward computation (see also [31]) shows that
f(h) =
works.
The quadric Q, equipped with and w is (thus) a Calabi-Yau manifold. Recall that Q is diffeomorphic to the tangent bundle TS2 by Q
X+iYu
TS2
x iivii2
In this way, what we have got is the structure of a Calabi-Yau manifold on the
tangent (or cotangent) bundle of the sphere S2. It is possible (but a little more complicated) to do the same for the cotangent bundles of all the spheres S" and more generally for those of all "rank-I symmetric spaces" (see [31]).
II. In symplectic and Calabi-Yau manifolds
64
Remark 11.4.4. Recall that we have identified C2 with the skew field H of quaternions (in § I.5.a). Similarly, the surface Q has the structure of a "quaternionic" or "hyperkähler" manifold. Call I the complex structure defined on Q by that of C3 (this is the multiplication by i) and notice that the symmetric bilinear form that is an equation for Define an operator on Q is still non degenerate when restricted to 21 = the tangent space by the fact that is the unique vector in that is orthogonal to Z for the complex bilinear form and such that
=
11Z112.
This is an almost complex structure since
detc(z, thus
=
— Id.
UZH2
This is an isometry since
IIJZII2 = detc(z, JZ, Moreover,
—Z) =
J2(Z)) = detc(z, JZ, —Z) =
IZH2.
J "anti-commutes" with I:
detc(z,IZ,JIZ) =
111Z112 = 11Z112 on the one hand idetc(z, Z, JIZ) by linearity.
We thus have
detc(z, Z, JIZ) =
—i
11Z112
= — detc(z, Z, IJZ)
so that JI = —IJ. Hence I, J and IJ form a quaternionic structure on Q. we
thus have
On Q,
• the Kähler form w, • the complex structure I defined by multiplication by i in C3, • the associated Riema.nnian metric g, so that w(X, IY) = g(X, Y), • the "holomorphic symplectic form" fi,
• the complex structure J defined in such a way that associated with the same metric g.
be a J-Kähler form,
It is said that Q is hyperkähler. See Exercise 11.7 for a kind of converse statement. Let us give now a few examples of compact Calabi-Yau manifolds.
11.4.
65
Calabi-Yau manifolds
Elliptic curves
The quotient M of C by a lattice A is an elliptic curve. The two forms 1
w = -dz A
and
2z
= dz
give it the structure of a dimension-i Calabi-Yau manifold. One can, more genby a lattice. It is time for a remark: no other erally, perform the quotient of "explicit" example of compact Calabi-Yau manifold is known. In all the known examples, the existence of the Kähler metric with all the desired properties is obtained as a consequence of the Yau theorem 114.b). Hypersurfaces Recall
that complex elliptic curves can be considered as degree-3 curves in P2 (C),
thanks to the Weierstrass p-function. They are thus the n = 1 case in the next theorem.
(C) is a dimension-n Calabi- Yau
Theorem 11.4.5. A degree-d manifold if and only if d = n + 2.
Proof. The condition on the degree is necessary, as we show it now by the computation of the first Chern classes. We want that the bundle M be trivializable, we must thus have ci(T*M) = —c1(TM) = 0. Calling j the inclusion of M in pn+1(C), we have
+ j*c(0(d)) = j*ci(Tpn+l(C)) since the normal bundle of M in the hyperplane section in H2
is 0(d). Denoting by t the dual class to (C))
,
we have
(n+2_d)j*t=0 so that d = n + 2. Assume conversely that d = n + 2. Let us construct explicitly a holomorphic n-form on M. Let F be a degree.-(n + 2) homogeneous polynomial that describes the hypersurface M. Every point of M lies in an affine chart 0 of (C). In affine coordinates zk = there is an index j such that
z3
since M is smooth. The formula "
OF
II. In syrnplectic and Calabi- Yau manifolds
66
defines a homogeneous holomorphic n-form on M that is nowhere zero. This is a consequence of the theorem of Yau 11.4.b) that there is, indeed, in the same cohomology class as the standard Kähler form w, another Kähler form + iOi9ço giving a Calabi-Yau structure on M. 0 Remark 11.4.6. The form the n + 1-form on
above is defined as "Poincaré residue4" starting from with poies along M defined by
(
in
the affine chart
1)
0.
Remark 11.4.7. Calabi-Yau manifolds of dimension 2 are hyperkahler. The proof of this fact is the subject of Exercises 11.6 and 11.7.
II.4.d Special Lagrangian submanifolds An immersion f : V
M from a manifold of real dimension n into a CalabiYau manifold M of complex dimension n is said special Lagrangian if it satisfies = = 0 and = 0. As in the case of the form is then a volume form.
11.5
Special Lagrangians in real Calabi-Yau manifolds
1L5.a
Real manifolds
A complex analytic manifold is real if it is endowed with a "real structure", that is, with an anti-bolomorphic involution S: an involution such that, for any holomorphic function f over an open subset U of M, 10 S is a holomorphic function. For example, on the algebraic submanifolds of P" (C) described by real polynomial equations, the complex conjugation is an anti-holomorphic involution. These manifolds are thus real manifolds. In particular, the projective space pN (C) itself is a real manifold. The real part, or set of real points of a real manifold is, by definition, the set of fixed points of S. For example, the real part of the real manifold PN(C) is p N (R). Notice that there exists respectable real manifolds that have no real point at all, as is, for example, the "Euclidean quadric" N+I
in 4See (151 p. 147.
11.5.
Special Lagrangians in real Calabi-Yau manifolds
67
Proposition 11.5.1. The real part of a real manifold of complex dimension n, if it is non empty, is a siibmanifold all connected components of which have dimension n.
Proof. The connected components of the set of fixed points of the action of a finite group (here the order-2 group generated by S) are always submanifolds. The
tangent space at x to such a component is the subspace of fixed points of the R-linear involution a = The fact that S is a real structure implies that foa is a complex linear form for any complex linear form f on the tangent space at x. We have to check that the fixed subspace of a has dimension n. To do this, we simply that the eigensubspaces associated with the eigenvalues 1 and —1 are isomorphic. Indeed, if or(X) = X, then for any complex linear form 1' we have
f o a(iX) =
if o a(X) = if(X) = f(—iX).
For any complex linear form f, we thus have
f(a(iX)) = f(—iX) so that cr(iX) = —iX. Hence, there are "as many" eigenvectors for the eigenvalue —1 than there are for the eigenvalue 1.
U
II.5.b Real Calabi-Yau manifolds A Calabi-Yau manifold is real if it is both a Calabi-Yau manifold and a real manifold, with a couple of compatibility conditions S*w = —w and
(similarly to what happens in forms Il and w).
=
with the complex conjugation and the two usual
• The affine quadric = 1 of C3, endowed with the complex conjugation of coordinates is a real manifold. It is also clear that this is a real Calabi-Yau manifold. Its real part is simply the unit sphere S2 C R3. If we consider Q as the tangent bundle to S2, notice that the complex conjugation is the multiplication by —1 on the fibers and that the real part is the zero section.
Examples 11.5.2.
• A real hypersurface of degree n+2 in P'"1(C) is a real Calabi-Yan manifold. S* This is checked by computing S the involution induced by the real structure (complex conjugation) of and w as in the proof of Theorem 11.4.5.
IL In syxnplectic and Calabi-Yau manifolds
68
II.5.c The example of elliptic curves Let us come back to the example of F = C/A where A is a lattice that we assume here to have the form
A={m+nrlm,nE Z} for some fixed r such that 0 Re(r) c 1 et Im('r) > 0. To define a real structure on C/A from the complex conjugation in C, it is necessary that A be invariant, that is, that = m + n'r for some m, it e Z. Considering the real and imaginary parts of r, it is seen that = 2Re(r), thus Re(r) = or 0.
/17 ReQr)
=
Re(r) = 0
Figure 11.1: Real elliptic curves
In the second case, the real part of F has two connected components, but in the first case, it has only one, as can be seen solving the equation 2
= z + m + n-i-
in both cases. These components are depicted in bold on Figure 11.1.
Notice (although this is a trivial remark) that the lines that are parallel to the x axis constitute a real foliation of C/A by circles (dimension-i tori) that are special Lagrangian submanifolds of F, represented by dotted lines on Figure 11.1. The space of these special Lagrangian submanifolds is parametrized by the axis generated by r or rather by its image in F, a circle. We shall see more generally in § 11.6 that the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold is, in the neighbourhood of a submanifold V, a manifold whose dimension is the first Betti number of V (here V is a circle and its first Betti number is 1).
II.5.d Special Lagrangians in real Calabi-Yau manifolds Assume now that M is a real Calabi-Yau manifold. We know that
=
—w
and S*Q =
69
Special Lagrangians in real Calabi-Yau manifolds
11.5.
Assume now that the real part MR is not empty. Call j the inclusion of Mft into M. We have So j = j and in particular j*(_w) = = (Soj)*w hence
=
0.
Similarly
= thus
f
13 = 0.
=
(S
=
=
We have proved:
Proposition 11.5.3. Let M be a real Calabi- Yau manifold. The real part of M, if it 0 is not empty, i,s a special Lagrangian submanifold of M. Let us describe now a few examples of this situation. The affine quadric
The sphere is a special Lagrangian submanifold of the affine quadric Q E C3. In other words, with the Calabi-Yau structure on TS2 defined in § II.4.c , the zero section is a special Lagrangian submanifold. In the next examples, we consider a smooth hypersurface defined by a real with its real Calabi-Yau homogeneous polynomial of degree n + 2 in structure. Effiptic curves
The n = 1 case, that of plane cubics, is isomorphic to the example of quotients of C by lattices II.5.c). The real part of a plane cubic has zero, one or two connected components (see Figure 11.1). All eomponents are (topologically) circles. Cubics are foliated by special Lagrangian circles, drawn in dotted lines on Figure 11.1. Degree-4 surfaces
Consider now real algebraic surfaces of degree 4 in P3(C) (the real part of this subject has been investigated and explained in [211). Here is an example from [9]. Consider the real polynomial
\_4
4
4
4
that describes a smooth surface M which has a non empty real part MR. This real
part is
= {(xo,x1,x2,x3)
R4
—
{0}
I
+ 4 = 4 + 4}/(s
Normalize the non zero vectors of R4 by the choice, in each real line through zero, of one of the two vectors such that +4
+4+4 = 2.
II. In symplectic and Calabi-Yau manifolds
70
Then MR is the quotient
{(xo,xi),(x2,x3)ER.2
It is clear that the curve C described in R2 by the equation x4 + y4 =
1
is
diffeomorphic to a circle (radially). Eventually
MR = (Cx C)/((u,v)
(—u
—
v))
is diffeomorphic to a torus. We have thus found a special Lagrangian torus in the Calabi-Yau surface M.
Moduli space of special Lagrangian submanifolds
11.6
We want now an analogue of Theorem 11.3.8, more precisely a description of a neighbourhood of a given special Lagrangian subma.nifold in the space of all special Lagrangian submanifolds.
Theorem 11.6.1 (McLean [25]). Let V be a compact manifold. The modiili space of the special Lagrangian embeddings of V in the Calabi- Yau manifold M is a manifold of finite dimension b1 (V) = dim H' (V; R). Its tangent space at a given point is isomorphic to the vector space of harmonic 1-forms on V. As usual, it is understood that the empty set is a manifold of any dimension — this is not an existence theorem.
Remark 11.6.2. There are two main differences between this statement and Theorem 11.3.8. The first one is that the moduli space here has finite dimension. The second one is that the condition "to be special Lagrangian" is no longer linear, so that this is indeed the tangent space that is identified to a space of differential forms.
Example 11.6.3. Let us come back to the example of the Calabi-Yau structure on TS2 described in §II.4.c. We have said in §II.5.d that the zero section is a special Lagrangian submanifold. As there are no non zero harmonic 1-forms on S2, the theorem of McLean asserts that the zero section is "rigid", that is, it cannot be deformed. In the moduli space of special Lagrangian submanifolds, this is an isolated point. Proof of Theorem 11.6.1. Using the tubular neighbourhood theorem (here in § II.3.b), replace M by a tubular neighbourhood of the submanifold V that is isomorphic to a neighbourhood of the zero section in the normal bundle of V (as are all tubular neighborhoods) and to a neighbourhood of the zero section in the cotangent T*V. We will use the structures induced by those of M on this neighbourhood, keeping their names, for example
= + i/3.
11.6.
71
Moduli space of special Lagrangian submanifolds
We have said in § H3.c that the space of Lagrangian submani folds can be identified with a neighbourhood of 0 in the space Z' (V) of the closed 1-forms on V. The special Lagrangian submanifolds are described, in this space, by the equation = 0 where
F:Z'(V) =
is the mapping defined by
Although these spaces are infinite dimen-
sional, the strategy of the proof is to show that F is submersive at 0. It is thus better to restrict, as much as possible, its target space. Notice first: Lemma 11.6.4. The image of F is contained in the subspace
of exact
n-forms on V.
T*V is the Proof of Lemma 11.6.4. If is the zero form, the mapping ij: V inclusion of the zero section, a special Lagrangian, thus F(0) = 0. Given a form joining it to the zero form... it is possible to consider the path (segment) and giving a homotopy from the section to the zero section. The cohomology class of the closed form does not depend on t, thus it is identically zero, and that means, indeed, that is an exact form. We thus consider F as a mapping
F: Z'(V) and compute its differential at 0. Lemma 11.6.5. The differential of F at 0 is the mapping
(dF)o(ij) = the Hodge star operators associated with the metric defined by the Cal abi- Yau structure on the special Lagrangian V. where * denotes
Proof of Lemma 11.6.5. To compute (dF)0(,7), one chooses a path of forms
whose tangent vector at 0 is the form ii. Let
be
a 1-form on V and X be
the vector field that corresponds to it via the metric on V, that is, the vector field such that g(X,.) = Let Y = JX be the vector field normal to V. This is the vector corresponding to under the isomorphism NV V. The vector field Y is only defined along V, we extend it (arbitrarily) in a vector field Y on the tubular neighbourhood under consideration. Call the flow of Y, so that is a diffeomorphism defined for t small enough. The restriction cot of to V is an embedding of V into NV (one pushes V using Wt)• For t = 0, this is the zero section. Hence for t small enough, this is still a section of NV. We have, for all x in V, d 5See Exercise 11.5,
= Y(x).
IL In symplectic and Calabi-Yau manifolds
72
Under the identification NV T*V, the section of NV corresponds to a section of T* V which is a path of forms, whose tangent vector at 0 is the form Consider now the (n, 0)-form Il, still on our neighbourhood of the zero section in NV. We have = applying Cartan formula together with the fact that Il is closed. For the embedding NV, we thus have V
=
= since 11 is C-linear. We then have
=
= Im(id(txIl)) = Red(cxQ) = The (n. — 1)-form
We still have to convince ourselves that txa =
is
the
unique form satisfying
=
A
But, as the (n + 1)-form is also zero and we have for any 1-form
A (txci) =
A ci
is zero, its interior product by X
=
by definition of X and of the metric g on the space of 1-forms.
Notice that this implies in particular that the differential dF0 is onto: if
0 is
an (ri — 1)-form, * a). du = To end the proof of the theorem, we need to precise what kind of implicit function theorem we use to go from "differential is surjective" to "inverse image is a submanifold". The simplest here is to use the standard implicit function theorem
for Banach spaces (see, for example, [10]). We need to endow the spaces of forms Z' (V) and with structures of Banach spaces. Let us precise the regularity of the forms we use. We consider forms of class in Z' (V) and of class in The Holder norm used here on forms is deduced from the usual Holder norm on functions: recall that (U) is the space of functions of class e" on the open set U of all the derivatives (of order 5 k) of which have a finite Holder norm huh6 (for EJO, 1]), with
hulL = sup
u(x)
—
lix — YhI
+ sup iu(x)I. xEU
11.7.
Towards mirror symmetry?
73
The implicit function theorem gives the fact that F-' (0) is a submanifold in a neighbourhood of 0, whose tangent space at 0 is the kernel (V) of It is important here that this kernel has finite dimension. The isomorphism between Hj,R(V) and the space (V) of harmonic 1-forms is the contents in degree 1 of the Hodge theorem, see [15). 0 Remark 11.6.6. The vector space to so that
(V) is isomorphic to the vector space dual
Hj,R(V) e has a natural symplectic structure (see Exercise 1.1), here A
=
— o' A
The space of harmonic 1-forms is a Lagrangian subspace, by
W(V)
—b (o
I
,
(this is the graph of the mapping *, which is symmetric with respect to the metric... see Exercise 1.8). If V W is a special Lagrangian submanifold, call the moduli space in a neighbourhood of jo• We thus have a Lagrangian subspace
H' "i"
T
•7()
DR V
and it is possible to "integrate" it in a Lagrangian embedding (see [19))
Iji
rV..2) See
L7fl1
also [11] for a description of all these structures by symplectic reduction.
11.7
Towards mirror symmetry?
The "mirror conjecture" asserts the existence, for any Calabi-Yau manifold M, of another Calabi-Yau manifold M* of the same dimension, related with M in a way we briefly describe now, sending the readers to [33] for missing detail. Call the space of isomorphism classes of
• a complex structure
deforming the complex structure J of M
• a "complexified Kähler class" on (M, J,), namely a cohomology class of the form + if3, for some Kähler class (for E and some element Z).
II. In symplectic and Calabi-Yau manifolds
74
Notice that, locally, +i/3 varies in an open subset of H2(M; C), so that the is, locally, a product. The manifold M and its "mirror" partner M* space should be related by an isomorphism of the moduli spaces
that exchanges the factors of this local decomposition as a product. Using in an essential way the symplectic structure of the loop space of M and techniques that go far beyond the level of these notes, Givental has proved the conjecture in [14], following a series of previous papers, the references of which can be found in [14] and [33]. Special Lagrangian submanifolds have been a few years ago the central object of another approach to mirror symmetry, more speculative and having given so far very few results — but a very beautiful approach indeed, that I intend to describe very briefly here.
II.7.a Fibrations in special Lagrangian submanifolds We are no more interested in a single special Lagrangian submanifold but in a whole family. More precisely, we consider a compact Calabi-Yau manifold M and a differential mapping
p:M
to a manifold B, whose general fibers are special Lagrangian submanifolds. The
dimension of B, as that of the fibers of p, must be n. It is not required that p be everywhere regular. Some of the fibers may be singular. The other ones, who correspond to regular values of p, are called general fibers. We know (see § L6.c and [4]) that in any proper Lagrangian fibration, the
general fibers are unions of tori, so this must be the case here. The first Betti number of a torus of dimension ii is precisely n, so that it can be expected that B "looks like" the moduli space of special Lagrangian submanifolds. So, let b E B be a regular value of p and let V C p'(b) be a connected component of the fiber (b). If X E TbB is a tangent vector, there exists a unique vector field Y normal to V in M and such that, for all x in V,
= X. To this field Y corresponds a harmonic 1-form on V, as in the proof of the theorem of McLean (here Theorem 11.6.1). As B has dimension n, starting from n independent vectors X1,. . , in TbB, one constructs n fields Y1,. . , that are normal to V and linearly independent at each point of V. Dually, we thus have n harmonic 1-forms ifl,. , that form a basis of (V) and are linearly independent at each point of V. In order that such a fibration p M —* B exists in a neighbourhood of a special Lagrangian tors V C M, it is necessary that, for the metric induced by the .
.
- .
11.7.
75
Towards mirror symmetry?
Calabi-Yau structure on V, there exists a basis of 5-C' (V) consisting of forms that are independent at each point of V. it is time to mention that (except in dimension 1) there is no known example having all the properties mentioned here.
• Notice first that, abstractly, a basis of harmonic 1-forms that are independent having this at each point exists on the flat torus, the basis dx,,. . , property. The metrics that are close enough to the flat metric thus have the same property. .
• We have seen in II.5.c that the situation of a Calabi-Yau manifold foliated by special Lagrangians submanifolds occurs in dimension 1.
• In dimension 2, on a special Lagrangian torus, one always has a basis of harmonic 1-forms as expected. We have seen that a special Lagrangian submanifold in dimension 2 is simply a complex curve (for a different complex structure). Assuming the submanifold is a torus, it must be an elliptic curve and it has a nowhere vanishing holomorphic form. Actually, the real and imaginary part of this form are harmonic forms on V and they are independent at every point.
II.7.b Mirror symmetry The Strominger, Yau and Zaslow approach to mirror symmetry [32] is to associate, to a Calabi-Yau manifold M endowed with a fibration in special Lagrangian tori
(assuming it exists), another Calabi-Yan manifold M*. The latter should be the "extended" moduli space of special Lagrangian submanifolds of M equipped with a fiat unitary line bundle. Call, as above, the moduli space of special Lagrangian submanifolds in the neighbourhood of V. Locally, the extended moduli space is M* =
x
H'(V;R/Z).
Its tangent space at a point m is TmM* = H'(V;R)@ H'(V; R)
H'(V;R)Ø C.
Thus, M* has a natural almost complex structure, it is even Kähler:
Theorem 11.7.1 (Hitchin [19]). The complex structure on M* is integrable, the metric of H' (V; R) defines a Kãhler metric on M*. We have seen (Remark 11.6.6 above) that
is a Lagrangian submanifold
of (V) e (V), a symplectic vector space endowed by the metric of an almost complex structure (see Exercise 1.12). It can be shown (see [19]) that M* is a Calabi-Yau manifold if is... a special Lagrangian submanifold in this complex vector space. See [11, 20].
II. In syrnplectic and Calabi-Yau manifolds
76
Exercises Exercise
IL 1.
Check that the Liouville form A of the cotangent T* V satisfies (w,/3)) = cx(w)
— 8(v)
(see Exercise 1.1).
Exercise 11.2. Let
L —b
L be a diffeomorphism. Prove that the formula —
defines a diffeomorphism of T*L into itself. Determine 4)*A and prove that 4' preserves the symplectic from.
Exercise 11.3. Let w be a non degenerate 2-form on a manifold W. Define the Hamiltonian vector fields and Poisson brackets as above (this does not use the fact that w is closed). Express Z)
when
X, Y et Z are tangent vectors to W at x that are the values at x of the
Haxniltonian vector fields of three functions f, g and h. Prove that w is a closed form if and only if the Poisson bracket it defines satisfies the Jacobi identity. 11.4. Assume X and Y are two "locally Hamiltonian" vector fields on a symplectic manifold, namely that txw et tyw are closed forms. Prove that their
Exercise
Lie bracket [X, Y) is a globally Haniiltonian vector field, namely that exact form.
is an
Exercise 11.5 (The Hedge star operator). Let V be an n-dimensional oriented manifold endowed with a Riemannian metric g and let a be the Riemannian volume form. Check that the formula g(u1 defines an metric on APT* V... and
that the map
*:APT*V
A
(*v) = g(u, v)a
for all u E APT* V defines, indeed, an operator, the Hodge star operator, which is an isometry. Check that
** = (_1)P(fl1)
77
Exercises
Exercise 11.6 (Multilinear algebra in R4). Consider the vector space R4, with its Euclidean structure g(X,Y) = (X,Y) azid canonical basis (el,e2,e3,e4), and the vector space A = A2(R4)*
of alternated bilinear forms on R4. (1) What is the dimension of A? Check that A is isomorphic to the vector space of skew-symmetric endomorphisms of R4.
(2) Endow A with the Euclidean structure (, ) induced by that of R4, namely c} I
The open submanifold of M given by {p E M I p(p) > c} embeds as an open dense submanifold into Exercise 31 Show that the reduced space
is the
of M at g.
This global description of blow-up for hamiltonian S'-spaces is due to Lerman [31], as a particular instance of his cutting technique. Symplectic cutting is the application of symplectic reduction to the product of a hamiltonian S'-space with the standard C as above, in a way that the reduced space for the original hamiltonian S1-space embeds symplectically as a codimension 2 submanifold in a symplectic manifold. As it is a local construction, the cutting operation may be more generally performed at a local minimum (or maximum) of the moment map p. There is a remaining S'-action on the cut space := '(c)/S' induced by
r: S' —i Diff(M xC),
rt(p,z) = (pt(p),z) In fact, r is a hamiltonian S'-action on M x C which commutes with descends to an action
S1 —*
thus
1.3. Moment Poiytopes Exercise
127
32
Show that
is hamiltoman by describing a moment map.
Loosely speaking, the cutting technique provides a hamiltonian way to close > e}, by using the reduced space at level e, We may similarly close (p e M jz(p) <e}. The resulting hamiltonian S1-spaces are called cut spaces, and denoted and Of course, if another
the open manifold (p e M
I
group G acts on M in a hamiltonian way which commutes with the S'-action, then the cut spaces are also hamiltonian G-spaces.
Chapter II
Algebraic Viewpoint 11.1
Toric Varieties
The goal of this lecture is to explain toric manifolds as a special class of projective varieties. The first five sections contain a crash course on notions and basic facts about algebraic varieties, mostly in order to fix notation. The combinatorial flavor of tone varieties is postponed until Lecture 5.
11.1.1
Affine Varieties
Let C[zj,. .. , Zn] be the algebra of polynomials in the n complex coordinate functions on Throughout this lecture, we consider the Zariski topology on a (Zariski) closed set in is a set of common zeros of a finite number of polynomials from C[zi,... , Zn]; naturally, the complement of a Zariski closed set is called a (Zariski) open set. The fact that any infinite intersection of closed sets is indeed a closed set follows from the stabilization property for sets given as zero sets of polynomials: any decreasing sequence of such sets X1 j X2 ... stabilizes, i.e., there exists an integer r such that X,. = = .... This is a restatement of Hilbert's basis theorem which says that any ideal in C[zi,. . , is finitely generated; see, for instance, [50, §IV-1]. Notice that any nonempty open set is dense (i.e., its closure is the full space), hence the Zariski topology is not Hausdorif. .
Definition 11.1.1. An afline variety is a nonempty closed set in a C".
The Zariski topology on an affine variety X C C" is the topology on X which declares to be (Zariski) open (respectively, closed) every set which is the intersection of X with an open (resp., closed) subset of C". An affine variety X is irreducible if it cannot be written as the union of two proper closed subsets. On an irreducible affine variety, any nonempty open subset is dense.
II. Algebraic Viewpoint
130
Example. The zero locus in C2 of the polynomial Z2 is not irreducible; its irreducible components are given by the lines z1 = 0 and Z2 0. Exercise 33
Let X C Ctm be the affine variety defined as the zero locus of polynomials Pi.• .. ,Pr E C(zj,. . , zn]. Show that X is irreducible if and only if the ideal generated by pi,. . . ,i-'r is prnne (an ideal I C CEzj,..., Zn) prime if whenever u,VECIZI .
Let X
C C72 be an
affine variety presented as the zero locus of polynomials
Pi,".,Pr Definition 11.1.2. A regular function on X is a function X —# C which is the restriction to X of a polynomial function in Ctm. The ring of regular functions on X is denoted C[X]. Exercise 34
Show that the ring Clxi generated by pi Pr
Let X C
is
isomorphic to Cizi,.
.
.
,
where I is the ideal
C Ctm be affine varieties.
Definition 11.1.3. A regular map from X to X' is a map cm. restriction of a polynomial map
: X —+
X' which is the
Exercise 35
Show that a map w: X —. X' is regular if and only if it pulls back regular functions on X' to regular functions on X.
Definition 11.1.4. An isomorphism from X to X' is a regular map X —' X' which is invertible by a regular map. The symbol indicates an isomorphism. The group
of isomorphisms X X is denoted by Isom(X). The affine varieties X and X' are isomorphic when there eDists an isomorphism between them. Exercise 36 Show that X and X' are isomorphic if and only if the associated rings of regular functions, C[X) and C[X'], are isomorphic.
Example. Consider the variety X = = 1 ,i 1,... ,n} C for n = 1, this is the (complex) hyperbola with {zi = O} and {z2 0] as asymptotes. On X the functions z1,. , E C[X1 are invertible by regular functions: = . .
11.1.
131
Toric Varieties
Hence the ring of regular functions on X is the ring of Laurent polynomials in n variables: The projection C2" —# C" onto the first vi components maps X isomorphically
onto the n-dimensional algebraic torus
=C"\ hyperplanes z1 =0 ,i= 1,...n.
(C*)hl := (C\{0})"
Hence we have
X
(Zl,...,zn,zn+l,...,Z2n)
(zi,zj',. ..
(Cs)"
*—
(z1,...
This shows that the torus (C* )fl is an affine variety. Note that the coordinates whereas they were not are invertible by regular functions on invertible by regular functions on C". - .
. ,
11.1.2
Rational Maps on Afline Varieties
If X C C" and X' C Ctm are affine varieties, the symbol X --÷ X' denotes a map defined on some open subset of X. Example. If Pl,P2
C[zi,.
. .
then the rational function defined on the set
Let X C C" be an irreducible affine variety. Definition 11.1.5. A rational function on X is a map X --÷ C which is the restriction of a rational function on C" whose denominator does not vanish identically on X. The ring of rational functions on X is denoted Ox-
Let C(X) denote the field of fractions of C[X}, and let Ix be the ideal in Ox formed by the rational functions on C" whose numerator vanishes identically on X. Exercise 37 Show that the field C(X)
is
isomorphic to the quotient Ox/Ix
An irreducible affine variety X is normal if its ring of regular functions C[XJ is integrally closed in its field of fractions, that is, for any I E C(X), if f satisfies an equation of the form
fm+gifm_l+...+9rn0 with coefficients g,
C[X], then f E C[X]. Any smooth variety is normal and the set of singular points of a normal
variety has codimension at least 2 [44].
II. Algebraic Viewpoint
132
Examples.
1. On the curve X C C2 defined by y2 = x2 + x3, the rational function t = C(X) is integral over C[X] since t2 — 1 — x = 0, but t C[X], hence X is not normal.
2. The cone X C C3 given by x2 + y2 =
z2
is normal [44], though it has a
singular point at the origin C Ctm be slIme varieties.
Let X C
Definition 11.1.6. A rational map from X to X' is a map w: X --. X' which is the restriction of a rational map C'1 --÷ Ctm whose denominator does not vanish identically on X. 38
X - -. X' is rational if and only if it pulls back rational Show that a map functions on X' to rational functions on X.
Definition 11.1.7. A birational equivalence from X to X' is a rational map X --s X' which is invertible by a rational map. The affine varieties X and X' are birationally equivalent when there exists a birational equivalence between them. ExercIse 39
Show that X and X' are birationally equivalent if and only if the associated fields of rational functions, C(X) and C(X'), are isomorphic.
Any is the zero set of one polynomial in C[zj,.. . , A hypersurface in affine variety is birationally equivalent to a hypersurface of some space cm; for a proof see, for example, [44].
11.1.3
Projective Varieties
vanishes at a point two : ... e P'1 We say that a polynomial p C(zo,.. , A e C*. Notice that this condition implies that each if p(Awo,... , homogeneous component of p vanishes. We consider the Zariski topology on a (Zariski) closed set in P" is a set of common zeros of a finite number of polynomials from C[zo, . . . ,z,,]; as usual, the complement of a Zariski closed set is called a (Zariski) open set. By considering homogeneous components, we may assume that each of those polynomials is homogeneous. A hypersurface in is the zero set of one (reduced) homogeneous polynomial in C[zo,. .. , \ {0}, the degree of which is called the degree of the .
11.1.
133
Toric Varieties
hypersurface'. A hypersurface of degree 2,3,4,... is traditionally when it is called a conic), a cubic, a quartic, etc.2 (except in
called
a quadric
Definition 11.1.8. A projective variety is a nonempty closed subset of some projective space
is the topology on X The Zariski topology on a projective variety X C which declares to be (Zariski) open (respectively, closed) every set which is the intersection of X with an open (resp., closed) subset of r. A projective variety X is irreducible if it cannot be written as the union of two proper closed subsets. Examples. 1.
The product of two projective spaces is a projective variety. This can be seen via the Segre embedding
S:
P" x
[z®wI.
([zI,[w])
The homogeneous coordinates of an image point [z ® to] are
yjj:=zjw,, i=O,...,n,j=O,...,m. The image of S
is
the set cut out by the system of equations
Ii,k_—O,...,n
YijYkt = ykjYit
S
is a nonempty closed subset of jpnrn+fl+rn In x x thus particular, 5(1?' x IF') is the subset of points [yoo 110, : 111° y,,] E determined by the single quadratic equation = YoiYio
YooYii hence
S(P' x p1) is a nondegenerate quadric in
2. The set of lines in IPhl through the point [0: ... 0: 1] E IF" (or, similarly, through any other point), is naturally identified with Jl" via :
1
IF"—' [to]
{ lines in 1?" through [0: ... : 0: 1]} L1,,,1
:= {[Awo: ...
:
:
y]
E C2 \ {(0,0)}}
'A polynomial is reduced if each of its irreducible factors has multiplicity 1. For some applications it is convenient to allow for nonreduced polynomials and then consider that some components of the hypersurface have multiplicities. 2A hypersurface of degree 1 is called a line when in P2, a plane when in in higher projective spaces.
and a hyperplane
II. Algebraic Viewpoint
134
The blow-up ofr at [0: ... :0:
1],
:
... :0: 1]), is the subset of
defined by the incidence relation
x
B(lr, [0: ... 0: 1]) := {([w], [z]) :
pn._l x
I
[z] E
defined by the
x
This can be translated as the closed subset of system of equations
+n I,
as a projective variety in x IF" The Segre embedding exhibits hence B(P'2, [0: ... : 0: 1]) is a projective variety.
Since the point [0 B(P'2, [0
:
...
:
0
:
...
:
0
:
1] belongs to any line through it, the set
1]) decomposes as the disjoint union of the so-called
:
exceptional divisor,
E:= {([w],[0: ... :0:1])
11w]
with
S:= {([w),[w:y])
\{[0: ... :0:
, yE C)
[w]
1])
Let X C r be a projective variety. C which, locally near Definition 11.1.9. A regular function on X is a function X each point x E X, may be written as a quotient of two homogeneous polynomials
of the same degree such that the denominator does not vanish at x. The ring of regular functions on X is denoted C[X]. Any regular function on P'2 is constant. More generally, it can be shown that CLX] = C whenever X is an irreducible projective variety [44]. Therefore, the ring
C[X] will not give much information.
Let X C P'2 and X' C projective space
standard charts (k = 0.. Vk := {[z0
IPtm
be projective varieties. Recall that complex
comes equipped with n + 1 affine neigbborhoods given by the
.
. .
, n):
z,,]
I
0}
[zo:...:z] '2
C'2 i—i
'
'
'
X —i X' such that X there exists a neighborhood U of x and an affine neighborhood V of ço(x) for which ço(U) C V and cc : U V is given by m regular functions.
Definition 11.1.10. A regular map from X to X' is a map
for each x
11.1.
135
Toric Varieties Exercise 40
Check that the regularity condition at a point z
X is independent of the
choice of the afline neighborhood containing f(x).
pN where Example. A Veronese embedding of degree d is a map V N= is mapped to the point with homogeneous ... — 1 and [z0 :
coordinates given by the various monomials + ... + being nonnegative integers such that
the exponents A0,. ..
. . .
,
= d.
Exercise 41 Check that the set of all homogeneous polynomials of degree d in n+ 1 variables z,, forms a vector space of dimension 20
In particular, if n = 1 and d = 2, the Veronese embedding is simply
V: [zo : z1]
'—÷
: z0z1 : z?]
Exercise 42
Check that V is a regular map.
The Veronese embedding of degree d allows to translate the study of some into the case of hyperplanes problems concerning hypersurfaces of degree d in inlPN.
X Definition 11.1.11. An isoinorphism from X to X' is a regular map indicates an isomorphism. The which is invertible by a regular map. The symbol group of isomorphisms X —÷ X is denoted by Isom(X). The projective varieties X and X' are isomorphic when there erists an isomorphism between them. 11.1.4
Rational Maps on Projective Varieties
are projective varieties, the symbol X --. X' still denotes and X' c If X C a map defined on some open subset of X.
Let X C
be an irreducible projective variety.
Definition 11.1.12. A rational function on X is a function - - s C whose restriction to each affine neighborhood is a rational function on whose numerator and denominator have the same degree and whose denominator does not vanish identically on X. The ring of rational functions on X is denoted Ox. Let C(X) denote the field of fractions of CIX], and let Ix be the ideal in Ox formed by the rational functions on whose numerator vanishes identically on X.
II. Algebraic Viewpoint
136 ExercIse 43 Show
Let
that the field C(X)
Xc
ffsfl
and X' c
is
Ptm
isomorphic to the quotient Cxlix.
be projective varieties.
is a map X --+ Ptm which Definition 11.1.13. A rational map from X to by m + 1 rational functions on X. A is given in homogeneous coordinates for rational map from X to X' is the restriction to X' of a rational map : X ---. is regular and ço(U) C X'. such that there is an open set U C X where Definition 11.1.14. A birational equivalence from X to X' is a rational map X -- + X' which is invertible by a rational map. The projective varieties X and X' are birationally equivalent when there exists a birational equivalence between them. Exercise 44
that X and X' are birationally equivalent if and only if the associated fields of rational functions, C(X) and C(X'), are isomorphic. Show
11.1.5
Quasiprojective Varieties
The notions before make sense in the broader class of quasiprojective varieties, which encompasses both affine and projective varieties. Definition 11.1.15. A quasiprojective variety is a nonempty open subset of a projective variety. Zariski topology, regular function, regular map, isomorphism, rational function, rational map and birational equivalence are defined for quasiprojective varieties analogously to projective varieties. For instance, a (Zariski) closed set of a quasiprojective variety is the intersection of the variety with a closed subset of projective space. From now on, the term variety (without specifying affine or projective) refers to a quasiprojective variety. It is a fact that two irreducible varieties X and X' are birationally equivalent if and only if they contain isomorphic open subsets U C X and U' C X' [44]. Example. The tautological line bundle L defined in Section 1.3.4 is a quasiprojective variety (yet not affine, nor projective). In fact, the inclusion of in as the open set of points (zo: ...: Zn] with 0 induces an inclusion of the blow-up of at the origin as an open subset of the projective variety [0: ... : 0: 1]); cf. Section 11.1.3.
The blow-down map /3 of quasiprojective varieties.
L —p
is a birational equivalence in the category
Blowing-up at a point is a local operation which extends to any quasiprojective variety modeled on the blow-up of at the origin or of at [0: ... : 0: 1].
11.1.
137
Toric Varieties Exercise 45
C), where P(L C) was discussed in [0: ... 0: 1]) Check that Section 1.3.4, and that the blow-down map f3: P(L C) —. 1?" is a birational equivalence in the category of projective varieties. :
An irreducible projective variety X
is
normal if every point has a normal
affine neighborhood.
A normalization of an irreducible variety X is an irreducible normal variety X, together with a regular map v : X —, X which is a finite birational equivalence. X' between varieties is finite, if any point x E X' has an affine X A map '(V) is afline and the restriction neighborhood V such that the preimage U := V is a finite map, that is every point has a finite number of preimages. U presents Any variety is a finite union of irreducible varieties [44]. If X = 2(3 for all i j, then X as a finite union of irreducible closed subsets and X2 the Xe's are called irreducible components of X.
One can define a normalization of an arbitrary variety X as a disjoint union of normalizations for each of the irreducible components of X.
Toric Varieties
11.1.6
is a 2n-dimensional Lie group The n-complex-dimensional algebraic torus under multiplication of complex numbers. The weight lattice of (C* is the lattice is a group homomorphism A character (or a Laurent monomial) of —'
There is a bijective correspondence between weights and characters of (C5
A =(A1,...,Afl)
E
An action of a torus (C5
A: w = (tv1,.
C5 . . ,
:=
...
on a variety X is a group homomorphism
Isom(X).
Definition 11.1.16. A toric variety is an irred'ucible variety3 X equipped with an action of an algebraic torus having an open dense orbit. Definition H.1.17. Two toric varieties are equivalent if there exists an equivariant isomorphism between them. Toric varieties are easy to construct, as the following examples illustrate. 3Toric varieties are usually required to be normal and normality is always the case for smooth varieties, which interest us mostly. However, our discussion does not require normality and is shortened without this assumption.
II. Algebraic Viewpoint
138
Examples.
1. Let A = {A('),. an action of w
[21
:
Associated to A, there is
be a finite subset of on the projective space
. .
...
:
Zk] =
[W
21
:
...
defined by: Zk]
:
,
* for w E (C
'1
Let XA be the closure of the (Ct)'1-orbit through [1:...: 1]. Then XA is a tone variety.
2. Let A = {A('),... ,A(k)} be a finite subset of
The action of
on the
vector space Ck associated to A is defined by:
forwe (C *
zl,...,W
w-(zl,...,Zk)=(W
through (1,..
Let YA be the closure of the tone variety.
,
1).
Then YA is a
3. Suppose that we have an action of a torus (C*)fl on some variety V. Let v E V. The closure of the (C* )'1-orbit through v is a toric variety. The important example XA has the following particular instances. Examples.
1. For a fixed positive integer d, let A1
= =
{A = (A0,.. {A(°>,.
.
,
A,2) e
> 0 all i , Ao + ... +
I
= d}
(which corresponds to the set of all Laurent monomials in n variables of
degree d containing no negative powers). Then
XA, = closure of
where V :
:
:
...
w E (C*)n+l} = V(P'2)
:
pN is the Veronese embedding of degree d defined in
Section 11.1.3.
2. Similarly, for a fixed positive integer d, let A2
=
P"
{A(°),. .. ,A(")}
As before XA2 = V(IP'1)
P'1. For
n
2 and d = 3, the set A2 is:
11.1.
139
Toric Varieties
I I •
II. Algebraic Viewpoint
140
11.2
Classification
Recall that a toric variety is an irreducible quasiprojective variety equipped with
an action of an algebraic torus having an open dense orbit. In this lecture we begin by reviewing the language of spectra used for classifying affine toric varieties. Arbitrary normal toric varieties are classified by combinatorial objects called fans.
11.2.1
Spectra
Let A be a finitely generated C-algebra without zero divisors. An ideal I in A is prime if
u,vEAanduvEl An ideal I in A is maximal if I is I itself.
uEIorvEI.
A and the only proper ideal in A containing I
Exercise 46
Regard A simply as a commutative ring with unity. Show that the ideal I is prime if and onJy if the quotient ring A/I is an integral domain (i.e., A/I has no zero divisors), and that the ideal I is maximal if and only if A/I is a field.
Exercise 47
Check that every maximal ideal is prime. Give an example of a prime ideal which Is not maximal.
Exercise 48
Let I be an ideal in A, and let p: A —. A/I be the surjective ring homoinorphism given by taking an element to its coset in the quotient ring A/I. Check that there exists a bijective correspondence between the ideals J of A which contain I, and the ideals J of A/I, given by J = (J).
p'
The spectrum of the algebra A is the set Spec A
{ prime ideals in A }
equipped with the Zariski topology, which declares to be closed a subset of Spec A consisting of all prime ideals containing some subset of A. The maximal spectrum of A is the set SPecm A
{ maximal ideals in A }
equipped with the Zariski topology, which declares to be closed a subset of consisting of all maximal ideals containing some subset of A.
A
11.2.
141
Classification
Examples.
Associated to the point E x there is a maximal ideal I,, C A consisting of all polynomials which vanish at x, that is, is the ideal generated by the monomials z1 — x1,. , z,-, —
1. Let A = C[zi,.. ,z,,] and let x = (x1,.
.
.
By Hubert's Nullstellensatz (see, for instance, [5] or [20]), any maximal ideal for some x C". Moreover, the corresponof C[z1,. .. , z,,j is of the form dence
C" c[c"I is
a homeomorphism for the Zariski topology, where we stress that the poly-
nomial ring C[zj,... , z,,J is the ring of regular functions on Exercise 49 Show that there exists a bijective correspondence between Spec CEz1,.. and the set of irreducible subvarieties in C".
flint: Any prime ideal in C[zi,..
,
z,,]
is finitely generated.
2. Let A = .,z,,,z;'] and let x = (x],. to the point x there is the maximal ideal in A: .
r
._
/
—1
,
—1
.
,x,,) E (C*)fl. Associated —1
X1
—.1
— yr,...,) is the full algebra A since it contains (z1 —xi)zj' +xi(zj' —yj) E Hubert's
By observing that the ideal (z1 — x1, when Yl
Nullstellensatz implies that any maximal ideal of C[zi, the form Moreover, the correspondence
.
, z,,,
z; 1] is of
C[Zi,Zi',...,Zn,Z'] is a homeomorphism for the Zariski topology, where Cf(C*)fJ is the ring of regular functions on cf. Section 11.1.1. Exercise 50
that Spec CIa1, Show
there
exists a
a
bijective correspondence between of irreducible subvarieties in
and the set
Let X be an affine variety in C" defined by polynomials p1,... ,Pr from C[zi,. ,z,,]. Let I = (pi,. . ,Pr) be the ideal generated by those polynomials. . .
.
II.
142
Then
Algebraic Viewpoint
X= = {
ideals
13
c C[zi,..
,
1
= { maximal ideals Jc = { maximal ideals in C[zi,. = { maximal ideals in C[X]
1
ç
}
J} .
.
,
z,,]/I
}
}
SPeCm
is a homeomorphism for the Zariski topol-
The correspondence X ogy.
More generally, if A is a finitely generated C-algebra without zero divisors, the set XA = SPeCm A is called an abstract affine variety. While maximal ideals in A play the role of points in XA, arbitrary prime ideals are thought of as irreducible subvarieties, by analogy with the ring of regular functions on an affine variety. The dimension of XA is defined to be
c
sup{3 chain
dimXA
nEZ
c ... C
of distinct prime ideals}
When A = C[X) is the ring of regular functions on an irreducible affine variety X, the dimension dim XA coincides with the (complex) dimension dim X since both are equal to chain X0 C X1 C nEZ
11.2.2
...
C
X,. = X of distinct irreducible subvarieties}
Toric Varieties Associated to Semigroups
Let S be a commutative semigroup. Its seinigroup algebra C[S] is the C-algebra generated as a complex vector space by the symbols z° with o E S and multiplication defined by the rule = In particular, a C-algebra.
generators
for S as a semigroup yield generators z° for C[SI as
Examples.
1. If S =
then
= C[zi,... ,
is the algebra of polynomials in n
variables.
2. If S =
then C[S] = C[zi,zj',...
polynomials in n variables.
is the algebra of Laurent
11.2.
143
Classification
Notice that in the previous two examples, the maximal spectrum of the semigroup algebras are toric varieties: and
SPeCm
SpeCm
In fact, we have the following general result: be a finitely generated Proposition 11.2.1. Let S ç mal spectrum SpeCm C[SJ is an affine tone varzety.
Then the maxi-
Proof. By shrinking the lattice if necessary, we may assume that S generates V2 as an abelian group, in which case Specm C[SI has dimension n. The inclusion of semigroups S ç V2 and hence of semigroup algebras C[S] C C[ZThJ gives an embedding of the torus (C*)fl
=
Specm
Specm
C[S].
Let 0 be the image of this embedding. The torus (C* )fl acts on C[S1 by
wz°:=vfz°, This action induces an action of (CS)fl on Spec1,, C[S]. By considering the dimen-
sion, the set 0
is an open orbit for this action. We conclude that its 0 closure 0 must be the full Spec,,, C[SI and hence this is a toric variety. Example. The complex curve inC2 with equation yk affine toric variety with given by t
(k = 1,2,3,...) is an
(x,y) =
It may be obtained as Spec,,, C[S] for the semigroup S =
generatedby{k,k+1,...,2k—1}.
\ {1,2,. . . ,k
—
1}
Exercise 51
Show that the variety in the previous example is not normal for k > 1, and that its normalization is the affine line X.
11.2.3
Classification of Affine Toric Varieties
Theorem 11.2.2. (Classification of alfine toric varieties) Any affine toric variety is eQuivalent to one of the form Spec,,., C(SJ for some finitely generated semigroup
S C Z" (n 0).
II. Algebraic Viewpoint
144
Proof. Let X be an affine toric variety for the torus (C*)rn. Let 0 be the open orbit for the (C* )m-action on X. Then 0 may be identified with the quotient of (C* by the stabilizer of some point in 0. Since this quotient is itself a (possibly acting on X. smaller dimensional) torus, we can regard 0 itself as a torus By irreducibility of X, we have an embedding C[X] C C[OJ = C[Z'2]. The subring C[X] c C[0] is (Ct)'2-invariant with respect to the induced actions of (C*)'2 on C[X] and on C[(C*)'2]. As a representation of an algebraic torus, the space C[XJ decomposes into one-dimensional weight spaces. The weight spaces are generated by monomials as C-algebras, hence the vector space C[X] itself is generated by Q monomials, i.e., it is a semigroup algebra. Example. Recall the construction of the affine toric variety YA from a finite set A= .. , C Z'2, described in Section 11.1.6. By the previous theorem, we must have YA Specm C[SJ for some finitely generated semigroup S C Z'2. It is not hard to see that S is the semigroup generated by A. In fact, the ring C[YA] of regular functions on YA is generated by the restrictions to YA of the coordinate functions on C'2. Since YA is the closure of {
(zAp>
. , I
z E (C*)'2}
the ring C[YAJ is generated by the monomials
,...,z
z
i.e., is the semigroup algebra of the semigroup in Z'2 generated by A.
Remark. The only smooth affine toric varieties are products of the form (C*)P x This follows from the classification of affine toric varieties (Theorem 11.2.2), the classification of normal toric varieties (Theorem 11.2.10) and the study of conditions for smoothness (Exercise 53). See the remark at the end of Section 11.3.4.
11.2.4
Fans
Definition 11.2.3. A (convex polyhedral) cone in R'2 is a set of the form
C = {aivi + . .. +
ER" I a1,... ,a,. 0}
for some finite set of vectors v1,.. , ii,. R'2, then called the generators of the cone C. The cone C is rational if it admzts a set of generators in Z'2. The cone C is smooth if it admits a set of generators which is part of some 7L-basis of Z'2. .
The dimension of a cone C is the dimension of the smallest R-subspace containing C (which is the vector space C + (— C)).
11.2.
145
Classification
Definition 11.2.4. The dual of a cone C C I
f(x) 2 0 Vx
E C}.
Farkas' theorem states that the dual of a rational cone is a rational cone [21]. From the theory of convex sets [20], it follows that (C* ) * = C. is a hyperplane of the form A supporting hyperplane for a cone C C
Hf:={xER'2If(x)=0}
forsomef€C*\{0}.
A face of a cone C c is either C itself (a nonproper face) or the intersection of C with any supporting hyperplane (proper faces). A face of a cone is itself a cone; indeed the face C fl H1 (with f E C*) is generated by those vectors in a set of generators for C such that 1(v2) = 0. Exercise 52
Show that a cone C has only finitely many faces and that any intersection of faces is also a face. Hint: The face C fl Hf is generated by those vectors v2 in a generating set for
C such that 1(v2) =
0.
If 0 is a face of C, then C is called strongly convex; this is the case precisely when C contains no one-dimensional R-subspaces, that is, when C fl (—C) = {0}.
If C is strongly convex, then its dual regardless of the dimension of C.
Let C be a rational cone in Lemma 11.2.5.
is n-dimensional (i.e., C*
and let C* be its dual,
+ (_C*) =
also
rational.
The intersection C* fl (V' ) * is a finitely generated semigroup.
Proof. Let v1,... ,v,. be generators of C* and let K = 0 0 for all a C, the integral (f0 fa(b) da) is positive and a
da = is
1.
indeed
0
nowhere zero.
From now on we will always assume that whenever a group actions preserves a contact structure it also preserves a contact form defining this structure. This concludes the proof of Theorem 1.31. It follows from the Theorem that if an n-torus C acts effectively on the punctured cotangent bundle M = T*Tfl 0 preserving the symplectic form and commuting with dilations then it acts on the quotient B = M/IR S*Tfl preserving the corresponding contact structure. Note that 2 dim G = dim B + 1.
Definition 1.37. An effective action of a torus C on a manifold B preserving a contact structure is completely integrable if 2 dim G = dim B + 1. A contact toric C-manifold is a co-oriented contact manifold with a completely integrable action of a torus C. We are now in position to reduce Theorem 1.4 to a statement about contact toric manifolds. Consider again the action of C = on M = T*Tfl '.. 0 preserving the symplectic form and commuting with dilations. As was remarked previously M is a symplectic cone over B = S*Tfl = x By Theorem 1.31 the action of C on M induces an effective action on B. Moreover B has a C-invariant contact structure making (B, into a compact connected contact toric C-manifold. Clearly if the action of C on B is free then the original action of C on M was free as well. Therefore a proof of Theorem 1.4 reduces to
Theorem 1.38. Let (B, = ker ci) be a compact connected contact toric C-manifold and suppose the action of C is not free. Then B is not homotopy equivalent to T" x n = dim C. The rest of the notes will be occupied with a proof of Theorem 1.38. The proof uses heavily contact moment maps which are discussed in the next chapter. We end this section with a partial converse to Theorem 1.31. We have seen that given a contact form ci on a manifold B the form d(etct) on B x R is symplectic. The pair (B x R, d(eta)) is called the symplectization5 of (B, ci). It is clearly a symplectic cone. Different contact forms on B give rise to different symplectic forms on B x lit However there is a symplectic cone that depends only on the contact structure 5Sometimes (B x
is
called the symplectificatiou of (B,cx).
1.2.
Symplectic cones and contact manifolds
191
= ker o (and its co-orientation) and not on a particular choice of a contact form: Let denote the component of NO giving its co-orientation (cf. Exercise 1.23). It is not hard to check that is a symplectic submanifold of the cotangent bundle
TB with its standard symplectic structure. The action of R on T*B N 0 by dilations preserves
is a contact form BxR
and makes it into a symplectic cone. A section fi of
B
on B with = ker/3. The form /3 defines a trivialization (b,t) = et/3b. The map pulls back the symplectic form on
to d(et/3).
Finally given a symplectic cone (M, w, X) with the base B, the orbit map B and the induced contact structure on B, there is an R equivariant symplectomorphism w: M defined as follows: for a point m M let be the covector in T,,(m)B such that
w: M
= (t(X)W)m(dSb(V)) for 811 V E Tv,(m)B and a section s of w: M —+ B.
The discussion above can be summarized as: Lemma 1.39. Let (B, = ker o) be a contact manifold, let be a component of N 0, the annihilator of in B minus the zero section. The principal R bundle B is a symplectic cone. If (M, w, X) is a symplectic cone with the base B and is the induced contact structure on B, then is isomorphic to (M, w, X) as a symplectic cone.
Chapter II
Contact group actions and contact moment maps Moment maps exist in the category of contact group actions. In fact moment maps
exist for all contact actions. This is because a contact form defines a bijection between contact vector fields and smooth functions.
Definition 11.1. A vector field X on a contact manifold (B, = ker is contact if Hence for any consists of contactomorphisms. In particular C its flow B. B, the Lie bracket [X, vi is again a section of section v of the bundle Thus for a contact action of a Lie group C on (B, the vector fields induced by elements of the Lie algebra g of C are contact. Exercise 11.2. Prove that a Reeb vector field is contact. More generally prove that
a vector field X on a contact manifold (B, = ker is contact if and only if = hcx for some function h (h can have zeros). A choice of a contact form on a contact manifold (B, identifies contact vector fields with smooth functions.
Proposition 113. Let (B, =
ker be a contact manifold. The linear map from fX a(X) is one-to-one contact vector fields to smooth functions given by X and onto.
Proof. Note that the Reeb vector field corresponds to the function 1. For any vector field X on B the vector field X — Cz(X)Ya is in the kernel of which iS the contact distribution Since dale is non-degenerate, X — Q(X)Ya is uniquely determined by t(X — For any section v of B and any vector field X we have
0=
+
II. Contact group actions and contact moment maps
194
Assume now that X is contact. Then a([X, vi) = 0. Therefore, by Cartan's formula
0 = (dt(X)a + t(X)da)(v). Hence for any section v off, dcx(X,v) = —d(a(X))(v) = _dfx(v). And, of course, da(X,v) = da(X — fXYa,V) for all v. We conclude that
=
—
(111)
for any contact vector field X. Thus if X is contact, its component in the direction of the Reeb vector field is fXYC, and its component in the direction of the contact distribution is uniquely determined by (11.1). Conversely, given a function I on B there is a unique section of such
that
= The vector field X1 :=
(11.2)
is a contact vector field with a(Xf) = f.
+
0
Exercise 11.4. Given a function / on a manifold B with contact form a check that the vector field X1 := X + where is defined by (11.2), is contact. That is, show that a = ha for some function h (cf. Exercise 11.2).
Suppose now that a Lie group C acts on a manifold B preserving a contact 1-form a. Then for any vector A in the Lie algebra g of C, the induced vector field AB satisfies1 LABa = 0 and, in particular, is contact. Since the contact form a defines a 1-1 correspondence between contact vector fields and functions, it makes sense to define the a-moment map :B by
=ab(AB(b))
(11.3)
for all b E B and all A E g. Note that by Exercise 1.14 the a-moment map 'IIQ is C-equivariant. The a-moment map, as the name suggests, depends rather strongly on a: if / is any positive C-invariant function on B, then ía is another C-invariant contact form and clearly
= In particular, unlike in the symplectic case, the image of a moment map is not an invariant of the action and of the contact structure. However the moment cone C('i'a) defined by
=
E
I
e W,,(B), t E [0, oo)}
is an invariant of the action of Con 'Recall that the vector field AB is defined by
= kera). =
exp(t.4)
b.
195
Remark 11.5. Suppose a Lie group G acts on a manifold B preserving a contact form a. The action of C lifts to an action on the cotangent bundle T*B which preserves the annihilator of the contact structure = ker a. Moreover, the action T*B. of 0 which contains the image of a B preserves the component The action of C on T* B is Hamiltonian with a natural moment map T*B g5 given by
= (p,AB(b)) for all b C B, p C
is a C-invariant symplectic is a moment map for the action of
A E g. Since the submanifold
submanifold of T* B the restriction 'F: It is not hard to check that C on
=
'F
as the moment For this and other reasons it makes sense to think of 'F = U {0}. We will Note also that map for the action of G on (B, thus denote the moment cone C('F,,) by C('F). Consequently and by analogy with symplectic toric manifolds we will think = kera,W ga). of contact toric C-manifolds as triples :
:
2Note that since 4' is G-equivariant and a B TB is G-equivariant, is O-equivariant as well. This gives us an alternative proof that the a-moment map is equivariant.
Chapter III
Proof of Theorem 1.38 The rest of the lecture notes will be devoted to a proof of Theorem 1.38. Right from the beginning the proof will bifurcate into two cases: the contact manifold B is 3-dimensional and dim B > 3. If dim .8 = 3 we will argue directly using slices that the orbit space B/C is homeomorphic to a closed interval [0, 1] and then use this to compute the integral cohomology of B. This will show that B cannot be homeomorphic to S*T2 = T3. We will then consider the case where dim B > 3. In this case we have a connectedness and convexity theorem of Banyaga and Molino (see [BM1, 13M2]; for a different proof see [L2J):
be a compact connected contact Theorem 111.1. Let = '4' : toric C-manifold. Suppose dim B > 3. Then the fibers of the moment map W are C-orbits (and in particular are connected) and the moment cone C(W) is a convex Moreover if the action of C is not free. polyhedral cone in
Our proof will then bifurcate again. We will consider separately the case where the moment cone contains a linear subspace of dimension k, 0 < k 0, with moment map
M) =
M —÷
(0,0)},
such that (111.2)
We then argue that (111.2) implies that M is G-equivariantly syniplectomorphic Note that M is homotopy equivalent to Tk x to We start with the definition of a symplectic slice representation (c.f. proof of Lemma 111.7), which is essential for understanding the local structure of symplectic toric manifolds.
Definition 111.16. Let (M, w) be a symplectic manifold with a Hamiltonian action of a torus G. Then an orbit C . m is an isotropic submanifold of (M,w).6 The symplectic slice representation at m is the representation of the isotropy group 6For a proof of this easy fact see, for example, [CS].
III. Proof of Theorem 1.38
206
Tm(G m)w/Tm(G m). Here, Gm of m on the symplectic vector space V as usual, Tm(C m)" denotes the symplectic perpendicular to Tm(G m) in the symplectic vector space (TmM,
The equivariant isotropic embedding theorem (see for example [CS], Theorem 39.1) asserts that a neighborhood of an orbit G . m is determined (up to equivariant symplectomorphisms) by the symplectic slice representation at m. In fact, the topological normal bundle of the orbit G is its Lie algebra (op. cit.). is Remark 111.17. In the case of symplectic tone manifolds the dimension of the symplectic slice V at m is twice the dimension of the isotropy group Gm. Hence by Lemma 111.6 the group Cm is connected and the representation of Cm Ofl V is determined by a set of weights {v,} which forms a basis of the weight lattice of Cm. Note that the image of V under the moment map :V defined by the representation is the cone
=
0).
In particular the edges of the cone are spanned by the weights. Alternatively, the isotropy group Cm is isomorphic to Tt for some 1 and the slice representation p : Gm —i Sp(V) is isomorphic to the standard representation of Tt on With a little more work one can prove the following two propositions (their proofs can be found, for example, in [D]). PropositIon 111.18. Let (M, w, '1': M —* g') be a symplectic tome manifold, m E
M a point and U a neighborhood of C m in M. Then for a sufficiently small ball 0 about = in the set (M) by such balls. Then for each index i we have an isomorphism :
Let
=
fl 03. Define
4)1(Q\
— Jj
f
J3
etc.; we for the restriction (to keep the notation manageable we wrote will continue to omit restrictions iii the rest of the section). It is easy to see that
= id,
o
= id, and gij
o
ogki = id
(111.3)
wherever these equations make sense. and (M,w, c1) allow us to reconstruct (M',w', i"). The data
on M by ? x2 Define a relation Equations (111.3) imply that '—' is an equivaif x3 = x3 E is a symplectic toric manifold. Moreover lence relation. It follows that M/ the map F: M M' defined by = descends to a well defined map F: M/ M'. The map F is an isomorphism. is another collection of isomorphisms. Suppose of2. Clearly Let = ohT' (111.4) =
Indeed, let M =
o where E We also get a different set of data if we choose a different cover. However, all cohomology with these sets of data define one object a class in the first
coefficients in a certain sheaf. Let us now review the notions of sheaves and tech cohomology. There are many good references for this material. I will be following cwW].
Definition 111.27. A sheaf of groups S on a topological space X is an assignment
5: {opens sets in X}
groups, U
S(U)
satisfying two conditions:
1. For a pair of open sets U c W in X there is a restriction map 5(U) such that for any three sets U C W C V of X
w v_v
Pu °Pw—Pu-
:
5(W)
III. Proof of Theorem 1.38
210
Elements of 8(U) are called sections. Given a section
E 8(W) we write
calu for 2.
Given an open cover {U2} of an open set U (so that U = U collection of sections
E
and a
8(U2) such that 'PiI(JiflU)
=
E 8(U) such that
for all indices i, j there is a unique section = çoj
for all i.
The sheaf S is abelian if 8(U) is an abelian group for all open sets U.
Three examples of sheaves will be important to us. Check that they are indeed sheaves. Example 111.28. Let (M, w, 4') be a good symplectic tone C-manifold. The assignment iso: U '—i Iso(U) (U c 4'(M) open) is a sheaf on 4'(M). The group operation is composition.
Example 111.29 (Locally constant sheaf). Let H be a group and X a topological space. The assignment that associates the group H to every open connected subset of X is a sheaf, called a locally constant sheaf. It is denoted by ff. Thus ff(U) = H for every connected open set U C X. The group operation is the multiplication in H. Example 111.30. Let (M, w, 4') be a good symplectic toric C-manifold. Define a sheaf Con 4'(M) by C(U) =
G-invariant smooth functions on 4'1(U).
The group operation is addition of functions. Definition 11131. Let 81,82 be sheaves on a topological space X. A map of sheaves —* S2 is a family of group homomorphisms r:
UCXopen compatible with the restrictions: (p2)utY °
fw
fu °
for all pairs U C W of open sets in X.
Example 111.32. Consider the sheaves Iso and C defined in Examples 111.28 and 111.30 above. A section f C(U) is a C-invariant function on Its time t flow preserves the fibers of 4 and is a G-equivariant symplectoinorphism of 4'1(U). Hence is a section of Iso(U). This gives us for each open set U C 4'(M) a map ru : C(U) Iso(U), ru(f) = Moreover, any two functions 11, C(U) +12 Poisson commute [prove thisj. Hence = and therefore is a group homomorphism. Thus we get a map of sheaves i- C —* Iso. :
Uniqueness of symplectic tone manifolds
111.3.
Given a map of sheaves r
S2
211
one can define the sheaves the kernel and
image sheaves kerr and imr: for an open set U (kerT)(U) = kerru, (imr)(U) = im rU. Hence it makes sense to say that a map of sheaves is onto and more generally
talk about exact sequences of sheaves.10 Proposition 111.33. Let (M, w, be a good syrnplectic tone manifold. The map of sheaves r : C —' Iso defined above is onto. Hence Iso is an abelian sheaf.
The kernel of r is the locally constant sheaf R x Z0.
We
thus have a short
exact sequence of abelian sheaves: Iso—'O.
The proposition is due to Boucetta and Molino [BoMI. See also [LT].
111.3.1
Cech cohomology
In this subsection we "review" the notion of Cech cohomology with coefficients in an abelian sheaf. There are many good references, such as [WW], for the nontrivial facts that we list below without proofs. Let X be a topological space {U1} an open locally finite cover of X and S an abelian sheaf on X. A 0 Cech cochain is a function that assigns to each index i an element of 8(U2), i.e., the group of 0-cochains C°({U2},S) is the product flS(U1). A 1 Cech cochain assigns to an ordered pair of indices ii an element of 8(U23) where = U. n U3. Moreover we require that gj, (we now think of the groups 8(U) additively). More generally a p-cochain assigns to an ordered p + 1 tuple of indices io. . . an element where E = U20 fl. . and is skew-symmetric in the indices. The coboundary operator o: C"({U1},S) —+ is defined by
= where means that the index is omitted, and where we omitted writing the restrictions of the terms on the right hand side to One proves that
=
The cohomology of the complex denoted by !f*({U2},S) is called the Cech cohomology of the cover { U,) with coefficients in the sheaf 8. Given a refinement { } of the cover {U2 } the restrictions give rise to a chain map CP({U2}, S)— CP({1'},S), which in turn gives rise to a map in cohomology H*({U2},S) H*({V,},S). Taking the direct limit over all locally finite covers we get a well-defined cohomology group 0.
ff*(X,S) = 1LmR*({Ut},8), the
cohomology of X with coefficients in the sheaf S.
'°Warning: the map r: S2 being onto does not mean that rjj is onto for every open set U. See [WWJ or any other good book on sheaves for more details.
III. Proof of Theorem L38
212
g*) be a good symplectic toric G-manifold and Now let (M, 4': M } E a iocaliy finite cover of 4'(M) by sufficiently small balls. If = 0 means that for all triples of indices ijk we have is a 1-cochain, then
+ gi,, which is (111.3) in additive notation (where on the right hand side we omitted the E C'({01},Iso) are two 1-cochains Similarly if {gj3}, restrictions to then for some 0-cochain that differ by 0=
=
—
gjy
=
—Øjk +
+ h,
hence
=
— + which is (111.4) in additive notation. Thus the discussion above shows that to ev}, Iso) there corresponds a good symplectic toric manifold ery element of R' g*) locally isomorphic to (M,w, 4' : M —' g*). More gener(M',w', 4" : M ally one can check that there is a one-to-one correspondence between cohomology classes in H1 (4'(M), Iso) and isomorphism classes of good symplectic toric mang*). Thus to complete the proof of ifolds locally isomorphic to (M, 4' : M
Proposition 111.26 (and thereby Theorem 111.15) it remains to show that the group
H'(4'(M), Iso) is trivial for any good symplectic tone manifold M. For this we use Proposition 111.33, two properties of cohomology and a property of the sheaf C defined in Example 111.30. The first property of Cech cohomology that we need is Theorem 111.34. A short exact sequence of abelian sheaves 0 Si on a space X induces a long exact sequence in Cech cohomology
IP'(X,S3)
-+
83
0
-+
The second property that we will use is
Theorem 111.35. Let X be a simply connected topological space and H an abelian group. The Cech cohomology H* (X, H) of X with coefficients in the locally constant sheaf H is isomorphic to the singular cohomology H* (X, H) of X with coefficients in the abelian group H. We will use the following property of the sheaf C (cf. [LTJ, Proposition 7.3) Lemma 111.36. The sheaf C defined in Example 111.30 is acyctic, that is,
=
0
for all q >
0.
Now putting Theorem 111.34, Lemma 111.36 and Proposition 111.33 together we see that if (M, w, 4') is a good symplectic tone G-manifold and Iso the sheaf defined in Example 111.28 then the cohomology group (4'(M), Iso) is isomorphic
to H2(4'(M), IR x Zc). The latter group is isomorphic to the singular cohomology group x R) by Theorem 111.35. But 4'(M) is contractible, so H2(4'(M),ZG x R) = 0. Therefore H'(4'(M),Iso) = 0, which proves Proposition 111.26 and thereby Theorem 111.15.
111.4.
111.4
213
Proof of Theorem 1.38, part three
Proof of Theorem 1.38, part three
The goal of this section is to prove be a compact connected contact —b Theorem 111.37. Let (B, = ker 'I' toric G-manifold with dim B 3. Suppose there is a vector X in the Lie algebra g of C such that the function (W, X) is strictly positive on B. Then dim H1 (B, R) n = dimC = x = 1. In particular B is not the co-sphere bundle The proof of Theorem 111.37 above will complete our proof of Theorem 1.38. Since Theorem 1.38 implies the main result of the notes, Theorem 1.4, this, in turn,
will finish the proof of the main result. As was sketched out at the beginning of Chapter 4 our proof of Theorem 111.37 has several steps. The first one is a theorem implicit in a paper of Boyer and Galicki [BG]:
be a compact connected contact = kerct, 'I! toric C-manifold with dim B 3. Suppose there is a vector X in the Lie algebra g of G such that the function ('p, X) is strictly positive on B. Then there exists on B a locally free circle action so that the quotient M = B/S' is a (compact)
Theorem 111.38. Let
symplectic toric orbifold.
The second step is the argument that if M is a compact connected symplectic (M, IR) = 0 for all odd degrees q. This step uses Morse theory toric orbifold then on orbifolds. Let us now see why these two steps give us a proof of Theorem 11137. Consider the circle action produced by Theorem 111.38 and the corresponding S' orbit map B —+ M. If the circle action is actually free, then ir is a circle fibration and we have the Gysin sequence
0=
H'(M,R)
H'(M,JR)
H'(B,R)
H°(M,JR)
H2(M,R) —* (111.5)
If the action of 5' is locally free, the long exact sequence (111.5) still exists. The reason is that the Gysin sequence arises from a collapse of the Leray-Serre spectral sequence for a sphere bundle. For locally free S' actions the orbit map ir: B M is not a fibration, but the corresponding spectral sequences still collapses if we use real coefficients.11 Now, since H' (M, R) = 0 by the second step, it follows from (111.5) that dim H'(B,R)
