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0 and b.(1'ljJ12) ::; O. This implies
,t
o ~ b.(
'ljJik'ljJik)
t,k=l n
(8)
n
L
~ 2
b.'ljJik'ljJik = 4
L
(Ricij gkl - Rijkl) 'ljJik 'ljJjl
i,j,k,l=l
i,k=l
at the point p. In order to analyze the curvature term on the right hand side, we write n = 2m. We can find an orthonormal basis {VI, WI, V2, W2, ... , V m , w m } of TpM and real numbers AI, ... , Am such that 'ljJ(vo,w{3)
= A0 60 {3
'ljJ(vo, v(3) = 'ljJ(wo, w(3) = 0
for 1 ::; ex, (3 ::; m. Using the first Bianchi identity, we obtain n
L
(Ricij i l - Rijkl) 'ljJik 'ljJjl
i,j,k,l=l m
= L A~ [Ric(vo, va) + Ric(wo, wo)] 0=1 m
L
- 2
Ao A{3 [R(vo, V{3, Wm w(3) - R(vo, W{3, wo, v(3)]
0,{3=1 m
=
L
A~ [R(vo, V{3, va' v(3)
+ R(vo, W{3, va' w(3)]
0,{3=1 m
+
L
A~ [R(wo,v{3,wo,v{3)
+ R(wo,w{3,wo,w(3)]
0,{3=1 m
L
-2
AoA{3R(vo,wo,v{3,w{3).
0,{3=1
This implies n
L
(Ricij i l - Rijkl)'ljJik'ljJjl
i,j,k,l=l
=
L A~[R(vo, v{3, va' v(3) + R(vo, W{3, va' w(3)] + L A~[R(wo, V{3, WO, v(3) + R(wo, W{3, Wo, w(3)] - 2 L AoA{3R(vo,wo,v{3,w{3). 0#{3
0#{3
0#{3
S. BRENDLE AND R. SCHOEN
60
Since M has positive isotropic curvature, we have R(va ,v(3,Va ,V(3) + R(v a ,w(3, Va, W(3) + R(wa, V{3, Wa, V(3) + R(wa, W{3, Wa, W(3)
for a
=1=
{3. Since 2:::~1 n
L
(Ricij II
i,j,k.l=1
>2
A; > 0, it follows that
- Rijkl) 1jJik 1jJjl
L A; IR(va,wmv(3,w(3)I- 2 L IAaII A(3IIR(vm wa,v(3,w(3)1 ai-(3
=
> 2IR(va, Wa, V/3, w(j)l·
ai-(3
L (IAal -
IA(3I)2I R (v a , Wa, v(3, w(3)1
2': 0
ai-/3
at the point p. This contradicts (8).
In odd dimensions, the following result was established by M. Berger: THEOREM 2.8 (M. Berger [4]). Let M be a compact Riemannian manifold of dimension n 2': 5. Suppose that n is odd and .M has pointwise l~t-::.:19pinched sectional curvatures. Then the second Betti number of 111 vanishes. PROOF OF THEOREM 2.8. Suppose that 1jJ is a non-vanishing harmonic two-form on M. The Bochner formula implies that !l1jJik =
n
n
n
j=1
j=1
j,l=1
L RiC; 1jJjk + L Ric{ 1jJij - 2 L Rijkl1jJjl.
As above, we fix a point p E M where the function 11jJ1 2 attains its maximum. At the point p, we have 11jJ1 2 > 0 and !l(11jJ12) :::; O. From this, we deduce that
(9)
n
L
n
i,k=1
i,j,k,l=1
(Ricij II
- Rijkl) 1jJik 1jJjl
at the point p. We now write n = 2m + 1. We can find an orthonormal basis { U, VI, WI, V2, W2, ... , Vm" wm,} of TpM and real numbers AI, ... , Am, such that 1jJ( U, va) = 1jJ( U, Wa) = 0 1jJ(va, w(3) = Aa 6a(3 1jJ(Va,V(3)
= 1jJ(Wa,W(3) = 0
61
SPHERE THEOREMS IN GEOMETRY
for 1
~ 0:, (3 ~
m. This implies n
L (Ricij II i,j,k,l=1
-
Rijkl) 'l/Jik 'l/Jjl
m
=
L A~ [R( U, Va, U, Va) + R( U, Wa , U, Wa)] a=1
+ L A~ [R(va, V,B, Va, v,B) + R(va, W,B, Va, w,B)] al-,B
+ LA; [R(wa, V,B, Wa, v,B) + R(wa, W,B, Wa, w,B)] al-,B - 2L
Aa A,B R(va, Wa , V,B, w,B).
al-,B By assumption, M has pointwise ~:=~-pinched sectional curvatures. After rescaling the metric if necessary, we may assume that all sectional curvatures of Mat p an lie in the interval (1, ~:=~]. Using Berger's inequality (cf. [47]), we obtain 2m-1 IR(va, Wa, v,B, w/3)1 < m _ 1 . Since
2::=1 A; > 0, it follows that
~
~ (Ric zJ g 00
i,j,k,l=1
kl
°kl ~ 2 4m - 2 ~ RZJ ) 'l/Jik 'l/Jjl > (4m - 2) ~ Aa - m _ 1 ~ IAaIIA,B1 a=1 al-,B 0
-
= 2m - 1
~(IAal-IA,BI)2 2
m-1 ~ al-/3
0
at the point p. This contradicts (9). We note that the pinching constant in Theorem 2.8 can be improved for
n
= 5 (see [5]). 3. The differentiable sphere theorem
The Topological Sphere Theorem provides a sufficient condition for a We next address the Riemannian manifold M to be homeomorphic to question of whether M is actually diffeomorphic to Various authors have obtained partial results in this direction. The first such result was established in 1966 by D. Gromoll [26] and E. Calabi. Gromoll showed that a simply connected Riemannian manifold whose sectional curvatures lie in the interval The pinching constant 8(n) depends only (1, 8(~)] is diffeomorphic to on the dimension, and converges to 1 as n -+ 00. In 1971, M. Sugimoto, K. Shiohama, and H. Karcher [72] proved the Differentiable Sphere Theorem
sn. sn.
sn.
S. BRENDLE AND R. SCHOEN
62
with a pinching constant 15 independent of n (15 = 0.87). The pinching constant was subsequently improved by E. Ruh [65] (15 = 0.80) and by K. Grove, H. Karcher, and E. Ruh [32] (15 = 0.76). Ruh [66] proved the Differentiable Sphere Theorem under pointwise pinching assumptions, but with a pinching constant converging to 1 as n ~ 00. Grove, Karcher, and Ruh [31],[32] established an equivariant version of the Differentiable Sphere Theorem, with a pinching constant independent of the dimension (15 = 0.98). The pinching constant was later improved by H. 1m Hof and E. Ruh: THEOREM 3.1 (H. 1m Hof, E. Ruh [46]). There exists a decreasing sequence of real numbers i5(n) with limn--+oo i5(n) = 0.68 such that the following statement holds: if M is a compact, simply connected i5(n)-pinched Riemannian manifold and p is a group homomorphism from a compact Lie group G into the isometry group of M, then there exists a diffeomorphism F : M ~ and a homomorphism (J : G ~ O(n + 1) such that F 0 p(g) = (J(g) 0 F for all g E G.
sn
In 1982, R. Hamilton [36] introduced fundamental new ideas to this problem. Given a compact Riemannian manifold (M,go), Hamilton studied the following evolution equation for the Riemannian metric: (10)
g(O)
= go.
This evolution equation is referred to as the Ricci flow. Hamilton also considered a normalized version of Ricci flow, which differs from the unnormalized flow by a cosmological constant: (11)
~ g(t) = -2 Ricg(t) + ~n rg(t) g(t),
ut
g(O) = go.
Here, rg(t) is defined as the mean value of the scalar curvature of g(t). The evolution equations (10) and (11) are essentially equivalent: any solution to equation (10) can be transformed into a solution of (11) by a rescaling procedure (cf. [36]). R. Hamilton [36] proved that the Ricci flow admits a shorttime solution for every initial metric go (see also [21]). Moreover, Hamilton showed that, in dimension 3, the Ricci flow deforms metrics with positive Ricci curvature to constant curvature metrics: THEOREM 3.2 (R. Hamilton [36]). Let (M, go) be a compact threemanifold with positive Ricci curvature. Moreover, let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with initial metric go. Then the rescaled metrics 4(f-t) g(t) converge to a metric of constant sectional curvature 1 as t ~ T. In particular, !vI is diffeomorphic to a spherical space form.
SPHERE THEOREMS IN GEOMETRY
63
In [37], Hamilton developed powerful techniques for analyzing the global behavior of the Ricci flow. Let (M, go) be a compact Riemannian manifold, and let g(t), t E [0, T), be the unique solution to the Ricci flow with initial metric go. We denote by E the vector bundle over M x (0, T) whose fiber over (p, t) EM x (0, T) is given by E(p,t) = TpM. The vector bundle admits a natural bundle metric which is defined by (V, W)h = (V, W)g(t) for V, WE E(p,t). Moreover, there is a natural connection D on E, which extends the Levi-Civita connection on T M. In order to define this connection, we need to specify the covariant time derivative D.£... Given two sections V, W of E, at we define
(12)
(D.£.. V, W)g(t) = at
(~llt V, W)g(t) -
Ricg(t) (V, W).
Note that the connection D is compatible with the bundle metric h. Let R be the curvature tensor of the evolving metric g(t). We may view R as a section of the vector bundle E* ® E* ® E* ® E*. It follows from results of R. Hamilton [37] that R satisfies an evolution equation of the form (13) Here, D.£.. denotes the covariant time derivative, and ~ is the Laplacian at with respect to the metric g(t). Moreover, Q(R) is defined by n
(14)
Q(R)ijkl
=
L p,q=l
n
~jpq Rklpq + 2
L p,q=l
n
Ripkq Rjplq -
2
L
~plq Rjpkq.
p,q=l
Hamilton established a general convergence criterion for the Ricci flow, which reduces the problem to the study of the ODE 9tR = Q(R) (see [37], Section 5). As an application, Hamilton proved the following convergence theorem in dimension 4: THEOREM 3.3 (R. Hamilton [37]). Let (M, go) be a compact fourmanifold with positive curvature operator. Moreover, let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with initial metric go. Then the rescaled metrics 6(T~t) g(t) converge to a metric of constant sectional curvature 1 as t -+ T. Consequently, M is diffeomorphic to 8 4 or ~ .
H. Chen [20] showed that the conclusion of Theorem 3.3 holds under the weaker assumption that (M, go) has two-positive curvature operator. (That is, the sum of the smallest two eigenvalues of the curvature operator is positive at each point on M.) Moreover, Chen proved that any fourmanifold with pointwise 1/4-pinched sectional curvatures has two-positive curvature operator. This implies the following result (see also [2]): 3.4 (H. Chen [20]). Let (M, go) be a compact four-manifold with pointwise 1/4-pinched sectional curvatures. Let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with initial metric go. Then THEOREM
64
S.
BRENDLE AND
R.
SCHOEN
the rescaled metrics 6(f-t) g(t) converge to a metric of constant sectional curvature 1 as t ~ T. The Ricci flow on manifolds of dimension n :2 4 was first studied by G. Huisken [45J in 1985 (see also [50],[58]). To describe this result, we decompose the curvature tensor in the usual way as Rijkl = Uijkl + Vijkl + Wijkl, where Uijkl denotes the part of the curvature tensor associated with the scalar curvature, \!ijkl is the part of the curvature tensor associated with the tracefree Ricci curvature, and Wijkl denotes the Weyl tensor. THEOREM 3.5 (G. Huisken [45]). Let (M, go) be a compact Riemannian manifold of dimension n :2 4 with positive scalar curvature. Suppose that the curvature tensor of (M, go) satisfies the pointwise pinching condition
1V12 + IWI 2 < .2 R1414
+ Ji R 2323 + >.2p,2 R2424 -
2)..J..lR1234 2: c:scal > 0
for all orthonormal four-frames {el,e2,e3,e4} and all >',J..l E [-1,1]. Then M is compact. PROOF OF THEOREM 7.4. We argue by contradiction. Suppose that M is non-compact. By work of Shi, we can find a maximal solution to the Ricci flow with initial metric go (see [11], Theorem 1.1). Let us denote this solution by g(t), t E [0, T). Using Proposition 13 in [10], one can show that there exists a positive constant 8 with the following property: for each t E [0, T), the curvature tensor of (M,g(t)) satisfies
(26)
R1313 + )..2 R1414 + J..l2 R2323
+ >.2 J..l2 R2424 -
2>.J..l R1234 2: 8 scal
80
S. BRENDLE AND R. SCHOEN
for all orthonormal four-frames {el,e2,e3,e4} and all A,p E [-1,1]. The constant 6 depends on c and n, but not on t. In particular, the manifold (M,g(t)) has positive sectional curvature for all t E [O,T). By a theorem of Gromoll and Meyer, the injectivity radius of (M, g(t)) is bounded from below by
inj(M,g(t)) Z
M' N(t)
where N(t) = sUPpE lIf scalg(t)(p) denotes the supremum of the scalar curvature of (lvI, g(t)). There are three possibilities: Case 1: Suppose that T < 00. Let F be a pinching set with the property that the curvature tensor of g(O) lies in F for all points P E M. (The existence of such a pinching set follows from Proposition 17 in [10].) Using Hamilton's maximum principle for systems, we conclude that the curvature tensor of g(t) lies in F for all points P E M and all t E [0, T). Since T < 00, we have SUPtE[O,T) N(t) = 00. Hence, we can find a sequence of times tk E [0, T) such that N(tk) ----1 00. Let us dilate the manifolds (M, g(tk)) so that the maximum of the scalar curvature is equal to 1. These rescaled manifolds converge to a limit manifold !v! which has pointwise constant sectional curvature. Using Schur's lemma, we conclude that if has constant sectional curvature. Consequently, M is compact by Myers theorem. On the other hand, M is non-compact, since it arises as a limit of non-compact manifolds. This is a contradiction. Case 2: Suppose that T = 00 and SUPtE[O,oo) t N(t) = 00. By a result of Hamilton, there exists a sequence of dilations of the solution (JI,f, g( t)) which converges to a singularity model of Type II (see [43], Theorem 16.2). We denote this limit solution by (M,g(t)). The solution (M,g(t)) is defined for all t E (-00,00). Moreover, there exists a point Po E M such that
scalg(t)(p) ~ scalg(o) (po)
=1
for all points (p, t) E M x (-00,00). The manifold (M,g(O)) satisfies the pinching estimate (26), as (26) is scaling invariant. Moreover, it follows from the strict maximum principle that scalg(o)(p) > 0 for all p E M. Therefore, the manifold (M,g(O)) has positive sectional curvature. Since (M,g(O)) arises as a limit of complete, non-compact manifolds, we conclude that (AI, g( 0)) is complete and noncompact. By a theorem of Gromoll and Meyer [28], the manifold if is diffeomorphic to IRrt. It follows from Proposition 6.4 that (M,g(O)) is a steady gradient Ricci soliton. By Proposition 7.2, the scalar curvature of (M,g(O)) decays exponentially. Hence, a theorem of A. Petrunin and W. Tuschmann implies that (M,g(O)) is isometric to IRrt (see [63], Theorem B). This contradicts the fact that scalg(o)(po) = 1.
SPHERE THEOREMS IN GEOMETRY
81
Case 3: Suppose that T = 00 and SUPtE[O,oo) t N(t) < 00. By a result of Hamilton, there exists a sequence of dilations of the solution (M, g(t)) which converges to a singularity model of Type III (see [43], Theorem 16.2). We denote this limit solution by (M,g(t)). The solution (M,g(t)) is defined for all t E (-A, 00 ), where A is a positive real number. Moreover, there exists a point Po E M such that
(A + t) . scalg(t)(p) SA· scalg(o) (po)
=
A
for all points (p, t) E M x (-A, 00). As above, the manifold (M,g(O)) satisfies the pinching estimate (26). Moreover, the strict maximum principle implies that scalg(o)(p) > 0 for all P E M. Consequently, the manifold (M,g(O)) has positive sectional curvature. Moreover, the manifold (M, 9(0)) is complete and non-compact, since it arises as a limit of complete, non-compact manifolds. Therefore, M is diffeomorphic to IRn (see [28]). By Proposition 6.5, the manifold (M,g(O)) is an expanding gradient Ricci soliton. Hence, Proposition 7.3 implies that the scalar curvature of (M, g(O)) decays exponentially. By Theorem Bin [63], the manifold (M, g(O)) is isometric to IRn. This contradicts the fact that scalg(o) (po) = 1. This completes the proof of Theorem 7.4. COROLLARY 7.5. Let (M, go) be a complete Riemannian manifold of dimension n ~ 4 with bounded curvature. Suppose that there exists a positive constant c such that 0 < K (7r1) < (4 - c) K (7r2) for all points P E M and all two-planes 7r1, 7r2 C TpM. Then M is compact.
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CA 94305
Surveys in Differential Geometry XIII
Geometric Langlands and non-abelian Hodge theory R. Donagi and T. Pantev
CONTENTS
1. 2. 3.
Introduction A brief review of the geometric Langlands conjecture Higgs bundles, the Hitchin system, and abelianization 3.1. Higgs bundles and the Hitchin map 3.2. Using abelianization 4. The classical limit 4.1. The classical limit conjecture 4.2. Duality of Hitchin systems 5. Non-abelian Hodge theory 5.1. Results from non-abelian Hodge theory 5.2. Using non-abelian Hodge theory 6. Parabolic Higgs sheaves on the moduli of bundles 6.1. Wobbly, shaky, and unstable bundles 6.2. On functoriality in non-abelian Hodge theory References
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89 94 94
97 104 104 105 107 107 109 111 111 113 113
1. Introduction
The purpose of this survey is to explain some aspects of the geometric Langlands Conjecture and the main ideas relating it to non abelian Hodge theory. These developments are due to many mathematicians and physicists, but we emphasize a series of works by the authors, starting from the outline in [Don89], through the recent proof of the classical limit conjecture in [DP06j, and leading to the works in progress [DP09j, [DPS09bj, and [DPS09aj. The Langlands program is the non-abelian extension of class field theory. The abelian case is well understood. Its geometric version, or geometric ©2009 International Press
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class field theory, is essentially the theory of a curve C and its Jacobian J = J(C). This abelian case of the Geometric Langlands Conjecture amounts to the well known result that any rank one local system (or: line bundle with fiat connection) on the curve C extends uniquely to J, and this extension is natural with respect to the Abel-Jacobi map. The structure group of a rank one local system is of course just the abelian group (Cx = G L1 (q. The geometric Langlands conjecture is the attempt to extend this classical result from (Cx to all complex reductive groups G. This goes as follows. The Jacobian is replaced by the moduli Bun of principal bundles V on C whose structure group is the Langlands dual group LG of the original G. The analogues of the Abel-Jacobi maps are the Hecke correspondences llecke C Bun x Bun xC. These parametrize quadruples (V, V', x,;3) where x is a point of C, while V, V' are bundles on C, with an isomorphism j3 : VJc-x ~ Vj~-x away from the point x having prescribed order of blowing up at x. (In case G = (Cx these become triples (L, L', x) where the line bundle L' is obtained from L by tensoring with some fixed power of the line bundle Oc(x). By fixing L and varying x we see that this is indeed essentially the Abel-Jacobi map.) For GL(n) and more complicated groups, there are many ways to specify the allowed order of growth of j3, so there is a collection of Hecke correspondences, each inducing a Hecke operator on various categories of objects on Bun. The resulting Hecke operators form a commutative algebra. The Geometric Langlands Conjecture says that an irreducible G-Iocal system on C determines a V-module (or a perverse sheaf) on Bun which is a simultaneous eigensheaf for the action of the Hecke operators - this turns out to be the right generalization of naturality with respect to the Abel-Jacobi map. Fancier ver~ions of the conjecture recast this as an equivalence of derived categories: of V-modules on Bun versus coherent sheaves on the moduli Coc of local systems. Our discussion of the geometric Langlands conjecture occupies section 2 of this survey. There are many related conjectures and extensions, notably to punctured curves via parabolic bundles and local systems. Some of these make an appearance in section 6. Great progress has been made towards understanding these conjectures [DriSO, DriS3, DriS7], [LauS7], [BD03], [Laf02], [FGKV9S], [FGVOl], [GaiOl], [Lau03], including proofs of some versions of the conjecture for GL2 [DriS3] and later, using Lafforgue's spectacular work [Laf02], also for GL n [FGVOl, GaiOl]. The conjecture is unknown for other groups, nor in the parabolic case. Even for GL(n) the non-abelian Hodge theory machinery promises a new concrete construction of the non-abelian Hecke eigensheaves. This construction is quite different from most of the previously known constructions except perhaps for the work of Bezrukavnikov-Braverman [BB07] over finite fields, which is very much in the spirit of the approach discussed in this survey.
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The work surveyed here is based on an abelianization of the geometric Langlands conjecture in terms of Higgs bundles. A Higgs bundle is a pair (E,O) consisting of a vector bundle E on C with a we-valued endomorphism o: E -+ E 0 we, where we is the canonical bundle of C. More generally, a G-Higgs bundle is a pair (E,O) consisting of a principal G-bundle E with a section 0 of ad(E) 0 we, where ad(E) is the adjoint vector bundle of E. Hitchin [HitS7b] studied the moduli 1iiggs of such Higgs bundles (subject to an appropriate stability condition) and showed that it is an algebraically integrable system: it is algebraically symplectic, and it admits a natural map h : 1iiggs -+ B to a vector space B such that the fibers are Lagrangian subvarieties. In fact the fiber over a general point bE B (in the complement of the discriminant hypersurface) is an abelian variety, obtained as Jacobian or Prym of an appropriate spectral cover Cb. The description in terms of spectral covers is somewhat ad hoc, in that it depends on the choice of a representation of the group G. A uniform description is given in terms of generalized Pryms of cameral covers, cf. [Don93, Fa193, Don95, DG02]. The results we need about Higgs bundles and the Hitchin system are reviewed in section 3.l. In old work [DonS9], we defined abelianized Hecke correspondences on 1iiggs and used the Hitchin system to construct eigensheaves for them. That construction is described in section 3.2. After some encouragement from Witten and concurrent with the appearance of [KW06], complete statements and proofs of these results finally appeared in [DP06]. This paper also built on results obtained previously, in the somewhat different context of large N duality, geometric transitions and integrable systems, in [DDP07a, DDP07b, DDD+06]. The case of the groups GL n , SLn and lPGL n had appeared earlier in [HT03], in the context of hyperkahler mirror symmetry. The main result of [DP06] is formulated as a duality of the Hitchin system: There is a canonical isomorphism between the bases B, L B of the Hitchin system for the group G and its Langlands dual LG, taking the discriminant in one to the discriminant in the other . Away from the discriminants, the corresponding fibers are abelian varieties, and we exhibit a canonical duality between them. The old results about abelianized Hecke correspondences and their eigenseaves then follow immediately. These results are explained in section 4 of the present survey. It is very tempting to try to understand the relationship of this abelianized result to the full geometric Langlands conjecture. The view of the geometric Langlands correspondece pursued in [BD03] is that it is a "quantum" theory. The emphasis in [BD03] is therefore on quantizing Hitchin's system, which leads to the investigation of opers. One possibility, discussed in [DP06] and [Ari02, AriOS], is to view the full geometric Langlands conjecture as a quantum statement whose "classical limit" is the result in [DP06]. The idea then would be to try to prove the geometric Langlands conjecture by deforming both sides of the result of [DP06] to higher and
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higher orders. Arinkin has carried out some deep work in this direction [Ari02, Ari05, Ari08]. But there is another path. In this survey we explore the tantalizing possibility that the abelianized version of the geometric Langlands conjecture is in fact equivalent, via recent breakthroughs in non-abelian Hodge theory, to the full original (non-abelian) geometric Langlands conjecture, not only to its O-th order or "classical" approximation. Instead of viewing the solution constructed in [DP06] as a classical limit of the full solution, it is interpreted a..'(,6) : p>'(V) ~ p>'(V') 0 OC( (J-L, A)x).
These stacks are equipped with natural projections LHecke
/
~
)/
LBun x C
LBun
LHeckel1
~
LBun
LBun xC
where p(V, V', x,,6) := V, q(V, V', x,,6) := V', and plL and qlL are the restrictions of p and q to LHeckell . Moreover • plL, qlL are proper representable morphisms which are locally trivial fibrations in the etale topology; • LHecke'L is smooth if and only if J-L is a minuscule weight of G; • LHecke is an ind-stack and is the inductive limit of all LHeckelL's; • p and q are formally smooth morphisms whose fibers are indschemes, the fibers of q are all isomorphic to the affine Grassmanian for LG. The Hecke functor L Hil is defined as the integral transform L HIl:
Dcoh ( L Bun, V) --,..~ Dcoh ( L Bun, V)
where L III is the Goresky-MacPherson middle perversity extension j!* ( C [dim LHeckell ]) of the trivial rank one local system on the smooth part j : (LHeckellrmooth '---t LHeckelL of the Hecke stack. 2.2. Similarly we can define Hecke operators L HIl,x labeled by a cocharacter J-L E cochar+(LG) and a point x E C. To construct these operators we can repeat the definition of the L HIl'S but instead of L Ill, we need to use the intersection cohomology sheaf on the restricted Hecke correspondence REMARK
LHeckell,x := LHeckell
X LBunxLBunxC
(LBun x LBun x {x}) .
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The operators LHJ.t,x are known to generate a commutative algebra of endafucntors of Dcoh(LBun, V) [BD03], [GaiOlJ. In particular it is natural to look for V-modules on LBun that are common eigen-modules of all the
LHJ.t,x. A V-module ~ on LBun is a Heeke eigen module with eigenvalue V E Cae if for every p E char+ (G) we have
LHJ.t(~) = ~ [8J pJ.t(V). This setup explains all the ingredients in (GLC). According to the conjecture (GLC) the derived category of coherent V-modules on Cae is equaivalent to the derived category of coherent V-modules on LBun. Moreover this equivalence transforms the skyscraper sheaves of points on Cae into Hecke eigen V-modules on LBun. EXAMPLE 2.3. Suppose G = GLnUC). Then LG = GLn(C) and Cae can be identified with the stack of rank n vector bundles C equipped with an integrable connection. In this case the algebra of Hecke operators is generated by the operators Hi given by the special Hecke correspondences
.
{
Hecke%:=
V and V' are locally free sheaves of rank} (V, V', x) n such that V C V' c V(x) and length(V' IV) = i.
The operators Hi correspond to the fundamental weights of GLn(C) which are all minuscule. In particular all Heckei,s are smooth. The fibers of the projection qi : Heckei -+ Bun x C are all isomorphic to the Grassmanian Gr(i, n) of i-dimensional subspaces in an n-dimensional space. 2.4. The categories related by the conjectural geometric Langlands correspondence admit natural orthogonal decompositions. For instance note that the center of G is contained in the stabilizer of any point V of Cae and so Cae is a Z(G)-gerbe over the full rigidification Loc := CaejZ(G) = Locj7fo(Z(G)) of Cae. (In fact by the same token as in Remark 2.1, the stack Loc is generically a variety.) Furthermore the stack Cae is in general disconnected and REMARK
7fo(Cae) = 7fo(Loc) = H 2 (C, 7f l(Ghor) = 7fl(Ghor where 7fl (G) tor C 7fl (G) is the torsion part of the finitely generated abelian group 7fl(G). Thus we get an orthogonal decomposition
(1)
Dcoh(Cae, V)
=
II
("n)E1rl (G}tor xZ(G)/\
where Z(G)/\ = Hom(Z(G), (CX) is the character group of the center and Dcoh(Loc" V; a) is the derived category of a-twisted coherent V-modules on the connected component Loc,.
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Similarly the group of connected components 1fo(Z(LG)) is contained in the stabilizer of any point of LBull and so is a 1fo(Z(LG))-gerbe over LBull := LBull/1fo(Z(LG)). Also the stack LBull can be disconnected and
Hence we have an orthogonal decomposition
II
(2)
where Dcoh(LBullQ , V; ,) is the derived category of ,-twisted coherent V-modules on the connected component LBullQ • Finally, observe that the group theoretic Langlands duality gives natural identifications 1f1(LG) = Z(G)/\ ZO(LG) =
(1fl(G)freet
1fo(Z(LG)) = ( 1fl(G)tort,
where again 1f1 (G) tor C 1f1 (G) is the torsion subgroup, 1fl (G)free = 1fl (G) / is the maximal free quotient, and Z(LG) is the center of LG, and Zo(LG) is its connected component. In particular the two orthogonal decompositions (1) and (2) are labeled by the same set and one expects that the conjectural equivalence c from (GLC) idenitifies Dcoh(Loc" 0; -n) with Dcoh(LBull V; ,). The minus sign on n here is essential and necessary in order to get a duality transformation that belongs to 8L2('£.)' This behavior of twistings was analyzed and discussed in detail in [DP08].
1f1(Ghor
Q ,
EXAMPLE 2.5. Suppose G ~ GLI(CC) ~ LG. Then BUll = Pic(C) is the Picard variety of C. Here there is only one interesting Heeke operator
HI : Dcoh(Pic(C), V) ---+ Dcoh(C x Pic(C), V)
which is simply the pull-back HI := aj* via the classical Abel-Jacobi map
aJ:
C x Picd(C) ~ Picd+I(C)
(x, L)
I
)
L(x).
In this case the geometric Langlands correspondence c can be described explicitly. Let lL = (L, V") be a rank one local system on C. Since 1f1 (Picd(C)) is the abelianization of 1f1 (C) and the monodromy representation of lL is abelian, it follows that we can view lL as a local system on each component
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Picd(C) of Pic(C). With this setup we have the unique translation invariant) rank one local system on Pic( C) ( c(lL):= whose restriction on each component . Picd(C) has the same monodromy aslL The local system c(lL) can be constructed effectively from lL (see e.g. [Lau90]): • Pullback the local system lL to the various factors of the d-th Cartesian power xd of C and tensor these pullbacks to get rank one local system lLl8ld on xd ; • By construction lLl8ld is equipped with a canonical Sd-equivariant structure compatible with the standard action of the symmetric group Sd on xd . Pushing forward lLl8ld via gd : xd -+ C(d) = xd /Sd and passing to Sd invariants we get a rank one local system (gd*lL l8ld )Sd on C(d); • For d > 2g-2 the Abel-Jacobi map ajd : C(d) Picd(C) is a projective bundle over Picd(C) and so by pushing forward by ajd we get a rank one local system which we denote by c(lL)IPicd(C)' In other words
c
c
c
c
c
• Translation (.) ® wc by the canonical line bundle transports the local systems c(lL)IPicd(C) to components Picd(C) of Pic (C) with d:::; 2g - 2. The rough idea of the project we pursue in [DP06, DP09, DPS09a, DPS09bj is that one should be able to reduce the case of a general group to the previous example by using Hitchin's abelianization. We will try to make this idea more precise in the remainder of the paper. First we need to introduce the Hitchin integrable system which allows us to abelianize the moduli stack of Higgs bundles.
3. Higgs bundles, the Hitchin system, and abelianization 3.1. Higgs bundles and the Hitchin map. As in the previous section fixing the curve C and the groups G, LC allows us to define moduli stacks of Higgs bundles: lliggs, Llliggs: the moduli stacks of wc-valued G, LC Higgs bundles (E, 0 iff it is diffeomorphic to one of the manifolds listed in Theorem 1. Similarly, it admits an Einstein metric g with A ~ a iff it is diffeomorphic to one of the manifolds listed in Theorem 2.
The proofs of these theorems proceed on two distinct fronts: existence results for Einstein metrics; and obstructions to the existence of Einstein metrics. We will first discuss the relevant existence results. The main ideas needed for these arise from Kahler geometry and conformal geometry. Recall that a Riemannian metric on a connected 2m-manifold M is Kahler iff its holonomy group is (conjugate to) a subgroup of U(m) C O(2m). This is equivalent to saying there exists an almost complex structure J E r(End (TM)), J2 = -1, with V' J = a and g(J., J.) = g. When this happens, J is integrable, and (M, J) thus becomes a complex manifold. Moreover, the J-invariant 2-form w defined by w = g(J., .), called the Kahler form of (M, g, J), satisfies dAJJ = O. In particular, w is a a symplectic form on M, meaning that it is a closed 2-form of maximal rank. One of the magical features of Kahler geometry is that the 2-form defined by ir(J·,·) is exactly the curvature of the canonical line bundle K = Am,O, where m is the complex dimension. Note that m = 2 in the n = 4 case that will concern us here. We will also need some rudiments of conformal geometry. Recall that two Riemannian metrics g and h are said to be conformally related if g = f h for some smooth function f : M -+ ~+. If h is also a Kahler metric, we will then say that the metric g is conformally Kahler. When the complex dimension m is at least two, and if f is non-constant, then g and h can then never be Kahler metrics adapted to the same complex structure J. However, it is worth pointing out that there are some rare but interesting examples with m = 2 where g and h are both Kahler metrics, but are adapted to different complex structures J and J.
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Many of the existence results needed here are supplied by the theory of Kahler-Einstein metrics (that is, of Einstein metrics that happen to be Kahler). The foundations of this theory were laid by Calabi [7], who translated the problem into a non-linear scalar PDE, called the complex Monge-Ampere equation, and conjectured that a compact complex manifold of Kahler type with C1 lR = 0 would admit a unique Ricci-fiat Kahler metric in each Kahler class. Yau's proof [42, 43] of this conjecture remains a major landmark of modern differential geometry. It predicts, in particular, that both K3 and the Enriques surface K3/'1L2 admit KahlerEinstein metrics with A = o. Of course, T4 and its relevant quotients also admit Ricci-fiat metrics, but in these cases the metrics are actually fiat, and so can be constructed directly, without the use of any sophisticated machinery. The theory of Kahler-Einstein metrics is considerably more subtle when A> 0, but case-by-case investigations by Siu [37] and Tian-Yau [40] did reveal that there exist A > 0 Kahler-Einstein metrics on CJP>2#kCJP>2 for each k E {3, ... ,8}. Of course, CJP>2 and 8 2 x 8 2 also admit such metrics, but in these cases the relevant metrics are just the obvious homogeneous ones. By contrast, however, CJP>2#CJP>2 and CJP>2#2CJP>2 cannot admit KahlerEinstein metrics. This refiects an important observation due to Matsushima [29]. Namely, if a compact complex manifold (M, J) admits a KahlerEinstein metric 9 with A> 0, then its biholomorphism group Aut(M, J) must be a reductive Lie group, since the identity component Isomo(M,g) of the isometry group is then a compact real form for Auto(M, J). Since CJP>2#CJP>2 and CJP>2#2CJP>2 have non-reductive automorphism groups, this therefore implies that they cannot admit Kahler-Einstein metrics. Nonetheless, in what was long thought to be an entirely unrelated development, Page [33] had succeeded in constructing an explicit A> 0 Einstein metric on CJP>2#CJP>2 by a very different method. The Page metric is of cohomogeneity one, meaning that its isometry group has a family of hypersurfaces as orbits. This feature allowed Page to construct his metric by solving an appropriate ODE. While none of this seemed to have anything to do with Kahler geometry, Derdzinski [12] later discovered that the Page metric is actually conformally KiLhler, and, in the same paper, then went on to prove a number of fundamental results concerning conformally Kahler, Einstein metrics on 4-manifolds. Recently, in joint work [9] with Xiuxiong Chen and Brian Weber, the present author managed to prove the existence of a companion of the Page metric. Namely, there is a conformally Kahler, A > 0 Einstein metric 9 on CJP>2#2CJP>2. This metric is toric, and so of co homogeneity two, but it is not constructed explicitly. Roughly speaking, the metric is found by first minimizing the functional A(h)
=
r 1M
82
dp,h
C. LEBRUN
140
on the space of all Kahler metrics h compatible with the fixed complex structure J, where s denotes the scalar curvature of h. Here it is crucial that the Kahler class [w] of h is allowed to vary in this problem. If, by contrast, we fixed [w], and only considered Kahler metrics with Kahler form in this fixed de Rham class, we would instead be talking about Calabi's problem for extremal Kahler metrics [8]. Thus, the problem under discussion here really amounts to minimizing A( h) among extremal Kahler metrics h. One thus proceeds by restricting A to the set of extremal Kahler metrics, and showing that a critical point h exists for this problem. This preferred extremal Kahler metric turns out to have scalar curvature s> 0, and one is therefore able to define a new Riemannian metric by setting 9 = s-2h. The punch line is that this conformally Kahler metric 9 then actually turns out to be Einstein, with..\ > O. To explain this seeming miracle, we will need a bit more background regarding 4-dimensional Riemannian geometry. The special nature of dimension four basically stems from the fact that the bundle A2 of 2-forms over an oriented Riemannian 4-manifold (M, g) decomposes, in a conformally invariant manner, into a direct sum
of the self-dual and anti-self-dual 2-forms; here A± are by definition the (±l)-eigenspaces of the Hodge star operator. Since the Riemann curvature tensor may be thought of as a self-adjoint linear map
it can therefore be decomposed into irreducible pieces
w++ (2)
0
t2
r
R= 0
r o
W_
+ 12 8
where s is the scalar curvature, r= r - ~g is the trace-free Ricci curvature, and where W ± are the trace-free pieces of the appropriate blocks. The tensors W± are both conform ally invariant, and are respectively called the selfdual and anti-self-dual Weyl curvature tensors. Their sum W = W + + W _ is called the Weyl tensor, and is exactly the conformally invariant part of the curvature tensor R. We can now consider the conformally invariant functional
EINSTEIN METRICS
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whose gradient on the space of metrics is represented [4] by the Bach tensor B, which is the traceless divergence-free tensor field given by Bab := (\7c\7d
+ ~fCd)Wacbd.
This tensor automatically vanishes for any conformally Einstein metric, since an Einstein metric is certainly a critical point of both non-Weyl contributions to the 4-dimensional Gauss-Bonnet formula
X(M)
= -1
871"2
1(IWI2+ -s2 - -If12) 24
M
2
dp,.
But since the signature
T(M) =
~ 1271"
r (IW+1 2- IW_12) dp, 1M
is also a topological invariant, W differs from twice the functional W+(g)
=
1M IW+1 dp,g 2
by only a constant, and the Bach tensor can correspondingly also be expressed as Bab := 2(\7c\7d
+ ~fCd)(W+)acbd.
Now, both of these last observations have rather dramatic consequences in the Kahler context. First, since S2
IW+12 = 24 for any Kahler metric on a 4-manifold, the critical points of the functional
A coincide with the critical points of the restriction of W to the space of Kahler metrics, and are therefore precisely those extremal Kahler metrics h for which the Bach tensor B is L 2 -orthogonal to all infinitesimal variations through Kahler metrics. Second, because W + of a Kahler metric can be written in terms of the scalar curvature and Kahler form, the Bach tensor of an extremal Kahler metric h can explicitly be expressed [9, 12] as B
=
112
[Sf + 2 Hesso(s)]
and therefore corresponds to a primitive harmonic (1, I)-form 'lj; = B(J·,·) = 112 [sp + 2i88s
L·
This implies that B is actually tangent to a curve of Kahler metrics h + tB. Hence the critical points of the functional A are exactly the Bach-fiat Kahler metrics, meaning those Kahler metrics for which B = O. Since multiplying a 4-dimensional metric by u 2 alters its traceless Ricci tensor by f.,.,...
j. =
f -
2uHesso(u-l)
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C. LEBRUN
we also see that, for any extremal Kahler metric h on a complex surface, the conformally related metric g = 8- 2 h will have traceless Ricci curvature ;. = 128- 1 B
where B is the Bach tensor of h. Thus, any Bach-flat Kahler metric will be conformal to an Einstein metric, at least on the open set where 8 =1= o. Fortunately, the A-energy of an extremal Kahler metric is a function of the Kahler class [w] which can be calculated a priori, without even knowing whether or not the extremal metric actually exists; namely it is given by
where :F is Futaki invariant [13]. This allows one, at the very outset, to locate the target Kahler class [w] where the minimizer h ought to live. The intimate relationship between the Futaki invariant and the scalar curvature 8 also allows one to show that, if the target extremal Kahler metric h exists, then it has 8 > 0, so our Einstein metric g = 8- 2 h really will then be defined on all of M = CIP2#2CIP2. Now a gluing argument of Arezzo, Pacard, and Singer [1] implies that CIP2#2CIP2 does admit some extremal Kahler metrics, albeit near the edge of the Kahler cone and far from the target class. On the other hand, a quite general implicit-function-theorem argument [25] shows that the Kahler classes of extremal Kahler metrics form an open subset of the Kahler cone. To prove the existence of the preferred extremal metric h, it therefore suffices to choose a nice path in the Kahler cone from a class where one has existence to the target class [w], and show that the the set of classes along this path with extremal representatives is closed as well as open. To do this, one appeals to a weak compactness result for extremal Kahler metrics [10], which allows one to conclude that sequences of such metrics have subsequences which Gromov-Hausdorff converge to orbifolds, once uniform Sobolev and energy bounds have been established. Smooth convergence is then established by ruling out all possible bubbling modes, using energy bounds and topological arguments. Finally, toric geometry is used to show that the limit Kahler metric is compatible with the original complex structure, and belongs to the expected Kahler class. These existence results suffice to prove one direction of implication in Theorems 1, 2, and 3. To prove the converse statements, one instead needs to consider obstructions to the existence of Einstein metrics. The first such result that we will need is the Hitchin- Thorpe inequality [19]. This is obtained by observing that the Gauss-Bonnet and signature formulas together imply that
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Since Einstein metrics are characterized by r = 0, the existence of such a metric would make the integrand in the above expression non-negative, so a smooth compact oriented 4-manifold can only admit an Einstein metric g if (2X+3T)(M) ~ 0, with equality iff g is Ricci-flat and anti-self-dual (W+ == 0). The latter happens, however, iff (M,g) has reduced holonomy c SU(2). If M admits a complex or symplectic structure, this then implies [24] that the relevant structure has ~ 0, with equality iff M is diffeomorphic to a complex surface with C1 torsion and b1 even. For the purpose of proving Theorems 1, 2, and 3, one may thus assume henceforth that ci(M) > O. The rest of the proof depends on Seiberg-Witten theory, which allows one to imitate certain aspects of Kahler geometry when discussing nonKahler metrics on appropriate 4-manifolds. One can't hope to generalize the [) operator in this setting, but [) + [)* does have a natural generalization, namely as a spinc Dirac operator. Thus, suppose that JI;[ is a smooth compact 4-manifold which admits an almost-complex structure J, which we then use to orient M. Let L = AO,2 be the anti-canonical line bundle of J. For any metric g on M, the bundles
ci
v+ =
A0,0 EEl A0,2
V_ =AO,1
can then formally be written as
where §± are the left- and right-handed spinor bundles of g. Each unitary connection A on L then induces a spinc Dirac operator
generalizing [) + [)*. The Seiberg-Witten equations [41] are the coupled system
for the unknowns A and E r(V +), where Ft denotes the self-dual part of the curvature of A. These equations are non-linear, but become elliptic once one imposes the 'gauge-fixing' condition d*(A - A o) = 0
to eliminate automorphisms of L --+ M. Because the Seiberg-Witten equations imply the Weitzenbock formula
144
C. LEBRUN
one can show that the moduli space of solutions is compact. In the presence of the assumption that ci(M, J) > 0, one can define the Seiberg-Witten invariant by counting solutions of the Seiberg-Witten equations, modulo gauge equivalence and with appropriate multiplicities. This count is then independent of the metric. However, if there exists a metric 9 of scalar curvature s 2': 0, and if ci(M, J) > 0, the above Weitzenbock formula forces the non-existence of solutions for the given metric, so the Seiberg-Witten invariant must then vanish. By contrast, the Seiberg-Witten invariant would be non-zero for a complex surface of general type [22, 32, 41], so the Kodaira classification [3] allows us to conclude that a complex surface with > 0 can therefore only admit a Riemannian metric of non-negative scalar curvature if it is deformation equivalent to a Del Pezzo surface. The converse directions in Theorems 1 and 2 now follow. In the symplectic case, one may reach the analogous conclusion by appealing to a result of Liu [26]. Liu's argument rests in part on a result of McDuff [30], which characterizes rational symplectic manifolds by the presence of a pseudo-holomorphic 2-sphere of positive selfintersection. The other crucial ingredient is a theorem of Taubes [38], which produces pseudo-holomorphic curves from solutions of perturbed versions of the Seiberg-Witten equations for appropriate spine structures. The converse direction in Theorem 3 thus also follows, as advertised. While we now know that all the manifolds listed in Theorem 2 actually admit Einstein metrics, there are still open questions regarding the moduli of such metrics. Our understanding is quite complete in the cases of K3, T 4 , and their quotients, as these spaces saturate the Hitchin-Thorpe inequality; every Einstein metric on any such manifold is therefore locally hyper-Kahler, and one can therefore [3] in particular show that the moduli space of Einstein metrics on any of these manifolds is connected. But the Del Pezzo cases are quite a different story. For example, while we do have a reasonable understanding of the moduli of Kahler-Einstein metrics on Del Pezzo surfaces [39], nothing we know precludes the existence of other components of the moduli space; however, when a Kahler-Einstein metric exists, it is at least known [17] that any non-Kahler Einstein metric would necessarily have strictly smaller Einstein-Hilbert action. By contrast, the Page and Chen-LeBrun-Weber metrics are not even currently known to have such a maximizing property. Indeed, the uniqueness of the latter metric has not really been conclusively demonstrated even among conformally Kahler metrics, although computer-based calculations [28] lend enormous credibility to such an assertion. What about the A < 0 case? The Aubin/Yau existence theorem [2, 42] constructs Kahler-Einstein metrics with A < 0 on a profusion of minimal complex surfaces of general type. But in the converse direction, we only have some partial results. If (M, J) is a compact complex surface, and if the underlying smooth 4-manifold M admits an Einstein metric g, then it is easy to show, using the Hitchin-Thorpe inequality and the Kodaira cla..'>sification,
ci
EINSTEIN METRICS
145
that either M appears on the list in Theorem 2, or else that (M, J) is of general type. What remains unknown is whether the underlying 4-manifold of a non-minimal complex surface of general type can ever admit an Einstein metric. The best we can currently say is that a surface of general type which admits an Einstein metric cannot be 'too' non-minimal, in the following numerical sense [23]: if X is a minimal complex surface of general type, then its k-point blow-up X #H:W2 cannot admit Riemannian Einstein metrics if k ~ cr(X)/3. Analogous results can also be proved in the symplectic setting. But, basically, our knowledge of the ,\ < 0 realm remains frustratingly incomplete, even though it is precisely here that most of the known examples reside. Perhaps what we really need now is some major progress in constructing Einstein metrics that have nothing to do with Kahler geometry!
References [1] C. AREZZO. F. PACARD, AND M. SINGER, Extr-emal metric8 on blow ups. e-print math.DG/070l028, 2007. [2] T. AUBIN, Equat'ions du type Monge-Ampere sur les varietes kahleriennes compactes, C. R. Acad. Sci. Paris, 283A (1976). pp. 119-121. [3] W. BARTH, C. PETERS, AND A. VAN DE VEN, Compact complex surfaces, vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1984. [4] A. L. BESSE, Einstein manifolds, vol. 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1987. [5] C. BOHM, Inhomogeneous Einstein metr'ics on low-dimensional spheres and other low-dimensional spaces, Invent. Math., 134 (1998), pp. 145-176. [6] C. P. BOYER AND K. GALICKI, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. [7] E. CALABI, On Kahler manifolds with vanishing canonical class, in Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N.J., 1957, pp. 78-89. [8] - - , E:1:tremal Kahler metrics, in Seminar on Differential Geometry, vol. 102 of Ann. of Math. Stud .. Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290. [9] X. X. CHEN, C. LEBRUN, AND B. WEBER, On conformally Kahler, Einstein manifolds, J. ArneI'. Math. Soc., 21 (2008), pp. 1137-1168. [10] X. X. CHEN AND B. WEBER, Moduli spaces of critical Riemannian metrics with L"/2 norm curvatur'e bounds. e-print arXiv:0705.4440, 2007. [11] M. DEMAZURE, Surfaces de del Pezzo, II, III, IV, V, in Seminaire sur les Singularites des Surfaces, vol. 777 of Lecture Notes in Mathematics, Berlin, 1980, Springer, pp. 21-69. [12] A. DERDZINSKI, Self-dual Kahler manifolds and Einstein manifolds of dimension fonr, Compositio Math., 49 (1983), pp. 405-433. [13] A. FUTAKI AND T. MABUCHI, Bilinear forms and extremal Kahler vector fields associated with Kahler classes, Math. Ann., 301 (1995), pp. 199-210. [14] G. GAMOW, My World Line; an Informal Autobiography, Viking Press, New York, NY, 1970. [15] .1. W. V. GOETHE, Maximen 'Il.nd Refiektionen, 1833/1840. Republished on-line at http://www.wissen-im-netz.info/literatur/goethe/maximen.
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[16] P. GRIFFITHS AND J. HARRIS, Principles of Algebraic Geometry, Wiley-Interscience, New York, 1978. [17] M. J. GURSKY, The Weyl functional, de Rham cohomology, and Kahler-Einstein metrics, Ann. of Math. (2), 148 (1998), pp. 315-337. [18] R. HAMILTON, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17 (1982), pp. 255-306. [19] N. J. HITCHIN, On compact four-dimensional Einstein manifolds, J. Differential Geom., 9 (1974), pp. 435-442. [20] B. KLEINER AND J. LOTT, Notes on Perelman's papers. e-print math.DG /0605667. [21] K. KODAIRA, On the structure of compact complex analytic surfaces. I, Amer. J. Math., 86 (1964), pp. 751-798. [22] C. LEBRUN, Four-manifolds without Einstein metrics, Math. Res. Lett., 3 (1996), pp. 133-147. [23] - - , Ricci curvature, minimal volumes, and Seiberg- Witten theory, Inv. Math., 145 (2001), pp. 279-316. [24] C. LEBRUN, Einstein metrics, complex surfaces, and symplectic 4-manifolds, Math. Proc. Cambr. Phil. Soc., 147 (2009), pp. 1-8. e-print arXiv:0803.3743[math.DG]. [25] C. LEBRUN AND S. R. SIMANCA, On the Kahler classes of extremal metrics, in Geometry and Global Analysis (Sendai, 1993), Tohoku Univ., Sendai, 1993, pp. 255-271. [26] A.-K. LIU, Some new applications of general wall crossing formula, Gompi's conjecture and its applications, Math. Res. Lett., 3 (1996), pp. 569-585. [27] Y. I. MANIN, Cubic Forms: Algebra, Geometry, Arithmetic, North-Holland Publishing Co., Amsterdam, 1974. Translated from the Russian by M. Hazewinkel. [28] G. MASCHLER, Uniqueness of Einstein metrics conformal to extremal Kahler metricsa computer assisted approach, AlP Conf. Proc., 1093 (2009), pp. 132-143. On-line at http://link.aip.org/link/? APCPCS /1093/132/1. [29] Y. MATSUSHIMA, Sur la structure du groupe d'homeomorphismes d'une certaine variete Kahlerienne, Nagoya Math. J., 11 (1957), pp. 145--150. [30] D. McDUFF, The structure of rational and ruled symplectic 4-manifolds, J. Amer. Math. Soc., 3 (1990), pp. 679-712. [31] C. W. MISNER, K. S. THORNE, AND J. A. WHEELER, Gravitation, W. H. Freeman and Co., San Francisco, Calif., 1973. [32] J. MORGAN, The Seiberg- Witten Equations and Applications to the Topology of Smooth Four-Manifolds, yo!. 44 of Mathematical Notes, Princeton University Press, 1996. [33] D. PAGE, A compact rotating gravitational instant on, Phys. Lett., 79B (1979), pp. 235-238. [34] G. PERELMAN, The entropy formula for the Ricci flow and its geometric applications. e-print math.DG/0211159. [35] - - , Finite extinction time for the solutions to the Ricci flow on certain threemanifolds. e-print math.DG/0307245. [36] - - , Ricci flow with surgery on three-manifolds. e-print math.DG/0303109. [37] Y. SIU, The existence of Kahler-Einstein metrics on manifolds with positive anticanonical line bundle and suitable finite symmetry group, Ann. Math., 127 (1988), pp. 585-627. [38] C. H. TAUBES, The Seiberg- Witten and Gromov invariants, Math. Res. Lett., 2 (1995), pp. 221-238. [39] G. TIAN, On Calabi's conjecture for complex surfaces with positive first Chern class, Inv. Math., 101 (1990), pp. 101-172. [40] G. TIAN AND S. T. YAU, Kahler-Einstein metrics on complex surfaces with Cl > 0, Comm. Math. Phys., 112 (1987), pp. 175-203. [41] E. WITTEN, Monopoles and four-manifolds, Math. Res. Lett., 1 (1994), pp. 809-822.
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[42] S. T. YAU, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. USA, 74 (1977), pp. 1789-1799. [43] - - , On the Ricci curvature of a compact Kahler manifold and the complex MongeAmpere equation. I, Comm. Pure Appl. Math., 31 (1978), pp. 339-411. DEPARTMENT OF MATHEMATICS, SUNY AT STONY BROOK, STONY BROOK, NY 11794-3651 E-mail address: claude@math. sunysb. edu
Surveys in Differential Geometry XIII
Existence of Faddeev knots in general Hopf dimensions Fengbo Hang, Fanghua Lin, and Yisong Yang
ABSTRACT. In this paper, we present an existence theory for absolute minimizers of the Faddeev knot energies in the general Hopf dimensions. These minimizers are topologically classified by the Hopf-Whitehead invariant, Q, represented as an integral of the Chern-Simons type. Our method involves an energy decomposition relation and a fractionally powered universal topological growth law. We prove that there is an infinite subset § of the set of all integers such that for each N E § there exists an energy minimizer in the topological sector Q = N. In the compact setting, we show that there exists an absolute energy minimizer in the topological sector Q = N for any given integer N that may be realized as a Hopf-Whitehead number. We also obtain a precise energy-splitting relation and an existence result for the Skyrme model.
1. Introduction
In knot theory, an interesting problem concerns the existence of "ideal knots", which promises to provide a natural link between the geometric and topological contents of knotted structures. This problem has its origin in theoretical physics in which one wants to ask the existence and predict the properties of knots "based on a first principle approach" [N]. In other words, one is interested in determining the detailed physical characteristics of a knot such as its energy (mass), geometric conformation, and topological identification, via conditions expressed in terms of temperature, viscosity, electromagnetic, nuclear, and possibly gravitational, interactions, which is also known as an Hamiltonian approach to realizing knots as field-theoretical stable solitons. Based on high-power computer simulations, Faddeev and Niemi [FNl] carried out such a study on the existence of knots in the Faddeev quantum field theory model [Fl]. Later, Faddeev addressed the existence problem and noted the mathematical challenges it gives rise to ©2009 International Press
F. HANG, F. LIN, AND Y. YANG
150
[F2]. The purpose of the present work is to develop a systematic existence theory of these Faddeev knots in their most general settings. Recall that for the classical Faddeev model [BSl, BS2, Fl, F2, FNl, FN2, Su] formulated over the standard (3+ 1)-dimensional Minkowski space of signature (+ - - - ), the Lagrangian action density in normalized form reads
(1.1 ) where the field u = (Ul' U2, U3) assumes its values in the unit 2-sphere and
(1.2) is the induced "electromagnetic" field. Since u is parallel to oJ.tu /\ ovu, it is seen that FJ.tv(u)FJ.tV(u) = (oJ.tu/\ov'u)· (oJ.tu/\OVu), which may be identified with the well-known Skyrme term [El, E2, MRS, SI, S2, S3, S4, ZB] when one embeds 8 2 into 8 3 ~ 8U(2). Hence, the Faddeev model may be viewed as a refined Skyrme model governing the interaction of baryons and mesons and the solution configurations of the former are the solution configurations of the latter with a restrained range [C]. We will be interested in the static field limit of the Faddeev model for which the total energy is given by
(1.3)
E(u) =
L, {~18;UI'
+ ~ j~1IFjk(U)12} dx.
Finite-energy condition implies that u approaches a constant vector U oo at spatial infinity (of JR3). Hence we may compactify JR3 into 8 3 and view the fields as maps from 8 3 to 8 2 . As a consequence, we see that each finite-energy field configuration u is associated with an integer, Q(u), in 7r3(82) = Z (the set of all integers). In fact, such an integer Q(u) is known as the Hopf invariant which has the following integral characterization: The differential form F = Fjk(u)dxj /\ dx k (j, k = 1,2,3) is closed in JR3. Thus, there is a one form, A = Ajdxj so that F = dA. Then the Hopf charge Q(u) of the map u may be evaluated by the integral (1.4)
Q(u) = 161 2 7r
f A /\ F, iw,,3
due to J. H. C. Whitehead [Wh]. The integral (1.4) is in fact a special form of the Chern-Simons invariant [CSl, CS2] whose extended form in (4n - 1) dimensions (cf. (2.2) below) is also referred to as the HopfWhitehead invariant. The Faddeev knots, or rather, knotted soliton configurations representing concentrated energy along knotted or linked curves, are realized as the solutions to the minimization problem [F2], also known as the Faddeev knot problem, given as
(1.5)
EN == inf{E(u) I E(u)
E(u; B2R \ BR)
= {
JB2\B1
{1\7 y u R(y)14n-l
+ l(u R)*(WS2n)(y)12 R- 1 + R4n-llu R(y) - n1 2 } dy. Consequently, we have (3.4) 2dr { dS {l\7u R I 4n-l e> r
r/ Jl
JaB,.
+ l(u R )*(WS2n)12R-l + R4n-lluR _ nI 2 }.
Hence, there is an r E (1,3/2) such that (3.5)
{
JaB,.
{1\7u R I 4n - 1 + l(u R )*(WS2n)12R-l
+ R 4n - 1 1uR - n1 2 }
dSr
< 2e.
In what follows, we fix such an r determined by (3.5). Consider a map v R : ]R4n-l --t ]R2n defined by (3.6)
~vR
= 0 in B2 \ B r.,
(3.7) Then, for p bound
= (4n - 1)2/(4n - 2), we have, in view of (3.6) and (3.7), the
(3.8)
which in terms of (3.5) leads to
(3.9)
1
(471.-1)2 4n-1 l\7v RI (471.-2) ~ G1e4n-2.
B2\Br
Since (4n-1)2 > 4n(4n- 2), we have p conjugate exponents sand t gives us
> 4n. So the HOlder inequality with
EXISTENCE OF FADDEEV KNOTS
where 4n8 = p = (4n _1)2/(4n - 2) and t in view of (3.9) and (3.10),
159
= 8/(1- s). Therefore, we have,
(3.11)
Recall that, since R ~ 1, we also have JaBr lu R - nl 2 dSr < 2e. Hence, for any q > 2, we have JaBr lu R - nl q dSr :s: C JaBr lu R - nl 2 dSr :s: Cle. Since the ball is in lR. 4n - 1 , we see that for q = 4n(4n-2)/(4n-1) (of course, q > 2), we have (3.12) Therefore, we have seen that (vR-n) has small W 1 ,4n(B2 \ Br)-norm. Using the embedding wl,4n(B2 \ Br) ---t C(B2 \ Br) (noting that dim(B2 \ Br) = 4n - 1 < 4n), we see that (v R - n) has small C(B2 \ Br)-norm. As a consequence, we may assume n .v
(3.13)
Since v R is harmonic, Iv R Hence
R
> -1 on B2 \ B r . 2
nl 2 is subharmonic, ~lvR - nl 2 ~ 0, on B2 \ B r .
(3.14) To get a map from B2 \ B r , we need to normalize v R, which is ensured by (3.13). Thus, we set R_ v R
(3.15)
w - IvRI
on B2 \ Br.
Then w R E s2n. We can check that IwR-nl < in view of (3.13). Therefore we have (3.16)
r
R 4n - 1 1w R
lB2\Br
(3.17)
nl 2 :s: 8CE,
r R- 1(w R )*(WS2n)12 :s: r lV'v R l4n :s: CIE4~~1, lB2\Br lB2\Br 1
r
(3.18)
-
4lv R-nl and IOjWRI < 410jv R I
l~\Br
C
lV'w R I4n-l
:s: C2
r
lV'v R I4n-l
lB2\Br
:S:C2IB2\Brlt(
r
lB2\Br
1
lV'vRI4n)S,
160
F. HANG, F. LIN, AND Y. YANG
where t = 8/(8 -1) and 8 = 4n/(4n -1). The bounds (3.11) and (3.18) may be combined to yield
r IVw R I4n-1 ::; C E. lB2\B,.
(3.19)
3
Thus, we can summarize (3.16), (3.17), and (3.19) and write down the estimate
r
W
{IVw RI4n-1 + R- 1 I(w R)*(WS2n + R 4n - 11w R lB2\Br On 8B2, w R = n; on 8Br , w R = uR/luRI = u R . Define (3.20)
(3.21)
u(X)=wR(~x)
forxEB2R\BrR;
u(x)=u(x)
n1 2 } < CEo
forxEBrR.
We see that the statements of the lemma in the first case are all established. The proof can be adapted to the case of the interchanged boundary conditions u = u on B2R and u = n on BR. Hence, all the statements of the lemma in the second case are also established.
4. Integer-valuedness of the Hopf-Whitehead integral As the first application of the technical lemma established in the previous section, we prove 4.1. ffu: 1R4n - 1 - t s2n is of finite energy, E(u) < 00, where the energy E is as given in {2.5}, then the Hopf-Whitehead integral {2.3} with 8v = 0 is an integer. THEOREM
Let the pair u, v be given as in the theorem and {Ej} be a sequence of positive numbers so that Ej - t 0 as j - t 00 and {Rj} be a corresponding sequence so that Rj - t 00 asj - t 00 and E(u; 1R4n - 1\ BRJ < Ej, j = 1,2,···. Let {Uj} be a sequence of modified maps from 1R4n - 1 to s2n produced by the technical lemma so that Uj = U in BRj and Uj = n on 1R4n - 1 \ B2Rj' Then
(4.1) is a sequence of integers. We prove that Q( Uj) - t Q( u) as j - t 00. We know that {IUj(WS2n)l} is bounded in L2(1R4n-1) and L 41~;;-1 (1R4n-1) due to the structure of the knot energy (2.5), the definition of Uj, and the relation (2.12). By interpolation, we see that the sequence is bounded in LP(1R4n - 1) for all p E [47~~1,2]. From the relations dVj = Uj(WS2n) and 8vj = 0, we see that {IVv.il} is bounded in LP(1R4n - 1) for all p E [4~~1, 2] as well. Using the Sobolev inequality
(4.2)
C(m,p)llfll q
::;
IIVfll p
EXlSTENCE OF FADDEEV KNOTS
161
in ~m with q = mp/(m-p) and 1 < p < m, we get the boundedness of {Vj} in Lq(~4n-1) for q = (4n - l)p/( 4n - 1 - p) with 4~~1 :::; P :::; 2, which gives the range for q,
2(4n-l) ( ) _4n-l = 2n - 1 :::; q:::; 4n - 3 .
(4.3)
qn
To proceed, we consider the estimate
Is 2nI2IQ(u) - Q(uj)1
=
IJ~4n-l r (v 1\ U*(WS2n) -
:::; I J~4n-l r (v 1\ U*(WS2n) -
(4.4)
r
Vj
Uj(WS2n)) I
v 1\ Uj(WS2n)) I
+ I J~4n-l (v 1\ Uj(WS2n) -
I?) + IY). To show that I?) 0 as j
1\
Vj
1\
uj(WS2n))1
==
---t
---t 00,
we look at the bottom numbers (for
example) for which (4.5) for p
=
4~~1 so that the conjugate ofp is pi
I?)
= pS
= ~~=~ = q(n), as defined
---t 0 immediately follows from (4.5). in (4.3). Hence the claim On the other hand, since q(n) > 2, we see that {Vj} is bounded in W 1,2(B) for any bounded domain B in ~4n-1. Using the compact embedding W 1,2(B) ---t L2(B) and a subsequence argument, we may assume that {Vj} is strongly convergent in L2(B) for any bounded domain B. Thus, we have
It is not hard to see that the quantity E( Uj; ~4n-1 \ B) may be made uniformly small. Indeed, for any E > 0, we can choose B sufficiently large so that E( u; ~4n-1 \ B) < E. Let j be large enough so that BR j ~ B. Then (4.7)
E( Uj; ~4n-1 \ B) :::; E( u; ~4n-1 \ B)
+ E( Uj; B 2R
j \
BR j
)
:::;E+CEj,
Iy)
in view of Lemma 3.1. Using (4.7) in (4.6), we see that ---t 0 as j ---t 00. Consequently, we have established Q( Uj) ---t Q( u) as j ---t 00. In particular, Q(u) must be an integer because Q(Uj)'s are all integers.
F. HANG, F. LIN, AND Y. YANG
162
5. Minimization for the Nicole-Faddeev-Skyrme model Consider the minimization problem (2.21) where the energy functional E is defined by (2.5). Let {Uj} be a minimizing sequence of (2.21) and set
h{x) = (IVujI4n-l
(5.1)
+ IUj(WS2n) 12 + In -
UjI2)(x).
Then we have 4n-l
IIhlll ~ CINI4il,
(5.2)
and IlhliI SEN + 1 (say) for all j. Use B(y, R) to denote the ball in 1R4n - 1 centered at y and of radius R > O. According to the concentration-compactness principle of P. L. Lions [Ll, L2j, one of the following three alternatives holds for the sequence {h}: (a) Compactness: There is a sequence {Yj} in 1R4n c > 0, there is an R> 0 such that
h(x)
sup (
(5.3)
j
JR4n-l\B(Yj,R)
dx
0,
(5.4)
.lim ( J-+OO
sup YER4n-l
J(B(y,R) h(x) dx)' =
O.
(c) Dichotomy: There is a sequence {Yj} C 1R4n - 1 and a positive number t E (0,1) such that for any c > 0 there is an R > 0 and a sequence of positive numbers {Rj} satisfying limj-+oo Rj = 00 so that
IJ(B(Yj,R) h(x) dx - tllhliIl < c,
(5.5)
(5.6)
I{
JR4n-l \B(Yj,Rj)
h(x)dx - (1- t)llhlhl < c.
We have the following. LEMMA 5.1. The alternative (b) (or vanishing) stated in (5.4) does not happen for the minimization problem when N =1= O.
Let B be a bounded domain in lRm and recall the continuous embedding W 1,P{B) - L:!p (B) for p < m. We need a special case of this at p = 1: PROOF.
(5.7)
EXISTENCE OF FADDEEV KNOTS
163
Hence, for any function w, we have
c( l,w + llw l + l'vw,m) Iwl k (k :S c(1,w ,2 + llvw,m).
:S
(5.8)
,k
(if
Now taking m
(k-1)m':':1
~ 2,
is bounded,
= 4n - 1 so that
m~1
!~=~
=
1)m~1 ~ 2,
then)
> 1, k = 4, w = Uj - n, and
B = B(Yj, R), we have from (5.8) the inequality
(5.9)
r
IUj _
nI2~~-':-II) :s c( r
) B(Yj ,R)
We now decompose lR 4n -
1
r
IUj _ nl2 +
) B(Yj ,R)
IVUjI4n-1) 1+
4nl_2.
) B(Yj ,R)
into the union of a countable family of balls,
(5.10) so that each point in lR4n - 1 lies in at most m such balls. Then define the quantity
Thus the alternative (b) (vanishing) implies
r
)"&.4n-l
IUj -
nI2~~-.:-/) :s
f r
IUj _
aj ----+
0 as j
----+ 00.
Therefore
nI2~~~-':-11)
i=1 ) B(Yi,R)
:sar~2cf(r
(5.12)
i=l
:sma4n~IC( J
r
IUj- n I2+
) B(Yi,R)
IVUjI4n-l)
) B(Yi,R)
r 4n-l (luj-n I2 + IVUjI4n-1)) )"&.
1
:S Define the set Aj (5.13)
=
mar-1 CE (Uj)
----+
0
as j
----+ 00.
{x E lR4n - 1 IIUj(X) - nl ~ I} (say). Then (5.12) implies
lim IAjl = 0, J->OO
164
F. HANG, F. LIN, AND Y. YANG
where IAjl denotes the Lebesgue measure of Aj . Since Q(Uj) = N =1= 0, we see that Uj(1l~4n-1) covers s2n (except possibly skipping n). The definition of Aj says uj(A j ) contains the half-sphere below the equator of s2n. Consequently,
i
(5.14)
IUj(WS2n)1 dx 2: IUj(Aj)1 2:
~IS2nl,
J
where Is 2nl is the total volume of s2n. However, the Schwartz inequality and (5.13) give us
{ IUj(WS2n)1 dx:::;
JAj
(5.15)
IAjl~
(rJ
1
IUj(WS2n)12) 2'
IR4n-l
1
1
:::; IAjl2'(EN + 1)2' as j
--t
00,
--t
0,
which is a contradiction to (5.14).
o
Suppose that (a) holds. Using the notation of (a), we can translate the minimizing sequence {Uj} to (5.16)
{Uj(' - Yj)} = {Uj(')}
so that {Uj} is also a minimizing sequence of the same Hopf charge. Passing to a subsequence if necessary, we may assume without loss of generality that {Uj} weakly converges in a well-understood sense over lR,4n-1 to its weak limit, say u. Of course, (5.17)
E(u) :::; liminf{E(uj)} = EN. J~OO
5.2. The alternative {a} {or compactness} stated in {5.3} implies the preservation of the Hopf charge in the limit described in {5.17}. In other words, Q( u) = N so that U gives rise to a solution of the direct minimization problem {2.21}. LEMMA
PROOF. Let c and R be the pair stated in the alternative (a). Then (5.18) Besides, for the weak limit (5.19) and (5.20)
U
of the sequence {Uj}, we have
EXISTENCE OF FADDEEV KNOTS
165
where
(5.21)
It is not hard to see that the quantities J and K j are small with a magnitude of some power of E. In fact, (2.5) and (2.12) indicate that luj(wS2n)1 is uniformly bounded in V(lR4n-1) for p E [4~~1, 2]. Then the relation dVj = uj(wS2n),5Vj = 0, and the Sobolev inequality (4.2) imply that Vj is uniformly bounded in Lq(lR4n - 1 ) for q E [i~=~, 2~~-=-31)] (see (4.3)). Using (2.12) again, we have (5.22)
K·J
< Ilv'll 4n-l Ilu*(w 2n)11 L-rn-(IR4n-I\B 4n-1 J LTri"=T(IR4n-I\B R ) J S R)
-
s:; CE( Uj; IR.
4
1 n- \
2n 2n BR) 4n-1 s:; CE4n-1.
By the same method, we can show that the quantity J obeys a similar bound as well. For Ij , we observe that since Uj(WS2n) converges to U*(WS2n) weakly in L2(BR) and Vj converges to v strongly in L2(BR), we have I j -----t 0 as j -----t 00. Summarizing the above results, we conclude that Q(Uj) -----t Q(u) as j -----t 00. D In the next section, we will characterize the alternative (c) (dichotomy).
6. Dichotomy and energy splitting in minimization Use the notation of the previous section and suppose that (c) (or dichotomy) happens. Then, after possible translations, we may assume that there is a number t E (0,1) such that for any E > 0 there is an R> 0 and a sequence of positive numbers {Rj} satisfying limj--->oo R j = 00 so that
(6.1)
(6.2)
I
LR fj(x) dx - tE(Uj)1
2R for all j. Therefore, from the decom posi tion
166
F. HANG, F. LIN, AND Y. YANG
and (6.1), (6.2), we have
E(uj; B2R \ BR) S E(uj; BRj \ BR) < 2c:,
(6.4)
E(uj; BRj \ B Rj / 2) S E(uj; BRj \ BR) < 2c:.
Using Lemma 3.1, we can find maps u?) and u;2) from 1R4n -
1
to s2n such
= Uj in BR, U)1) = n in 1R4n - 1 \ B 2R, and E(uY); B2R \ B R ) < Cc:; . 1lJ>4n-l \ B Rj' U(2) = n III . B Rj/2, an d E( U(2) ; B Rj \ B Rj/2 ) < C c:. Uj(2) = Uj III ~ j j Here C > is an irrelevant constant. that u;1)
°
Use the notation F(u) = v 1\ U*(WS2n). Since F(u) depends on U nonlocally, we need to exert some care when we make argument involving truncation. In view of the fact that Uj and U)l) coincide on BR and Uj and u?) coincide on 1R4n - 1 \ BRj' we have
(6.5)
r
. IUj(WS2n) - (u;1))*(wS2n) -
lJR4n-l
S C(E(uj; BRj sCc:.
\
(u?))*(wS2n)14~;;-1
B R ) + E(u?); B2R \ B R ) + E(u?); B Rj \ B Rj / 2))
Consequently, using the relations dVj
= Uj(WS2n), 6vj = 0, dvji) = (uY))*
(wS2n),6vY) = 0, i = 1,2, we have in view of (6.5) and (4.2) with p = (4n - 1)/2n and q = (4n - 1)/(2n -1) that
(6.6) IIvj - v)(l) - v)\2)1I4n_l
S Clluj(WS2n) - (u)(1))*(WS2n) - (u)(2))*(WS2n)114n-l
2n-l
2n S C1C: 4n - 1 •
2n
Since the numbers p, q above are also conjugate exponents, we obtain from (6.6) the bound
(6.7) 2n
S CC: 4n - 1 .
167
EXISTENCE OF FADDEEV KNOTS
Applying (6.7), we have
IS 2n I2IQ(uj) _ (Q(u)l))
::; r
+ Q(u)2)))1
IF(uj)-F(u)1))-F(u)2))1
J BRU{IR4n-l\BRj}
(6.8)
+
1
IF(Uj)1
+
B Rj \BR 2n
::; C1E4n-l
1
IF(u?))1
B2R\BR
+ C2(E(Uj; BR
+ E(u j(2) ; BR
1
IF(u)2))1
B Rj \BRj/2
2n
j \
+
(1)
+ E(u j
BR)4n-1
2n
; B2R \ BR)4n-1
2n
j
\
B Rj / 2)4n-1)
2n
::; CE4n-1 . Since E > 0 can be arbitrarily small and Q(Uj), Q(u?\ Q(U)2)) are integers, the uniform bound (6.8) enables us to assume that (6.9) On the other hand, since (2.9) implies that (6.10)
. (1) (1) 4n-1 IQ(u j )I--;rn-::; CE(u j ) = C(E(Uj; BR) ::; CE(Uj)
+ E(u j(1) ; B2R \
BR))
+ ClE,
we see that {Q( U)l))} is bounded. We claim that Q(U)l)) 1= 0 for j sufficiently large. Indeed, if Q(U)l)) = 0 for infinitely many j's, then, by going to a subsequence when necessary, we may assume that Q(u?)) = 0 for all j. Thus we see that Q(U)2)) = N in (6.9) for all j and (6.11)
E(U)2))::; E(uj;
jR4n-l \
BRj)
+ CE =
r
iJ(x) dx + CEo
JIR4n-1\BRj
As a consequence, we have in view of (6.11) and (6.2) that
EN ::; limsupE(u)2)) ::; (1- t) lim E(uj) (6.12)
j
--+00
::; (1 - t)EN
+ E+ CE
J--+OO
+ C1E.
Since 0 < t < 1 and E is arbitrarily small, we obtain EN = 0, which con4",-1 tradicts the topological lower bound EN 2: CjNI--;rn- (N 1= 0) stated in (2.9). Similarly, we may assume that Q(U)2)) 1= 0 for j sufficiently large. Of course, {Q( U)2))} is bounded as well.
F. HANG, F. LIN, AND Y. YANG
168
Hence, extracting a subsequence if necessary, we may assume that there are integers N1 =1= 0 and N2 =1= 0 such that (6.13) Furthermore, for the respective energy infima at the Hopf charges N I , N 2 , N, we have
+ EN2 ~ E(u;l)) + E(u;2)) = E(uj; B R ) + E(uj; lR. 4n - 1 \ + E(u;2); BRj \ B Rj / 2) ~ E(uj) + 2Cc.
ENl
(6.14)
BRj)
+ E(u?); B2R \
Since c > 0 may be arbitrarily small, we can take the limit j to arrive at
BR)
---t 00
in (6.14)
(6.15) We can now establish the following energy-splitting lemma. LEMMA 6.1. If the alternative (c) (or dichotomy) stated in (5.5) and (5.6) happens at the Hopf charge N =1= 0, then there are nonzero integers N I , N2,'" ,Nk such that
(6.16)
and that the alternative (a) (or compactness) stated in (5.3) takes place at each of these integers Nb N 2, .. " Nk. If the alternative (c) happens at N =1= 0, we have the splitting (6.15). We may repeat this procedure at all the sublevels wherever the alternative (c) happen. Since (2.9) and (2.10) imply that there is a universal constant C > 0 such that E£ ~ C for any € =1= o. Hence the above splitting procedure ends after a finitely many steps at (6.16) for which the alternative (c) cannot happen anymore at N I , N2,' .. , Nk. Since the alternative (b) never happens because Ns =1= 0 (s = 1,2, ... , k) in view of Lemma 5.1, we see that (a) takes place at each of these integer levels. 0 PROOF.
The energy splitting inequality, (6.16), is referred to as the "Substantial Inequality" in [LY4] which is crucial for obtaining existence theorems in a noncompact situation.
7. Existence theorems We say that an integer N =1= 0 satisfies the condition (S) if the nontrivial splitting as described in Lemma 6.1 cannot happen at N. Define
(7.1)
§ =
{N E Z I N satisfies condition (S)}.
EXISTENCE OF FADDEEV KNOTS
169
It is clear that, for any N E §, the minimization problem (2.21) has a solution. As a consequence of our study in the previous sections, we arrive at
THEOREM 7.1. Consider the minimization problem {2.21} in which the energy functional is of the NFS type given in {2.5}. Then there is an infinite subset of Z, say §, such that, for any N E §, the problem {2.21} has a solution. In particular, the minimum-mass or minimum-energy Hopf charge No defined by No is such that ENo = min{EN I N =I- O}
(7.2)
is an element in §. Furthermore, for any nonzero NEZ, we can find N1, ... , Nk E § such that the substantial inequality {6.16} is strengthened to the equalities
(7.3)
EN
= EN! + EN2 + ... + Nk,
N
= N1 + N2 + ... + Nk,
which simply express energy and charge conservation laws of the model in regards of energy splitting.
PROOF. Use the Technical Lemma (Lemma 3.1) as in [LYl] to get (7.3). The rest may also follow the argument given in [LYl]. 0 Next, we show that, in the compact situation, the minimization problem (2.21) has a solution for any integer N. For this purpose, let E(u) denote the energy functional of the NFS type or the Faddeev type given as in (2.5) or (2.6) evaluated over s4n-l for a map u from s4n-l into s2n. Namely,
(7.4) (7.5)
The Hopf invariant Q(u) of u is given in (2.2). We have THEOREM 7.2. For any nonzero integer N which may be realized as a Hopf number, i.e., there exists a map u : s4n-l ----+ s2n such that Q(u) = N, the minimization problem EN = inf{E(u) I E(u) < 00, Q(u) = N} over s4n-l has a solution when E is given either by {7.4} or {7.5}. PROOF. Let {Uj} be a minimizing sequence of the stated topologically constrained minimization problem and Vj be the "potential" (2n - I)-form satisfying (7.6)
dVj = Uj(WS2n),
8vj = 0,
j = 1,2, ....
Passing to a subsequence if necessary, we may assume that there is a finiteenergy map u (say) such that Uj ----+ U, dUj ----+ u, and Uj(WS2n) ----+ U*(WS2n) weakly in obvious function spaces, respectively, as j ----+ 00, which lead us to the correct comparison E(u) ::; EN by the weakly lower semi-continuity
170
F. HANG, F. LIN, AND Y. YANG
of the given energy functional. To see that Q(u) = N, we recall that the sequence {Vj} may be chosen [Mo] such that it is bounded in W l ,2(s4n-l) by the L2(s4n-l) bound of {Uj(WS2n)}. Hence Vj ---t some v E Wl,2(S4n-l) weakly as j ---t 00. Therefore Vj ---t v strongly in L2(S4n-l) as j ---t 00. Of course, dv = U*(WS2n) and 6v = O. Consequently, we immediately obtain
(7.7) Q(u) = Is;nl2
irS
4n-l
J-->ooirS
v 11 U*(WS2n) = Is;nl2 lim
1Ij
4n-l
11 Uj(WS2n)
and the proof is complete.
= N, 0
Note that the existence of global minimizers for the compact version of the Nicole energy (2.4),
(7.8)
E(u) =
ir
Idul 4n - 1 dS,
S4n-l
was studied by Riviere [Ri] for n = 1. See also [L] and [DK]. In particular, he showed that there exist infinitely many homotopy classes from S3 into S2 having energy minimizers. We now address the general problem of the existence of critical points of (7.8) at the bottom dimension n = 1 whose conformal structure prompts us to simply consider it over ]R3. Thus we are led to the Nicole model. Specifically, for a map u : ]R3 ---t S2, the Nicole energy [Ni] is given by
E(u) =
(7.9)
r l\7uI iR3
3.
For convenience, we may use the stereographic projection of S2 ---t C from the south pole to represent u = (Ul' U2, U3) by a complex-valued function U = U1 + iU2 as follows,
(7.10) where U3 = ±y'1 - UI - u~ for u belonging to the upper or lower hemisphere, S1o. Following [AFZ] (see also [ASVW, HS]), we use the toroidal coordinates ('f}, ~,
r!~1'
a
E
U (JRr!) and da = ¢dX1 !\ ... !\ dx n . Hence
10. The Hopf-Whitehead invariant: integer-valuedness
In this section, we will prove that for a map with finite Faddeev energy, the Hopf-Whitehead invariant Q (u) is always an integer. This fact is not only needed for us to come up with a reasonable mathematical formulation for the Faddeev model but also plays a crucial role in understanding the minimizing sequences for the minimization problems. THEOREM 10.1. Assume that u E ~~; (JR 4n -
{ Idul 4n - 2 + IU*WS2nI2 iIR 4n - 1
1,
S2r!) such that
< 00,
183
EXISTENCE OF FADDEEV KNOTS
where WS2n is the volume form on s2n. Then du*wS2n = O. Let 1
f(x)- (4n - 3) IS4n-21IxI4n-3'
T=d*(f*u*WS2n),
where d* is the L2-dual of d, IS 4n- 2 is the area of s4n-2. Then T E L2 (JR4n-1), dT = U*WS2n, d*T = 0, and the Hopf- Whitehead invariant 1
Q(u) =
~ 2 IS nl
r
U*WS2n /\T
Jw,4n-1
is well defined and equal to an integer. To prove Theorem 10.1, we first show that dU*WS2n CLAIM
u*da =
J 0,
...
10.2. For any smooth 2n-form a on s2n, we have du*a
o.
PROOF. f
= O.
By linearity we may assume a
=
fodJI /\ ... /\ dhn, where
1,2no f E COO (JR2n+1 ,JR). Because u E w 1,4n-2(JR4n-1) C Wl (JR 4n - 1') ,2n C c
it follows from Lemma 9.2 that
du* (JIdh /\ ... /\ dhn) = u* (dJI /\ ... /\ dhn). Hence
du* (dJI /\ ... /\ dhn) = O. For any integer k, we write
------.. Addu) = du /\ ... /\ duo k times
Then IU*WS2nl =
IA2n (du)l. It follows that A 2n (du)
E L2 (JR4n-1). Hence
u* (dJI /\ ... /\ dhn) E L2 (JR4n-1). On the other hand, because fa 0 u E Loo (JR4n-l), d (fa (JR 4n - 1 ) C Ltoc(JR4n-l), it follows from Lemma 9.1 that
0
u)
du*a = d (fa 0 U· u* (dJI /\ ... /\ dhn))
= d (fa 0 u) /\ u* (dJI /\ ... /\ dhn) = u*da = O. 2n-1
2
o
Note that U*WS2n E L-n- n L where and in the sequel, we often omit the domain space when there is no risk of confusion. Hence, if we let Tf = f * U*WS2n, then dTf = 0, dd*Tf = tl.Tf = U*WS2n. Here f is the fundamental solution of the Laplacian operator on JR4n - 1, * means we convolute each component of U*WS2n with f and in tl.Tf, the tl. is equal to dd* + d*d (the Hodge Laplacian, it is the negative of the standard
F. HANG, F. LIN, AND Y. YANG
184
Laplacian when acting on functions). Let 7 = d*17. Then d7 = U*WS2n. It follows from the usual singular integral estimate that ([St]) 8n 2-6n±1 7 E L4n L ;ln±1 3
7 E L"2+C:
2(4n-l)
n L 4n-3,
n L6 , D7
E
2n-l D7 E L-n-
L1+c:
nL
2
when n
~
2;
n L2 when n = 1.
Here c is an arbitrarily small positive number. In particular, we always have 7 E L2 (1R4n-1) and Q(u) =
r
~ IS 2nl lffi. 4n -
U*WS2n 1\7 1
is well defined. To show it is an integer, we will first use an idea from [HR, Section 11.4] which would imply that Q (u) is equal to the usual HopfWhitehead invariant of another weakly differentiable map. Then we will apply ideas from [Sv, EM] to show that the invariant is an integer. CLAIM
10.3. Let U : :JR4n-1 U(x,y)=
X :JR4n-1 ---+
s2n
X
s2n
X
s4n-2 be given by
X-y) . (u(x),u(y)'lx_yl
Then U*WS2nxS2nxS4n-2 ELI and
Roughly speaking, the claim says the Hopf invariant of u is equal to the degree of U. This is a special case of a more general formula for rational homotopy in [HR, section H.4]. Since we will need the proof later on and for completeness, we present the argument in this special case. PROOF.
Let Ju
= IU*WS2n I be the Jacobian of u, then 1 (x U
). (x) I/L (y) dx). /L
A
(Xj - Yj)
j=o
x (dx/L) lOxj A dYl A··· A dYn. Hence
where
'" =
L (f * 1>.) dx).. ).
Hence
Q (u) = -
r
21 IS2nl IS4n- 2 1JIR4n- 1 XIR4n -
U*WS2nxS2nxS4n-2. 1
o It follows from Proposition 9.8 that there exists an integer-valued function du E L1 (s2n X s2n X s4n-2) such that for every I E L oo (s2n X
s2n
X
S4n-2),
r
I
A (Ix - Y
)*
x - YI
Here
Z
(u (x), u(Y), IX - YI) Y
(U*WS2n)
(x) A (U*WS2n) (y)
X -
JIR4n-1XIR4n-1
WS4n-2
=
r
J S2n xS2n XS4n - 2
I (z) du (z) dS(z') dS(z") dS(z"').
= (z', z", z"'). Denote
G1 =
r
21 du (z) dS (z') dS (z") dS (i") IS 2nl IS4n- 2 1JS2nxS2nxS4n-2
.
Once we know du == G1 , by choosing I = 1 in the above equation, it follows from Claim 10.3 that H (u) = -G1 is an integer. To show du == G1 , we only need to prove the following.
EXISTENCE OF FADDEEV KNOTS
f
10.4. For every
CLAIM
E
L oo (s2n
X
s2n
!s2nXS2nXS4n-2 f (z) du (z) dS (Z') = C1
X
dS
{ f (z) dS (Z') JS2n xS2n XS4n-2
187
s4n-2) ,
(Z")
dS
dS
(i')
(Z",)
dS
(i") .
By approximation we only need to verify the equality for
f (z) fI, hE (a)
=
fI (z') h (z") 13 (z",) ,
13 E Coo (s4n-2). To achieve this we only need to If JS4n-2 13 (z",) dS (z",) = 0, then
prove
Coo (s2n),
!s2nXS2nXS4n-2 fI (z') h (z") 13 (z",) du (z) dS (z') dS (z") dS (z",) (b) If
JS2n h (z") dS (z") = 0,
=
o.
then
{ fI (z') h (z") du (z) dS (z') dS (z") dS (z",) JS2n xS2n XS 4n -2 (c) If JS2n fI (z') dS (z') = 0, then
= O.
!s2nXS2nXS4n-2 fI (z') du (z) dS (z') dS (z") dS (z",) = o. Indeed, if (a)-(c) are true, then we have
!s2nXS2nXS4n-2 fI (z') h (z") 13 (i") du (z) dS (z') dS (z") dS (z"')
!-21 JS4n-2 ( 13 (z",) dS (z",)
= IS 4 X
!s2nXS2nXS4n-2 fI (z') h (z") du (z) dS (z') dS (z") dS (i")
= Is;n II S4!-21 X
!s2n h (z") dS (z") !s4n-2 13 (z",) dS (z",) .
!s2nXS2nXS4n-2 fI (z')
- _1_ 1 - IS2n121S4n-21 X
X
{
JS2n
du
(z) dS (z') dS (z") dS (z",)
f (z') dS (z') J{ S 2n 1
h (Z") dS (Z") { 13 (Z",) dS (Z",) . JS4n-2
!s2nXS2nXS4n-2 du (Z) dS (Z') dS (Z") dS (Z",)
= C1
(
JS2n XS2n XS4n-2
fI (Z') h (Z") 13 (Z",) dS (Z') dS (Z") dS (Z",).
F. HANG, F. LIN, AND Y. YANG
188
We start with (a). Since fS4n-2 Is (z",) dS (z",) = 0 we may find a smooth (4n - 3)-form , on s4n-2 such that d, = IsWS4n-2. Note that
r
) s2n xs2n XS 4n - 2
=
r
JJR4n-l
h (z') h (Z") Is (z",) du (z) dS (z') dS (Z") dS (z",) U*(hWS2n)(X)/\u*(hwS2n)(Y)/\ (IX-YI)* (d,). X -
xJR4n-l
2
2n-l
4n-l
Recall that A 2n (du) E L-n- n L c L 2'n. Let () =
8
2
8~:::3'
Y
Note that
C: =~I) *,1
lu* (hWS2n) (x) /\ u* (hWS2n) (y) /\
< clA2n (du) (x)IIA2n (du) (y)1 Ix _ y14n-3 4n-l
It follows from the fact A2n (du) E L 2'n, the Hardy-Littlewood-Sobolev inequality, and 2n() 4n - 1
+
2n(} 4n - 1
= 1 + _4n_-_l_----'--(4_n_-_3-,)_(} 4n - 1
that
IA2n (du) (x)lo IA2n (du) (y)lo (4n-3)0 Ix-y 1 Hence
u* (hWS2n) (x) /\ u* (hWS2n) (y) /\ CLAIM
ELI
(jR4n-l
C: =~I)
10.5.
d [u* (hWS2n) (x) /\ u* (hWS2n) (y) /\
X
jR4n-l).
* , E LO (jR4n-l
C: =~I) C: =~I)
Because ..!l.:::JL Ix-yl Lemma 9.2 that
E
W l ,4n-2 (jR4n-l loe
1R4n - l ) .
* ,]
= u* (hWS2n) (x) /\ u* (hWS2n) (y) /\ PROOF.
X
X jR4n-l)
'
* (d,).
it follows from
On the other hand, it follows from Claim 10.2 that d [u* (hWS2n)] = O.
By smoothing we may find a sequence of smooth 2n-forms on Q:i, such that 4n-l ( Q:i - t U* (hWS2n ) in L 2'n jR4n - 1)
jR4n-l,
namely
EXISTENCE OF FADDEEV KNOTS
189
and dai = O. Similarly we may find a sequence of smooth 2n-forms on namely f3i such that
f3i and df3i
=
/\ 8n-2 L8n-3
* (12WS2n ) in L 2;;4n-1 ( 4n 1) ffi. -
O. It follows from Hardy-Littlewood-Sobolev inequality that
ai (x) /\ f3i (y) /\
in
-t U
(
(ffi.4n -
(I: =~I) *
x-y Ix _
yl
as i
-t
as i
-t
00.
* (d'Y)
/\ C:=~I)* d [adX)/\f3i(Y)/\
-t
u* (f1 WS2n ) (x) /\ u* (12wS2n) (y)
'Y
C: =~I)
L1 (ffi. 4n - 1 X ffi.4n-1)
'Y
*
)
1 X ffi.4n-1)
ai (x) /\ f3i (y) /\
in
ffi. 4n - 1 ,
Similarly -t
u* (!IWS2n) (x) /\ u* (12wS2n) (y)
(d'Y)
00.
Taking limit in the equality
C:=~I)* 'Y] =adx)/\f3i(Y)/\ C:=~I)* (d'Y) , D
we prove the claim.
< ~~=~ < !~=~
It follows from Claim 10.5, Lemma 9.9, and the fact 1 that
r
}rrt4n-1 xrrt4n-1
)*
u* (f1WS2n) (x) /\ u* (12wS2n) (y) /\ (Ix - YI x- Y
(d'Y) = O.
Part (a) follows. Next we check part (b). If 1S2n 12 (z") dS (z") = 0, then we may find a smooth (2n - I)-form 'Y on s2n such that d'Y = 12WS2n. We have
fs2n xS 2n XS 4n - 2 !I = =
r
}rrt4n-1 xrrt4n-1
(z') 12 (z") du (z) dS (z') dS (z") dS (z",) u*(!IwS2n)(x)/\u*(12wS2n)(Y)/\ (IX-YI)* WS4n-2 X - Y
-ls4n- 2 1r4n}rrt
u* (12wS2n) /\ T1. 1
Here
T1 = T = d* (f * u* (f1WS2n)). We have used the calculation in the proof of Claim 10.3 in the last step. 4n-1 By Claim 10.2, du* (f1WS2n) = O. This together with u* (flWS2n) E L 2;;implies
F. HANG, F. LIN, AND Y. YANG
190
Because u E
w 1,4n-2 (JR4n- 1), it follows from Lemma 9.2 that u* (hWS2n)
= u* (d-y) = du*'Y.
4n-l
Using u*'Y E L2, Tl E L2n-l , du*'Y = u* (hWS2n) E L2, dTl 4n-l 2 L 2n n L , it follows from Lemma 9.1 that
= u* (fIWS2n)
E
= du*'Y 1\ Tl - u*'Y 1\ dTl = du*'Y 1\ Tl - u*'Y 1\ u* (fIWS2n) = du*'Y 1\ Tl
d (u*'Y 1\ Tl)
= u* (hWS2n) 1\ Tl.
8n-2 Note that u*'VI 1\ Tl E L8n-3 and 1 we get
r
Jffi. 4n - 1
< 8n-2 4n-l. Applying Lemma 9.9, 8n-3 < 4n-2
u* (hWS2n) 1\ Tl = O.
Part (b) follows. Part (c) can be proved exactly in the same way as part (b). This finishes the proof of Claim 10.4 and hence Theorem 10.1. It is worth pointing out that there is freedom in the choice of T in Theorem 10.1. More precisely, we have PROPOSITION 10.6. Assume u E Wl~: (JR 4n -
r
J~4n-l
1,
s2n) such that
{lduI 4n - 2 + IU*WS2n 12}
0 such that IX1-x21 rl +r2, UI, U2 E X such that Ul (x) = for Ix - xII ~ rl, U2 (x) = for Ix - x21 ~ r2. Let
e
UI(X), XEBr1(XI), U (x) = { U2 (x), x E Br2 (X2) , e, otherwise. Then U E X and Q(u) = Q(ud +Q(U2).
>
e
196
F. HANG, F. LIN, AND Y. YANG
Hence
r
Q(u)=~ 2
IS nl 1~4n-1
= Q (Ul)
(UiWS2n +U2WS2n) 1\(71 +72)
+ Q (U2) + ~ 2
+ - -212 IS nl
r
IS nl 1~4n-1
l
~4n-l
UiWS2n 1\ 72
U2*W S 2n 1\ 71·
Fix a 8 > 0 such that rl + r2 + 28 < IXI - x21· Then d72 = 0 on BTl +0 (xt). It follows that 72 = d l2 for some 12 E W l ,2 (BTI+O (Xl)). Note that on BTl +0 (Xl),
Hence
r
1~4n-l
r
UiWS2n 1\72 =
UiWS2n 1\72 =
lBqH(xI)
=
r
1~4n-l
r
d(uiWS2n 1\12)
lBqH(xI)
d (UiWS2n 1\ 12)
=0
by Lemma 9.9. LEMMA 11.2. We simply deal with the case n::/:: 1,2,4. The case when n = 1,2,4 may be treated by similar methods. It follows from the previous facts that E-N = EN. Hence we may assume N > O. By [Hu, corollary 3.6 on p214] we may find avo E Coo (s4n-1, s2n) such that Q (vo) = 2 and Vol~n-l = n, the north pole of s2n. Let Uo (x) = Vo (1I"~1 (x)). + Here 11"n is the stereographic projection with respect to the north pole of s4n-l. For any even N, we may find a unique mEN such that PROOF OF
m 2 :S
N
2"
. du oo in £4n-2 (IR 4n - 1 ), and Ui WS2n --->. U~WS2n in £2 (IR 4n - 1 ).
204
F. HANG, F. LIN, AND Y. YANG
Indeed we may find a U oo E WI~4n-2 (lR4n - 1, s2n) such that, after passing to a subsequence, we have Ui ---7 U oo a.e. and dUi ......>. du oo in L 4n - 2 (lR4n - 1 ). Next we claim for every 1 :S k :S 2n, >. E A (2n + 1, k), dUi,Al 1\ .. . 1\ dUi,Ak
---7
dU OO ,Al 1\ ... 1\ dUOO,Ak'
in sense of distribution as i ---7 00. Here Ui,j is the jth component of the vector Ui. The claim is true for k = 1. Assume it is true for k - 1. Then for >. E A (2n + 1, k), since k - 1 :S 2n - 1 < 4n - 2, we see
IIdui,A2 1\ ... 1\ dUiAk II L Ii=T 4n-2 :S c(n, A) . , (lR4n-l) Combining with the induction hypothesis, we get du oo .A2 1\ ... 1\ 4n-2 L k-l (lR4n-l) and dUi,A2 1\ ... 1\ dUi,Ak
......>.
dU OO ,A2 1\ ... 1\ dUOO,Ak
in L
dUOO,Ak E
(lR4n- 1 )
4n-2 k-l
.
Hence Ui,Al dUi,A2 1\ ... 1\ dUi,Ak
in L
U OO ,Al dU OO ,A2 1\ ... 1\ dUOO,Ak
......>.
4n-2 k-l
(lR4n- I ).
It follows from Lemma 9.2 that dUi,Al 1\ ... 1\ dUi,Ak
= d (Ui,Al dUi,A2 1\ ... 1\ dUi,Ak) d
---7
(UOO,Al dU OO ,A2 1\ ... 1\ dUOO,Ak)
= dUOO,Al 1\ ... 1\ dUOO,Ak
in sense of distribution. The claim follows. Using the fact
IIA2n (du) II L2 (lR4n- 1) :S lIu*wS2n 1IL2(lR4n-1) :S .../A, we see that, for>. E A (2n + 1, 2n), dU OO ,Al 1\ .. ·l\du OO ,A2n E L2 (lR4n-I) dUi,Al 1\ ... 1\ dUi,A2n
This together with the fact ui*WS2n
......>.
. L2 U* oo WS2n In
dU OO ,Al 1\ ... 1\ dU oo .A2n
......>.
Ui
---7
U oo
in
L2
a.e. implies U~WS2n E
(lIll4n-l) ~ as ~. ---7
00.
If we let
then
L
2(4n-l)
(lR4n-l)
T'z ......>. T. in 4n-3 o o ,
DTi
......>.
DT00
Ti""">' Too
Hence for all r
> O.
in
L2
(lR4n-l) ,
in W I ,2 (Br) for every r
> O.
and
(lR4n-l) . L2
(lR4n-l) and
EXISTENCE OF FADDEEV KNOTS
205
PROOF OF THEOREM 13.1. Since N i= 0, it follows from Lemma 11.1 that 4n-l EN ~ c (n) INI4n > O. We may assume that i is large enough such that E (Ui) ~ 2EN. Let E > 0 be a tiny number to be fixed later. It follows from Lemma 12.1 that we may find some R = R(n,E,EN) > 0, Yi E QR/4' and integers ""i,{ for ~ E 2R'l}n-l, such that '~ " 1 ~ {E2RZ4n-l IS I
J[f
ui*W S 2n ATi -
""i,{
~ E.
QR({)+Yi
Here Ti = d* (r * uiWs2n). By translation we may assume Yi from the calculation in the proof of Lemma 11.1 that
= O.
It follows
Hence
~
4n c (n) Ef,F-l .
Hence
#
i= O}
{~E 2RZ4n - 1 1 ""i,{
4n
~ c(n) Ef.tn-l.
After passing to a subsequence we may assume
# {~E 2RZ4n - 1
i= O} = l. {~E 2RZ4n - 1 : ""i,{ i= O} and 1
""i,{
~il' .. . ,~il. After For each i, we may order passing to a subsequence we may assume for all 1 ~ j, k ~ l, limi---+oo I~ij ~ikl = 00 or limi---+oo (~ij - ~ik) = (jk E 2RZ4n - 1 exists. Passing to another subsequence we may assume for all 1 ~ j, k ~ l, either limi---+oo I~ij - ~ikl = 00 or ~ij - ~ik = (jk for all i. We may also assume that ""i,Ej = ""j for 1 ~ j ~ l and all i's. Let I = {I, ... , l}. We say that j, k E I are equivalent if ~ij - ~ik = (jk. This defines an equivalence relation on I. Let h,··· ,Im be the equivalent classes. For each 1 ~ a ~ m, we fix a ka E Ia. Let
Na =
L jE1a
""j
=
L ""i,{j jE1a
for all i. Then m
Nl
+ ... + N m
=
L ""i,{j = L j=l
{E2RZ4n-l
""i,E = Q (Ui) = N.
206
F. HANG, F. LIN, AND Y. YANG
Let
Yia = ~ika
as i
--t 00.
Let
E 2Rz4n-l. Then for 1 :S a, b :S m, a
Via (X)
= Ui (X -
Yia), Tia
i= b,
= d* (r * viaWs2n).
Then
After passing to a subsequence if necessary, by the discussion following the statement of the theorem, we may find Va E X such that as i --t 00, Via
--t
Va
a.e., dVia
viaWS2n -"
-"
V~WS2n
· L4n-2(TllAn-l) dVa In m.. ,
in L2(JR4n - 1),
and Tia -" Ta
Here
Ta =
in W 1 ,2 (Br) for every r > O.
d* (r * V~WS2n). In particular,
for all r > O. Note that it is clear that limi-+oo Moreover
Kil.+Yia
if ~ = (jk a for j E otherwise.
= Kt;,a
always exists.
la,
Hence
IQ (va) -
Nal
= Q (va)
-
I:
Kj
jE1a
This implies Q (va) = then
Na
if we choose c < 1. Moreover, if we choose c :S
!,
207
EXISTENCE OF FADDEEV KNOTS
i
Using the fact that Kja =1= 0, we see that QR IV~WS2n t\ Tal dx 2: c (n) > O. Hence the calculation in Lemma 11.1 implies E (va) 2: c (n) > O. Finally, fix r> O. Then for i large enough, we have
E(Ui) 2:
f1 f1 a=1
=
Br(Yi,a)
a=1
Letting i
---t
Letting r
---t
l
(l du n- 2 + IU;WS2nI2) dx (ld Vi,aI 4n - 2 + Ivi,a w s 2n
12) dx.
Br
00,
we see that
00,
we see that m
m
EN 2: LE(va) 2: LENa' a=1 a=1 Using E (va) 2: c (n) > 0, we see that m ::; c (n) EN. To finish the argument, we observe that it follows from Corollary 13.3 below that 2::=1 ENa 2: EN. Hence EN = 2::=1 ENa and ENa = E (va) for all a's. 0 LEMMA 13.2. For every U E X, there exists a sequence Ui sequence of positive numbers bi such that
Ui
---t
U a.e.,
dUi
---t
du in L 4n - 2 (jR4n-l) ,U;WS2n
---t
E
X and a
U*WS2n in L2 (jR4n-l)
and Ui (x', X4n-l) == const
for X4n-l < -bi·
Here x = (x', X4n-l) with x' representing the first 4n - 2 coordinates. To prove the lemma, we first introduce some coordinates on jR4n-l. Note that we may use the stereographic projection with respect to the north pole non s4n-2 to get
s4n-2\ {n}
---t
jR4n-2 : x
I---+~,
x'
~=---
1- X4n-l
In this way, we get a coordinate system on S4n-2\ {n}. For x E jR4n-l\ {(O, a) : a 2: O}, we may take r = Ixl and ~ as the stereographic projection of I~I with respect to n. In this way, we get a coordinate system (r, ~). The Euclidean metric is written as
208
F. HANG, F. LIN, AND Y. YANG
We will use freely the coordinates x and (r, ~). For a > 0, we denote
0 < r < 00, I~I < a} C jR4n-1
Va = {(r,~):
as the corresponding cone with origin as the vertex. Note that VI =
{x E jR4n-1 : X4n-1 < o}.
To continue we define a function
We also write for 0 < r
0, by vertical translation we may assume
r
(lduI 4n- 2 + IU*WS2n 12) dx
< c.
lV32
Then for the above constructed v, we have
r
lJR4n-l
(Idv - dul 4n - 2 + IV*WS2n - U*WS2n 12) dx
~ c (n)
r
(lduI 4n - 2 + IU*WS2n 12) dx
~ c (n) c.
lV32
Lemma 13.2 follows. COROLLARY
and
13.3. For N I , N2
E
Z, if XN1 , XN2
=f. 0,
then
XNl +N2
=f. 0
EXISTENCE OF FADDEEV KNOTS
211
Indeed, for any c > 0 small, it follows from Lemma 13.2 that we can find Ul E XN 1 , U2 E XN2 such that E (Ul) < ENI + c, E (U2) < EN2 + c, Ul (x', X4n-l) = -n for X4n-l < 0 and U2 (x', X4n-l) = -n for X4n-l > o. Here n is the north pole of s2n. Define U
(x) = {
Ul
U2
(x), when (x), when
X4n-l X4n-l
> 0, < o.
Then clearly U E X and E (u) = E (ud + E (U2) < ENI will show that Q (u) = Nl + N 2 . It follows that EN1+N2 S Letting c -4 0+, we get the corollary. Indeed, denote i : lR4n - 2
-4
lR4n - 1 : x'
f---t
+ EN2 + 2E. We ENI + EN2 + 2c.
(x', 0) 2(4n-l)
as the natural put in map. Since UiWS2n E L 4n+l and UiWS2n lR~n-l, it follows from the Hodge theory that we may find 71 E L2
= 0 on (lRtn-l)
2(4n-l)
with D71 E L 4n+l (lRtn-l) and i*71 = O. Let 71 = 0 on lR~n-l. Then the same argument as in the proof of Claim 12.3 shows that d71 = UiWS2n on lR4n-1. Similarly we may find 72 E L2 (lR4n-l) such that d72 = u2WS2n and 721IR4n-1 = O. Note that +
It follows from Proposition 10.6 that
Q (u) =
r
~ U*WS2n IS 2nl JIR4n-l
/\
(71
+ 72)
13.1. Some discussion. Here we describe some consequences of Theorem 13.1. For n = 1,2,4, we know for all NEZ, XN =1= 0 and C
(n)-
1
4n-l
INI4n
SEN S
C
(n)
4n-l
INI4n .
In particular, one can find No > 0 with ENo
= inf {EN I N
E N}
and ENo is attainable. Let §
Then for every N Nl + ···+Nm and
=1=
= {N
E Z : EN is attainable} .
0, there exist nonzero N 1 , ... , N m E § with N
212
F. HANG, F. LIN, AND Y. YANG
4n-l
It follows from this and the fact EN :::; c (n) INI~ that § must be infinite (otherwise EN would grow at least linearly). The situation for n =1= 1,2,4 is more subtle. In this case, we do not know whether XN =1= 0 when N is an odd integer (see Conjecture 1). If Conjecture 1 is verified, then similar conclusions as above are true with all N's being even. On the other hand, if XN =1= 0 for some odd integer N, then it follows from Lemma 13.2 and the proof of Lemma 11.2 that for all integers N, XN =1= 0 and (n)-
C
Again the set
I
4n-l
INI~:::; EN :::;
c (n)
4n-l
INI~
.
{N E Z 1EN is attainable} must be infinite.
§ =
14. Skyrme model revisited In this section, we will prove a similar subadditivity property for the Skyrme energy spectrum (Corollary 14.2). As a consequence, the substantial inequality derived in [El, E2, LYl] is improved to an equality (Theorem 14.3). Recall that for a map u E Wl~ (lR.3 , 8 3 ), the Skyrme energy is given by E (u)
=
Denote
x
= {
u
E
L3
(l du l2 + Idu 1\ dul2) dx.
Wl~; (lR.3 , 8 3 ) 1 E (u) < 00 }
.
The main aim of this section is to prove the following. 14.1. For every u E X, there exists a sequence ui sequence of positive numbers bi such that LEMMA
Ui
---t U
a.e.,
dUi
---t
du in L2
(lR.3 )
,
dUi 1\ dUi
---t
For NEZ, we let XN
= {U E X I deg(u) = 1;31
L3
U*WS3
and (14.2) A simple corollary of the lemma is the following COROLLARY
14.2. For N I , N2
E
Z,
EN1 +N2 :::; EN!
+ EN2'
X and a
du 1\ du in L2
and
(14.1)
E
=
N}
(lR.3 )
EXISTENCE OF FADDEEV KNOTS
213
14.3. Assume N is an nonzero integer and Ui that E (Ui) --t EN. Then there exists an integer m with 1 ~ m nonzero integers N I , ... , N m and Yil, . .. , Yim E ]R3 such that THEOREM
• N
XN such ~ c . EN, m E
= Nl + ... + N m ·
• !Yij - Yik! --t 00
--t 00 for 1 ~ j, k ~ m, = Ui (x - Yij) for 1 ~ j ~
as i
• If we set Vij (x) Vj E X such that
Vij --t Vj
dv·· lJ
as i
--t 00
--->.
j =I- k. m, then there exists a
a. e.
dv·J in L2 (]R3) '
and
•
m
EN
= LENj" j=l
In particular, if EN < EN' + EN" for N = N' EN defined in (14.2) is attainable.
+ Nil,
N', Nil =I- 0, then
This theorem follows from similar arguments for Theorem 13.1 (see [EI, E2, LYI]). Unlike the integral formula for the Hopf-Whitehead invariant, the formula for the topological degree given in (14.1) is purely local and it makes the discussion relatively simpler. Now we turn to the proof of Lemma 14.1. First we introduce some coordinates on ]R3. Note that we may use the stereographic projection with respect to (0,0,1) on 8 2 to get
2
8 \ {(O, 0, I)}
2x
--t]R :
f--t
~,
c_
'" -
(Xl
X2)
--, - -
1-
X3
1-
•
X3
In this way, we get a coordinate system on 8 2\ {(O, 0, I)}. For x E ]R3\ { (0, 0, a) : a ~ O}, we may use coordinate r = !x I and ~ as the stereographic projection of I~I with respect to (0,0, -1). In this way, we get a coordinate (r, 6, 6)· The Euclidean metric is written as 4r2
91R3 =dr0dr+
(1 + !~!2)
2(d60 d 6+ d60 d6)·
We will use freely the coordinates x and (r, ~). For a > 0, we denote
Va = {(r,~) :
°< r
R.
Then
f
(ldw l2 + Idw!\ dwl2) dx ::; c {Xl Ig (rW dr + c (,° 19' (r)1 2 r 2dr
iR3\BR
iR
::; c
f
iV1
iR
(ldul2 + Idu!\ dul2) dx.
Finally, we let {
v (x) =
V2
(x) ,
7l'~1 (w (x)),
J xi + x~ -
if X3 ~ if X3 ::; Jxi
+ x~ -
2R, 2R.
Then, it follows from the construction, that v EX,
f
(ldv l2 + Idv !\ dvl2) dx ::; c
f
(ldul2
+ Idu!\ dul2) dx,
i V32
iV32
and VIR3\V32
= u,
V (XI, X2, X3)
= (0,0,0, -1) for
X3 ::;
-4R.
For every e > 0, after a vertical translation, we may assume
f (ldul2 + Idu!\ dul2) dx < e. i V32 Then for the above constructed v, we have
k3
(Idv - dul 2 + Idv!\ dv - du!\ dul2) dx
::; c
f (l du l2 + Idu!\ dul2) dx ::; ceo i V32
Lemma 14.1 follows.
15. Conclusions
In this paper, we have carried out a systematic study of the Faddeev type knot energies in the most general Hopf dimensions governing maps from jR4n-l into s2n. These maps are topologically stratified by the HopfWhitehead invariant, Q, which may be represented by a Chern-Simons type integral invariant. Two different types of energies are considered. The first type, referred to as the Nicole-Faddeev-Skyrme (NFS) model, contains a potential energy term and a conformally invariant kinetic energy term and
EXISTENCE OF FADDEEV KNOTS
217
allows a direct resolution in the spirit of the concentration-compactness principle due to the validity of an energy-cutting lemma. The second type, referred to as the Faddeev model, does not contain a potential energy term or a conformally invariant kinetic term and challenges a direct approach in a similar fashion. Nevertheless, we are able to show that both models follow the same energetic and topological decomposition relations in a global minimization process which closely resemble the energy conservation and charge conservation relations observed in a nuclear fission process. Furthermore, both types of models obey the same fractionally-powered universal growth laws relating knot energy to knot topology. These results lead us to the conclusion that, for either the NFS model or the Faddeev model, there is an infinite set of integers, §, such that for each N E §, there exists a global energy minimizer among the maps in the topological class given by Q = N. Besides, in the compact setting where the domain space is s4n-l, both models allow the existence of a global energy minimizer among the topological class Q = N at any realizable Hopf-Whitehead number N. Acknowledgements. F. Hang was supported in part by NSF under grant DMS-0647010 and a Sloan Research Fellowship. F. Lin was supported in part by NSF under grant DMS-0700517. Y. Yang was supported in part by NSF under grant DMS-0406446 and an Othmer senior faculty fellowship at Polytechnic University. References [AS] C Adam, J Sanchez-Guillen, Symmetries of generalized soliton models and submodels on target space 8 2 , J. High Energy Phys. 0501 (2005) 004. [ASVW] C. Adam, J. Sanchez-Guillen, RA. Vazquez, A. Wereszczynski, Investigation of the Nicole model, J. Math. Phys. 47 (2006) 052302. [AA1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. 72 (1960) 20-104. [AA2] J. F. Adams and M. F. Atiyah, K-theory and the Hopf invariant, Quarterly J. Math. 17 (1966) 31-38. [AFZ] H. Aratyn, L. A. Ferreira, and A. H. Zimerman, Exact static soliton solutions of (3+ 1)-dimensional integrable theory with nonzero Hopf numbers, Phys. Rev. Lett. 83 (1999) 1723-1726. [BS1] R A. Battye and P. M. Sutcliffe, Knots as stable solutions in a three-dimensional classical field theory, Phys. Rev. Lett. 81 (1998) 4798-4801. [BS2] R A. Battye and P. M. Sutcliffe, Solitons, links and knots, Proc. Roy. Soc. A 455 (1999) 4305-4331. [BT] R Bott and L. W. Th, Differential Forms in Algebraic Topology, Springer, New York, 1982. [BFHW] S. Bryson, M. H. Freedman, Z. X. He, and Z. H. Wang, Mobius invariance of knot energy, Bull. Amer. Math. Soc. (N.S.) 28 (1993) 99-103. [B] G. Buck, Four-thirds power law for knots and links, Nature 392 (1998) 238-239. [CKS1] J. Cantarella, R Kusner, and J. Sullivan, Tight knots deviate from linear relation, Nature 392 (1998) 237.
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F. HANG, F. LIN, AND Y. YANG
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EXISTENCE OF FADDEEV KNOTS
COURANT INSTITUTE, NEW YORK UNIVERSITY, NY
251
MERCER STREET, NEW YORK,
251
MERCER STREET, NEW YORK,
10012 E-mail address: fengbotDcims.nyu.edu COURANT INSTITUTE, NEW YORK UNIVERSITY,
NY
221
10012 E-mail address: linftDcims .nyu. edu DEPARTMENT OF MATHEMATICS, POLYTECHNIC UNIVERSITY, BROOKLYN, NY
11201
(ADDRESS AFTER SEPTEMBER 1, 2008: DEPARTMENT OF MATHEMATICS, YESHIVA UNIVERSITY, NEW YORK, NY 10033)
E-mail address: yyangbath.poly.edu
Surveys in Differential Geometry XIII
Milnor K2 and field homomorphisms Fedor Bogomolov and Yuri Tschinkel ABSTRACT. We prove that the function field of an algebraic variety of dimension ~2 over an algebraically closed field is completely determined by its first and second Milnor K-groups.
CONTENTS
1. Introduction 2. Background 3. Functional equations 4. Reconstruction 5. Milnor K-groups References
223 225 228 239
241 243
1. Introduction In this paper we study the problem of reconstruction of field homomorphisms from group-theoretic data. A prototypical example is the reconstruction of function fields of algebraic varieties from their absolute Galois group, a central problem in "anabelian geometry" (see [9], [6], [5], [7]). Within this theory, an important question is the "section conjecture", i.e., the problem of detecting homomorphisms of fields on the level of homomorphisms of their Galois groups. In the language of algebraic geometry, one is interested in obstructions to the existence of points of algebraic varieties over higher-dimensional function fields, or equivalently, rational sections of fibrations. Here we study group theoretic objects which are dual, in some sense, to small pieces of the Galois group, obtained from the abelianization of the absolute Galois group and its canonical central extension. This connection will be explained in Section 2. Date: February 27, 2009. ©2009 International Press
223
F. BOGOMOLOV AND
224
Y.
TSCHINKEL
We now formulate the main results. In this paper, we work in characteristic zero. An element of an abelian group is called primitive, if it cannot be written as a nontrivial multiple in this group. DEFINITION 1. Let k be an infinite field. A field K will be called geometric over k if
(1) k
c K;
(2) for each f E K* \ k*, the set {f + ti;}KEk has at most finitely many elements whose image in K* / k* is non primitive. If X is an algebraic variety over an algebraically closed field k of characteristic zero then its function field K = k(X) is geometric over k. There exist other examples, e.g., some infinite algebraic extensions of k(X) are also geometric over k.
THEOREM 2. Let K, resp. L, be a geometric field of transcendence degree ~
2 over an algebraically closed field k, resp. 1, of characteristic zero. Assume
that there exists an injective homomorphism of abelian groups 'l/Jl : K* /k* -+ L* /1* such that (1) the image of'l/Jl contains one primitive element in L* /1* and two elements whose lifts to L * are algebraically independent over 1; (2) for each f E K* \ k* there exists agE L such that 'l/Jl (k(f)*/k*nK*/k*)
~l(g)*/l*nL*/l*.
Then there exists a field embedding 'l/J: K-+L which induces either 'l/Jl or 'l/J11 .
REMARK 3. An analogous statement holds in positive characteristic. The final steps of the proof in Section 4 are more technical due to the presence of pn-powers of "projective lines". Let K be a field. Denote by KfI (K) the i-th Milnor K-group of K. Recall that Kf1(K) = K* and that there is a canonical surjective homomorphism UK :
Kf1 (K) ® Kf1 (K) -+ K~ (K)
whose kernel is generated by x ® (1 - x), for x E K* \ 1 (see [4] for more background on K-theory). Put KfI (K) := KfI (K)/infinitely divisible elements,
i = 1,2.
MILNOR K2 AND FIELD HOMOMORPHISMS
225
The homomorphism CfK is compatible with reduction modulo infinitely divisible elements. As an application of Theorem 2 we prove the following result. THEOREM 4. Let K and L be function fields of algebraic varieties of dimension 2: 2 over an algebraically closed field k, resp. l. Let
(1.1) be an injective homomorphism of abelian groups such that the following diagram of abelian group homomorphisms is commutative
Kr(K) ®Kr(K)
'l/Jl®'l/Jl
.. Kr(L) ®Kr(L)
!UL
UK! K~(K)
'l/J2
.. K~(L).
Assume further that 'l/Jl (K* / k*) is not contained in E* / k* for any i-dimensional subfield EeL. Then there exist a homomorphism of fields 'lj;: K-tL,
and an r E Q such that the induced map on K* /k* coincides with the r-th power of'l/Jl. In particular, the assumptions are satisfied when 'lj;1 is an isomorphism of abelian groups. In this case, Theorem 4 states that a function field of transcendence degree 2: 2 over an algebraically closed ground field of characteristic zero is determined by its first and second Milnor K-groups. Acknowledgments: The first author was partially supported by NSF grant DMS-0701578. He would like to thank the Clay Mathematics Institute for financial support and Centro Ennio De Giorgi in Pisa for hospitality during the completion of the manuscript. The second author was partially supported by NSF grant DMS-0602333. We are grateful to B. Hassett, M. Rovinsky and Yu. Zarhin for their interest and useful suggestions.
2. Background The problem considered in this paper has the appearance of an abstract algebraic question. However, it is intrinsically related to our program to develop a skew-symmetric version of the theory of fields, and especially, function fields of algebraic varieties. Let K be a field and OK its absolute Galois group, i.e., the Galois group of a maximal separable extension of K. It is a compact profinite group. We
226
F. BOGOMOLOV AND Y. TSCHINKEL
are interested in the quotient
and its maximal topological pro-i-completion
91U, , i
~
char(K).
The group 9Ke is a central pro-i-extension of the pro-i-completion of the abelianization '9 K of 9K . We now assume that K is the function field of an algebraic variety over an algebraically closed ground field k. In this case, 9Ke is a torsion-free topological pro-i-group which is dual to the torsion-free' abelian group K* /k*, i.e., there is a canonical identification
9K,e = Hom(K* /k*, Ze(l)), via Kummer theory. The group 9Kf admits a simple description in terms of one-dimensional subfields of K, i.e.', subfields of transcendence degree lover k. For each such subfield E C K, which is normally closed in K, we have a surjective homomorphism 9Kf. -t g e' where the image is a free central pro-i-extension of the group 9~ f. ' Our main goal is to establish a functorial correspondence between function fields of algebraic varieties K and L, over algebraically closed ground fields k and l, respectively, and corresponding topological groups 9K, resp. 9Kf· We are aiming at a (conjectural) equivalence between homomorphisms of function fields
e
~:K-tL
and homomorphisms of topological groups
It is clear that ~ induces (but not uniquely) a homomorphism Wi as above. The problem is to find conditions on Wi such that it corresponds to some ~. In particular, Wi would give rise to homomorphisms of the full Galois groups 9K -t 9L. REMARK 5. By a theorem of Stallings [8J, a group homomorphism that induces an isomorphism on HI(-,Z) and an epimorphism on H2(-,Z) induces an isomorphism on the lower central series. Thus we expect that 9 K,f is in some sense the maximal pro-i-group with given HI and H2.
MILNOR K2 AND FIELD HOMOMORPHISMS
227
Consider the diagram
OIu~OLe , ,
The group OK e can be identified with a closed subgroup in the direct product of free central pro-t'-extensions
where the product runs over all normally closed one-dimensional subfields E of K. The homomorphisms OK e -+ Ok e are induced from certain homomorphisms of abelian quotients O~ e -+ ofe, namely those which commute with surjective maps of 01O,ni>O
Ciy)ni
=
II (Plq2 j>O,mj>O
djP2QI)m j ,
(x II i>O,niO,mjO ni i,j;::: 1. By (AI),
= 0 or Lj>o mj = 0 then ni = mj = 0 for all X no -_
pmop-mo 1 2 .
By assumption (ii), R is nonconstant. Hence no power of x, contradicting (i). We can now assume (3.5) i>O
i>O
=1=
O. It follows that p is a
232
F. BOGOMOLOV AND Y. TSCHINKEL
It follows that
( mo, -mO
-L m)) "# (0,0)
and
»0
(mo-s,s-mo- Lm;)"# (0,0). \
»0
On the other hand, by (i), combined with (AI) and (A2), one of the terms in each pair is zero. We have the following cases: (1) mo "# 0, mo = - I:j>o mj, mo = sand xno = pT'0, qf = yT-n o- Li>O ni;
(2) mo=O, s= I:j>o mj and xno =p~ Lj>omj =P2 s , q2s=yT-no-Li>0 ni. We turn to (A3), with J ~ 1 and ni, mj replaced by Inil, Imjl. From (AI) we know that Pl(X) = x a or P2(X) = xa, for some a E N. Similarly, from (A2) we have ql(y) = yb or q2(y) = yb, for some bEN. All irreducible components of the divisor of
are of the form x = CiY, i.e., these divisors are homogeneous with respect to
(x,y) t-+ (AX,AY), It follows that
Ii
A E k*.
is homogeneous, of some degree
rj
EN. If
then fj has a nonzero constant term, contradiction. Lemma 10 implies that either (3.6) or (3.7)
It follows that all rj are equal, for j ~ l. The cases are symmetric, and we first consider (3.6). Note that equation (3.6) is incompatible with Case mo = 0 and equation (3.7) with the Case m "# O. By Lemma 10, P2(X) = P2,jXTj + P2(0) Q2(y) = Q2,jyTj + Q2(0), with (3.8)
P2(0), Q2(0)
"# 0,
and
Q2,j - djP2,j = O.
By assumptions (i), Q2,j and P2,j are nonzero. The coefficients dj were distinct, thus there can be at most one one such equation, i.e., J = 1.
233
MILNOR K2 AND FIELD HOMOMORPHISMS
To summarize, we have the following cases: (1) mo -I- 0, mo = -m1 = sand
with coefficients satisfying q2,1 - d1P2,1 = 0,
=
II(x - ciyt i
(q1(0)X T1 - d1P2(0) yTl )-s.
i~1
= ml =
It follows that 1= rl and that ni
c. = ri d1/ T1
Z '>Tl
-s, for i 2 1. We have
'
with d = -dI/P2(0)/ql(0). This yields r = no = rIB. We can rewrite equation (3.4) as yTl
(~)Tl Y
IT (~ _
P.
i=1
q
Ci) -1 =
Y
which is the same as (3.3) with s
8 qs
= 0, m1 =
P
B,
-1
q,
q
= 1 and r = rl. We have
= (q-I_dIP-I)-S -
(2) mo
(p. _dl )
(
xTlyTl ) ql(0)X T1 - d1P2(0)yTl
S
and
(x) = PI,lXT1 + PI (0) x TI '
II (x - ciyt
i
=
(PI (O)yTl - d1q2(0)XT1 )S.
i~1
We obtain I=r1,ni=s, for i21,no= - rls=r, and Ci=(:l = d1q2(0)/PI (0). We can rewrite Equation (3.4) as
d1/ T1 , with d
We have
234
F. BOGOMOLOV AND Y. TSCHINKEL
o
This concludes the proof of Proposition 11.
13. Let Xl, X2 E K* be algebraically independent elements and let h E k(Xi), i = 1,2. Assume that Id2 E k(X1X2). Then there exists an a E Q such that li(Xi) = xf, in K* /k*. LEMMA
PROOF. Assume first that Ii E k(Xi) and write
li(Xi) = I1(Xi -
Cijt ij
•
j
By assumption,
i,j
r
However, the factors are coprime, unless Cij = 0, dr = 0, for all i, j, r. Now we consider the general case: Ii E k(Xi). We have a diagram of field extensions
The Galois group Gal( k(xl, x2)/k(X1, X2)) preserves k(X1X2). We have f := Gal( k(X1) k(X2)/k(X1, X2)) = f1 x f2,
with fi acting trivially on k(Xi). Put 13 := Id2 and consider the action of 1'1 := b1' 1) E f on
It follows that and
k(X1) 3 hh1(f2) = 1311'1(13) E k(X3)' Hence each side is in k. The action of 1'1 has finite orbit, so that 1'1 (h) = (nh and 1'1(f2) = (~h for some n-th roots of 1. Note that f acts on iI, h, and 13 through a finite quotient. It follows that for some mEN, we have lim E k(Xi), for i = 1,2,3, and we can apply the argument above. 0
MILNOR
K2
AND FIELD HOMOMORPHISMS
235
Let x, y E K* be algebraically independent over k. We want to determine the set of solutions of the equation
(3.9)
Ry
= Sq,
where
R E k(x/y), q E k(y), p E k(x), S E k(p/q). We assume that X,p, y, q are multiplicatively independent in K* /k* and that Sand Rare nonconstant. We will reduce the problem to the one solved in Proposition 11. LEMMA 14. There exists an n(p) EN such that pn(p) E k(x/y) k(y). PROOF. The function S E k(p/q) n k(x/y) k(y) is nonconstant. The Galois group r := Gal(k(x, y)/k(x/y) k(y)) acts trivially on q E k(y) and S. Thus k(p/q) = -=-k(-;-'Y-;-(p""7)-;-/q""7). Assume that "I E r acts nontrivially on p E k(x). It follows that
'Y(p)/p
E
k(p/q)
n k(x) =
k,
by assumption on these I-dimensional fields. Thus 'Y(p) = (p, where ( is a root of 1. Since r acts on p via a finite quotient and since each "I E r acts by multiplication by a root of 1, pn(p) E k(x/y) k(y), for some n(p) EN. 0 LEMMA 15. There exists an N
= N(p)
EN such that
pn(p) E k(x l / N ). PROOF. The intersection k(x) n k(x/y) k(y) is preserved by action of x r y. Its elements are fixed by any lift of
r = r x/y
a :
y
x/yo
H
to the Galois group r. All such lifts are obtained by conjugation in r x/y x r y. Hence (1, "I) acts as (O'b), 1). The group homomorphism
r x/y x r y -+ r x
:=
Gal(k(x)/k(x))
has abelian image since bl' 1) and (1,"12) commute and generate r. Every abelian extension of k(x) is described by the ramification divisor. It remains to observe that the only common irreducible divisors of k(y), k(x/y) and k(x) are x = 0 or x = 00. 0 LEMMA 16. There exists an n E N such that
Sn
E
k(x l / N , y)
and
qn
E
k(y).
F. BOGOMOLOV AND Y. TSCHINKEL
236
PROOF. Let r~
c rx
Gal(k(x)/k(xl/N)) be the subgroup of elements acting trivially on k(x 1/ N ). Let =
'"'( = h~, 1) E rx x r x / y ,
'"'(~ E r~.
Then
Ry
= Sq = '"'((Sh(q)
and
S/,",((S)
= '"'((q)/q.
We also have
ph(q) = qh(q) p/q with
S E k(p/q), ph(q), '"'((S) E k(ph(q)), qh(q) E k(y). By Lemma 13, if we had k(p/q) n k(p/'"'((q)) = k then S = p/q. However, equation Ry = p and Lemma 13 imply that R = x/y, contradicting the assumption that x and p are multiplicatively independent. Thus we have k(p/q) = k(ph(q)). The equality S/,",((S) = (qh(q))-l implies that both sides are constant. Hence there exists an n E N such that sn E k(x1/N,y), and qn E k(y). 0 LEMMA 17. There exists an n(R) such that Rn(R) E k( Vx/y). PROOF. We have that
Rnyn = snqn with qn E k(y) and sn E k(x1/N,y). Thus Rn E k(x/y) n k(x1/N)k(y). Applying a nontrivial element '"'( E Gal(k(xl/N,y)/k(xl/N,y)) we find that Rn /,",((R n) E k*, and is thus a root of 1. As in the proofs above, we find that there is a multiple n(R) of n such that Rn(R) E k( Vx/y). 0 We change the coordinates
x := xl/N,
jj:= yl/N.
LEMMA 18. There exist
P E k(x),q E k(jj) such that (3.10)
F := k(p/q) n k(x, jj) = k(pjq).
PROOF. Every sub field of a rational field is rational. In particular, F = k(s) for some s E k(x, jj). Since p E k(x), q E k(y) they are both in k(x, jj) so that p(x)/q(x) E F = k(s). By Lemma 6, F = k(pjq), as claimed. 0
MILNOR
K2
AND FIELD HOMOMORPHISMS
237
COROLLARY 19. There exists an mEN such that
8m
E
k(p/ii),
with
P E k(x) Moreover,
q=
q E k(jj).
and
qT, for some r E Q.
PROOF. We apply Lemma 13: since
P E k(x) c k(x) = k(p),
l/q
E
k(y)
= k(l/q)
and
p/ii E k(p/q), by (3.10),
k(p/ii) = k(8) = k(P/q), we have
o
for some a E Q.
We have shown that if R, 8 satisfy equation (3.9) then for all sufficiently divisible mEN we have (3.11) with
S := 8 m
R := Rm E k(x/f;)
E k(p/ii),
and q := qm E k(y)
c k(jj).
Choose a smallest possible m such that s := m/a E Z and put r = mN. Equation 3.11 transforms to s R- y-T = 8--q.
In the proof of Proposition 11 we have shown that sir and that either
_ (X)TiS Ti (xII
R=
-
_
Y
i=I
-I'" , '-'l
)-S
,
Y
with rls = r or
_ (X)-TiS Ti (x-_ -c·z)S , -_ II
R=
Y
i=I
Y
with -rls = r. We have obtained that every nonconstant element in the intersection (3.12)
k(xjy)* . y n k(p/q)* . q,
F. BOGOMOLOV AND Y. TSCHINKEL
238
is of the form or
(3.13)
with b = rI/N, N E N, and "', ",' E k*. The corresponding solutions, modulo k*, are
with respectively,
with
,
'/' "'x·
'" = "'y
By equation (3.9), we have (for s E Z)
It follows that bs = 1. 20. The pair (x, y) satisfies the following condition: if both xb, yb E K* then b E Z. ASSUMPTION
This assumption holds e.g., when either x, y or xy is primitive in K* /k*. 21. Assume that the pair (x, y) satisfies Assumption 20. Fix a solution (3.13) of Condition (3.12). Assume that the corresponding P""x,b,m is in K*, for infinitely many "'x, resp. "'~. Then b = ±1 and m = ±l. LEMMA
PROOF.
By the assumption on the pair (x, y) and K,
is primitive in K* / k*, for infinitely many "'x. It follows that m = ± 1. To deduce that b = ±1 it suffices to recall the definitions: on the one hand, b = rI/N E Z, with N E N, rl EN, and r = ±N. Thus, b = ±rI/r E Z. On the other hand, ±rls = r, with sEN. 0
MILNOR K2 AND FIELD HOMOMORPHISMS
239
After a further substitution 8 = -b, we obtain: THEOREM 22. Let x, y E K* be algebraically independent elements sat--* --* isfying Assumption 20. Let p E k(x) , q E k(y) be rational functions such that x, y,P, q are multiplicatively independent in K* jk*. Let 1 E k(xjy)* . y be such that there exist infinitely many p, q E K* j k* with
1 E k(xjy)* . y
n k(pjq)* . q.
Then, modulo k*,
(3.14) with
K
E k* and
8=
±1.
The corresponding p and q are given by
Plt x,l(X)
-
X+Kx,
Plt x,-l ( X)
-
( X-1
+ Kx )-1 ,
qlty,l(y) qlt x,-l(y)
-
with
4. Reconstruction In this section we prove Theorem 2. We start with an injective homomorphisms of abelian groups
'l/J1 : K* /k* -+ L* /l*. Assume that Z E K* is primitive in K* j k* and that its image under 'l/J1 is also primitive. Let x E K* be an element algebraically independent from z and put y = z/x. By Theorem 22, the intersection k(xjy)* . y n k(p/q)* . q
c K* jk*
with infinitely many corresponding pairs (p, q) elements I It ,8(x, y) given in (3.14). Note that
For 8 = 1, each I It ,l determines the infinite sets
as the corresponding solutions (p, q). The set
c
K* x K*, consists of
240
F. BOGOMOLOV AND Y. TSCHINKEL
forms a projective line. On the other hand, for 8 = -1, we get the set t(l,x) =
{I, _/+ } . x
K
K,Ek
Note that this set becomes a projective line in JP>k(K), after applying the automorphism K* jk* -+ K* jk*
f
t-t
f-l.
We can apply the same arguments to 'lfJl (x), 'l/Jl (y) = 'l/Jl (z) j 'l/Jl (x). Our assumption that 'l/Jl maps multiplicative groups of I-dimensional subfields of K into multiplicative groups of I-dimensional subfields of L and Theorem 22 imply that 'l/Jl maps the projective line ((1, x) c JP>k(K) to either the projective line ((I,'l/Jl(X)) C JP>1(L) or to the set t(I,'l/Jl(X)). Put
C:= {x E K* l'l/Jl(r(I,x))
= ((I,'l/Jl(X))}
R:= {x E K* l'l/Jl(r(I,x))
= t(I,'l/Jl(X))}.
Note that these definitions are intrinsic, i.e., they don't depend on the choice of z. By the assumption on K, both ((I,'l/Jl(X)) and t(I,'l/Jl(X)) contain infinitely many primitive elements in L * jl*, whose lifts to L * are algebraically independent from lifts of 'l/Jl(Z). We can use these primitive elements as a basis for our constructions to determine the type of the image of ((1, z') for every z' E k(z)* n K*. Thus CUR
= K*jk*,
CnR
=
1 E K*jk*.
LEMMA 23. Both sets C and R are subgroups of K* j k*. In particular, one of these is trivial and the other equal to K* j k* .
PROOF. Assume that x, yare algebraically independent and are both in C. We have Indeed, fix elements
p(x) = x + Kx E ((1, x)
and
q(y) = y + Ky E ((1, y)
so that x, y, p, q satisfy the assumptions of Theorem 22. Solutions of
R(xjy)y
= S(pjq)q
map to solutions of a similar equation in L. These are exactly
MILNOR K2 AND FIELD HOMOMORPHISMS
241
for some A E l*. This implies that
'l/JI(x/y - "") = 'l/Jl(X/Y) - A E L* /l*, i.e., x/y E C. Now we show that if x E C then every x' E k(x)* /k* n K* /k* is also in C. First of all, l/x E C. Next, elements in the ring k[x], modulo k*, can be written as products of linear terms x + ""i. Hence
Let
f
be integral over k[x] and let
r + ... + ao(x)
E
k[x]
be the minimal polynomial for f, where ao(x) (j. k. Replacing f by f + "", if necessary, we may assume that f is not a unit in the ring k[x]. Then f (j. n, since otherwise we would have ao (x) E n, contradiction. Finally, any element of k(x)* is contained in the integral closure of some k[l/g(x)], with g(x) E k[x]. The same argument applies to once we composed with 'l/Jll, to show that both C and are subgroups of K* / k*. An abelian group cannot be a union of two subgroups intersecting only in the identity. Thus either C or has to be trivial. 0
n,
n
n
The set JID(K) = K* /k* carries two compatible structures: of an abelian group and a projective space, with projective subspaces preserved by the multiplication. The projective structure on the multiplicative group JID(K) encodes the field structure: PROPOSITION 24. [2, Section 3] Let K/k and L/l be geometric fields over k, resp. l, of transcendence of degree 2: 2. Assume that 'l/Jl : K* /k* -+ L * / l* maps lines in JID( K) into lines in JID( L). Then 'l/J1 is a morphism of projective structures, 'l/Jl (JID( K)) is a projective subspace in JID( L), and there exist a subfield L' eLand an isomorphism of fields
'l/J : K -+
i/,
which is compatible with 'l/Jl. Lemma 23 shows that either 'l/Jl or sition 24. This proves Theorem 2.
'l/J11 satisfies the conditions of Propo-
5. Milnor K-groups Let K = k(X) be a function field of an algebraic variety X over an algebraically closed field k. In this section we characterize intrinsically infinitely
242
F.
BOGOMOLOV AND Y. TSCHINKEL
divisible elements in Kj"f (K) and K~ (K). For (5.1)
Ker2(f) := {g E K* jk* = Rj"f (K)
LEMMA 25. An element f E K* only if f E k*. In particular, (5.2)
f
E K* put
I (f, g)
= Kj"f (K)
= 0 E R~ (K)
}.
is infinitely divisible if and
Rj"f(K) = K*jk*.
PROOF. First of all, every element in k* is infinitely divisible, since k is algebraically closed. We have an exact sequence 0-+ k* -+ K* -+ Div(X).
The elements of Div(X) are not infinitely divisible. Hence every infinitely divisible element of K* is in k*. 0 LEMMA 26. Given a nonconstant
fl
E K* jk*, we have
where E = k(fl) n K. PROOF. Let X be a normal projective model of K. Assume first that fl,12 E K \ k lie in a I-dimensional subfield E C K that contains k and is normally closed in K. Such a field E defines a rational map 1f : X -+ C, where C is a projective model of E. By the Merkurjev-Suslin theorem [3], for any field F containing n-th roots of unity one has
Br(F)[n] = K~ (F)j(K~ (F)t, where Br(F)[n] is the n-torsion subgroup of the Brauer group Br(F). On the other hand, by Tsen's theorem, Br(E) = 0, since E = k(C), and k is algebraically closed. Thus the symbol (fl, h) is infinitely divisible in K~ (E) and hence in K~ (K). Conversely, assume that the symbol (fl, h) is infinitely divisible in K~ (K) and that the field k(fl, h) has transcendence degree two. Choosing an appropriate model of X, we may assume that the functions fi define surjective morphisms 1fi : X -+ JID} = JlDl, and hence a proper surjective map 1f : X -+ JlDi x JID~. For any irreducible divisor D C X the restriction of the symbol (fl, h) to D is well-defined, as an element of Kj"f(k(D)). It has to be infinitely divisible in Kj"f (k(D)), for each D. For j = 1,2, consider the divisors div(fj) = 'LnijDij, where Dij are irreducible. Let Du be a component surjecting onto JlDi x O. The restriction
MILNOR K2 AND FIELD HOMOMORPHISMS
243
of 12 to Dl1 is nonconstant. Thus Dl1 is not a component in the divisor of 12 and the residue
It remains to apply Lemma 25 to conclude that the residue and hence the symbol are not divisible. This contradicts the assumption that k(h, h) has transcendence degree two. 0 COROLLARY
27. Let K and L be function fields over k. Any group homo-
morphism -M
-M
'l/Jl : Kl (K) -+ Kl (L) satisfying the assumptions of Theorem 4 maps multiplicative subgroups of normally closed one-dimensional subfields of K to multiplicative subgroups of one-dimensional subfields of L. We now prove Theorem 4. Step 1. For each normally closed one-dimensional subfield E exists a one-dimensional sub field EeL such that
c K there
'l/Jl(E* jk*) c E* jl* Indeed, Lemma 26 identifies multiplicative groups of I-dimensional normally closed subfields in K: For x E K* \ k* the group k(x)* c K* is the set of all Y E K*jk* such that the symbol (x,y) E :Rr(K) is zero. Step 2. There exists an r E N such that 'l/Ji/r (K* j k*) contains a primitive element of L*jl*. Note that L*jl* is torsion-free. For f,g E K*jk* assume that 'l/Jl (f), 'l/Jl (g) are n f' resp. n g , powers of primitive, multiplicatively independent elements in L * j l*. Let M := ('l/Jl (f), 'l/Jl (g)) and let Prim( M) be its primitivization. Then Prim(M)jM = 'Ljn EB 'Ljm, with n I m, i.e., n = gcd(nj, n g ). Thus, we can take r to be is the smallest nontrivial power of an element in 'l/Jl (K* jk*) c L* jl*. Step 3. By Theorem 2 either 'l/Ji/r or 'I/J~l/r extends to a homomorphism of fields. References [1] F. A. BOGOMOLOV - "Abelian subgroups of Galois groups", Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, p. 32-67. [2] F. BOGOMOLOV and Y. TSCHlNKEL - "Reconstruction of function fields", Geom. Funct. Anal. 18 (2008), no. 2, p. 400-462. [3] A. S. MERKURJEV and A. A. SUSLIN - "K-cohomology of Severi-Brauer varieties and the norm residue homomorphism", Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, p. 1011-1046,1135-1136. [4] J. MILNOR - Introduction to algebraic K -theory, Princeton University Press, Princeton, N.J., 1971, Annals of Mathematics Studies, No. 72.
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[5] S. MOCHIZUKI ~ "The local pro-p anabelian geometry of curves", Invent. Math. 138 (1999), no. 2, p. 319~423. [6] ___ , "Topics surrounding the anabelian geometry of hyperbolic curves", Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., vol. 41, Cambridge Univ. Press, Cambridge, 2003, p. 119~165. [7] F. POP ~ "On Grothendieck's conjecture of birational anabelian geometry", Ann. of Math. (2) 139 (1994), no. 1, p. 145~182. [8] J. STALLINGS ~ "Homology and central series of groups", J. Algebra 2 (1965), p. 170~181. [9] A. TAMAGAWA ~ "The Grothendieck conjecture for affine curves", Compositio Math. 109 (1997), no. 2, p. 135~194. COURANT INSTITUTE, NEW YORK UNIVERSITY, NEW YORK, NY 10012, USA E-mail address: bogomolotOcims. nyu. edu COURANT INSTITUTE, NEW YORK UNIVERSITY, NEW YORK, NY 10012, USA E-mail address: tschinkeltOcims. nyu. edu
Surveys in Differential Geometry XIII
Arakelov inequalities Eckart Viehweg
Introduction The proof of the Shafarevich Conjecture for curves of genus 9 ~ 2 over complex function fields K = C(Y), given by Arakelov in [AR71J, consists of two parts, the verification of "boundedness" and of "rigidity". In order to obtain the boundedness, Arakelov first constructs a height function for K-valued points of the moduli stack Mg of stable curves of genus g. In down to earth terms, he chooses a natural ample sheaf A on the coarse moduli scheme Mg. Then, extending the morphism Spec(K) -+ Mg to Y -+ Mg he chooses as height deg(cp* A). Secondly, still assuming that cp is induced by a genuine family j : X -+ Y of stable curves, he gives an upper bound for this height in terms of the curve Y and the discriminant S = Y \ Yo for Yo = cp-l(Mg). Finally the rigidity, saying that Xo = j-l(yO) -+ Yo does not extend to a family f : Xo -+ Yo x T in a non-trivial way, easily follows from the deformation theory for families of curves. The boundedness part of Arakelov's proof was extended by Faltings [Fa83] to families of abelian varieties, using Deligne's description of abelian varieties via Hodge structures of weight one. He chooses a suitable toroidal compactification Ag of the coarse moduli scheme of polarized abelian varieties and A E Pic(Ag) 0 Q to be the determinant of the direct image of relative one forms, hence the determinant of the Hodge bundle of bidegree (1,0) in the corresponding variation of Hodge structures. Then A is semiample and ample with respect to the open set Ag (as defined in Definition 1.2), which is sufficient to define a height function. He proves an upper bound for the height, hence the finiteness of deformation types, and gives a criterion for infinitesimal rigidity. A family of 8-dimensional abelian varieties gives an example that contrary to the case of curves the rigidity fails in general. Deligne [De87] takes up Faltings approach. He obtains more precise inequalities and his arguments extend to C-variations of Hodge structures This work has been supported by the DFG-Leibniz program and by the SFB/TR 45 "Periods, moduli spaces and arithmetic of algebraic varieties" . ©2009 International Press
246
E. VIEHWEG
of weight one. Peters proved similar inequalities for variation of Hodge structures of higher weight. Unfortunately his results (improved by Deligne in an unpublished letter) were only available years later (see [PeOO]), shortly after the subject was taken up by Jost and Zuo in [JZ02]. Since then the results for families of curves or abelian varieties over curves have been extended in several ways. Firstly the definition and the bounds for height functions have been extended to moduli schemes of canonically polarized manifolds or of polarized minimal models (see [BVOO], [VZ01], [VZ04a], [Vi05], and [KL06], for example). We sketch some of the results in Section 1. However we will not say anything about rigidity and strong boundedness properties, discussed in [VZ02] and [KL06]. Secondly generalizations of the Arakelov inequalities are known for variations of Hodge structures of higher weight over curves, and for weight one over a higher dimensional bases. In both cases the inequalities are optimal, i.e. there are families where one gets equality. As we recall in Section 1 such an equality should be rare for families of varieties of positive Kodaira dimension. Except for abelian varieties and for K3-surfaces the geometric interpretation of such an equality is still not understood (see [Li96], [STZ03], [VZ02], [LTYZ], [VZ03], [VZ04a], and [VZ05] for some results pointing in this direction). Finally the Arakelov inequalities have a topological counterpart, the Milnor-Wood inequalities for the Toledo invariant, for certain local systems on projective curves and on higher dimensional projective manifolds (see [BGG06], [KM08a], and [KM08b], for example). Again the equality has consequences for the structure of the local system (or its Higgs bundle). We will state this (in)equalities in very special cases in Section 5 and in Section 8 and compare it with the Arakelov inequality. The main theme of this survey is the interplay between stability of Higgs bundles and the stability of the Hodge bundles for variations of Hodge structures of weight k (see Section 2 for the basic definitions). As we try to explain in Section 3 for all k in the curve case, and in Section 6 for k = lover certain higher dimensional varieties, the Arakelov inequalities are translations of slope conditions for polystable Higgs bundles, whereas the Arakelov equalities encode stability conditions for the Hodge bundles. In Sections 4 and 7 we indicate some geometric consequences of Arakelov equalities for k = 1 or for families of abelian varieties.
Acknowledgments. This survey is based on a series of articles coauthored by Kang Zuo, by Martin Moller or by both of them. Compared with those articles there are only minor improvements in some arguments and no new results. Martin Moller pointed out some ambiguities in the first version of this article, and the idea for the simplified proof of Claim 6.7, needed for Theorem 6.4, is taken from his letter explaining the "r = 2" -case. I am gratefull to Oscar Garda-Prada, Vincent Koziarz and Julien Maubon for their
247
ARAKELOV INEQUALITIES
explanations concerning "Milnor-Wood" inequalities over a one or higher dimensional base. 1. Families of manifolds of positive Kodaira dimension
Let f : X -+ Y be a semistable family of n-folds over a complex projective curve Y, smooth over Yo = Y \ S and with X projective. We call f semistable if X is non-singular and if all fibres f-l(y) of f are reduced normal crossing divisors. We write Xo = f- I (Yo) and fo = flxo' THEOREM 1.1 ([VZOl], [VZ06], and [MVZ06]). Assume that f:X -+ Y is semistable. Then for all v ~ 1 with f*w'X/y i= 0
(1.1)
n .v 1 deg(f*w'X/y) k(f 1/ ) : : ; -2- . deg(Oy(log S)). r *w x / y
The morphism f is called isotrivial if there is a finite covering Y' -+ Y and a birational Y' morphism X x YY' --+ F x y'.
For projective manifolds F with WF semiample and polarized by an invertible sheaf with Hilbert polynomial h, there exists a coarse quasiprojective moduli scheme M h . Hence if wXo/Yo is fo-semiample fo induces a morphism 'Po : Yo -+ Mh· If wxo/Yo is fo-ample, or if w'Xo/Yo is for some v > 0 the pullback of an invertible sheaf on Yo, then the birational non-isotriviality of f is equivalent to the quasi-finiteness of 'Po. In this situation the left hand side of (1.1) can be seen as a height function on the moduli scheme. In fact, choosing v > 1 with h(v) i= 0 in the first case, and or v ~ 1 with wp = OF in the second one, by [Vi05] there exists a projective compactification M h of the moduli scheme Mh and some
with: • A is nef and ample with respect to Mh . • Let 'P : Y -+ M h be the morphism induced by f. Then det (f*w'X/y) = 'P* A. For moduli of abelian varieties one can choose the Baily-Borel compactification and there A is ample. By [Mu77] on a suitable toroidal compactification of Ag the sheaf A is still semi-ample, but for other moduli functors we only get weaker properties, as defined below. DEFINITION
and dense.
1.2. Let Z be a projective variety and let Zo
C
Z be open
E. VIEHWEG
248
A locally free sheaf:F on Z is numerically effective (nef) if for all morphisms p : C -+ Z, with C an irreducible curve, and for all invertible quotients N of p* :F one has deg(N) ~ O. ii. An invertible sheaf £ on Z is ample with respect to Zo if for some l/ ~ 1 the sections in HO(Z, £1/) generate the sheaf £1/ over Zo and if the induced morphism Zo -+ JP'(HO(Z, £1/)) is an embedding. 1.
For non-constant morphisms p : C -+ Z from irreducible projective curves one finds in Definition 1.2, ii) that deg(p*(£)) > 0, provided p(C) n Zo =I 0. Moreover, fixing an upper bound c for this degree, there are only finitely many deformation types of curves with deg(p*(£)) < c. Applying this to birationally non-isotrivial families f : X -+ Y whose general fibre F is either canonically polarized or a minimal model of Kodaira dimension zero, one finds the left hand side of (1.1) to be positive, hence n~(1og S) = wy(S) must be ample. The finiteness of the number of deformation types is more difficult and it has been worked out in [KL06] just for families of canonically polarized manifolds. Roughly speaking, one has to show that morphisms from a curve to the moduli stack are parameterized by a scheme. This being done, one finds that for a given Hilbert polynomial h and for a given constant c there are only finitely many deformation types of families f : X -+ Y of canonically polarized manifolds with deg(n~(logS)) :::; c. For smooth projective families fo : Xo -+ Yo over a higher dimensional quasi-projective manifold Yo with wXo/Yo semiample, some generalizations of the inequality (1.1) have been studied in [VZ02] (see also [VZ04a]). There we assumed that S = Y \ Yo is a normal crossing divisor and that the induced map 'Po : Yo -+ Mh is generically finite. Then for some p, » 0 there exists a non-trivial ample subsheaf of SIL(n~ (log S)). However neither p, nor the degree of the ample subsheaf have been calculated and the statement is less precise than the inequality (1.1). In this survey we are mainly interested in a geometric interpretation of equality in (1.1), in particular for l/ = 1. As explained in [VZ06] and [MVZ06] such equalities should not occur for families with pg(F) > 1 for the general fibre F. Even the Arakelov inequalities for non-unitary subvariat ions of Hodge structures, discussed in Section 3 should be strict for most families with F of general type. As recalled in Example 4.6, for curves "most" implies that the genus 9 of F has to be 3 and that the "counter-example" in genus 3 is essentially unique. So what Arakelov equalities are concerned it seems reasonable to concentrate on families of minimal models of Kodaira dimension zero.
2. Stability DEFINITION 2.1. Let Y be a projective manifold, let S E Y be a normal crossing divisor and let :F be a torsion-free coherent sheaf on Y.
ARAKELOV INEQUALITIES
249
i. The degree and slope of F are defined as deg(F) = cl(F).Cl(Wy(S))
d· (Y) 1 1m
-
and
J-L(F)
=
J-Lwy(S) (F)
=
deg(F) rk(F) .
9 c F with rk(Q) < rk( F) one has J-L(Q) < J-L( F). iii. The sheaf F is J-L-semistable if for all non-trivial subsheaves 9 c F one has J-L(Q) ~ J-L(F). iv. F is J-L-polystable if it is the direct sum of J-L-stable sheaves of the same slope. ii. The sheaf F is J-L-stable if for all subsheaves
This definition is only reasonable if dim(Y) = 1 or if Wy (S) is nef and big. Recall that a logarithmic Higgs bundle is a locally free sheaf E on Y together with an Oy linear morphism () : E -+ E ® n} (log S) with () A () = o. The definition of stability (poly- and semistability) for locally free sheaves extends to Higgs bundles, by requiring that
J-L
(F) = deg(F)
rk(F) < J-L
(E) = deg(E) rk(E)
(or J-L(F) ~ J-L(E)) for all subsheaves F with ()(F) c F®n}(logS). If dim(Y) > 1, for the Simpson correspondence in [Si92] and for the polystability of Higgs bundles, one takes the slopes with respect to a polarization of Y, i.e replacing wy(S) in Definition 2.1, i) by an ample invertible sheaf. However, as we will recall in Proposition 6.4, the Simpson correspondence remains true for the slopes J-L(F) in 2.1, i), provided wy(S) is nef and big. Our main example of a Higgs bundle will be the one attached to a polarized C variation of Hodge structures V on Yo of weight k, as defined in [DeS7] , and with unipotent local monodromy operators. The F-filtration of Fo = V ®c OYo extends to a locally splitting filtration of the Deligne extension F of Fo to Y, denoted here by
Fk+l
C
Fk
C ... C
;:0.
We will usually assume that Fk+l = 0 and ;:0 = F, hence that all nonzero parts of the Hodge decomposition of a fibre Vy of V are in bidegrees (k - m, m) for m = 0, ... ,k. The Griffiths transversality condition for the GauB-Manin connection V' says that
V'(p) C p-l ® n}(log S). Then V' induces a Oy linear map
()p,k-p : EP,k-p = P / pH --+ EP-l,k-p+l = p-l / p ® n}(log S).
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250
We will call
(E
= E9 EP,k- p, fJ = E9 fJp,k-p) p
the (logarithmic) Higgs bundle of V, whereas the sheaves Ep,q are called the Hodge bundles of bidegree (p, q). DEFINITION
2.2. For the Higgs bundle (E, fJ) introduced above we define:
i. The support supp(E, fJ) is the set of all m with Ek-m,m =f. O. ii. (E, fJ) has a connected support, if there exists some mo :::; ml E Z with
supp(E, fJ)
= {m; mo :::; m:::; md
fJk-m,m =f. 0
for
and if
mo:::; m :::; ml - 1.
iii. (E, fJ) (or V) satisfies the Arakelov condition if (E, fJ) has a connected support and if for all m with m, m + 1 E supp(E, fJ) the sheaves Ek-m,m and Ek-m-1,m+l are p,-semistable and
3. Variations of Hodge structures over curves Let us return to a projective curve Y, so S = Y\Yo is a finite set of points. The starting point of our considerations is the Simpson correspondence: THEOREM 3.1 ([Si90]). There exists a natural equivalence between the category of direct sums of stable filtered regular Higgs bundles of degree zero, and of direct sums of stable filtered local systems of degree zero.
We will not recall the definition of a "filtered regular" Higgs bundle [Si90, page 717], and just remark that for a Higgs bundle corresponding to a local system V with unipotent monodromy around the points in S the filtration is trivial, and automatically deg(V) = O. By [De71] the local systems underlying a Z-variation of Hodge structures are semisimple, and by [De87] the same holds with Z replaced by C. So one obtains: 3.2. The logarithmic Higgs bundle of a polarized Cvariation of Hodge structures with unipotent monodromy in s E S is polystable of degree O. COROLLARY
In [VZ03] and [VZ06] we discussed several versions of Arakelov inequalities. Here we will only need the one for Ek,o, and we sketch a simplified version of the proof:
251
ARAKELOV INEQUALITIES
3.3. Let V be an irreducible complex polarized variation of Hodge structures over Y of weight k and with unipotent local monodromies in s E S. Write (E, 0) for the logarithmic Higgs bundle of V and assume that EP,k-p = o for p < 0 and for p > k. Then one has: LEMMA
a. p,(Ek,o) :S b.
k
2' deg(n} (log S)).
o:s p,(Ek,O)
and the equality implies that V is unitary or equivalently that 0 k c. The equality p,(Ek,o) = 2' deg(n} (log S)).
= O.
implies that the sheaves Ek-m,m are stable and that
Ok-m,m : Ek-m,m -----+ E k- m- 1 ,m+1
@
n}(log S)
is an isomorphism for m = 0, ... , k - 1. PROOF. Let Gk,o be a subsheaf of Ek,o, and let Gk-m,m be the (k m, m) component of the Higgs subbundle G = (Gk,O), generated by Gk,o. By definition one has a surjection
G k- m+1,m-l -----+ Gk-m,m @ n}(log S). Its kernel K m - 1 , together with the O-map is a Higgs subbundle of (E,O), hence of non-positive degree. Remark that
So one finds (3.1)
+ rk(Gk-m,m) . deg(nHlog S)) :S deg(Gk-m,m) + rk(G k- 1,1) . deg(n} (log S)).
deg(G k- m+1,m-1):s deg(Gk-m,m)
~
Iterating this inequality gives for m (3.2)
1
deg(Gk,o) :S deg(Gk,o) - deg(Ko) = deg( G k - 1 ,1)
+ rk( G k - 1,1) . deg(nHlog S))
:S deg(Gk-m,m)
+ m· rk(G k- 1,1). deg(nHlogS))
and adding up
(k + 1) deg(Gk,o) :S (k + 1) deg(Gk,o) - k· deg(Ko) k
:S
L
k
deg(Gk-m,m)
m· rk(G k- 1,1). deg(nHlogS))
m=l
m=O
= deg(G) +
+L
k·(k+1) 2
. rk(G k- 1,1). deg(n}(logS)).
252
E. VIEHWEG
Since G is a Higgs subbundle, deg( G) ::; 0, and (3.3)
°
k deg( Gk,O) k 1 f.-l(G' ) ::; rk(Gk-l,l) ::; "2' deg(ny(logS)).
Taking Gk,o = Ek,o one obtains the inequality in a). If this is an equality, as assumed in c), then the right hand side of (3.3) is an equality. Firstly, since the difference of the two sides is larger than a positive multiple of deg( G) = 0, the latter is zero and the irreducibility of V implies that G = E. Secondly the two inequalities in (3.2) have to be equalities. The one on the right hand side gives rk(Ek-m,m) = rk(Ek-1,1) for m = 2, ... , k. The one on the left implies that deg(K o) = 0 and the irreducibility of V shows that this is only possible for Ko = 0 hence if rk(Ek,O) = rk(E k- 1,1). All together one finds that the surjections
Ek,o -+ Ek-m,m ® n}(log s)m are isomorphisms, for 1 ::; m ::; k. On the other hand the equality in c) and the inequality (3.3) imply that for all subsheaves Gk,o
f.-l(Gk,o) ::;
~ . deg(nHlog S)) = f.-l(Ek,o),
If this is an equality, then deg(G) = 0 and (G, Ble) c (E, B) splits. The irreducibility implies again that (G,Ble) = (E,B), hence Ek,o as well as all the Ek-m,m are stable. The sheaf Ek,o with the O-Higgs field is a Higgs quotient bundle of (E, B), hence of non-negative degree. If deg(Ek,O) = 0, then the surjection of Higgs bundles (E, B) -+ (Ek,O, 0) splits. The irreducibility of V together with Theorem 3.1 implies that both Higgs bundles are the same, hence that B = 0 and V unitary. So b) follows from a). 0 COROLLARY
3.4. In Lemma 3.3 one has the inequality
(3.4)
The equality in Lemma 3.3, c) is equivalent to the equality (3.5)
In particular (3.5) implies that the sheaves Ek-m,m are stable and that Bk-m,m : Ek-m,m -+ E k- m- 1 ,m+1 ® n}(logS)
is an isomorphism for m = 0, ... , k - 1. For (3.4) one applies part a) of Lemma 3.3 to (E, B) and to the dual Higgs bundle (EV, BV). The equality (3.5) implies that both, (E, B) and (EV, BV) satisfy the Arakelov equality c) in Lemma 3.3. PROOF.
ARAKELOV INEQUALITIES
253
Finally assume that the equation c) in Lemma 3.3 holds for (E, 0). Then
Ek,o /-l(EVk,o)
~
EO,k 0 O}(log 8)k
= _/-l(EO,k) = k· deg(O} (log 8)) -
and
/-l(Ek,o)
= ~ . deg(O} (log 8)).
Adding this equality to the one in c) one gets (3.5).
o
The inequality in part a) of Lemma 3.3 is not optimal. One can use the degrees of the kernels Km to get correction terms. We will only work this out for m = O. What equalities are concerned, one does not seem to get anything new. VARIANT 3.5. In Lemma 3.6 one has the inequalities (3.6)
deg(Ek,O) k 1 rk(Ok,o) ~ 2" . deg(Oy(log 8)).
The equality in Lemma 3. 3, c) is equivalent to the equality (3.7)
deg(Ek,O) rk(Ok,o)
k
1
= 2" . deg(Oy(log 8)).
PROOF. The inequality is a repetition of the left hand side of (3.3) for Ck,o = Ek,o. If Ok,O is an isomorphisms, hence if rk(Ek,O) = rk(Ok,o), the two equalities (3.7) and c) in Lemma 3.3 are the same. As stated in Lemma 3.3, the equality c) implies that Ok,O is an isomorphisms, hence (3.7). In the proof of Lemma 3.3 we have seen that the equality of the right hand side of (3.3) implies that C = E, hence that the morphisms
Ok-m,m : Ek-m,m
~
E k- m- 1 ,m+1 0 O}(log 8)
are surjective for m = 0, ... ,m - 1. Using the left hand side of (3.2), one finds that K o = 0 hence that Ok,O is an isomorphisms. So (3.7) implies the equality c). 0 Replacing Yo by an etale covering, if necessary, one may assume that #8 is even, hence that there exists a logarithmic theta characteristic £. By definition £2 ~ O} (log 8) and one has an isomorphism r:
£ ~ £ 0 O}(log8).
Since (£ EB £-1, r) is an indecomposable Higgs bundle of degree zero, Theorem 3.1 tell us that it comes from a local system IL, which is easily seen to be a variation of Hodge structures of weight 1. We will say that IL is induced by a logarithmic theta characteristic. Remark that IL is unique up to the tensor product with local systems, corresponding to two division points in pica (Y). By [VZ03, Proposition 3.4] one has:
254
E. VIEHWEG
ADDENDUM 3.6. Assume in Lemma 3.3 that #S is even and that IL is induced by a theta characteristic. k d. Then the equality /-l (Ek,o) = "2 . deg (nHlog S)) implies that there exists an irreducible unitary local system
V~
1['0
1['0
on Yo with
® Sk(IL).
REMARK 3.7. In Addendum 3.6 the local monodromies of 1['0 are unipotent and unitary, hence finite. So there exists a finite covering 7 : Y' ~ Y, etale over Yo such that 7*1['0 extends to a unitary local system 1[" on Y'. The property d) in Addendum 3.6 is equivalent to the condition c) in Lemma 3.3. In particular it implies that each Ek-m,m is the tensor product of an invertible sheaf with the polystable sheaf 1['0 ®c Oy. The Arakelov equality implies that the Higgs fields are direct sums of morphisms between semistable sheaves of the same slope. Then the irreducibility of V can be used to show that 1['0 ®c Oy and hence the Ek-m,m are stable. REMARK 3.8. Let us collect what we learned in the proof of Lemma 3.3. • Simpson's polystability of the Higgs bundles (E,O) implies the Arakelov inequality a) in Lemma 3.3 or inequality (3.4). • The equality in part c) of Lemma 3.3 implies that the Hodge bundles Ek-m,m are semistable and that the Higgs field is a morphism of sheaves of the same slope. • If one assumes in addition that V is irreducible, then the Ek-m,m are stable sheaves. As we will see in Section 6 the first two statements extend to families over a higher dimensional base (satisfying the positivity condition (*) in 6.2), but we doubt that the third one remains true without some additional numerically conditions. Assume that W is the variation of Hodge structures given by a smooth family fo : Xo ~ Yo of polarized manifolds with semistable reduction at infinity, hence W = Rk fo*C xo ' Let W = VI EB··· EB Ve be the decomposition of W as direct sum of irreducible local subsystems, hence of C irreducible variations of Hodge structures of weight k. Replacing V~ by a suitable Tate twist V~(v~), and perhaps by its dual, one obtains a variation of Hodge structures of weight k~ = k - 2 . V~, whose Hodge bundles are concentrated in bidegrees (k~ - m, m) for m = 0, ... , k~ and non-zero in bidegree (k~, 0). Applying Lemma 3.3 to V~(v~) one gets Arakelov inequalities for all the V~. If all those are equalities, each of the V~ will satisfy the Arakelov condition in Definition 2.2, iii, and for some unitary bundle 1['~ one finds V~ = 1['~ ® Sk-2·v, (1L)( -v~). We say that the Higgs field of W is strictly maximal in this case (see [VZ03] for a motivation and for a slightly different presentation of those results).
ARAKELOV INEQUALITIES
255
Let us list two results known for families of Calabi-Yau manifolds, satisfying the Arakelov equality. ASSUMPTIONS 3.9. Consider smooth morphisms fo ; Xo -+ Yo over a non-singular curve Yo, whose fibres are k-dimensional Calabi-Yau manifolds. Assume that fo extends to a semistable family f ; X -+ Y on the compactification Y of Yo. Let V be the irreducible direct factor of Rk fo*Cxo with Higgs bundle (E,O), such that Ek,o i= o. THEOREM 3.10 ([Bo97], [Vo93], and [STZ03], see also [VZ03]). For all k ~ 1 there exist families fo ; Xo -+ Yo satisfying the Assumptions 3.9, such that the Arakelov equality (3.5) holds for V. For families of K3surfaces, i.e. for k = 2, there exist examples with Yo = Y projective. For k = 1 those families are the universal families over elliptic modular curves, hence Yo is affine in this case. A similar result holds whenever the dimension of the fibres is odd. THEOREM 3.11 ([VZ03]). Under the assumptions made in 3.9 assume that k is odd and that V satisfies the Arakelov equality. Then S = Y\ Yo i= 0, i. e. Yo is affine. It does not seem to be known whether for even k ~ 4 there are families of Calabi-Yau manifolds over a compact curve with V satisfying the Arakelov equality. The geometric implications of the Arakelov equality for V in 3.9 or of the strict maximality of the Higgs field, are not really understood. The structure Theorem 3.6 can be used to obtain some properties of the Mumford Tate group, but we have no idea about the structure of the family or about the map to the moduli scheme Mh. The situation is better for families of abelian varieties. So starting from the next section we will concentrate on polarized variations of Hodge structures of weight one.
4. Arakelov equality and geodecity of curves in Ag ASSUMPTIONS 4.1. Keeping the assumptions from the last section, we restrict ourselves to variations of Hodge structures of weight one, coming from families fo ; Xo -+ Yo of abelian varieties. Replacing Yo by an etale covering allows to assume that fo ; Xo -+ Yo is induced by a morphism 'Po ; Yo -+ Ag where Ag is some fine moduli scheme of polarized abelian varieties with a suitable level structure, and that the local monodromy in s E S of WQ = R1fo*Qxo is unipotent. Let us fix a toroidal compactification A g , as considered by Mumford in [Mu77]. In particular Ag is non-singular, the boundary divisor SAg has non-singular components, and normal crossings, ~Al (log SA ) is nef and 9
9
w:A9 (SA) is ample with respect to A g • 9
E. VIEHWEG
256
In [MVOB] we give a differential geometric characterization of morphisms rpo : Yo -+ Ag for which the induced ((>variation of Hodge structures W contains a non-unitary C-subvariation V with Higgs bundle (E,O), satisfying the Arakelov equality (4.1)
/.L(E1,0)
1
= '2 . deg(O}(log S)).
To this aim we need: DEFINITION 4.2. Let M be a complex domain and W be a subdomain. W is a totally geodesic submanifold for the Kobayashi metric if the restriction of the Kobayashi metric on M to W coincides with the Kobayashi metric on W. If W = ~ we call ~ a (complex) Kobayashi geodesic. A map rpo : Yo -+ Ag is a Kobayashi geodesic, if its universal covering map ~o : Yo ~ ~ ---+ lHIg is a Kobayashi geodesic. In particular here a Kobayashi geodesic will always be one-dimensional. THEOREM 4.3. Under the assumptions made in tions are equivalent:
4.1
the following condi-
a. rpo : Yo -+ Ag is Kobayashi geodesic. 1 (log SA ) -+ O}(log S) splits. b. The natural map rp*O-A 9 9 c. W contains a non-unitary irreducible subvariation of Hodge structures V which satisfies the A rakelov equality (4.1). The numerical condition in Theorem 4.3 indicates that Kobayashi geodesic in Ag are "algebraic objects". In fact, as shown in [MVOB] one obtains: COROLLARY 4.4. Let rpo : Yo -+ Ag be an affine Kobayashi geodesic, such that the induced variation of Hodge structures WQ is Q-irreducible. Then rpo : Yo -+ Ag can be defined over a number field. Geodesics for the Kobayashi metric have been considered in [Mo06] under the additional assumption that fo : Xo -+ Yo is a family of Jacobians of a smooth family of curves. In this case rpo(Yo ) is a geodesic for the Kobayashi metric if and only if the image of Yo in the moduli scheme Mg of curves of genus 9 with the right level structure is a geodesic for the Teichmiiller metric, hence if and only if Yo is a Teichmiiller curve. In particular Yo will be affine and the irreducible subvariation V in Theorem 4.3 will be of rank two. By Addendum 3.6 it is given by a logarithmic theta characteristic on Y. Using the theory of Teichmiiller curves (see [McM03]), one can deduce that there is at most one irreducible direct factor V which satisfies the Arakelov equality.
ARAKELOV INEQUALITIES
257
The Theorem 4.3 should be compared with the results of [VZ04b]. Starting from Lemma 3.3 and the addendum 3.6 it is shown that under the assumptions 4.1 Yo (or to be more precise, an etale finite cover of Yo) is a rigid Shimura curve with universal family fo : Xo -+ Yo if the Arakelov equality holds for all irreducible C-subvariations of Hodge structures of Rl fo*Cxo' Recall that "rigid" means that there are no non-trivial extensions of fo to a smooth family f : Xo -+ T x Yo with dim T > O. If one allows unitary direct factors, and requires the Arakelov equality just for all non-unitary subvariations V, then Yo C Ag is a deformation of a Shimura curve or, using the notation from [Mu69], the family fo : Xo -+ Yo is a Kuga fibre space. In [Mo05] it is shown (see also [MVZ07, Section 1]), that for all Kuga fibre spaces and all non-unitary irreducible VeRI fo*Cxo the Arakelov equality holds. In [MVZ07] this was translated to geodecity for the Hodge (or Bergman-Siegel) metric, and we can restate the main result of [VZ04b] in the following form: THEOREM 4.5. Keeping the notations and assumptions introduced in 4.1, the following conditions are equivalent: a. ... > J.t(F;'o) > 0 o > J.t( E~,l) > J.t( 11,1) > ... > J.t( F~,l).
B. J.t(E:'o)
PROOF. By (6.3) (EL = EZ'o EB E?'l, OlE.) is a Higgs subbundle of (E, 0). So A) follows from Proposition 6.3. Since (E~,1, 0) is a Higgs subbundle of (E,O) and since (F;'o,O) is a quotient Higgs bundle, one also obtains J.t(F;'o) > 0 > J.t(E~,l). The slope inequalities
J.t(GjjGj,-d > J.t(Gj,+l/Gj.}
J.t(Gj/Gj:_d > J.t(Gj:+l/Gj),
and
together with (6.4) and (6.5), imply the remaining inequalities in B). CLAIM
6.6.
J.t(E1,o) - J.t(EO,l) ::; Max{J.t(F!'o) - J.t(F~,l);
/'i,
0
= 1, ... ,r}
and the equality is strict except if r = 1. Before proving Claim 6.6 let us finish the proof of Theorem 6.4. By (6.3) the Higgs field induces a non-zero map
°
Gj,_l+I/Gj'_l ---+ (Gj/Gj~_l) ® n~(logS).
(6.6)
The semistability of both sides of (6.6) implies that J.t(Gj'_l+l/Gj'_l) ::; J.t(Gj/Gj~_l)
+ J.t(n~(logS)).
By (6.4) and (6.5) one has (6.7)
J.t(Gj'_l+I/Gj'_l) ~ J.t(FL1,o)
and
J.t(FLO,l) ~ J.t(Gj:/Gj:_l).
and altogether
(6.8) J.t(FL1,o) - J.t(FLO,l) ::; J.t(Gj'_l+l/Gj'_l) - J.t(Gj:/Gj:_l) ::; J.t(n~(logS)). For j = r the first part of Claim 6.6 implies that J.t(E1,O) - J.t(EO,l) ::; J.t( n~ (log S)) as claimed in (6.2). This can only be an equality if r = 1, hence jl = i and ji = i'.
ARAKELOV INEQUALITIES
263
In addition, the equality in (6.2) can only hold if (6.8) is an equality. Then the two inequalities in (6.7) have to be equalities as well. By the definition of the Harder-Narasimhan filtration the equalities
J-L(Gd imply that
e=
= J-L(E 1,0)
and
J-L(EO,l)
= J-L(G~I/G~/_1)
= 1, hence that E 1,0 and EO,l are both J-L-semistable.
e'
0
PROOF OF CLAIM 6.6. We will try to argue by induction on the length of the filtration, starting with the trivial case r = 1. Unfortunately this forces us to replace the rank of the F2,1 by some virtual rank. We define: (1) Ii
= c1(Fd·C1(wy(s))dim(Y)-1.
(2) J-Lf,q = J-L(Ff'q) and ~i = J-L;'o - J-L?,1. (3) P;'o = rk(Fi1,0) and p?,l = rk(Fio,l) (4) For 0
O. Recall that the condition B) in Claim 6.4 says that -J-L~'o > -J-L;'o and OIL"lor'/, < K,. Th"IS Impl'les J-Li' 1 > J-LK,' K, K, s~,o . p~,l . J-L~,1 = P;'o . p~,l . J-L~,1 = P;'o . p~,o . (- J-L~'o) i=l i=l K, K,
°
L
L
-> "'" L.J Pi1,0 . PK,1,0 . ( -J-Li1,0) -_ "'" L.J Pi0,1 . PK,1,0 . J-Li0,1 i=l i=l K, O 1 110,1 = sO,l . p1,0 . 110,1 -> "'" L.J pz ,l . pK, ,0 . r'K, K, K, r'K,' i=l
01 IS . negat'lve, one ge t s · Smce J-LK,' . 1 tl 1,0 0,1 (6) SK,1,0 . pK,0,1 < _ SK,0,1 . pK,1,0 or eqmva en y SK,-l . PK,0,1 < _ SK,-l . PK,1,0 .
The induction step will use the next claim.
264
E. VIEHWEG
CLAIM 6.7. For 0 < /'i, ::::; f one has 8", ::::; Max{8",_I, ~"'}, with equality 'f ~ i\ 1 and only' f 1 u,..-1 = U", an d P,..1,0 . S",0,1 = p,..0,1 . S",1,0 .
1 t A = S",_1 1,0 . S"'_I' 0,1 B = P",1,0 . p""0,1 C = S",_1 1,0 . p,..0,1 an d P R00 F. "tXT vve e D = p~,o . S~'~I' By (6) one has D - C 2 O. Then
,.. sl,O . sO,1 .8 '"
,..
= '"
~ ~
(11fA't1,0 . pt1,0 . sO,1 ,..
_
1I?,1 . pO ,1 . sl,O) = 1/1,0 . pl,O . sO,1 t ,.. fA'''' ,.. ,..
fA't
i=1 ",-1
_ J.l~,1 . p~,1 . s~,o
+L
(J.l:'o . P:'o . s~,1 _ J.l?,l . p?,1 . s~,O)
i=1 =
B . ~,.. + A . 8",-1 + C . (Y~~~\ - J.l~,1) + D . (J.l~,o - Y2~1)
=
B·~,.. + A· 8",-1 + C· (8,..-1 + ~,..) + (D-C) . (J.l~,o - Y2~1)'
1 ° < yl",'-1 ° and < J.li'1 ° £ or .~ < /'i, one fi nds J.l",' (A + B + C + D) ·8,.. ::::; B . ~'" + A· 8",-1 + C· ~'" + D· 8,..-1.
· 1° Smce J.l",'
This implies the inequality in Claim 6.7. If the equality holds, ~,.. = 8",-1 and
o=
D - C = pl,O . s°,l _ s1,0 . pO,1 ,.. ",-1 ",-1,..
= pl,O . sO,1 _ ,.. ,..
D
sl,O . pO,1 ,.. ",'
6.S. One has the inequality J.l(E 1,0) - J.l(EO,1) ::::; 8r and the equality can only hold for '"Y1 = ... = '"Yr = O. CLAIM
J.l(E 1,0) = y;"o it remains to verify that J.l(EO,1) 2 y~,I.
PROOF. Since
As a first step, r (6.9) p?,1) - rk(Eo,1) i=1
(L
r
=L
r
(p?,l - rk(Fio,1)) =
i=1
L
-o:~
J.li r r-l 0,1 i -'"Yi (~ ~ J.li+1 (~ = Q,l' ~ '"Yi) + ~ 0,1 . ~ '"Yj). J.lr i=l i=1 J.li+1 j=1 0,1 0,1 " i d I e d ' J.li J.li+1 . · S mce ~j=1 '"Yj ::::; 0 an equa to zero lor i = r, an smce 01 01 IS J.li' . J.li+ 1 positive, one obtains i=1 0,1 J.li 0,1. J.li
r
L p?,1 ::::; rk(E°,1). i=1 Then
J.l(EO,1)
=
"r
0,1
k(pO,1) .r i rk(EO,1)
~i=1 J.li
=
"r
0,1 "r 0,1 . Pi + ~i=1 '"Yi rk(EO,1) rk(EO,1) "r 0,1 . pO,1 "r 0,1 0,1 ~i=1 J.li i > ~i=1 J.li . Pi rk(EO,1) "r 0,1 ~i=l Pi 0,1
~i=1 J.li
= yO,1 r
,
ARAKELOV INEQUALITIES
265
as claimed. The equality implies that the expression in (6.9) is zero, which is only possible if ')'1 = ... = ')'r = O. 0 Using the Claims 6.7 and 6.8 one finds that J-L(E 1,0) - J-L(EO,l)::; 8r ::; Max{8r-1,~r}::; Max{8r-2,~r-1,~r}::;
... ::; Max{ ~1" .. ,~r-1' ~r}. The equality implies that for all K, the inequalities in Claims 6.7 and 6.8 are equalities. The second one implies that for all K, one has ')'K = 0, hence p~,l = rk(F~,l), and the first one that
o=
p1,0 . sO,l _ pO,l . s1.0 = rk(F 1,0) . SO,l _ rk(F o,l) . Sl,O KKK KKK K K'
o
As for variation of Hodge structures over curves, the Arakelov inequality (6.2) is a direct consequence of the polystability of the Higgs bundle (E,O). The Arakelov equality J-LeV) = J-L( O~ (log S)) allows to deduce the semistability of the sheaves E 1,0 and EO,l. However, we do not know whether one gets the stability, as it has been the case over curves (see 3.4). Although we were unable to construct an example, we do not expect this. So it seems reasonable to ask, which additional conditions imply the stability of the sheaves E 1,0 and EO,l.
7. Geodecity of higher dimensional subvarieties in Ag Let us recall the geometric interpretation of the Arakelov equality, shown in [VZ07] and [MVZ07]. 7.1. We keep the assumptions and notations from Section 6. Hence Y is a projective non-singular manifold, and Yo C Y is open with S = Y\Yo a normal crossing divisor. We assume the positivity condition (*) and we consider an irreducible polarized C-variation of Hodge structures V of weight one with unipotent monodromies around the components of S. As usual its Higgs bundle will be denoted by (E,O). ASSUMPTIONS
The first part of Yau's Uniformization Theorem ([Ya93], discussed in [VZ07, Theorem 1.4]) was already used in the last section. It says that the Assumption (*) forces the sheaf OHlog S) to be J-L-polystable. The second part gives a geometric interpretation of stability properties of the direct factors. Writing
(7.1) for its decomposition as direct sum of J-L-stable sheaves and ni = rk(Oi), we say that Oi is of type A, if it is invertible, and of type B, if ni > 1 and if for all f > 0 the sheaf Sf(Oi) is J-L-stable. In the remaining cases, i.e. if for some f> 1 the sheaf Sf(Oi) is J-L-unstable, we say that Oi is of type C.
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266
Let 7f : Yo -+ Yo denote the universal covering with covering group r. The decomposition (7.1) of OHlog S) gives rise to a product structure
Yo = MI
X .••
x Ms,
where ni = dim(Mi). The second part of Yau's Uniformization Theorem gives a criterion for each Mi to be a bounded symmetric domain. This is automatically the case if Oi is of type A or C. If Oi is of type B, then Mi is a ni-dimensional complex ball if and only if (7.2)
[2. (ni
+ 1)· C2(Oi) -
ni' C1(Oi)2] .c(wy(s))dim(Y)-2 = O.
DEFINITION 7.2. The variation of Hodge structures V is called pure (of type i) if the Higgs field factors like EI,o ---+ E O,1 ® Oi
c E o,1 ® O~(log S)
(for some i = i(V)).
If one knows that Yo is a bounded symmetric domain, hence if (7.2) holds for all direct factors of type B, one obtains the purity of Vasa consequence of the Margulis Superrigidity Theorem: THEOREM 7.3. Suppose in 7.1 that Then V is pure.
Yo
is a bounded symmetric domain.
SKETCH OF THE PROOF. Assume first that Yo = UI X U2. By [VZ05, Proposition 3.3] an irreducible local system on V is of the form priV I ® pr2V2, for irreducible local systems Vi on Ui with Higgs bundles (Ei' ()i). Since V is a variation of Hodge structures of weight 1, one of those, say V2 has to have weight zero, hence it must be unitary. Then the Higgs field on Yo factors through EO,1 ® 0hl' By induction on the dimension we may assume that V I is pure of type /.. for some /.. with M~ a factor of [h. Hence the same holds true for V. So we may assume that all finite etale coverings of Yo are indecomposable. By [Zi84] § 2.2, replacing r by a subgroup of finite index, hence replacing Yo by a finite unramified cover, there is a partition of {I, ... , s} into subsets h such that r = Ilk rk and rk is an irreducible lattice in IliElk Gi· Here irreducible means that for any normal subgroup N c IliElk Gi the image of rk in IliEh Gd N is dense. Since the finite etale coverings of Yo are indecomposable, r is irreducible, so It = {I, ... ,s}. If s = 1 or if V is unitary, the statement of the proposition is trivial. Otherwise, G := Ilf=l Gi is of real rank ~ 2 and the conditions of Margulis' superrigidity theorem (e.g. [Zi84, Theorem 5.1.2 ii)]) are met. As consequence, the homomorphism r -+ Sp(V, Q), where V is a fibre of V and where Q is the symplectic form on V, factors through a representation p : G -+ Sp(V, Q). Since the Gi are simple, we can repeat the argument from [VZ05, Proposition 3.3], used above in the product case: p is a tensor product of representations, all of which but one have weight O. D
267
ARAKELOV INEQUALITIES
The next theorem replaces the condition that domain by the Arakelov equality. THEOREM
Yo is a bounded symmetric
7.4. Suppose in 7.1 that V satisfies the Arakelov equality p,(V) = p,( D~ (log S)).
Then V is pure.
The two Theorems 6.4 and 7.4 imply that the Higgs field of V is given by a morphism
El,o --+ EO,l ® Di between p,-semistable sheaves of the same slope. If Di is of type A or C this implies geodecity (for the Hodge or Bergman metric) in period domains of variation of Hodge structures of weight one. THEOREM 7.5. Suppose in Theorem 7.4 that for i = i(V) the sheaf Di is of type A or C. Let M' denote the period domain for V. Then the period map factors as the projection Yo --t Mi and a totally geodesic embedding }vIi --t AI'.
If Di is of type B we need some additional numerical invariants in order to deduce a similar property. Let (F, T) be any Higgs bundle, not necessarily of degree zero. For = rk(Fl,O) consider the Higgs bundle
e
£
£
£-1
i=O
i=O
1\ (F, T) = ( EB F£-i,i, EB T£-i,i) £-m
with
m
1\ (Fl,o) ® 1\ (FO,l) and with £-m m £-m-l m+l T£-m,m: 1\ (FI,o) ® I\(FO,I) --+ 1\ (Fl,D) ® 1\ (FO,l) ® D~(logS)
(7.3)
F£-m,m =
induced by T. Then F£'o = det(FI,O) and (det(Fl,O)) denotes the Higgs subbundle of I\£(F,T) generated by det(Fl,O). Writing
T(m) = T£-m+l ,m-l
0 ... 0
T£ ,0,
we define as a measure for the complexity of the Higgs field ~((F,T)):= Max{m E N; T(m)(det(Fl,o)) =1= O}
= Max{m
E N;
(det(Fl,o))£-m,m
For the Higgs bundle (E,O) of V, we write
~(V)
=1=
a}.
= ~((E, 0)).
E. VIEHWEG
268
LEMMA 7.6. Suppose in 7.1 that V satisfies the Arakelov equality and, using the notation from Theorem 7.4, that for i = i(V) the sheaf n i is of type B (or of type A). Then
(7.4)
2. PROPOSITION 8.3. In 7.1 one has the Milnor- Wood type inequality
(8.5)
(1
+ n)· /-L(El,o)
::; n· /-L(O}(logS)).
The equality implies that p = q . n and hence that (8.5) coincides with the Arakelov (in)equality. If SlI(OHlog S)) is stable for all v > 0, and if (8.5) is an equality, then the universal covering M of U is the complex ball SU (1, n) / K, and V is the tensor product of a unitary representation with the standard representation of SU(l, n).
PROOF. Let us repeat the argument used in [KM08aJ in the special case of a variation of Hodge structures of weight one, allowing logarithmic poles of the Higgs bundles along the normal crossing divisor S. As in the proof of Theorem 6.4 one starts with the maximal destabilizing /-L-semistable subsheaf G of El,o. Let G' be the image of G ® Ty( -log S) in EO,l. Then the /-L-semistability of G ® Ty ( -log S) and the choice of G imply
(8.6) (8.7)
/-L(G') 2': /-L(G) - /-L(O} (log S)), and
/-L(G) 2': /-L(El,o),
rk( G') ::; rk( G) . n.
Since (G EB G', OIGEBGI) is a Higgs subbundle of (E, 0) one finds
o 2': deg( G) + deg( G') =
rk( G) . /-L( G)
+ rk( G') . /-L( G')
2': (rk(G) + rk(G'))· /-L(G) - rk(G')· /-L(O}(logS)), hence
/-L(OHlogS)) 2':
(1 +
;:(~:)) . /-L(G)
2': (1
+~) . /-L(El,o),
ARAKELOV INEQUALITIES
273
as claimed. If n· j.t(n} (log S)) = (1 +n)·j.t(EI,O) one finds that all the inequalities in (8.6) and (8.7) are equalities. The first one and the irreducibility of V imply that G = EI,o and that G' = EO,1, whereas the last one shows that p = n· q for q = rk(EI,O) and p = rk(EO,1). Then p. q
(p+ q). n
q
n
+1
and the equality is the same as the Arakelov equality. Finally Lemma 8.1 allows to apply Theorem 7.7, in case that is j.t-stable and of type A or B.
n} (log S) 0
The situation considered in [KM08a] and [KM08b] is by far more general than the one studied in Proposition 8.3. Nevertheless the comparism of the inequalities (8.3) and (8.4) seems to indicate that an optimal MilnorWood inequality for for representations in SU(q,p) with q,p > 2 should have a slightly different shape. As said in Remark 7.8, it is likely that an interpretation of the equality will depend on a second numerical condition.
III. The proof of the Arakelov inequality (3.4) for k > 1 and the interpretation of equality break down if the rank of n} (log S) is larger than one. In the proof of Theorem 6.4 we used in an essential way that the weight of the variation of Hodge structures is one. For the Milnor-Wood inequality for a representation of the fundamental group of a higher dimensional manifold of general type with values in SU(p, q) one has to assume that Min{p, q} ::; 2, which excludes any try to handle variations of Hodge structures of weight k > 1 using methods, similar to the ones used in Example 5.3. So none of the known methods give any hope for a generalizations of the Arakelov inequality to variations of Hodge structures of weight k > lover a higher dimensional base. We do not even have a candidate for an Arakelov inequality. On the other hand, in the two known cases the inequalities are derived from the polystability of the Higgs bundles and the Arakelov equalities are equivalent to the Arakelov condition, defined in 2.2, iii). So for weight k > 1 over a higher dimensional base one should try to work directly in this set-up. Even for k > 1 and dim(Y) = 1, as discussed in Section 3, we do not really understand the geometric implications of the Arakelov equality (3.5), even less the possible implications of the Arakelov condition over a higher dimensional base. Roughly speaking, the Addendum 3.6 says that the irreducible subvariations of Hodge structures of weight k over a curve, which satisfy the Arakelov equality, look like subvariations of the variation of Hodge structures of weight k for a family of k-dimensional abelian varieties. However we do not see a geometric construction relating the two sides. IV. Can one extend the results of [MV08], recalled in Section 4, to higher dimensional bases? For example, assume that Ag is a Mumford compactification of a fine moduli scheme Ag with a suitable level structure and that 'P : Y -+ Ag is an embedding. Writing SAg for the boundary, assume
274
E. VIEHWEG
that (Y,8 = cp-1(8:;;{ )) satisfies the condition (*) in Assumption 6.1. So one 9 would like to characterize the splitting of the tangent map
in terms of the induced variation of Hodge structures, or in terms of geodecity of Yin A g • References [AR71] Arakelov, A.: Families of algebraic curves with fixed deneracies. Math. U.S.S.R. Izv.5 (1971) 1277-1302. [BVOO] Bedulev, E., Viehweg, E.: On the Shafarevich conjecture for surfaces of general type over function fields. Invent. Math. 139 (2000) 603-615. [BCHM] Birkar, C., Cascini, P., Hacon, C.-D., McKernan, J.: Existence of minimal models for varieties of log general type. Preprint, (2006). arXiv:math.AG/0610203 [Bo97] Borcea, C.: K3 surfaces with involution and mirror pairs of Calabi- Yau manifolds. In Mirror Symmetry II, Ams/IP Stud. Advanced. Math. 1, AMS, Providence, Rl (1997) 717-743. [BGG06] Bradlow, S.E., Garda-Prada, 0., Gothen, P.E.: Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces. Geom. Dedicata 122 (2006) 185-213. [De71] Deligne, P.: Theorie de Hodge II. LH.E.S. Pub!. Math. 40 (1971) 5-57. [De87] Deligne, P.: Un theoreme de finitude pour la monodromie. Discrete Groups in Geometry and Analysis, Birkhiiuser, Progress in Math. 67 (1987) 1-19. [Fa83] Faltings, G.: Arakelov's theorem for abelian varieties. Invent. math. 73 (1983) 337-348. [JZ02] Jost, J., Zuo, K: Arakelov type inequalities for Hodge bundles over algebraic varieties. 1. Hodge bundles over algebraic curves. J. Alg. Geom. 11 (2002) 535-546. [KL06] Kovacs, S., Lieblich, M.: Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevichs conjecture. Preprint, (2006). arXiv:math/0611672 [KM08a] Koziarz, V., Maubon, J.:Representations of complex hyper'bolic lattices into rank 2 classical Lie Groups of Hermitian type. Geom. Dedicata 137 (2008) 85-111. [KM08b] Koziarz, V., Maubon, J.: The Toledo invariant on smooth varieties of general type. Preprint, (2008). arXiv:081O.4805 [Li96] Liu, K: Geometric height inequalities. Math. Res. Lett. 3 (1996) 693-702. [LTYZ] Liu, K, Todorov, A., Yau, S.-T., Zuo, K: Shafarevich conjecture for CYmanifolds 1. Q. J. Pure App!. Math. 1 (2005) 28-67. [McM03] McMullen, C.: Billiards and Teichmiiller curves on Hilbert modular surfaces. Journal of the AMS 16 (2003) 857-885. [Mo06] Moller, M.: Variations of Hodge structures of Teichmiiller curves, J. Amer. Math. Soc. 19 (2006) 327-344. [Mo05] Moller, M.: Shimura and Teichmiiller curves. Preprint, (2005). arXiv:math/0501333 [MV08] Moller, M., Viehweg, E.: Kobayashi geodesics in Ag. Preprint, (2008). arXiv:0809.1018 [MVZ06] Moller, M., Viehweg, E., Zuo, K: Special families of curves, of Abelian varieties, and of certain minimal manifolds over curves. In: Global Aspects of Complex Geometry. Springer Verlag 2006, pp. 417-450. [MVZ07] Moller, M., Viehweg, E., Zuo, K: Stability of Hodge bundles and a numerical characterization of Shimura varieties. Preprint, (2007). arXiv:0706.3462
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[Mu69] Mumford, D.: A note of Shimura's paper: Discontinuous groups and Abelian varietes. Math. Ann. 181 (1969) 345-35l. [M u 77] Mumford, D.: Hirzebruch's proportionality theorem in the non-compact case. Invent. math. 42 (1977) 239-272. [PeOO] Peters, C.: Arakelov-type inequalities for Hodge bundles. Preprint, (2000). arXiv:math/0007102. [Si90] Simpson, C.: Harmonic bundles on noncompact curves. Journal of the AMS 3 (1990) 713-770. [Si92] Simpson, C.: Higgs bundles and local systems. Publ. Math. I.H.E.S. 75 (1992) 5-95. [STZ03] Sun, X.-T., Tan, S.L., Zuo, K.: Families of K3 surfaces over curves reaching the Arakelov- Yau type upper bounds and modularity. Math. Res. Lett. 10 (2003) 323-342. [Vi05] Viehweg, E.: Compactifications of smooth families and of moduli spaces of polarized manifolds. Annals of Math., to appear, arXiv:math/0605093 [VZ01] Viehweg, E., Zuo, K.: On the isotriviality of families of projective manifolds over curves. J. Alg. Geom. 10 (2001) 781-799. [VZ02] Viehweg, E., Zuo K.: Base spaces of non-isotrivial families of smooth minimal models. In: Complex Geometry (Collection of Papers dedicated to Hans Grauert) 279-328 Springer, Berlin Heidelberg New York (2002) [VZ03] Viehweg, E., Zuo, K.: Families over curves with a strictly maximal Higgs field. Asian J. of Math. 7 (2003) 575-598. [VZ04a] Viehweg, E., Zuo, K.: Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks. Survey in differential geometry VIII 337-356, International Press, 2004. [VZ04b] Viehweg, E., Zuo, K.: A characterization of certain Shimura curves in the moduli stack of abelian varieties. J. Diff. Geom. 66 (2004) 233-287. [VZ05] Viehweg, E., Zuo, K.: Complex multiplication, Griffiths- Yukawa couplings, and rigidy for families of hypersurfaces. J. Alg. Geom. 14 (2005) 481-528. [VZ06] Viehweg, E., Zuo, K.: Numerical bounds for semistable families of curves or of certain higher dimensional manifolds. J. Alg. Geom. 15 (2006) 771-79l. [VZ07] Viehweg, E., Zuo, K.: Arakelov inequalities and the uniformization of certain rigid Shimura varieties. J. Diff. Geom. 77 (2007) 291-352. [Vo93] Voisin, C.: Miroirs et involutions sur les surface K3. Journees de geometrie algebrique d' Orsay, Asterisque 218 (1993) 273-323. [Ya93] Yau, S.T.: A splitting theorem and an algebraic geometric characterization of locally Hermitian symmetric spaces. Comm. in Analysis and Geom. 1 (1993) 473-486. [Zi84] Zimmer, R.J.: Ergodic theory and semisimple groups. Birkhauser (1984). UNIVERSITAT DUISBURG-ESSEN, MATHEMATIK, 45117 ESSEN, GERMANY E-mail address:viehweg l. As Lian-Yau [199] showed, mirror maps in some way can be thought of as generalization of modular functions. The precise conditions under which it is is a modular function were determined by Doran in [85]. It is easy to see that the elliptic modular function j(T) is nothing but the mirror map for elliptic curves. j (T) satisfies a Schwarzian differential equation {j(T),T} = Q(j), where Q(j) is a certain rational function. And in fact, j can be uniquely determined by the differential equation. For certain families of K3 surfaces, Clingher-Doran-Lewis-Whitcher [68] derived the Schwarzian differential equation directly from geometry by studying the Picard-Fuchs equations over modular curves. Indeed, modularity of the mirror map implies integrality, and hence results for families of elliptic curves and K3 surfaces of generic Picard rank 19. However, only a handful of specially constructed families of Calabi-Yau three folds have classically modular mirror maps. Klemm-Lian-Roan-Yau [160] have also shown that mirror maps too satisfy similar, but higher order, nonlinear differential equations. These equations can be used to study divisibility property of the instanton numbers of Calabi-Yau threefolds. For example, it was shown that the instanton number nd predicted by the CDGP formula is divisible by 125 (at least for all d coprime to 5). If nd correctly counts the number of smooth rational curves in a general quintic, as expected, then the divisibility property of nd above supports a conjecture of Clemens. On another front, the mirror principle, developed by Lian-Liu-Yau [195, 196, 197] also has important application in birational geometry. For example, Lee-Lin-Wang [177] have used the mirror principle recently to study local models of Calabi-Yau manifolds in their study of analytic continuations of quantum cohomology rings under flops. Arithmetic properties of algebraic Calabi-Yau manifolds defined over finite fields and their mirrors have been studied. Focusing on the oneparameter 't/J family of Fermat quintic threefolds X1/J, Candelas, de la Ossa and Rodriguez-Villegas [53, 54] showed that the number of lFp-rational points can be computed in terms of the periods of the holomorphic threeform. They also found a closed form for the congruence zeta function which counts the number N r (X1/J) of lFpr -rational points. The zeta function is a rational function and the degrees of the numerator and denominator are exchanged between the zeta functions of X1/J and their mirror Y1/J' Interestingly, Wan [282] has proved that N r (X1/J) = N r (Y1/J) (mod pr) for arbitrary dimension Fermat Calabi-Yau manifolds and has conjectured that such relations should hold for all mirror pair Calabi-Yau manifolds in general. 5.5. Donaldson-Thomas invariants. Another duality on Calabi-Yau threefolds is based on the invariants introduced by Donaldson-Thomas [84]. Paired with the holomorphic three-forms on Calabi-Yau threefolds,
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Donaldson-Thomas introduced and studied the holomorphic Chern-Simons functional on the space of connections on vector bundles over Calabi-Yau threefolds. Their study leads to a collection of new invariants of Calabi-Yau threefolds, modulo some analytical technicality. These technicality can be by-passed in algebraic geometry using the moduli of stable sheaves and their virtual cycles. A special case is the moduli of rank one stable sheaves. This leads to the virtual counting of ideal sheaves of curves, which are referred to as Donaldson-Thomas invariants. (These invariants based on ideal sheaves of curves can be generalized to all smooth threefolds.) In [215], based on their explicit computation of such invariants for toric threefolds, MaulikNekrasov-Okounkov-Pandharipande (MNOP) conjectured that (the rank one version of) Donaldson-Thomas invariants is, in explicit form, equivalent to the GW invariants of the same varieties. Henceforth, DonaldsonThomas invariants provide integers underpinning for the rational GW invariants. Recently, Pandharipande and Thomas [234, 235] found a third curvecounting theory involving stable pairs. In order to define how to count these, one must think of curves as defining elements in the derived category of coherent sheaves, where they differ from the ideal sheaves of [215] by a wall crossing in the space of stability conditions [39]. The more transparent geometry has made this curve-counting easier to study, leading to progress [235] on a mathematical definition of the remarkable BPS invariants of Gopakumar-Vafa [112, 113], which give perhaps the best integer description of GW theory for threefolds. The interaction of the MNOP duality with mirror symmetry is a little mysterious. It relates GW invariants, which belong to the A-model of mirror symmetry, to counting objects of the derived category (which describes the B-model) on the same manifold rather than its mirror. The point is that these latter invariants are independent of complex structures (they are deformation invariant), but depend on the stability conditions, one would hope that such invariants are symplectic invariants in nature, like GW invariants. A purely symplectic construction of the gauge-theoretic invariants of Donaldson-Thomas would be an important advance in our understanding. Mirror symmetry would then relate this derived category picture to the Fukaya category of the mirror. Counting stable sheaves gets replaced by counting special Lagrangians, as proposed by Joyce [151]. His counts are invariant under deformations of symplectic structures, but undergo wall crossings as the complex structure varies. From physical considerations, Denef and Moore [73] have independently found formulas describing the wall crossing phenomena. They are important for the counting of BPS D-branes bound states in string theory. Specifically, Donaldson-Thomas invariants have been identified with the counting of bound states of a single D6-brane with D2- and DO-branes. Wall crossings are also relevant for making precise the Ooguri-Strominger-Vafa conjecture
296
S.-T. YAU
[227] which relates the topological string partition function with BPS Dbranes/black holes degeneracies. At the moment, wall crossing is a subject of much interest in both mathematics and physics, see for example [168, 106].
5.6. Stable bundles and sheaves. Stable holomorphic bundles and sheaves are important geometric objects on Calabi-Yau manifolds and give interesting invariants (e.g. Donaldson-Thomas invariants). Stable principal G-bundles are also necessary data for heterotic strings on Calabi-Yau manifolds and for various duality relations in string theory. The stability condition of Mumford-Takemoto and of Gieseker on sheaves ensures that the moduli space is quasi-projective. By the results of Narasimhan-Seshadri [226] for Riemann surfaces, and Donaldson [79], Uhlenbeck-Yau [277] for higher dimensions, there exist on stable (and poly-stable) bundles connections that solve the Hermitian-Yang-Mills equations. These equations are important for physical applications and requires that the (2,0) and (0,2) part of the curvature two-form vanish and the (1,1) part is traceless. In dimension one, the classification of vector bundles on an elliptic curve was due to Atiyah [7]. The set of isomorphism classes of indecomposable bundles of a fixed rank and degree is isomorphic to the elliptic curve. For general structure groups, Looijenga [205] and Bernstein-Shvartsman [25] showed that the moduli space of semistable G bundles for any simply-connected group G of rank r is a weight projective space of dimension r. In dimension two, Mukai [224, 225] studied in depth the moduli space MH (v) of Gieseker-semistable sheaves F on a smooth projective K3 surface (5, H). He showed that in case the moduli space MH (v) is smooth, it is symplectic. His insight also led to the powerful Fourier-Mukai transformation. Friedman-Morgan-Witten [95, 96, 97] constructed stable principal G-bundles on elliptic Calabi-Yau threefolds (see also Donagi [77] and Bershadsky-Johansson-Pantev-Sadov [28].) The construction is based on spectral covers [78] introduced on curves by Hitchin [131, 132]. The spectral data consists of a hypersurface and a line bundle over it. The spectral cover construction can be interpreted in terms of a relative Fourier-Mukai transformation and have been used extensively in string theory (see, for example [31, 36, 4] and references therein). Thomas [267], Andreas, Hernandez Ruiperez and Sanchez Gomez [5] have constructed stable bundles on K3 fibration Calabi-Yau threefolds. 5.7. Yau-Zaslow formula for K3 surfaces. In 1996, Yau and Zaslow [298] discovered a formula for the number of rational curves on K3 surfaces in terms of a quasi-modular form. Their method was inspired by string theory considerations. Let X be a K3 surface. Suppose C is a holomorphic curve in X representing a cohomology class [C]. We write its self-intersection number as [C] . [C] = 2d - 2 and its divisibility, or index, as r. If C is a smooth curve, then d is equal to the genus of C and also to the dimension of the linear
A SURVEY OF CALABI-YAU MANIFOLDS
297
system of G. If we denote the number of genus 9 curves in X representing [GJ as N g (d, r). Then the Yau-Zaslow formula says that when 9 = 0 they are given by the following formula,
The Yau-Zaslow formula was generalized by G6ttsche [114J to arbitrary projective surface. The universality for having such a formula for all surfaces was analyzed by Liu [200J using Seiberg-Witten theory which is related to the curve counting problem by the work of Taubes on GW = SW. The conjecture originated from a study by Yau and Zaslow on the BPS states in string theory on complex two dimensional Calabi-Yau manifolds, which are K3 surfaces. Shortly after the paper by Yau-Zaslow, Beauville [19J, and later Fantechi-G6ttsche-van Straten [92], rephrased and clarified the argument of Yau-Zaslow in algebraic geometry for primitive class. Chen [65] in 2002 proved that rational curves of primitive classes in general polarized K3 surfaces are nodal. Combined, these prove the Yau-Zaslow formula for primitive classes. The Yau-Zaslow formula is for all index r 2: 1. Following the original approach of Yau-Zaslow, Li-Wu [188] proved the conjecture for nonprimitive classes of index at most five under the assumption that all rational curves are nodal. Via a different approach, Bryan and Leung [41] proved the formula for the primitive case by considering elliptic K3 surfaces with section by computing the family GW invariants for the twistor family. These invariants are typically difficult to compute and they used a clever matching method to transport it to an enumerative problem for rational surfaces and then used Cremona transformations to further simplify it. Their method is more powerful than the sheaf-theoretic approach in that it works for any genus as well. Using a degeneration for the family GW invariants, J.H. Lee-Leung settled the r = 2 case of the Yau-Zaslow formula [174] and the genus one formula [175J. Recently Klemm, Maulik, Pandharipande and Scheidegger [159J proved the Yau-Zaslow formula for any classes by studying a particular Calabi-Yau threefold M with a K3 fibration. The Yau-Zaslow number can be related to the GW invariants on M representing fiber classes. Using localization techniques to compute these threefold invariants they proved the Yau-Zaslow formula. 5.B. Chern-Simons knot invariants, open strings and string dualities. Calabi-Yau geometry is the central object iIi string duality to unify different types of string theory. Mirror symmetry is just the duality between lIA and lIB string theory as discussed above. Using string duality
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between the large N Chern-Simons theory and the topological string theory of non-compact to ric Calabi-Yau manifolds, string theorists have made many striking conjectures about the moduli spaces of Riemann surfaces, ChernSimons knot invariants and GW invariants. Of note are two which have been rigorously proven. First, the Marino-Vafa conjecture [212] which expresses the generating series of triple Hodge integrals on moduli spaces of Riemann surfaces for all genera and any number of marked points in terms of the Chern-Simons knot invariants was proved by C.-C. Liu-K. Liu-Zhou in [201]. Second, the Labastilda-Marino-Ooguri-Vafa conjecture [229, 173, 172] which predicts integral and algebraic structures of the generating series of the SU(N) Chern-Simons quantum knot invariants was proved by Liu-Peng [203]. GW invariants for all genera and all degrees can be explicitly computed for non-compact toric Calabi-Yau manifolds via the theory of topological vertex. In [2], Aganagic, Klemm, Marino and Vafa proposed a theory to compute GW invariants in all genera and all degrees of any smooth noncompact toric Calabi-Yau threefold. In that paper, they first postulated the existence of open GW invariants that count holomorphic maps from bordered Riemann surfaces to C 3 with boundaries mapped to Lagrangian submanifolds, which they called the topological vertex; they then argued based on a physically derived duality between Chern-Simons theory and GW theory that the topological vertex can be expressed in terms of the explicitly computable Chern-Simons link invariants. Then by a gluing algorithm, they derived an algorithm computing all genera GW invariants of toric CalabiYau threefolds. In [184]' J. Li, C.-C. Liu, K. Liu and J. Zhou (LLLZ) developed the mathematical theory of the open GW invariants for toric Calabi-Yau threefold. (In the case compact Calabi-Yau threefolds, open GW invariants have only been defined in the case where the Lagrangian sub manifold is the fixed point set of an antiholomorphic involution [259]. See [280, 233] for calculations of open GW invariants on the Calabi-Yau quintic.) The definition of LLLZ relies on applying the relative GW invariants of J. Li [182, 183] to formal toric Calabi-Yau threefolds. By degenerating a formal toric Calabi-Yau to a union of simple ones, they derived an algorithm that expresses the open GW invariants of any (formal) toric Calabi-Yau in terms of that of the simple one. Their results express the open GW invariants in terms of explicit combinatorial invariants related to the Chern-Simons invariants. In many cases their combinatorial expressions coincide with those of [2], and they conjectured that the two combinatorial expressions should be equal in general. Later, a proof of this conjecture appeared in the work of Maulik-OblomkovOkounkov-Pandharipande [216]. Combined, all genera GW invariant for toric Calabi-Yau threefolds is solved. By using the results of [184], Peng [237] was able to prove the integrality conjecture of Gopakumar-Vafa for all formal toric Calabi-Yau manifolds.
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When applying the mirror principle to certain toric Calabi-Yau manifolds, we get the local mirror formulas of Chiang-Klemm-Yau-Zaslow [66] which are closely related to geometric engineering in string theory [153]. This is an important technique to recover gauge theory such as the SeibergWitten theory at various singularities in the moduli space of string theory [154]. Chiang-Klemm-Yau-Zaslow [66] also studied the asymptotic growth of genus zero Gromov-Witten invariants as the degree runs to infinity. Computational evidences have suggested in many cases a relationship between these growth rates and special values of L-functions. These observations have now been geometrically explained by Doran-Kerr [86], who showed, using higher Abel-Jacobi maps, that they follow from the deep mathematical conjectures of Beilinson-Hodge and Beilinson-Bloch. 6. Homological mirror symmetry
The Homological Mirror Symmetry (HMS) conjecture was made in 1994 by Maxim Kontsevich [164]. This was a proposal to give an explanation for the phenomena of mirror symmetry. This conjecture, very roughly, can be explained as follows. Let X and Y be a mirror pair of Calabi-Yau manifolds. We view X as a complex manifold and Y as a symplectic manifold. The idea is that mirror symmetry provides an isomorphism between certain aspects of complex geometry on X and certain aspects of symplectic geometry on Y. More precisely, Kontsevich suggested that the bounded derived category of coherent sheaves on X is isomorphic to the Fukaya category of Y. The first object has been well-studied, and is known to capture a significant amount of information about the complex geometry on X, while the Fukaya category is a much less familiar object introduced by Fukaya [100] in a 1993 paper. This is not a true category, but something known as an Aoo cateogry: the composition of morphisms is not associative, but only associative up to homotopy. The Fukaya category captures information about the symplectic geometry of Y. Its objects are Lagrangian submanifolds of Y and morphisms come from intersection points of Lagrangian submanifolds. Compositions involve counting holomorphic disks, and essentially arise from the product in Floer homology. The homological mirror symmetry conjecture has remained an imposing problem. There have been a number of different threads of work devoted to this. Work of a number of researchers, especially Polishchuk and Zaslow [242] and Fukaya [101]' dealt with the simplest cases, namely mirror symmetry for elliptic curves and abelian varieties, respectively. Other work has been devoted to clarifying the conjecture: at first sight, the two categories cannot be isomorphic since the derived category is an actual triangulated category, while the Fukaya category is not an actual category and is not likely to be triangulated. There are various ways around these issues, and there are now precise rigorous statements. Most significantly, the work of Seidel [254] has proved the conjecture for quartic surfaces in projective three-space.
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The HMS conjecture implies that complex manifolds which have equivalent bounded derived categories are mirrored to the same manifold. These manifolds, related by Fourier-Mukai transforms, are called Fourier-Mukai partners. In complex dimension one, Orlov [230] has determined both the group of autoequivalences and the Fourier-Mukai partners of an abelian variety. Interesting results have also known for K3 surfaces. Mukai [224] long ago showed that the Fourier-Mukai partners of a given K3 surface is again a K3. The Fourier-Mukai transform induces a Hodge isometry of the "Mukai lattice" of K3 [231]. Bridgeland and Maciocia [40] have shown that the number of Fourier-Mukai partners of any given K3 is finite. Hosono, Lian, Oguiso, and Yau [142] have recently, given an explicit counting formula for this number. A similar formula was given for abelian surfaces and was used to answer an old question of T. Shioda [140]. They have also given a description for the group of autoequivalences of the bounded derived category of a K3 surface [141]. It turns out that the Fourier-Mukai number formula is closely related to the class numbers of imaginary quadratic fields of prime discriminants [142]. There is also a nice analogue for real quadratic fields. As shown in [143], the real case turns out to be crucial for classifying c = 2 rational toroidal conformal field theory in physics. The HMS conjecture for Calabi-Yau manifolds has been generalized to Fano varieties. For toric varieties, the work of Abouzaid [1] established part of the conjecture and was recently settled by Fang-Liu-Treumann-Zaslow [91]. Moreover, for surfaces, Auroux-Katzarkov-Orlov [9, 10] have proved the HMS conjecture for some toric surfaces (Le. weighted projective planes, Hirzebruch surfaces, and toric blowups of p2) and also non-toric del Pezzo surfaces. Another thread has been addressing the question of how more traditional aspects of mirror symmetry, such as holomorphic curve counting, would follow from homological mirror symmetry.
7. SYZ geometric interpretation of mirror symmetry 7.1. Special Lagrangian snbmanifolds in Calabi-Yan manifolds. By the Wirtinger formula for Kahler manifolds, every complex submanifold in X is absolute volume minimizing. This is a special case of calibration, a notion introduced by Harvey and Lawson [128] in analyzing area-minimizing subvarieties, and later on rediscovered in physics by Becker-BeckerStrominger [21] from supersymmetry considerations. Special Lagrangian submanifolds in Calabi-Yau manifolds form another class of examples of calibrated submanifolds. A real n-dimensional submanifold L in X is called special Lagrangian if the restrictions of both wand 1m n to L are zero:
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As calibrated submanifolds, special Lagrangian submanifolds are always absolute volume minimizing. 7.2. The SYZ conjecture - SYZ transformation. In string theory, each Calabi-Yau threefold X determines two twisted theories, one A -model and another B-model. The mirror symmetry between X and its mirror Y interchanges the two models between them. From the mathematical perspective, A-model is about the symplectic geometry of X and B-model is about the complex geometry of Y. A-model on X (symplectic geometry)
(
.
)
mIrror symmetry
B-model on Y (complex geometry)
The search for the underlying geometric root of this symmetry led Strominger, Yau and Zaslow to their conjecture. In 1996, Strominger, Yau and Zaslow [265] proposed that for a mirror pair (X, Y) that is near a large volume/complex structure limit, (1) both admit special Lagrangian torus fibrations with sections: T
dual tori T* (
)
t
t
X
Y
t
t
B
B*
(2) the two torus fibrations are dual to each other; (3) a fiberwise Fourier-Mukai transformation along fibers interchanges the symplectic (resp. complex) geometry on X with the complex (resp. symplectic) geometry on Y. This is called the SYZ mirror transformation. On the nutshell, it says that the mysterious mirror symmetry is simply a Fourier transform. The quantum corrections, for instance the GW invariants, come from the higher Fourier modes. The SYZ conjecture inspired a flourish of work to understand mirror symmetry, which include works of Gross (and with Siebert) [122, 123, 124, 125, 126], Joyce [150, 152], KontsevichSoibelman [166, 167], Vafa [278], Leung-Yau-Zaslow [180] and manyothers. On the other hand, it has led to new developments of other branches of mathematics, including the calibrated geometry of special Lagrangian submanifolds and the affine geometry with singularities. The work of Auroux has shed some lights on the phenomenon of quantum corrections [8]. 7.3. Special Lagrangian geometry. Special Lagrangian submanifolds coupled with unitary flat bundles are branes in A-model in string theory. These geometric objects are crucial to the understanding of the SYZ conjecture. So far, many examples were constructed using cohomogeneity one method by Joyce [150], using singular perturbation method by
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Butscher [44], Lee [176], Haskins-Kapouleas [129] and others. Their deformations are studied by McLean [217]; their moduli spaces by Hitchin [133]; their existence by Schoen-Wolfson [253] using variational approach and by Smoczyk and M.-T. Wang [261] using mean curvature flow. Thomas-Yau [268] formulated a conjecture on the existence and uniqueness of special Lagrangian submanifolds which is the mirror of the theorem of Donaldson, Uhlenbeck and Yau [79, 277] of the existence of unique Hermitian YangMills connection on any stable holomorphic vector bundle. 7.4. Special Lagrangian fibrations. SYZ conjecture predicts that mirror Calabi-Yau manifolds should admit dual torus fibrations whose fibers are special Lagrangian submanifolds, possibly with singularities. Lagrangian fibrations is an important notion in symplectic geometry as real polarizations, as well as in dynamical system as completely integrable systems. Their smooth fibers admit canonical integral affine structures and therefore they must be tori in the compact situation. Toric varieties JP>.6., for instance CJP>n+l, are examples of Lagrangian fibrations in which the fibers are orbits of an Hamiltonian torus action and the base is a convex polytope ~. A complex hypersurface X = {f = O} in CJP>n+l is a Calabi-Yau manifold if deg f = n + 2. The most singular ones is when X is a union of coordinate hyperplanes in CJP>n+1, which is an example of the large complex structure limit. Such limiting points on the moduli space are important and an explicit construction of them for Calabi-Yau toric hypersurfaces as T-fixed points on the moduli space has been given by Hosono-Lian-Yau [145]. A numerical criterion for the large complex structure limit in anyone parameter family of Calabi-Yau manifolds has also been given by Lian-Todorov-Yau [193]. At this most singular limit, X inherits a torus fibration from the toric structure on CJP>n+1. Thus one can try to perturb this to obtain Lagrangian fibration structures on nearby smooth Calabi-Yau manifolds. This approach was carried out by Gross [124], Mikhalkin [219]' Ruan [247, 248] and Zharkov [302]. This approach can be generalized to Calabi-Yau hypersurfaces X in any Fano toric variety JP>.6.. Furthermore, their mirror manifolds Yare CalabiYau hypersurfaces in another Fano toric variety JP>V' whose defining polytope is the polar dual to ~. The situation is quite different for Calabi-Yau twofolds, namely K3 surfaces, or more generally for hyperkahler manifolds. In this case, the CalabiYau metric on X is Kahler with respect to three complex structures I, J and K. When X admits a J-holomorphic Lagrangian fibration, then this fibration is a special Lagrangian fibration with respect to the Kahler metric WI, as well as WK. Furthermore, SYZ also predicts that mirror symmetry is merely a twistor rotation from I to K in this case. For K3 surfaces, there are plenty of elliptic fibrations and they are automatically complex Lagrangian fibrations because of their low dimension. Furthermore Gross and Wilson [127] described the Calabi-Yau metrics for generic elliptic K3 surfaces by using the singular perturbation method. They used model metrics which
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were constructed by Greene, Shapere, Vafa and Yau [121] away from singular fibers and by Ooguri and Vafa [228] near singular fibers. 7.5. The SYZ transformation. Recall that SYZ conjecture says that mirror symmetry is a Fourier-Mukai transformation along dual special Lagrangian torus fibrations. We also need to include a Legendre transformation on the base affine manifolds. This SYZ transformation was generalized to the mirror symmetry for local Calabi-Yau manifolds by Leung-Vafa [179]. On the mathematical side, Leung-Yau-Zaslow [180] and Leung [178] used the SYZ transformation to verify various correspondences between symplectic geometry and complex geometry between semi-fiat Calabi-Yau manifolds when there is no quantum corrections. To include quantum corrections in the SYZ transformation for Calabi-Yau manifolds is a more difficult problem. In the Fano case, there are recent results on applying the SYZ transformation with quantum corrections by Auroux [8], Chan-Leung [63] and Fang [90]. 7.6. The SYZ conjecture and tropical geometry. Work of Joyce [152] forced a rethinking of the SYZ conjecture in a limiting setting. The SYZ mirror transformation is now believed to be applicable near the large complex structure limit points. Two groups of researchers, Gross and Wilson [127] on the one hand and Kontsevich and Soibelman [166] on the other, suggested that near a large complex structure limit of n-dimensional Calabi-Yau manifolds, the Ricci-flat metric on the Calabi-Yau manifold converges (in a precise sense known as Gromov-Hausdorff convergence) to an n-dimensional sphere. For example, in the simplest case of an elliptic curve (a real two-dimensional torus), the torus gets thinner as the large complex structure limit is approached, until it converges to a circle. Therefore, the idea is that in the large complex structure limit, the SYZ fibration is expected to be better behaved though the fibers of the SYZ fibration will collapse, with its volume going to zero in the limit. In any event, once one has this picture of a collapsing fibration, one can ask for a description of the behavior of holomorphic curves in the fibration as the fibres collapse. The expectation is that a holomorphic curve converges to a piecewise linear graph on the limiting sphere. This graph should satisfy certain conditions which turn this graph into what is now known as a "tropical curve." This terminology arises from the "tropical semiring", which is the semiring consisting of real numbers, with addition given by maximum and multiplication given by the usual addition. Tropical varieties are then defined by polynomials over the tropical semiring, and the "zeroes" of a tropical polynomial are in fact points where the piecewise linear function defined by the tropical polynomial is not smooth. This gives rise to piecewise linear varieties, and tropical curves arising as limits of holomorphic curves are examples of such.
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This picture began to emerge in the works of Fukaya [102]' Kontsevich and Soibelman [166] around 2000. In particular, Kontsevich's suggestion that one could count holomorphic curves by counting tropical curves was realized in 2003 by Mikhalkin [220], when he showed that curves in toric surfaces could be counted using tropical geometry. For the purposes of mirror symmetry, it is then important to understand how tropical geometry arises on the mirror side. The initial not so rigorous work of Fukaya in 2000 gave some suggestions as to how this might happen in two dimensions. This was followed by the work of Kontsevich and Soibelman [167] in 2004, again in two dimensions, and the work of Gross and Siebert [126] in 2007 in all dimensions, which demonstrate that the geometry of Calabi-Yau manifolds near large complex structure limits can be described in terms of data of a tropical nature. This provides the clearest link to date between the two sides of mirror symmetry.
8. Geometries related to Calabi-Yau manifolds 8.1. Non-Kahler Calabi-Yau manifolds. Given a smooth three dimensional complex manifold X with trivial canonical line bundle, i.e. Kx ~ Ox. When X is Kahler, Yau's theorem [295] provides a unique Ricci-flat Kahler metric in each Kahler class. A large class of such three folds which are non-Kahler are obtained by Clemens [67] and Friedman [94] from Calabi-Yau threefolds by an operation called extremal transition or its inverse. An extremal transition is a composition of blowing down rational curves and smoothing the resulting singularity. It has the effect of decreasing the dimension of H2 (X, JR) and increasing the dimension of H3 (X, JR) while keeping their sum fixed. For example, the connected sum of k copies of 8 3 x 8 3 for any k ~ 2 can be given a complex structure in this way. Based on this construction, Reid [244] speculated that any two Calabi-Yau threefolds are related by deformations, extremal transitions and their inverses, even though their topologies are different. This speculation demonstrates the potential role of non-Kahler complex manifolds. It is important to construct canonical metrics on such non-Kahler manifolds which are counterparts of Ricci-flat Kahler metrics on Calabi-Yau manifolds. In 1986, Strominger proposed for supersymmetric compactification in the theory of heterotic string a system of a pair (w, h) of a Hermitian metric w on a complex three-dimensional manifold X with a non-vanishing holomorphic three form n and a Hermitian metric h on a vector bundle V on X. The Strominger system is such a pair satisfying the elliptic system of differential equations,
d(llnllw w2 ) = 0, F 1\ w2 = 0 ,
F 2 ,0
= FO,2 = 0 ,
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where R (resp. F) is the curvature of w (resp. h). The first equation is equivalent to the existence of a balanced metric, also the same as the existence of supersymmetry. The system of equations in the second line is the Hermitian-Yang-Mills equations. When V is the tangent bundle Tx and w is Kahler, the system is solved by the Calabi-Yau metric. Using perturbation method, J. Li and S.-T. Yau [189] constructed smooth solutions to a class of Kahler Calabi-Yau with irreducible solutions for vector bundles with gauge group 8U (4) and 8U (5). The first existence result for solutions of Strominger system for a non-Kahler Calabi-Yau was due to Fu-Yau on a class of torus bundles over K3 surfaces [99, 20]. (The construction of the complex structure is called the Calabi-Eckmann construction [47] and was carried out by GoldsteinProkushkin [111]. Based on physical arguments of superstring dualities, the existence of such solutions was suggested in [71, 22].) Mathematical construction of balanced metrics on manifolds constructed by ClemensFriedman was recently carried out rigorously by Fu-Li-Yau [98].
8.2. Symplectic Calabi-Yau manifolds. Another generalization of Calabi-Yau manifolds are symplectic Calabi-Yau manifolds. Recall a symplectic manifold (X, w) is an even dimensional (real) manifold X with w a closed, non-degenerate 2-form on X. Examples of symplectic manifolds include Kahler manifolds. Using any compatible almost complex structure on X, we can define the first Chern class CI (X) for any symplectic manifold X. Symplectic Calabi-Yau manifolds are symplectic manifolds with CI (X) = O. In dimension four, we have the Kodaira-Thurston examples; the homological type of such symplectic manifolds are classified, due to the work of T.-J. Li [192]' and to Bauer [17], that their Betti numbers are in the range bl ::; 4, bt ::; 3 and b"2 ::; 19. To their smooth structures, it is conjectured that the diffeomorphism types of such manifolds are either Kahler surfaces with zero Kodaira dimension or oriented torus bundles over torus. In higher dimensions, Smith-Thomas-Yau [262] has constructed many such examples of symplectic Calabi-Yau manifolds. They contain structures which are mirror to complex non-Kahler Calabi-Yau structures on connected sums of 8 3 x 8 3 . As described in [262], the symplectic mirror of the Clemens-Friedman construction reverses the conifold transition by first collapsing Lagrangian three-spheres and then replacing them by symplectic two-spheres. If one can collapse all three-spheres, then such a process should result in symplectic Calabi-Yau structures on connected sums of CJP>3. As the Strominger-Fu-Yau geometry on complex non-Kahler Calabi-Yau manifolds plays an important role in string theory, it is expected to have a dual system on these symplectic Calabi-Yau manifolds which will also play an important role in string theory. One can also generalize the Ricci-flat condition in dimension four. Donaldson conjectured in [82] that an analogue of the Calabi-Yau theorem should hold on symplectic 4-manifolds. If it is true, there are interesting
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applications to symplectic topology in dimension four. So far relatively little is known about this conjecture, but some progress has been made in [290] and [276]. There it is shown that the conjecture holds when the manifold is nonnegatively curved, so for example on C]p>2 with a small perturbation of the standard Kahler structure.
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