Surveys in Differential Geometry, Vol. 13: Geometry, Analysis, and Algebraic Geometry

Volume XIII

Surveys in Differential Geometry Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry

edited by Huai-Dong Cao and Shing-Tung Yau

i?

International Press

Surveys in Differential Geometry, Vol. 13

International Press P.O. Box 43502 Somerville, MA 02143 www.intlpress.com Copyright © 2009 by International Press 2000 Mathematics Subject Classification: 53C44

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Huai-Dong Cao and Shing-Thng Yau, editors

ISBN: 978-1-57146-138-4 Typeset using the LaTeX system. Printed in the USA on acid-free paper.

Surveys in Differential Geometry XIII

Preface

Publication of the Journal of Differential Geometry began in 1967, and for the past 40 years it has flourished, coming to be regarded as the leading journal on the subject. In 1980, C.-C. Hsiung asked me to be chief editor of the JDG. I was reluctant, for many obstacles presented themselves. However, many great geometers came to the support of the journal. Almost immediately after establishing a new editorial board, we were able to obtain the highly regarded works of Freedman, Donaldson, Schoen, Uhlenbeck, Witten, etc. The editors continue to be deeply grateful for the tremendous support lent by these authors and many others. The quality of the JDG is as strong as ever. (A few years ago, in fact, we found it necessary to increase the journal's size, that we might do justice to the depth and excellence of the submissions received.) The JDG's 40th anniversary was celebrated at the the Seventh Conference on Geometry and Topology, held at Harvard University in May 2008. The outstanding survey papers making up this volume are the outcome of that conference. Professor C.-C. Hsiung has just passed away. We would like to dedicate this volume to the memory of him. He was the founding father of the journal, and editor-in-chief these past 40 years. His contribution is immeasurable. Shing-Tung Yau Harvard University June 2009

Surveys in Differential Geometry XIII

Contents

Preface.........................................................

v

Special Lagrangian fibrations, wall-crossing, and mirror symmetry Denis Auroux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Sphere theorems in geometry Simon Brendle and Richard Schoen. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Geometric Langlands and non-abelian Hodge theory R. Donagi and T. Pantev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Developments around positive sectional curvature Karsten Grove..................................................

117

Einstein metrics, four-manifolds, and conformally Kahler geometry Claude LeBrun..................................... ............

135

Existence of Faddeev knots in general Hopf dimensions Fengbo Hang, Fanghua Lin, and Yisong yang...................

149

Milnor K2 and field homomorphisms Fedor Bogomolov and Yuri Tschinkel. . . . . . . . . . . . . . . . . . . . . . . . . . . .

223

Arakelov inequalities Eckart Viehweg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

A survey of Calabi-Yau manifolds Shing-Tung Yau................................................

277

Surveys in Differential Geometry XIII

Special Lagrangian fibrations, wall-crossing, and mirror symmetry Denis Auroux ABSTRACT. In this survey paper, we briefly review various aspects of the SYZ approach to mirror symmetry for non-Calabi-Yau varieties, focusing in particular on Lagrangian fibrations and wall-crossing phenomena in Floer homology. Various examples are presented, some of them new.

CONTENTS

1. 2.

Introduction Lagrangian tori and mirror symmetry 2.1. Lagrangian tori and the SYZ conjecture 2.2. Beyond the Calabi-Yau case: Landau-Ginzburg models 2.3. Example: Fano toric varieties 3. Examples of wall-crossing and instanton corrections 3.1. First examples 3.2. Beyond the Fano case: Hirzebruch surfaces 3.3. Higher dimensions 4. Floer-theoretic considerations 4.1. Deformations and local systems 4.2. Failure of invariance and divergence issues 5. Relative mirror symmetry 5.1. Mirror symmetry for pairs 5.2. Homological mirror symmetry 5.3. Complete intersections References

2 3 3 6 10 12 12 16 24 30 30 34 38 38 41 44 45

This work was partially supported by NSF grants DMS-0600148 and DMS-0652630. ©2009 International Press

2

D.AUROUX

1. Introduction

While mirror symmetry first arose as a set of predictions relating Hodge structures and quantum cohomology for Calabi-Yau 3-folds (see e.g. [8, 13]), it has since been extended in spectacular ways. To mention just a few key advances, Kontsevich's homological mirror conjecture [26] has recast mirror symmetry in the language of derived categories of coherent sheaves and Fukaya categories; the Strominger-Yau-Zaslow (SYZ) conjecture [39] has provided the basis for a geometric understanding of mirror symmetry; and mirror symmetry has been extended beyond the Calabi-Yau setting, by considering Landau-Ginzburg models (see e.g. [23, 27]). In this paper, we briefly discuss various aspects of mirror symmetry from the perspective of Lagrangian torus fibrations, i.e. following the StromingerYau-Zaslow philosophy [39]. We mostly focus on the case of Kahler manifolds with effective anticanonical divisors, along the same general lines as [4]. The two main phenomena that we would like to focus on here are, on one hand, wall-crossing in Floer homology and its role in determining "instanton corrections" to the complex geometry of the mirror; and on the other hand, the possibility of "transferring" mirror symmetry from a given Kahler manifold to a Calabi-Yau submanifold. The paper is essentially expository in nature, expanding on the themes already present in [4]. The discussion falls far short of the level of sophistication present in the works of Kontsevich-Soibelman [28, 29], Gross-Siebert [18, 19], or Fukaya-Oh-Ohta-Ono [14, 15]; rather, our goal is to show how various important ideas in the modern understanding of mirror symmetry naturally arise from the perspective of a symplectic geometer, and to illustrate them by simple examples. Accordingly, most of the results mentioned here are not new, though to our knowledge some of them have not appeared anywhere in the literature. Another word of warning is in order: we have swept under the rug many of the issues related to the rigorous construction of Lagrangian Floer theory, and generally speaking we take an optimistic view of issues such as the existence of fundamental chains for moduli spaces of discs and the convergence of various Floer-theoretic quantities. These happen not to be issues in the examples we consider, but can be serious obstacles in the general case. The rest of this paper is organized as follows: in Section 2 we review the SYZ approach to the construction of mirror pairs, and the manner in which the mirror superpotential arises naturally as a Floer-theoretic obstruction in the non Calabi-Yau case. Section 3 presents various elementary examples, focusing on wall-crossing phenomena and instanton corrections. Section 4 discusses some issues related to convergent power series Floer homology. Finally, Section 5 focuses on mirror symmetry in the relative setting, namely for a Calabi-Yau hypersurface representing the anticanonical class inside a Kahler manifold, or more generally for a complete intersection.

SPECIAL LAGRANGIAN FIBRATIONS

3

Acknowledgements. The ideas presented here were influenced in a decisive manner by numerous discussions with Mohammed Abouzaid, Paul Seidel, and Ludmil Katzarkov. Some of the topics presented here also owe a lot to conversations with Dima Orlov, Mark Gross, and Kenji Fukaya. Finally, I am grateful to Mohammed Abouzaid for valuable comments on the exposition. This work was partially supported by NSF grants DMS-0600148 and DMS-0652630. 2. Lagrangian tori and mirror symmetry 2.1. Lagrangian tori and the SYZ conjecture. The SYZ conjecture essentially asserts that mirror pairs of Calabi-Yau manifolds should carry dual special Lagrangian torus fibrations [39]. This statement should be understood with suitable qualifiers (near the large complex structure limit, with instanton corrections, etc.), but it nonetheless gives the basic template for the geometric construction of mirror pairs. From this perspective, to construct the mirror of a given Calabi-Yau manifold X, one should first try to construct a special Lagrangian torus fibration f : X -+ B. This is a difficult problem, but assuming it has been solved, the first guess for the mirror manifold XV is then the total space of the dual fibration fV. Given a torus T, the dual torus TV = Hom(1f1 (T), 8 1) can be viewed as a moduli space of rank 1 unitary local systems (i.e., flat unitary connections up to gauge equivalence) over T; hence, points of the dual fibration parametrize pairs consisting of a special Lagrangian fiber in X and a unitary local system over it. More precisely, let (X, J,w) be a Kahler manifold of complex dimension n, equipped with a nonvanishing holomorphic volume form 0 E on,O(x). This is sometimes called an "almost Calabi-Yau" manifold (to distinguish it from a genuine Calabi-Yau, where one would also require the norm of 0 with respect to the Kahler metric to be constant). It is an elementary fact that the restriction of 0 to a Lagrangian submanifold LeX is a nowhere vanishing complex-valued n-form. DEFINITION 2.1. A Lagrangian submanifold LeX is special Lagrangian if the argument of OIL is constant. The value of the constant depends only on the homology class of L, and we will usually normalize 0 so that it is a multiple of 1f /2. For simplicity, in the rest of this paragraph we will assume that OIL is a positive real multiple of the real volume form volg induced by the Kahler metric g = w(·, J.). The following classical result is due to McLean [32] (in the Calabi-Yau setting; see §9 of [24] or Proposition 2.5 of [4] for the almost Calabi-Yau case): PROPOSITION 2.2 (McLean). Infinitesimal special Lagrangian deformations of L are in one to one correspondence with cohomology classes in H1(L,JR). Moreover, the deformations are unobstructed.

D.AUROUX

4

Specifically, a section of the normal bundle v E Coo (N L) determines a I-form a = -tvW E Ol(L,IR) and an (n -I)-form f3 = tvImO E on-l(L,IR). These satisfy f3 = 'ljJ *9 a, where 'ljJ E Coo(L, 1R+) is the ratio between the volume elements determined by 0 and g, i.e. the norm of 0 with respect to the Kahler metric; moreover, the deformation is special Lagrangian if and only if a and f3 are both closed. Thus special Lagrangian deformations correspond to "'ljJ-harmonic" I-forms -tvW E

1i~(L)

= {a E Ol(L,IR) Ida = 0,

d*('ljJa)

= o}.

In particular, special Lagrangian tori occur in n-dimensional families, giving a local fibration structure provided that nontrivial 'ljJ-harmonic I-forms have no zeroes. The base B of a special Lagrangian torus fibration carries two natural affine structures, which we call "symplectic" and "complex". The first one, which encodes the symplectic geometry of X, locally identifies B with a domain in HI (L, 1R) (L ~ Tn). At the level of tangent spaces, the cohomology class of -tvW provides an identification ofTB with Hl(L, 1R); integrating, the local affine coordinates on B are the symplectic areas swept by loops forming a basis of HI (L). The other affine structure encodes the complex geometry of X, and locally identifies B with a domain in Hn-l(L,JR). Namely, one uses the cohomology class of tvIm 0 to identify T B with H n - l (L, 1R), and the affine coordinates are obtained by integrating 1m 0 over the n-chains swept by cycles forming a basis of H n-l (L ). The dual special Lagrangian fibration can be constructed as a moduli space M of pairs (L, \7), where LeX is a special Lagrangian fiber and \7 is a rank I unitary local system over L. The local geometry of M is wellunderstood (cf. e.g. [20, 30, 17]); in particular we have the following result (cf. e.g. §2 of [4]): 2.3. Let M be the moduli space of pairs (L, \7), where L is a special Lagrangian torus in X and \7 is a fiat U(I) connection on the trivial complex line bundle over L up to gauge. Then M carries a natural integrable complex structure JV arising from the identification PROPOSITION

T(L,\l)M

= {(v, a)

E

Coo(NL) EB Ol(L,JR) I -

[,vW

+ ia E 1i~(L) 0

C},

a holomorphic n-form

ov ( (VI, aI), ... , (Vn, an)) = [( - t

v1

W + ial) /\ ... /\ ( -

[,v n

W + ian),

and a compatible Kahler form WV((vl,al), (v2,a2))

= [a2 /\ [,vl ImO -

(this formula for WV assumes that

al /\ [,v2 ImO

fL Re 0 has been suitably normalized).

SPECIAL LAGRANGIAN FIBRATIONS

5

In particular, M can be viewed as a complexification of the moduli space of special Lagrangian submanifolds; forgetting the connection gives a projection map jV from M to the real moduli space B. The fibers of this projection are easily checked to be special Lagrangian tori in the almost Calabi-Yau manifold (M,Jv ,w v ,f2 V). This special Lagrangian fibration on M is fiberwise dual to the one previously considered on X; they have the same base B, and passing from one fibration to the other simply amounts to exchanging the roles of the two affine structures on B. In real life, unless we restrict ourselves to complex tori, we have to consider special Lagrangian torus fibrations with singularities. The base of the fibration is then a singular affine manifold, and the picture discussed above only holds away from the singularities. A natural idea would be to obtain the mirror by first constructing the dual fibration away from the singularities, and then trying to extend it over the singular locus. Unfortunately, this cannot be done directly; instead we need to modify the complex geometry of M by introducing instanton corrections. To give some insight into the geometric meaning of these corrections, consider the SYZ conjecture from the perspective of homological mirror symmetry. Recall that Kontsevich's homological mirror symmetry conjecture [26] predicts that the derived category of coherent sheaves DbCoh(XV) of the mirror XV is equivalent to the derived Fukaya category of X. For any point p E Xv, the skyscraper sheaf Op is an object of the derived category. Since Ext * (Op, Op) ~ H*(Tnj C) (as a graded vector space), we expect that Op corresponds to some object £p of the derived Fukaya category of X such that End(£p) ~ H*(Tn). It is natural to conjecture that, generically, the object £p is a Lagrangian torus in X with trivial Maslov class, equipped with a rank 1 unitary local system, and such that HF*(£p,£p) ~ H*(Tn) (as a graded vector space). This suggests constructing the mirror XV as a moduli space of such objects of the Fukaya category of X. (However it could still be the case that some points of XV cannot be realized by honest Lagrangian tori in X.) In the Calabi-Yau setting, it is expected that "generically" (Le., subject to a certain stability condition) the Hamiltonian isotopy class of the Lagrangian torus £p should contain a unique special Lagrangian representative [40, 41]. Hence it is natural to restrict one's attention to special Lagrangians, whose geometry is richer than that of Lagrangians: for instance, the moduli space considered in Proposition 2.3 carries not only a complex structure, but also a symplectic structure. However, if we only care about the complex geometry of the mirror XV and not its symplectic geometry, then it should not be necessary to consider special Lagrangians. On the other hand, due to wall-crossing phenomena, the "convergent power series" version of Lagrangian Floer homology which is directly relevant to the situation here is not quite invariant under Hamiltonian isotopies

D.AUROUX

6

(see e.g. [10], and Section 4 below). Hence, we need to consider a corrected equivalence relation on the moduli space of Lagrangian tori in X equipped with unitary local systems. Loosely speaking, we'd like to say that two Lagrangian tori (equipped unitary local systems) are equivalent if they behave interchangeably with respect to convergent power series Floer homology; however, giving a precise meaning to this statement is rather tricky.

2.2. Beyond the Calabi-Yau case: Landau-Ginzburg models. Assume now that (X, J, w) is a Kahler manifold of complex dimension n, and that D c X is an effective divisor representing the anticanonical class, with at most normal crossing singularities. Then the inverse of the defining section of D is a section of the canonical line bundle Kx over X \ D, i.e. a holomorphic volume form n E nn,O(X \ D) with simple poles along D. We can try to construct a mirror to the almost Calabi-Yau manifold X\D just as above, by considering a suitable moduli space of (special) Lagrangian tori in X \ D equipped with unitary local systems. The assumption on the behavior of n near D is necessary for the existence of a special Lagrangian torus fibration with the desired properties: for instance, a neighborhood of the origin in C equipped with n = zk dz does not contain any compact special Lagrangians unless k = -1. Compared to X \ D, the manifold X contains essentially the same Lagrangians. However, (special) Lagrangian tori in X \ D typically bound families of holomorphic discs in X, which causes their Floer homology to be obstructed in the sense of Fukaya-Oh-Ohta-Ono [14]. Namely, Floer theory associates to £ = (L, \7) (where L is a Lagrangian torus in X \ D and \7 is a flat U(l) connection on the trivial line bundle over L) an element mo(£) E CF*(£, C), given by a weighted count of holomorphic discs in

(X,L). More precisely, recall that in Fukaya-Oh-Ohta-Ono's approach the Floer complex CF*(£, £) is generated by chains on L (with suitable coefficients), and its element mo(£) is defined as follows (see [14] for details). Given a class (3 E 1T"2(X, L), the moduli space Mk(L, (3) of hoi omorphic discs in (X, L) with k boundary marked points representing the class (3 has expected dimension n - 3 + k + f..L({3), where f..L({3) is the Maslov index; when LeX \ D is special Lagrangian, f..L({3) is simply twice the algebraic intersection number {3·[D] (see e.g. Lemma 3.1 of [4]). This moduli space can be compactified by adding bubbled configurations. Assuming regularity, this yields a manifold with boundary, which carries a fundamental chain [Mk(L, (3)]; otherwise, various techniques can be used to define a virtual fundamental chain [Mk(L, (3)]vir, usually dependent on auxiliary perturbation data. The (virtual) fundamental chain of M 1 (L, (3) can be pushed forward by the evaluation map at the marked point, ev: Ml(L,{3) -j. L, to obtain a chain in L: then one sets

(2.1)

mO(£)=

L ~E7r2(X,L)

z~(£)ev*[Ml(L,{3)]vir,

SPECIAL LAGRANGIAN FIBRATIONS

7

where the coefficient Z{3(C) reflects weighting by symplectic area: (2.2) or Z{3(C) = TJi3 w holv(8;3) E Ao if using Novikov coefficients to avoid convergence issues (see below). Note that z{3 as defined by (2.2) is locally a holomorphic function with respect to the complex structure JV introduced in Proposition 2.3. Indeed, recall that the tangent space to the moduli space M is identified with the space of complex-valued 1P-harmonic I-forms on L; the differential of log z{3 is just the linear form on 1i~(L) ® C given by integration on the homology class 8(3 E Hl(L). In this paper we will mostly consider weakly unobstructed Lagrangians, i.e. those for which mo(C) is a multiple of the unit (the fundamental cycle of L). In that case, the Floer differential on CF*(C, C) does square to zero, but given two Lagrangians C, C' we find that CF*(C, C') may not be well-defined as a chain complex. To understand the obstruction, recall that the count of holomorphic triangles equips CF*(C, C') with the structure of a left module over CF*(C, C) and a right module over CF*(C', C'). Writing m2 for both module maps, an analysis of the boundary of I-dimensional moduli spaces shows that the differential on CF*(C, C') squares to

m2(mO(C'),·) - m2(-, mo(C)). The assumption that mo is a multiple of the identity implies that Floer homology is only defined for pairs of Lagrangians which have the same mo. Moreover, even though the Floer homology group HF*(C,C) can still be defined, it is generically zero due to contributions of holomorphic discs in (X, L) to the Floer differential; in that case C is a trivial object of the Fukaya category. On the mirror side, these features of the theory can be replicated by the introduction of a superpotential, i.e. a holomorphic function W : XV -+ C on the mirror of X \ D. W can be thought of as an obstruction term for the B-model on Xv, playing the same role as mo for the A-model on X. More precisely, homological mirror symmetry predicts that the derived Fukaya category of X is equivalent to the derived category of singularities of the mirror Landau-Ginzburg model (XV, W) [25, 33]. This category is actually a collection of categories indexed by complex numbers, just as the derived Fukaya category of X is a collection of categories indexed by the values of mo. Given oX E C, one defines D~ing(W, oX) = DbCoh(W-1(oX))/ Perf(W-1(oX)), the quotient of the derived category of coherent sheaves on the fiber W- 1 (oX) by the subcategory of perfect complexes. Since for smooth fibers the derived category of coherent sheaves is generated by vector bundles, this quotient is trivial unless oX is a critical value of W; in particular, a point of XV defines a nontrivial object of the derived category of singularities only if it is a critical point of W. Alternatively, this category can also be defined in

D.AUROUX

8

terms of matrix factorizations. Assuming XV to be affine for simplicity, a matrix factorization is a Z/2-graded projective qXV]-module together with an odd endomorphism 8 such that 82 = (W - >.) id. For a fixed value of >., matrix factorizations yield a Z/2-graded dg-category, whose cohomological category is equivalent to D~ing(W, >.) by a result of Orlov [33]. However, if we consider two matrix factorizations (PI, 81) and (P2 , 6'2) associated to two values >'1, >'2 E C, then the differential on hom((P1, 6'1), (P2 , 82)) squares to (>'1 - >'2) id, similarly to the Floer differential on the Floer complex of two Lagrangians with different values of mo. This motivates the following conjecture: CONJECTURE 2.4. The mirror of X is the Landau-Ginzburg model (XV, W), where (1) XV is a mirror of the almost Calabi- Yau manifold X \ D, i.e. a (corrected and completed) moduli space of special Lagrangian tori in X \ D equipped with rank 1 unitary local systems;

(2) W : XV -t C is a holomorphic function defined as follows: if p E XV corresponds to a special Lagrangian C p = (L, \7), then W(p) =

(2.3)

(3E7r2(X,L),

p.({3)=2

wheren{3(Cp ) is the degree of the evaluation chainev*[M1(L,,8)]vir, i.e., the (virtual) number of holomorphic discs in the class,8 passing through a generic point of L, and the weight z{3(Cp ) is given by (2.2).

There are several issues with the formula (2.3). To start with, except in specific cases (e.g. Fano toric varieties), there is no guarantee that the sum in (2.3) converges. The rigorous way to deal with this issue is to work over the N ovikov ring

(2.4)

Ao

= {

y

ai TAi

I ai E C, >'i E lR~o, >'i -t +00 }

rather than over complex numbers. Holomorphic discs in a class ,8 are then counted with weight TJ(3 W hol.v( 8,8) instead of exp( - J{3 w) holv (8,8). Assuming convergence, setting T = e- 1 recovers the complex coefficient version. Morally, working over Novikov coefficients simultaneously encodes the family of mirrors for X equipped with the family of Kahler forms I\.W, I\. E lR+. Namely, the mirror manifold should be constructed as a variety defined over the Novikov field A (the field offractions of Ao), and the superpotential as a regular function with values in A. If convergence holds, then setting T = exp( -I\.) recovers the complex mirror to (X,l\.w); if convergence fails for all values of T, the mirror might actually exist only in a formal sense near the large volume limit I\. -t 00.

SPECIAL LAGRANGIAN FIBRATIONS

9

Another issue with Conjecture 2.4 is the definition of the numbers nf3(f:'p). Roughly speaking, the value of the superpotential is meant to be "the coefficient of the fundamental chain [L] in mo". However, in real life not all Lagrangians are weakly unobstructed: due to bubbling of Maslov index o discs, for a given class 13 with /-L(13) = 2 the chain ev* [M 1 (L, (3) ]vir is in general not a cycle. Thus we can still define nf3 to be its "degree" (or multiplicity) at some point q E L, but the answer depends on the choice of q. Alternatively, we can complete the chain to a cycle, e.g. by choosing a "weak bounding cochain" in the sense of Fukaya-Oh-Ohta-Ono [14], or more geometrically, by considering not only holomorphic discs but also holomorphic "clusters" in the sense of Cornea-Lalonde [12]; however, nf3 will then depend on some auxiliary data (in the cluster approach, a Morse function on L). Even if we equip each L with the appropriate auxiliary data (e.g. a base point or a Morse function), the numbers nf3 will typically vary in a discontinuous manner due to wall-crossing phenomena. However, recall that XV differs from the naive moduli space of Proposition 2.3 by instanton corrections. Namely, XV is more accurately described as a (completed) moduli space of Lagrangian tori LeX \ D equipped with not only a U(l) local system but also the auxiliary data needed to make sense of the Floer theory of L in general and of the numbers nf3 in particular. The equivalence relation on this set of Lagrangians equipped with extra data is Floer-theoretic in nature. General considerations about wall-crossing and continuation maps in Floer theory imply that, even though the individual numbers nf3 depend on the choice of a representative in the equivalence class, by construction the superpotential W given by (2.3) is a single-valued smooth function on the corrected moduli space. The reader is referred to §19.1 in [14] and §3 in [4] for details. In this paper we will assume that things don't go completely wrong, namely that our Lagrangians are weakly unobstructed except when they lie near a certain collection of walls in the moduli space. In this case, the process which yields the corrected moduli space from the naive one can be thought of decomposing M into chambers over which the nf3 are locally constant, and gluing these chambers by analytic changes of coordinates dictated by the enumerative geometry of Maslov index 0 discs on the wall. Thus, the analyticity of W on the corrected mirror follows from that of zf3 on the uncorrected moduli space. One last thing to mention is that the incompleteness of the Kahler metric on X \ D causes the moduli space of Lagrangians to be similarly incomplete. This is readily apparent if we observe that, since IZf31 = exp( - ff3 w), each variable zf3 appearing in the sum (2.3) takes values in the unit disc. We will want to define the mirror of X to be a larger space, obtained by analytic continuation of the instanton-corrected moduli space of Lagrangian tori (i.e., roughly speaking, allowing IZf31 to be arbitrarily large). One way to think of the points of Xv added in the completion process is as Lagrangian tori in X \ D equipped with non-unitary local systems; however this can lead to serious convergence issues, even when working over the Novikov ring.

10

D.AUROUX

Another way to think about the completion process, under the assumption that D is nef, is in terms of inflating X along D, i.e replacing the Kahler form w by Wt = W + t'fJ where the (1, I)-form 'fJ is Poincare dual to D and supported in a neighborhood of D; this "enlarges" the moduli space of Lagrangians near D, and simultaneously increases the area of all Maslov index 2 discs by t, i.e. rescales the superpotential by a factor of e- t . Taking the limit as t --+ 00 (and rescaling the superpotential appropriately) yields the completed mirror. 2.3. Example: Fano toric varieties. Let (X, w, J) be a smooth toric variety of complex dimension n. In this section we additionally assume that X is Fano, i.e. its anticanonical divisor is ample. As a Kahler manifold, X is determined by its moment polytope ~ C ~n, a convex polytope in which every facet admits an integer normal vector, n facets meet at every vertex, and their primitive integer normal vectors form a basis of zn. The moment map ¢ : X --+ ~n identifies the orbit space of the rn-action on X with ~. From the point of view of complex geometry, the preimage of the interior of ~ is an open dense subset U of X, biholomorphic to (c*)n, on which Tn = (8 1 )n acts in the standard manner. Moreover X admits an open cover by affine subsets biholomorphic to C n , which are the preimages of the open stars of the vertices of ~ (i.e., the union of all the strata whose closure contains the given vertex). For each facet F of ~, the preimage cp-1(F) = DF is a hypersurface in X; the union of these hypersurfaces defines the toric anticanonical divisor D = LF D F . The standard holomorphic volume form on (c*)n ~ U = X \ D, defined in coordinates by n = d log Xl 1\ ... 1\ d log X n , determines a section of Kx with poles along D. It is straightforward to check that the orbits of the Tn-action are special Lagrangian with respect to wand n. Thus the moment map determines a special Lagrangian fibration on X \ D, with base B = int(~); by definition, the symplectic affine structure induced on B by the identification T B ~ H1(L,~) is precisely the standard one coming from the inclusion of B in ~n (up to a scaling factor of 27r). Consider a Tn-orbit L in the open stratum X \ D ~ (c*)n, and a flat U(I)-connection \7 on the trivial bundle over L. Let

where CPj is the j-th component of the moment map, i.e. the Hamiltonian for the action of the j-th factor of Tn ,and 'I'j = [8 1 (rj)] E HI (L) is the homology class corresponding to the j-th factor in L = 8 1 (rd x ... x 8 1 (rn) C (c*)n. Then Zl, ... , Zn are holomorphic coordinates on the moduli space M of pairs (L, \7) equipped with the complex structure JV of Proposition 2.3. For each facet F of~, denote by v(F) E zn the primitive integer normal vector to F pointing into ~, and let a(F) E ~ be the constant such that the

SPECIAL LAGRANGIAN FIBRATIONS equation of F is (v(F), ¢)+a(F) = O. Moreover, given a we denote by za the Laurent monomial 1 ••• z~n .

zr

11

= (al,""

an) E zn

PROPOSITION 2.5. The SYZ mirror to the smooth Fano torie variety X is (c*)n equipped with a superpotential given by the Laurent polynomial (2.5)

W =

L

e- 27rcx (F) zll(F).

F facet

More precisely, the moduli space M of pairs (L, '\7) is biholomorphic to the bounded open subset of (c*)n consisting of all points (Zl' ... ,zn) such that each term in the sum (2.5) has norm less than 1; however, the completed mirror is all of (c*) n . Proposition 2.5 is a well-known result, which appears in many places; for completeness we give a very brief sketch of a geometric proof (see also [21, 11, 4, 15] for more details). SKETCH OF PROOF. Consider a pair (L, '\7) as above, and recall that L can be identified with a product torus Sl(rl) x ... x Sl(rn) inside (c*)n. It follows from the maximum principle that L does not bound any nonconstant holomorphic discs in (c*)n; since the Maslov index of a disc in (X, L) is twice its intersection number with the toric divisor D, this eliminates the possibility of Maslov index 0 discs. Moreover, since X is Fano, all holomorphic spheres in X have positive Chern number. It follows that the moduli spaces of Maslov index 2 holomorphic discs in (X, L) are all compact, and that we do not have to worry about possible contributions from bubble trees of total Maslov index 2; this is in sharp contrast with the non-Fano case, see §3.2. A holomorphic disc of Maslov index 2 in (X, L) intersects D at a single point, and in .particular it intersects only one of the components, say DF for some facet F of ~. Cho and Oh [11] observed that for each facet F there is a unique such disc whose boundary passes through a given point x O = (x~, ... ,x~) E L; in terms of the components (VI, .•• , vn ) of the normal vector v(F), this disc can be parametrized by the map (2.6)

(for W E D2 \ {O}; the point w = 0 corresponds to the intersection with D F)' This is easiest to check in the model case where X = C n , the moment polytope is the positive octant lR~o, and the normal vectors to the facets form the standard basis of zn. Namely, the maximum principle implies that holomorphic discs of Maslov index 2 with boundary in a product torus in Cn are given by maps with only one non-constant com:ponent, and up to reparametrization that non-constant component can be assumed to be linear. The general case is proved by working in an affine chart centered at a vertex of ~ adjacent to the considered facet F, and using a suitable change of coordinates to reduce t" "he previous case.

12

D.AUROUX

A careful calculation shows that the map (2.6) is regular, and that its contribution to the signed count of holomorphic discs is +1. Moreover, it follows from the definition of the moment map that the symplectic area of this disc is 27r((I/(F),¢(L)) + a(F)). (This is again easiest to check in the model case of cn; the general case follows by performing a suitable change of coodinates). Exponentiating and multiplying by the appropriate holonomy factor, one arrives at (2.5). Finally, recall that the interior of ~ is defined by the inequalities (l/(F), ¢(L)) +a(F) > 0 for every facet F; exponentiating, this corresponds exactly to the constraint that le- 27ro (F) zv(F) I < 1 for every facet F. However, adding the Poincare dual of tD to w enlarges the moment polytope by t/27r in every direction, i.e. it increases a(F) by t/27r for all facets. This makes M a larger subset of (c*)n; rescaling W by e t and taking the limit as t -t +00, we obtain all of (c*)n as claimed. 0

3. Examples of wall-crossing and instanton corrections 3.1. First examples. In this section we give two simple examples illustrating the construction of the mirror and the process of instanton corrections. The first example is explained in detail in §5 of [4], while the second example is the starting point of [2]; the two examples are in fact very similar. EXAMPLE 3.1.1. Consider X = C 2 , equipped with a toric Kahler form wand the holomorphic volume form 0 = dx 1\ dy/(xy - E), which has poles along the conic D = {xy = E}. Then X \ D carries a fibration by special Lagrangian tori

where /-lSI is the moment map for the S1-action ei(}. (x,y) = (ei(}x, e-i(}y) , for instance /-lsl(x,y) = !(JxI 2 -IYI2) for w = ~(dXl\dx+dyl\dy). These tori are most easily visualized in terms of the projection f : (x, y) I-t xy, whose fibers are affine conics, each of which carries a S1-action. The torus Tr ,>. lies in the preimage by f of a circle of radius r centered at E, and consists of a single S1-orbit inside each fiber. In particular, Tr ,>. is smooth unless (r,.\) = (lEI, 0), where we have a nodal singularity at the origin. One can check that Tr ,>. is special Lagrangian either by direct calculation, or by observing that Tr ,>. is the lift of a special Lagrangian circle in the reduced space Xred,>. = /-ls{(.\)/ S1 equipped with the reduced Kahler form Wred,>. and the reduced holomorphic volume form Ored,>. = i({)/{)(})#O = i dlog(xy - E); see §5 of [4]. As seen in §2, away from (r,.\) = (1e1,0) the moduli space M of pairs consisting of a torus L = Tr ,>. and a U(l) local system V' carries a natural complex structure, for which the functions z(3 = exp( - J(3 w) hoI,\? (8,8), ,8 E 7r2(X, L) are holomorphic.

13

SPECIAL LAGRANGIAN FIBRATIONS

X = I

xO

FIGURE

1. The special Lagrangian torus Tr ,>. in

((;2 \

D

Wall-crossing occurs at r = lEI, namely the tori 71€I,>' for>. > 0 intersect the x-axis in a circle, and thus bound a holomorphic disc contained in the fiber f- 1 (0), which has Maslov index O. Denote by a the relative homotopy class of this disc, and by w = Za the corresponding holomorphic weight, which satisfies Iwl = e->'. Similarly the tori 71€I,>' for>. < 0 bound a holomorphic disc contained in the y-axis, representing the relative class -a and with associated weight La = w- 1 . Since the projection f is holomorphic, holomorphic discs of Maslov index 2 in (((;2, T r ,>.) are sections of f over the disc of radius r centered at E. When r > lEI there are two families of such discs; these can be found either by explicit calculation, or by deforming Tr ,>. to a product torus Sl(r1) x Sl(r2) (by deforming the circle centered at E to a circle centered at the origin, without crossing E), for which the discs are simply D2 (r1) x {y} and {x} X D2 (r2)' Denote by f31 and f32 respectively the classes of these discs, and by Zl and Z2 the corresponding weights, which satisfy zI/ Z2 = w. In terms of these coordinates on M the superpotential is then given by W = Zl + Z2. On the other hand, when r < lEI there is only one family of Maslov index 2 discs in (((;2, T r ,>.). This is easiest to see by deforming Tr ,>. to the Chekanov torus Ixy-EI = r, Ixl = Iyl (if w is invariant under x ++ y this is simply Tr,o); then the maximum principle applied to y / x implies that Maslov index 2 discs are portions of lines y = ax, lal = 1. Denoting by f30 the class of this disc, and by u the corresponding weight, in the region r < lEI the superpotential is W = u. When we increase the value of r past r = lEI, for>. > 0, the family of holomorphic discs in the class f30 deforms naturally into the family of discs in the class f32 mentioned above; the coordinates on M naturally glue according to u = Z2, W = zI/ Z2. On the other hand, for>. < 0 the class f30 deforms naturally into the class f31, so that the coordinates glue according to u = Zl, W = zI/ Z2. The discrepancy in these gluings is due to the monodromy of our special Lagrangian fibration around the singular fiber 71€I,o, which acts nontrivially on 7T2(((;2, Tr ,>.): while the coordinate w is defined globally on M, Zl and Z2 do not extend to global coordinates.

D. AUROUX

14

There are now two issues: the complex manifold M does not extend across the singularity at (r, A) = (IEI,O), and the superpotential W is discontinuous across the walls. Both issues are fixed simultaneously by instanton corrections. Namely, we correct the coordinate change across the wall r = lEI, A > 0 to u = z2(1 + w). The correction factor 1 + w indicates that, upon deforming Tr ,).. by increasing the value of r past lEI, Maslov index 2 discs in the class /30 give rise not only to discs in the class /32 (by a straightforward deformation), but also to new discs in the class /31 = /32 + 0 formed by attaching the exceptional disc bounded by 1l€i,)..' Similarly, across r = lEI, A < 0, we correct the gluing to u = Zl (1 + w- 1 ), to take into account the exceptional disc in the class - 0 bounded by 1l€1,)..' The corrected gluings both come out to be u = Zl + Z2, which means that we now have a well-defined mirror Xv, carrying a well-defined superpotential W = u = Zl + Z2. More precisely, using the coordinates (u, w) on the chamber r < IEI, and the coordinates (v, w) with v = zi 1 and w = zd Z2 on the chamber r > lEI, we claim that the corrected and completed mirror is XV

= {(u,v,w)

E

c 2 x C*,

uv

= 1 +w},

W=u.

More precisely, the region r > lEI of our special Lagrangian fibration corresponds to IZ11 and IZ21 small, i.e. Ivllarge; whereas the region r < lEI corresponds to lui large compared to e-I€I. When considering M we also have lui < 1, as lui -+ 1 corresponds to r -+ 0, but this constraint is removed by the completion process, which enlarges X along the conic xy = E by symplectic inflation. It turns out that we also have to complete XV in the "intermediate" region where u and v are both small, in particular allowing these variables to vanish; for otherwise, the corrected mirror would have "gaps" in the heavily corrected region near (r, A) = (lEI, 0). Let us also point out that XV is again the complement of a conic in C 2. General features of wall-crossing in Floer theory ensure that, when crossing a wall, holomorphic disc counts (and hence the superpotential) can be made to match by introducing a suitable analytic change of coordinates, consistently for all homotopy classes (see §19.1 of [14] and §3 of [4]). For instance, if we compactified C 2 to Cp2 or Cp1 x CP1, then the tori Tr ,).. would bound additional families of Maslov index 2 holomorphic discs (passing through the divisors at infinity), leading to additional terms in the superpotential; however, these terms also match under the corrected gluing u = Zl + Z2 (see §5 of [4]). EXAMPLE 3.1.2. Consider C 2 equipped with the standard holomorphic volume form dlogxAdlogy (with poles along the coordinate axes), and blow up the point (1,0). This yields a complex manifold X equipped with the holomorphic volume form n = 7r* (d log x A d log y), with poles along the proper transform D of the coordinate axes. Observe that the 5 1-action eiO . (x, y) = (x, eiOy) lifts to X, and consider an 5 1-invariant Kahler form w for which the area of the exceptional divisor is €. Denote by /-lSI: X -+ ffi. the moment

SPECIAL LAGRANGIAN FIBRATIONS

15

map for the SI-action, normalized to equal 0 on the proper transform of the x-axis and E at the isolated fixed point. Then the SI-invariant tori L r,).. =

UTr*xl =

r,

/-lSI = .\}

define a special Lagrangian fibration on X \ D, with a nodal singularity at the isolated fixed point (for (r,.\) = (1, E)) [2]. The base of this special Lagrangian fibration is pictured on Figure 2, where the vertical axis corresponds to the moment map, and a cut has been made below the singular point to depict the monodromy of the symplectic affine structure. For r = 1 the Lagrangian tori L r ,).. bound exceptional holomorphic discs, which causes wall-crossing: for .\ > E, Ll,).. bounds a Maslov index 0 disc in the proper transform of the line x = 1, whereas for .\ < E, L 1,).. splits the exceptional divisor of the blowup into two discs, one of which has Maslov index O. Thus, we have to consider the chambers r > 1 and r < 1 separately. When r < 1, the Lagrangian torus L r ,).. bounds two families of Maslov index 2 discs. One family consists of the portions where /-lSI < .\ of the lines x = constant; we denote by 8 the homotopy class of these discs, and by z (= ZtS) the corresponding holomorphic coordinate on M, which satisfies Izl = e-)... The other family consists of discs intersecting the y-axis, and is easiest to see by deforming L r ,).. to a product torus, upon which it becomes the family of discs of radius r in the lines y = constant. (In fact, L r ,).. is typically already a product torus for r sufficiently different from 1, when it lies in the region where the blow-up operation does not affect the Kahler form.) We denote by f3 the class of these discs, and by u the corresponding holomorphic coordinate on M. The coordinates u and z on M can be thought of as (exponentiated) complexifications of the affine coordinates on the base pictured on Figure 2. On the other hand, when r > 1 the torus L r ,).. bounds three families of Maslov index 2 discs. As before, one of these families consists of the portions where /-lSI < .\ of the lines x = constant, contributing z = ZtS to 'wall

u v-lr-----------------~~

eEzv- 1r------------------(

L FIGURE 2. A special Lagrangian fibration on the blowup of (:2

D.AUROUX

16

the superpotentiaL The two other families intersect the y-axis, and can be described explicitly when L T ,>. is a product torus (away from the blown up region): one consists as before of discs of radius r in the lines y = constant, while the other one consists of the proper transforms of discs which hit the x-axis at (1,0), namely the family of discs z I--t (rz, p(rz - l)/(r - z)) for fixed Ipl. Denote by v the complexification of the right-pointing affine coordinate on Figure 2 in the chamber r > 1, normalized so that, if we ignore instanton corrections, the gluing across the wall (r = 1,..\ > f) is given by u = v-I. Then the two families of discs intersecting the y-axis contribute respectively v-I and e f zv- I to the superpotential; the first family survives the wall-crossing at r = 1, while the second one degenerates by bubbling of an exceptional disc (the part of the proper transform of the line x = 1 where J.LSl < ..\). This phenomenon is pictured on Figure 2 (where the various discs are abusively represented as tropical curves, which actually should be drawn in the complex affine structure). Thus the instanton-corrected gluing is given by u = v-I + efzv- I across the wall (r = 1,..\ > f); and a similar analysis shows that the portion of the wall where ..\ < f also gives rise to the same instanton-corrected gluing. Thus, the instanton-corrected and completed mirror is given by XV

= {( u, v, z)

E C 2 X C*, uv

= 1 + e z }, f

W=u+z.

(Before completing the mirror by symplectically enlarging X, we would impose the restrictions lui < 1 and Izl < 1.) The reader is referred to [2] for more details. Remark. The above examples are particularly simple, as they involve a single singularity of the special Lagrangian fibration and a single wall-crossing correction. In more complicated examples, additional walls are generated by intersections between the "primary" walls emanating from the singularities; in the end there are infinitely many walls, and hence infinitely many instanton corrections to take into account when constructing the mirror. A framework for dealing with such situations has been introduced by Kontsevich and Soibelman [29], see also the work of Gross and Siebert [18, 19]. 3.2. Beyond the Fano case: Hirzebruch surfaces. The construction of the mirror superpotential for toric Fano varieties is well-understood (see e.g. [21, 11, 4, 15] for geometric derivations), and has been briefly summarized in §2.3 above. As pointed out to the author by Kenji Fukaya, in the non-Fano case the superpotential differs from the formula in Proposition 2.5 by the presence of additional terms, which count the virtual contributions of Maslov index 2 configurations consisting of a disc of Maslov index 2. or more together with a collection of spheres of non-positive Chern number. A non-explicit formula describing the general shape of the additional terms has been given by Fukaya-Oh-Ohta-Ono: compare Theorems 3.4 and 3.5 in

SPECIAL LAGRANGIAN FIBRATIONS

17

[15]. In this section we derive an explicit formula for the full superpotential in the simplest example, using wall-crossing calculations. The simplest non-Fano toric examples are rational ruled surfaces, namely the Hirzebruch surfaces IFn = P( O]p>l EB O]p>l (n)) for n ~ 2. The mirror of IFn is still (C*)2, but with a superpotential of the form W = W o+ additional terms [15], where Wo is given by (2.5), namely in this case e-[w],[Sn] Wo(x, y) = x

(3.1)

+y+

e-[w]·[F]

+

xyn

y

,

where [F] E H2(lFn) is class of the fiber, and [Sn] is the class of a section of square n. The superpotential Wo has n + 2 critical points, four of which lie within the region of (C*)2 which maps to the moment polytope via the logarithm map. Discarding the other critical points (Le., restricting to the appropriate subset of (C*)2), homological mirror symmetry can be shown to hold for (a deformation of) Wo [5] (see also [1]). However, this is unsatisfactory for various reasons, among others the discrepancy between the critical values of Wo and the eigenvalues of quantum cup-product with the first Chern class in QH*(lFn) (see e.g. §6 of [4], and [15]). The approach we use to compute the full superpotential relies on the observation that, depending on the parity of n, IF n is deformation equivalent, and in fact symplectomorphic, to either lFo = Cpl X Cpl or lFi (the one-point blowup of CP2), equipped with a suitable symplectic form. Carrying out the deformation explicitly provides a way of achieving transversality for the Floer theory of Lagrangian tori in lFn, by deforming the non-regular complex structure oflF n to a regular one. In the case oflF2 and lF3 at least, the result of the deformation can be explicitly matched with IF0 or IF 1 equipped with a non-toric holomorphic volume form of the type considered in Example 3.1.1, which allows us to compute the superpotential W. In the case oflF2 the deformation we want to carry out is pictured schematically in Figure 3. PROPOSITION 3.1. The corrected superpotential on the mirror of IF 2 is the Laurent polynomial e-[W],[S+2]

(3.2)

W(x, y)

=

x

+y+

xy

2

e-[w].[F]

+

e-[W],[S-2] e-[w].[F]

+

y

.

y

This formula differs from Wo by the addition of the last term; geometrically, this term corresponds to configurations consisting of a Maslov index 2 a

a

01

~~ Fo a

10

~1fJ u x

+2

lFa

-2

o~o~-{~;~ +2

FIGURE 3. Deforming lFo to lF2

+2

D.AUROUX

18

disc intersecting the exceptional section S-2 together with the exceptional section itself. PROOF. Consider the family of quadric surfaces X = C Cp3 x C. For t f= 0, X t = {XOXI = (X2 + tX3)(X2

xn

{XOXI

= X~

-

- tX3)} C Cp3 is a smooth quadric, and can be explicitly identified with the image of the embedding of Cpl x Cpl given in homogeneous coordinates by t2

or, in terms of the affine coordinates x

= ~0/6 and y = 170/171,

it(x, y) = (x: y : ~(xy + 1) : tt(xy - 1)). For

t

= 0, the surface Xo =

{XOXI

xn is a cone with vertex at the point itself presents an ordinary double point =

(0 : 0 : 0 : 1), where the 3-fold X singularity. Denote by 7r : X, ---+ X a small resolution: composing with the projection to C, we obtain a family of surfaces XL such that Xi ~ Xt for t f= 0, while Xb is the blowup of Xo, namely Xb ~ lF2. Consider the family of anticanonical divisors

and equip X t with a holomorphic volume form nt with poles along D t . Observe that D t is the image by it of the lines at infinity 6 = 0 and 171 = 0, . Ct: <,,0170 (: l-t22 <,,117l, (: • • ffi coord'mates, xy = 1+t l-t 22 ' an d 0 f th e comc = 1+t I.e. mane Thus, for t = 1 the pair (Xt , D t ) corresponds to the toric anticanonical divisor in Cpl x Cpl, while for general t the geometry of (Xt , D t , nt ) resembles closely that of Example 3.1.1 (the only difference is that we have compactified C 2 to Cpl X Cpl ). Finally, Do C Xo is precisely the toric anticanonical divisor, consisting of the lines Xo = 0, Xl = 0 (two rays of the cone) and the conic X3 = 0 (the base of the cone). The quadrics Xt, the divisors D t and the volume forms nt are preserved by the Sl-action (xo : Xl : X2 : X3) H (xoe iO : xle- iO : X2 : X3), and so is the Kahler form induced by restriction of the Fubini-Study Kahler form on cp3. Moreover, for t = 1 the standard T2-action on Cpl x Cpl is induced by a subgroup of PU(4), so that the Kahler form is toric, and the configuration at t = 0 is also toric (with respect to a different T2-action!); however for general t we only have Sl-invariance. The Sl-action lifts to the small resolution, and the lifted divisors D~ = 7r- l (Dt ) C Xi and holomorphic volume forms n~ = 7r*nt are Sl-invariant. Additionally, X' can be equipped with a Sl-invariant Kahler form, whose cohomology class depends on the choice of a parameter (the symplectic area of the exceptional - 2-curve); restricting to XL we obtain a family of Sl-invariant Kahler forms wi. Moreover, careful choices can be made in the construction in order to ensure that the Kahler forms wb and w~ on Xb and X~ are invariant under the respective T2-actions.

SPECIAL LAGRANGIAN FIBRATIONS

19

From the symplectic point of view, the family (X:, w~) is trivial; however, the complex structure for t = 0 is non-generic. At t = 1 and t = 0 the anticanonical divisors are the toric ones for IF 0 and IF 2 respectively; deforming away from these values, we partially smooth the toric anticanonical divisor, which in the case of IF2 requires a simultaneous deformation of the complex structure because the exceptional curve 5-2 is rigid. The deformation from t = 1 to t = 0 is now as pictured on Figure 3, which the reader is encouraged to keep in mind for the rest of the argument. For t = 1, the mirror superpotential is given by the formula for the toric Fano case (2.5), namely

(3.3)

W = ZI

-1 + Z2 + e -A ZI-1 + e -B Z2'

where A and B are the symplectic area of the two Cpl factors; the first two terms correspond to discs contained in the affine chart with coordinates x and y we have considered above, while the last two terms correspond to discs which hit the lines at infinity. Deforming to general t, we have X: \ D~ ~ C 2 \ {xy = i+~~}. Even though the Kahler form w~ is not toric, the construction of an 5 1-invariant special Lagrangian fibration proceeds exactly as in Example 3.1.1. The discussion carries over with only one modification: when considered as submanifolds of Cpl x Cpl, the tori Tr ,>. = {Ixy - i+~~ 1 = r, /-LSi = .x} bound additional families of Maslov index 2 holomorphic discs (intersecting the lines at infinity). In the chamber r > 1i+~~ I, deforming T r ,>. to a product torus shows that it bounds four families of Maslov index 2 discs, and the superpotential is given by (3.3) as in the toric case. On the other hand, in the chamber r < 1i+~~ I, deforming Tr ,>. to the Chekanov torus shows that it bounds five families of Maslov index 2 holomorphic discs; explicit calculations are given in Section 5.4 of [4]. (In [4] it was assumed for simplicity that the two Cpl factors had equal symplectic areas, but it is easy to check that the discussion carries over to the general case without modification.) Using the same notations as in Example 3.1.1, the superpotential is now given by

(3.4) (see Corollary 5.13 in [4]). The first term u corresponds to the family of discs which are sections of f : (x, y) 1-7 xy over the disc ~ of radius r centered at i+~~; these discs pass through the conic xy = (1 - t 2 )j(1 + t 2 ) and avoid all the toric divisors. The other terms correspond to sections of f over Cpl \ ~. These discs intersect exactly one of the two lines at infinity, and one of the two coordinate axes; each of the four possibilities gives rise to one family of holomorphic discs. The various cases are as follows (see Proposition 5.12 in [4]):

D.AUROUX

20

class HI - /30 - a HI - /30 H2 - /30 H2 - /30 + a Here HI

X=O y=O x = CXJ Y = CXJ no yes no yes

yes no yes no

yes yes no no

no no yes yes

weight

e-A/uw e-A/u e-B/u e-Bw/u

= [Cpl X{pt}], H2 = [{pt} XCpl], and /30 and a are the classes in

71"2 (C2,T r ,>..)

introduced in Example 3.1.1. Here as in Example 3.1.1, it is easy to check that the two formulas for the superpotential are related by the change of variables w = zI/ Z2 and u = Zl +Z2, which gives the instant on-corrected gluing between the two chambers. The tori T r ,>.. with r < I~~;$I cover the portion of X~ \ D~ where Ixy -

~+~~ I < It~~; I, which under the embedding it corresponds to the

inequality

For t -7 0 this region covers almost all of Xi \ D~, with the exception of a small neighborhood of the lines {xo = 0, X2 = tX3} and {Xl = 0, X2 = tX3}. On the other hand, as t -7 0 the family of special Lagrangian tori T r ,>.. converge to the standard toric Lagrangian fibration on Xb = lF2, without any further wall-crossing as t approaches zero provided that r is small enough for Tr ,>. to lie within the correct chamber. It follows that, in suitable coordinates, the superpotential for the Landau-Ginzburg mirror to lF2 is given by (3.4). All that remains to be done is to express the coordinates X and y in (3.2) in terms of u and w. In order to do this, we investigate the limiting behaviors of the five families of discs contributing to (3.4) as t -7 0: four of these families are expected to converge to the "standard" families of Maslov index 2 discs in IF 2, since those are all regular. Matching the families of discs allows us to match four of the terms in (3.4) with the four terms in Woo The leftover term in (3.4) will then correspond to the additional term in (3.2). Consider a family of tori Tr(t),>..(t) in X~ which converge to a T 2-orbit in Xb = lF2' corresponding to fixed ratios Ixol/lx31 = Po and IXII/lx31 = PI (and hence IX21/lx31 = VPOPI)' Since the small resolution 71" : X' -7 X is an isomorphism away from the exceptional curve in Xb, we can just work on X and use the embeddings it to convert back and forth between coordinates on Xi ~ X t ~ Cpl x Cpl and homogeneous coordinates in Cp3 for t =/: O. Since

-lt + ll-lt+ _2t_1

_IX_21 IX31 -

XY xy - 1 -

xy-l

should converge to a finite non-zero value as t -7 0, the value of Ixy - 11 must converge to zero, and hence r(t) = Ixy - ~+~~ I must also converge to 0; in fact, an easily calculation shows that r(t) '" 21tl/ VPOPI' Therefore,

21

SPECIAL LAGRANGIAN FIBRATIONS

for t small, xy is close to 1 everywhere on Tr(t),)..(t). On the other hand, Ixl/IYI = Ixol/ixil converges to the finite value PO/PI; thus Ixl and Iyl are bounded above and below on Tr(t),)..(t). Now, consider a holomorphic disc with boundary in Tr(t),)..(t) , representing the class HI - /30. Since the y coordinate has neither zeroes nor poles, by the maximum principle its norm is bounded above and below by fixed constants (independently of t). The point where the disc hits the line x = 00 (i.e., 6 = 0) has coordinates (xo : Xl : X2 : X3) = (~01]1 : 0 : !~01]0 : ~~01]0) = (1 : 0 : y : y/2t), which given the bounds on Iyl converges to the singular point (0 : 0 : 0 : 1) as t -t o. Thus, as t -t 0, this family of discs converges to stable maps in Xb which have non-empty intersection with the exceptional curve. The same argument (exchanging X and y) also applies to the discs in the class H2 - /30. On the other hand, the three other families of discs can be shown to stay away from the exceptional curve. In lF2, the T 2-orbits bound four regular families of Maslov index 2 holomorphic discs, one for each component of the toric anticanonical divisor D~ = B+2 U B-2 U Fo U F I ; here B-2 is the exceptional curve, B+2 is the preimage by 7r of the component {X3 = O} of Do C Xo, and Fo and FI are two fibers of the ruling, namely the proper transforms under 7r of the lines Xo = 0 and Xl = 0 in Xo. The four families of discs can be constructed explicitly in coordinates as in the proof of Proposition 2.5, see eq. (2.6); regularity implies that, as we deform Xb = lF2 to X; for to:/:O small enough, all these discs deform to holomorphic discs in (X:, Tr(t),)..(t)). The term y in Wo corresponds to the family of discs intersecting the section B+2, which under the projection Xb -t Xo corresponds to the component {X3 = O} of the divisor Do). Thus, its deformation for to:/:O intersects the component {X3 = -tx2} of the divisor Dt , namely the conic C t in the affine part of Cpl x Cpl. Comparing the contributions to the superpotential, we conclude that y = u. Next, the term X in Wo corresponds to the family of discs intersecting the ruling fiber Fo, which projects to the line {xo = O} on Xo. Thus, for small enough to:/:O these discs deform to a family of discs in that intersect the component {xo = 0, X2 = tX3} of D~, i.e. the line at infinity 1]1 = O. There are two such families, in the classes H2-/30 and H2-/30+a; however we have seen that the discs in the class H 2-/30 approach the exceptional curve as t -t 0, which would give a contradiction. Thus the term X in Wo corresponds to the family of discs in the class H2 - /30 + a, which gives x = e-Bw/u. The proof is then completed by observing that the change of variables x = e-Bw/u, y = u identifies (3.2) with (3.4). (Recall that the symplectic areas of B+ 2, B-2, and the ruling fibers in][f2 are respectively A + B, A - B, and B). Note: as a quick consistency check, our change of variables matches the term e-(A+B) /xy2 in (3.2), which corresponds to discs in ][f2 that intersect the ruling fiber FI, with the term e- A /uw in (3.4), which corresponds to discs representing the class HI - /30 - a in X; and intersecting the line at infinity 6 = O. The remaining two terms in (3.4) correspond to discs representing

X:

D.AUROUX

22

the classes HI - 130 and H2 - 130 in XL whose limits as t -+ 0 intersect the exceptional curve 8-2, and can also be matched to the remaining terms in (3.2). []

A similar method can be applied to the case of 1F3, and yields: PROPOSITION 3.2. The corrected superpotential on the mirror of 1F3 is the Laurent polynomial

(3.5) e-[W).[S+3)

W(x,y) = x+y+

xy

3

+

e-[w).[F)

y

+

2e-[w),([S-3)+2[F])

y

2

e-[w),([S-3]+[F])x

+-----y

SKETCH OF PROOF. The deformation we want to carry out is now depicted on Figure 4. One way of constructing this deformation is to start with the family X' considered previously, and perform a birational transformation. Namely, let C' c X' be the proper transform of the curve C = {xo = Xl = 0, X2 = tX3} c X, and let X' be the blowup of X' along C'. (This amounts to blowing up the point X = Y = 00 in each quadric X: for t =1= 0, and the point where 8-2 intersects the fiber FI in Xb ~ 1F2)' Next, let Z c X' be the proper transform of the surface Z = {Xl = 0, X2 = tX3} c X. Denote by X" the 3-fold obtained by contracting Z in X': namely, X" is a family of surfaces each obtained from by first blowing up a point as explained above and then blowing down the proper transform of the line Xl = 0, X2 = tX3 (for t =1= 0 this is the line at infinity 6 = 0, while for t = 0 this is the ruling fiber FI)' One easily checks that X:, ~ 1FI for t =1= 0, while xg ~ IF3. Moreover, the divisors D~ c X: transform naturally under the birational transformations described above, and yield a family of anticanonical divisors D~' c X:,; for t = 0 and t = 1 these are precisely the toric anticanonical divisors in xg = IF3 and = IFI. In terms of the affine charts on ~ IF0 considered in the proof of Proposition 3.1, the birational transformations leading to (X:" D~') are performed "at infinity": thus D~' is again the union of the conic xy = i+~~ and the divisors at infinity, namely for general t we are again dealing with a compactified version of Example 3.1.1. Thus X:, \ D~' still contains an 8 1-invariant family of special Lagrangian tori TT,)', constructed as previously, and there are again two chambers separated by the wall r = Ii+~~ I; the only difference

X:',

X:

X:

-1 ~ 0

%1

O+%21F1 +1

-

~ .. -1 'i,;.W 11'\ 0 u . ·x ... WalJ ...... = +3

Xr

-~x..... ~-3 :" ..... 0 x 0 ~o~ +3

FIGURE 4. Deforming IFI to 1F3

L..-----;+"3---;:O"

SPECIAL LAGRANGIAN FIB RATIONS

23

concerns the superpotential, since the tori Tr ,). now bound different families of holomorphic discs passing through the divisors at infinity. These families and their contributions to the superpotential can be determined by the same techniques as in the cases of ClP'2 and ClP'l x ClP'l, which are treated in Section 5 of [4]. Namely, for r > I~~~~ I the tori Tr ,). can be isotoped to product tori, and hence they bound four families of Maslov index 2 discs, giving the familiar formula

C(A+B) (3.6)

W=Zl+Z2+ ZlZ2

e-B

+-, Z2

where B is the area of the ruling fiber in IF 1 and A is the area of the exceptional curve. Meanwhile, for r < I~~~~ I the tori Tr ,). can be isotoped to Chekanov tori; it can be shown that they bound 6 families of Maslov index 2 discs, and using the same notations as in Example 3.1.1 we now have

e-(A+B){l + w? e- B {l + w) 2 + . uw u The instanton-corrected gluing between the two chambers is again given by u = Zl + Z2 and w = zI/ Z2; in fact (3.7) can be derived from (3.6) via this change of variables without having to explicitly determine the holomorphic discs bounded by T r ,).. The strategy is now the same as in the proof of Proposition 3.1: as t -+ 0, the special Lagrangian fibrations on X:'\D~ converge to the standard fibration by T 2-orbits on xg ~ IF3, and the chamber r < I~~~~ I covers arbitrarily large subsets of X:, \ D~'. Therefore, as before the superpotential for the Landau-Ginzburg mirror to IF3 is given by (3.7) in suitable coordinates; the expression for the variables x and y in (3.5) in terms of u and w can be found by matching some of the families of discs bounded by T r ,). as t -+ 0 to the regular families of Maslov index 2 discs bounded by the T 2-orbits in lF3. Concretely, the term y in (3.1) corresponds to holomorphic discs in IF3 which intersect the section 8+3. By regularity, these discs survive the deformation to a small nonzero value of t, and there they correspond to a family of discs which are entirely contained in the affine charts. Hence, as before we must have y = u. Identifying which term of (3.7) corresponds to the term x in (3.1) requires more work, but can be done exactly along the same lines as for Proposition 3.1; in fact, we find that it is given by exactly the same formula x = e-Bw/u as in the case of IF 2 . A posteriori this is not at all surprising, since this family of discs stays away from the line at infinity 6 = 0 in and hence lies in the part of that is not affected by the birational transformations that lead to X:,. Applying the change of variables x = e-Bw/u, y = u to (3.7), and recalling that the symplectic areas of 8+3, 8-3 and the ruling fibers in IF3 are respectively A + 2B, A - B, and B, we arrive at (3.5), which completes the proof. 0 (3.7)

X:,

W = u+

X:

D.AUROUX

24

It is tempting to interpret the last two terms in (3.5) as the contributions of Maslov index 2 stable configurations that include the exceptional curve 8-3 as a bubble component. Namely, the next-to-Iast term should be a virtual count of configurations that consist of a double cover of a Maslov index 2 disc passing through 8_ 3, together with 8-3; and the last term should be a virtual count of configurations consisting of a Maslov index 4 disc which intersects both the ruling fiber Fo and the exceptional section 8-3, together with 8_ 3 . In general, Fukaya-Oh-Ohta-Ono show that the "naive" superpotential Wo should be corrected by virtual contributions of Maslov index 2 configurations for which transversality fails in the toric setting; moreover, they show that the perturbation data needed to make sense of the virtual counts can be chosen in a T2-equivariant manner [15J. In principle, different choices of perturbation data could lead to different virtual counts of holomorphic discs, and hence to different formulas for the corrected superpotential. Our approach here can be understood as an explicit construction of a perturbation that achieves transversality for holomorphic discs, by deforming the complex structure to a generic one. However, our perturbation is only 8 l -equivariant rather than T 2-equivariant, so it is not clear that our count of discs agrees with the virtual counts obtained by using Fukaya-Oh-OhtaOno's perturbation data (the latter have not been computed yet, in fact their direct computation seems extremely difficult). It is nonetheless our hope that the two counts might agree; from this perspective it is encouraging to note that open Gromov-Witten invariants are well-defined in the 8 l -equivariant setting, and not just in the toric setting [31J.

3.3. Higher dimensions. In this section we give two explicit local models for singularities of Lagrangian fibrations in higher dimensions and their instanton-corrected mirrors, generalizing the two examples considered in §3.1. The open Calabi-Yau manifolds underlying the two examples are in fact mirror to each other, as will be readily apparent. In complex dimension 3 these examples are instances of the two types of "trivalent vertices" that typically arise in the discriminant loci of special Lagrangian fibrations on Calabi-Yau 3-folds and appear all over the relevant literature (see e.g. [16]). These examples can also be understood by applying the general machinery developed by Gross and Siebert [18, 19J; nonetheless, we find it interesting to have a fairly explicit and self-contained description of the construction. 3.3.1. Consider X = en, equipped with the standard Kahler form wand the holomorphic volume form n = (IT Xi - 10)-1 dXl A··· A dx n , which has poles along the hypersurface D = {IT Xi = €}. Then X \ D carries a fibration by special Lagrangian tori Tr ,). = {(Xl, ... ,Xn ) E en, 1IT Xi - 101 = r, ILTn-l(Xl, ... ,Xn ) = .x}, where ILTn-l : en -+ R n - l is the moment map for the action of the group T n - l = {diag(ei61, ... ,ei6n), ~(}i = a}. More EXAMPLE

SPECIAL LAGRANGIAN FIBRATIONS

25

explicitly,

T

r ,>..

={(X1' ... ' xn) E cn, IQ Xi - EI = r,

!(l XiI 2 -l x nI 2 ) = Ai Vi = 1, ... , n-l}.

The tori T r ,>.. are Tn-I-invariant, and as in previous examples they are obtained by lifting special Lagrangian fibrations on the reduced spaces. As in Example 3.1.1, these tori are easiest to visualize in terms of the projection I : (Xl, ... , Xn) M Xi, with respect to which they fiber over circles centered at E; see Figure 1. The main difference is that 1-1(0) is now the union of the n coordinate hyperplanes, and Tr ,>.. is singular whenever it hits the locus where the Tn-I-action is not free, namely the points where at least two coordinates vanish. Concretely, Tr ,>.. is singular if and only if r = lEI and A lies in the tropical hyperplane consisting of those A = (AI, . .. , An-I) such that either min(Ad = 0, or min(Ai) is attained twice. (For n = 3 this is the union of the three half-lines 0 = Al :S A2, 0 = A2 :S AI, and Al = A2 :S 0.) By the maximum principle, any holomorphic disc in (C n , T r ,>") which does not intersect D = 1- 1 (E) must be contained inside a fiber of f. The regular fibers of I are diffeomorphic to (c*)n-1, inside which product tori do not bound any nonconstant holomorphic discs. Hence, T r ,>.. bounds nontrivial Maslov index 0 holomorphic discs if and only if r = kl. In that case, 'lIEI,>" intersects one of the components of 1-1 (0) (i.e. a coordinate hyperplane isomorphic to C n - l ) in a product torus, which bounds various families of holomorphic discs inside 1-1 (0). The wall r = lEI divides the moduli space of special Lagrangians into two chambers. In the chamber r > lEI, the tori Tr ,>.. can be be deformed into product tori by a Hamiltonian isotopy that does not intersect 1-1(0) (from the perspective of the projection I, the isotopy amounts simply to deforming the circle of radius r centered at E to a circle of the appropriate size centered at the origin). The product torus Sl(r1) x ... x Sl(rn) bounds n families of Maslov index 2 discs parallel to the Xl, ... ,Xn coordinate axes; denote their classes by f3l,··., f3n, and by Zi = exp( - J;3i w) hoI V' (8f3d the corresponding holomorphic weights. Thus we expect that Tr ,>.. bounds n families of Maslov index 2 holomorphic discs; these are all sections of lover the disc of radius r centered at E, and the discs in the class f3i intersect the fiber 1-1 (0) at a point of the coordinate hyperplane Xi = O. Since the deformation from Tr ,>.. to the product torus does not involve any wall-crossing, the count of discs in the class f3i is 1, and the superpotential is given by W = Zl + ... + Zn. Next we look at the chamber r < lEI. We first observe that the Chekanovtype torus Tr,o bounds only one family of Maslov index 2 holomorphic discs. Indeed, since Maslov index 2 discs have intersection number 1 with D = 1- 1 (E), they must be sections of lover the disc ofradius r centered at E, and hence they do not intersect any of the coordinate hyperplanes. However, on Tr,o we have IXII = ... = IXnl, so the maximum principle applied to xdxn implies that the various coordinates Xi are proportional to each other, i.e. all such holomorphic discs must be contained in lines passing through

n

D.AUROUX

26

the origin. One easily checks that this gives a single family of holomorphic discs; we denote by (30 the corresponding homotopy class and by u = z(3o the corresponding weight. Finally, since no exceptional discs arise in the deformation of Tr,o to Tr,A' we deduce that Tr,A also bounds a single family of holomorphic discs in the class (30, and that the superpotential in the chamber r < lEI is given by W = u. When we increase the value of r past r = lEI, with all Ai > 0, the torus Tr,A crosses the coordinate hyperplane Xn = 0, and the family of holomorphic discs in the class (30 naturally deforms into the family of discs in the class (3n mentioned above. However, the naive gluing u = Zn must be corrected by wall-crossing contributions. For r = lEI, Tr,A intersects the hyperplane Xn = 0 in a product torus. This torus bounds n - 1 families of discs parallel to the coordinate axes inside {xn = O}, whose classes we denote by al, ... , an-I; we denote by WI, .•. , Wn-l the corresponding holomorphic weights, which satisfy IWil = e- Ai • It is easy to check that, on the r > lEI side, we have ai = (3i - (3n, and hence Wi = Zi/ Zn; general features of wall-crossing imply that Wi should not be affected by instanton corrections. Continuity of the superpotential across the wall implies that the relation between u and Zn should be modified to u = Zl + ... + Zn = Zn (WI + ... + Wn-l + 1). Thus, only the families of Maslov index 0 discs in the classes a!, ... ,an-l contribute to the instanton corrections, even though the product torus in {xn = O} also bounds higher-dimensional families of holomorphic discs, whose classes are positive linear combinations of the ai. Similarly, when we increase the value of r past r = lEI, with some Ak = minPd < 0, the torus Tr,A crosses the coordinate hyperplane Xk = 0, and the family of discs in the class (30 deforms to the family of discs in the class (3k. However, for r = lEI, Tr,A intersects the hyperplane Xk = 0 in a product torus, which bounds n - 1 families of discs parallel to the coordinate axes, representing the classes ai - ak = (3i - (3k (i =/:. k, n), with weight Wiwkl = zd Zk, and -ak = (3n - (3k, with weight w k l = zn/ Zk. The instanton-corrected gluing is now u = Zk(zI/Zk + ... + Zn/Zk + 1) = Zl + ... + Zn. Piecing things together as in Example 3.1.1, we obtain a description of the corrected and completed SYZ mirror in terms of the coordinates u, v = z;;:l, WI,"" Wn-l: PROPOSITION

3.3. The mirror of X

= C n relatively to the divisor D =

{TIXi = E} is XV

= {(u, v, WI, ... , Wn-l)

E

C2

X

(C*t- I ,

UV

= 1 + WI

+ ... + Wn-l},

W=u. A final remark: one way to check that the variables Wi are indeed not affected by the wall-crossing is to compactify C n to (cpl)n, equipped now with the standard product Kahler form. Inside (cpl)n the tori Tr,A also bound families of Maslov index 2 discs that pass through the divisors at infinity. These discs are sections of f over the complement of the disc of

SPECIAL LAGRANGIAN FIBRATIONS

27

radius r centered at E, and can be described explicitly in coordinates after deforming Tr ,).. to either a product torus (for r > lEI) or a Chekanov torus Tr,o (for r < lEI). In the latter case, we notice that the discs intersect the divisor at infinity once and j-1 (0) once, so that in affine coordinates exactly one component of the map has a zero and exactly one has a pole. Each of the n 2 possibilities gives one family of holomorphic discs; the calculations are a straightforward adaptation of the case of CP1 x CP1 treated in Section 5.4 of [4]. The continuity of W leads to an identity between the contributions to the superpotential coming from discs that intersect the compactification divisor "Xk = 00" (a single family of discs for r > lEI, vs. n families for r < lEI): namely, denoting by A the area of Cp1, we must have

e- A

-

Zk

e- A

= -(WI + ... +Wn -1 + 1). UWk

This is consistent with the formulas given above for the gluing between the two chambers. EXAMPLE 3.3.2. This example is treated carefully in [2], where it is used as a standard building block to construct mirrors of hypersurfaces in toric varieties. Here we only give an outline, for completeness and for symmetry with the previous example. Consider C n equipped with the standard holomorphic volume form TI d log Xi, and blow up the co dimension 2 linear subspace Y x 0 = {Xl +... + Xn-1 = 1, Xn = O}. This yields a complex manifold X equipped with the holomorphic volume form n = 7r* (TI d log xd, with poles along the proper transform D of the coordinate hyperplanes. The Sl-action rotating the last coordinate Xn lifts to X; consider an Sl-invariant Kahler form w for which the area of the CP1 fibers of the exceptional divisor is E (E « 1), and which agrees with the standard Kahler form of C n away from a neighborhood of the exceptional divisor. Denote by PSI : X --+ R the moment map of the Sl-action, normalized to equal 0 on the proper transform of the coordinate hyperplane Xn = 0, and E at the stratum of fixed points given by the section "at infinity" of the exceptional divisor. The reduced spaces X).. = {PSI = A}/S1 (A ~ 0) are all smooth and diffeomorphic to Cn - 1 . They carry natural holomorphic volume forms, which are the pullbacks of dlogx1 A··· A dlogx n -1, and Kahler forms W)... While w).. agrees with the standard Kahler form for A » E, for A < E the form w).. is not toric; rather, it can be described as the result of collapsing a tubular neighborhood of size E - A of the hypersurface Y = {Xl + ... + Xn-1 = 1} inside the standard C n - 1 . Thus, it is not entirely clear that X).. carries a special Lagrangian torus fibration (though it does seem likely). Nonetheless, using Moser's theorem to see that w).. is symplectomorphic to the standard form on cn-1, we can find a Lagrangian torus fibration on the complement of the coordinate hyperplanes in (X).., w)..). Taking the preimages of these Lagrangians in {PSI = A}, we obtain a Lagrangian fibration on

D.AUROUX

28

x \ D, whose fibers are Sl-invariant Lagrangian tori L r ,>..; for A »

10

these

tori are of the form {i'7r*(Xi) I = ri VI ::; i ::; n - 1,

/-LsI

= A}.

The singularities of this fibration correspond to the fixed points of the Sl-action inside X \ D, namely the "section at infinity" of the exceptional divisor, defined by the equations {/-LsI = 10, 71"* Xl + ... + 71"* Xn-l = I}. In the base of the fibration, the discriminant locus is therefore of real codimension 1, namely the amoeba of the hypersurface Y, sitting inside the affine hyperplane A = 10 (see Figure 5 left). Moreover, L r ,>.. bounds nonconstant discs of Maslov index 0 if and only ifit contains points where 7I"*Xl + .. ·+7I"*Xn-l = 1. In that case, the Maslov index 0 discs are contained in the total transforms of lines parallel to the xn-axis passing through a point of Y x o. Thus, there are n + 1 regions in which the tori L r ,>.. are weakly unobstructed, corresponding to the connected components of the complement of the amoeba of Y. To analyze holomorphic discs in (X, L r ,>..) and their contributions to the superpotential, we consider tori which lie far away from the exceptional divisor and from the walls, Le. for r = (rl, ... , r n- d sufficiently far from the amoeba of Y; then L r ,>.. projects to a product torus in When all ri « 1 for all i, the maximum principle implies that holomorphic discs bounded by L r ,>.. cannot hit the exceptional divisor; hence L r ,>.. bounds n families of Maslov index 2 holomorphic discs, parallel to the coordinate axes. Denote by (31, ... , (3n-l, 6 the classes of these discs, and by Ul, ... , Un-I, Z the corresponding weights (Le., the complexifications of the affine coordinates pictured in the lower-left chamber of Figure 5 right). Next consider the case where rk » 1 and rk » ri Vi -=I k. Then we claim that L r ,>.. now bounds n + 1 families of Maslov index 2 holomorphic discs. Namely, since a Maslov index 2 disc intersects D exactly once, and the projections to the coordinates (Xl, ... , Xn) are holomorphic, at most one of 7I"*(Xl), ... ,7I"*(Xn-l) can be non-constant over such a disc. Arguing as in the 2-dimensional case (Example 3.1.2), we deduce that L r ,>.. bounds n families of discs parallel to the coordinate axes, and one additional family, namely the proper transforms of Maslov index 4 discs in which are parallel to the (Xk' xn)-plane and hit the hyperplane Xn = 0 at a point of Y. Denote by ul,(k), ... ,un-l,(k), z(k) the weights associated to the first n families of

en.

en

FIGURE 5.

e3 blown up along {Xl + X2 = 1,

X3

= O}

29

SPECIAL LAGRANGIAN FIBRATIONS

discs: then the contribution of the additional family to the superpotential is e€ z(k)uk,(k). Matching the contributions of the families of discs that intersect each component of D, we conclude that the instanton-corrected gluings are given by z = Z(k) , Ui = Ui,(k) for i =I k, and Uk = Uk,(k)(l + e€z). Let

UO,(k) =

(IT

Ui,(k))

-1 =

~=l

(IT

Ui) -1 (1 + e€ z).

~=l

Then the coordinate uO,(k) is independent of k, and we can denote it simply by Uo. The coordinates (Uo, ... , Un-I, z) can now be used to give a global description of the mirror (since forgetting one of the Ui gives a set of coordinates for each chamber, as depicted in Figure 5 right). Namely, after completion we arrive at: PROPOSITION 3.4 (Abouzaid,-,Katzarkov [2]). The SYZ mirror of the blowup of c n along {Xl + ... + Xn-l = 1, Xn = O} with anti canonical divisor the proper transform of the toric divisor is

= {(uo, ... ,Un-l,Z) E C n x W = Ul + ... + Un-l + z.

XV

C*, Uo ... Un-l

= 1 +e€z},

If instead we consider the blowup of (c*)n-l X C along the generalized pair of pants {Xl + .. ·+Xn-l = 1, Xn = O}, i.e. we remove all the components of D except the proper transform of the Xn = 0 coordinate hyperplane, then XV remains the same but the superpotential becomes simply W = z (since all the other terms in the above formula correspond to discs that intersect the coordinate hyperplanes that we are now removing). In [2], these local models are patched together in order to build mirrors of more complicated blowups. The motivation for such a construction comes from the observation that, if Y is a hypersurface in X, then the derived category of Y embeds into that of the blowup of X x C along Y x 0 (this follows from a more general theorem of Bondal and Orlov, see e.g. [7]); and, if Y deforms in a pencil, then the Fukaya categories of these two manifolds are also closely related (using Seidel's work; the key point is that Lefschetz thimbles for a pencil in X can be lifted to Lagrangian spheres in the blowup of X x C along Y). Thus, a mirror for the blowup of X x C along Y is almost as good as a mirror for Y. We illustrate this by considering one half of the homological mirror symmetry conjecture in a very simple example. Consider the case n = 3 of Proposition 3.4 and its variants where we remove various divisors from D. Consider the blowup of (C*)2 X C along {Xl + X2 = 1, Xl, X2 =I O} (a pair of pants, i.e. pI minus three points): then XV is as in Proposition 3.4, i.e. (solving for z as a function of Uo, Ul, U2) the complement of the hypersurface UOUI U2 = 1 inside C3 , and the superpotential is W = z = e-€(uoulU2 - 1), whose critical locus consists of the union of the three coordinate axes. Up to an irrelevant scaling of the superpotential, this

D.AUROUX

30

Landau-Ginzburg model is indeed known to be a mirror to the pair of pants (cf. work of Abouzaid and Seidel; see also [38]). If instead we consider the blowup of C* x C 2 along {Xl +X2 = 1, Xl 1= O} (~ C*), then the superpotential becomes W = U2+Z = (e-Euoul +1)u2-e-E; hence W has a Morse-Bott singularity along M = {UOUI = -e E , U2 = O} ~ C*, which is mirror to C*. Finally, if we compactify our example to consider the blowup of CP2 x C along a projective line (given by Xl + X2 = 1 in affine coordinates), then the mirror remains the same manifold, but the superpotential acquires an extra term counting discs that pass through the divisor at infinity, and becomes

W = e -A Uo

+ UI + U2 + Z = e -A Uo + UI + u2 + e

-E

UOUI u2 - e -E

where A is the area of a line in CP2. This superpotential has two isolated non-degenerate critical points at e-Auo = UI = U2 = e±i7f-j2 e (E-A)/2, which is reminiscent of the usual mirror of a Cpl with symplectic area A - E (to which our mirror can be related by Knorrer periodicity).

4. Floer-theoretic considerations 4.1. Deformations and local systems. There are at least three possible ways of deforming the Floer theory of a given Lagrangian submanifold L (for simplicity we assume L to be weakly unobstructed): (1) formally deforming the Floer theory of L by an element bE CF I (L, L); (2) equipping L with a non-unitary local system; (3) deforming L by a (non-Hamiltonian) Lagrangian isotopy and equipping it with a unitary local system. Our goal in this paragraph is to explain informally how these three flavors of deformation are related. In particular, the careful reader will notice that Fukaya-Oh-Ohta-Ono define the superpotential as a function on the moduli space of weak bounding cochains for a given Lagrangian [14, 15], following the first approach, whereas in this paper and in [4] we view it as a function on a moduli space of Lagrangians equipped with unitary local systems, following the last approach. Recall that there are several models for the Floer complex CF*(L, L). We mostly consider the version in [14], where the Floer complex is generated by singular chains on L, representing incidence conditions at marked points on the boundary of holomorphic discs. The k-fold product mk is defined by

(4.1)

mk(CI, ... ,Ck )

=

L

zf3(L)(e~o)*([Mk+I(L,~)]virnevicln ... nevkCk)'

f3E7r2 (X,L)

where [Mk+l (L, ~)]vir is the (virtual) fundamental chain of the moduli space of holomorphic discs in (X, L) with k+ 1 boundary marked points representing the class ~, evo, . .. , eVk are the evaluation maps at the marked points,

SPECIAL LAGRANGIAN FIBRATIONS

31

and z{3 is a weight factor as in (2.2); when k = 1 the term with f3 = 0 is replaced by a classical boundary term. Here it is useful to also keep in mind a variant where the Floer complex consists of differential forms or currents on L. The product mk is defined as in (4.1), which now involves pulling back the given forms/currents to the moduli space of discs via the evaluation maps eVl, •.. ,eVk and pushing forward their product by integration along the fibers of evo. This setup allows us to "smudge" incidence conditions by replacing the integration current on a submanifold Ci by a smooth differential form supported in a tubular neighborhood. Given bE CPl(L, L), Fukaya-Oh-Ohta-Ono [14] deform the Aoo-algebra structure on the Floer complex by setting

We will actually restrict our attention to the case where b is a cycle, representing a class [b] E Hl(L) (or, dually, in H n -l(L)). Working over the Novikov ring, the sum (4.2) is guaranteed to be welldefined when b has coefficients in the maximal ideal

(4.3) of Ao = {L: ai TA; Iai E C, Ai E lR.2:0, Ai --+ +oo}. However, it has been observed by Cho [10] (see also [15]) that, in the toric case, the sum (4.2) is convergent even when b is a general element of HI (L, Ao). Similarly, in favorable cases (at least for toric Fanos) we can also hope to make sense of (4.2) when working over C (in the "convergent power series" setting); however in general this poses convergence problems. The second type of deformation we consider equips L with a local system (a fiat connection), characterized by its holonomy holY', which is a homomorphism from 7[1 (L) to (the multiplicative group formed by elements of the Novikov ring with nonzero coefficient of TO) or C*. The local system modifies the weight z{3 for the contribution to mk of discs in the class f3 by a factor of holY' (8f3).

Ao

For any cycle b such that convergence holds, the deformation of the Aoo-algebra CP*(L, L) given by (4.2) is equivalent to equipping L with a local system with holonomy exp(b), i.e. such that holY'b) = exp([b]· bD for all "Y E 7[1(L). LEMMA 4.1.

32

D.AUROUX

The statement reduces to a calculation showing that, given a holomorphic disc U E Mk+1(L,!3) (or more generally an element of the compactified moduli space), the contribution of "refined" versions of u (with extra marked points mapped to b) to m~ is exp([b] . [8!3]) times the contribution of u to mk. This is easiest to see when we represent the class [b] by a smooth closed I-form on L. For fixed I = (lo, ... , lk), consider the forgetful map 7rL : Mk+l+1 (L,!3) -+ Mk+l (L,!3) which deletes the marked points corresponding to the b's in (4.2), and its extension 7rL to the compactified moduli spaces. The fiber of 7rL above u E Mk+1 (L,!3) is a product of open simplices of dimensions lo, . .. , lk, parametrizing the positions of the lo + ... + lk new marked points along the intervals separated by the k + 1 marked points of u on the boundary of the disc; we denote by fl.l the corresponding subset of (8D2)1. The formula for mk+l (b Q910 , C1, bQ9l!, ... ,-Ck, bQ9lk ) involves an integral over Mk+l+1 (L,!3), but this integral can be pushed forward to Mk+l (L,!3) by integrating over the fibers of 7r1; the resulting integral differs from that for mk(C1, ... , Ck) by an extra factor f7i" l l(u) TI evi b = L TI(UI8D2 0 pri)*b in the integrand. Note that this calculation assumes that the virtual fundamental chains have been constructed consistently, so that [Mk+l+l (L,!3) ]vir = ([Mk+l (L, !3)]vir) as expected. Achieving this property is in general a non-trivial problem. Next we sum over I: the subsets fl.L of (8D2)1 have disjoint interiors, and their union fl. is the set of alll-tupies of points which lie in counterclockwise order on the interval obtained by removing the outgoing marked point of u from 8D2. By symmetry, the integral of TI(UI8D2 0 pri)*b over fl. is l/l! times the integral over (8D2)1. Thus SKETCH OF PROOF.

hS:

7ft

2:L J7::.CL n(uopri)*b=-ll, J(8D2)1 r i==l

.

n(u opri )*b=-ll,( i==l

.

r

u*bY

J 8D2

([b] . [8!3])1 l! The statement then follows by summing over l.

o

One can also try to prove Lemma 4.1 working entirely with chains on L instead of differential forms, but it is technically harder. If we take b to be a co dimension 1 cycle in L and attempt to reproduce the above argument, the incidence constraints at the additional marked points (all mapping to b) are not transverse to each other. In fact, m~ will include contributions from stable maps with constant disc bubbles mapping to b. The difficulty is then to understand the combinatorial rule for counting such contributions, or more precisely, why a constant bubble with j marked points on it, all mapped to a same point of b, should contribute a combinatorial factor of 1jj!.

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The equivalence between the two types of deformations also holds if we consider not just L itself, but the whole Fukaya category. Given a collection of Lagrangian submanifolds L o, ... , Lk with Lio = L for some io, the Floer theoretic product mk : CF*(Lo, Lt} 0 ···0 CF*(Lk-l' Lk) -+ CF*(Lo, Lk) can again be deformed by a cycle b E CFl(L, L). Where the usual product mk is a sum over holomorphic discs with k + 1 marked points, the deformed product m~ counts discs with an arbitrary number of additional marked points, all lying on the interval of 8D 2 which gets mapped to L, and with inputs b inserted accordingly into the Floer product as in (4.2). By the same argument as above, if we represent b by a closed I-form on L, and consider discs with fixed corners and in a fixed homotopy class (3, the deformation amounts to the insertion of an extra factor exp(JaJ3nL b). Meanwhile, equipping L with a fiat connection \7 affects the count of discs in the class (3 by a factor holv(8{3nL). Thus, if we ensure that the two match, e.g. by choosing \7 = d + b, the two deformations are again equivalent. Next, we turn to the relation between non-unitary local systems and non Hamiltonian deformations. Consider a deformation of L to a nearby Lagrangian submanifold L 1 ; identifying a tubular neighborhood of L with a neighborhood of the zero section in T* L, we can think of Ll as the graph of a C 1-small closed form

34

D.AUROUX

system \7 with holonomy exp(JA w). One easily checks that the Lagrangians Ll and (L, \7) have well-defined and non-vanishing Floer homology, and the Aoo-algebras CF*(Ll' L 1 ) and CF*((L, \7), (L, \7)) are isomorphic (by the argument above). However, CF*((L, \7), LI) = 0 since Land Ll are disjoint, so (L, \7) and Ll cannot be isomorphic. (See also the discussion in §4.2).

Remark. Specializing (4.2) to k = 0, the identity mS = mo + ml(b) + m2(b, b) + ... offers a simple perspective into the idea that the derivatives of the superpotential W at a point C = (L, \7) encode information about the (symmetrized) Floer products mk on CF*(C, C), as first shown by Cho in [9]. In particular, one can re-derive from this identity the fact that, if C is not a critical point of the superpotential, then the fundamental class of L is a Floer coboundary and H F* (C, C) vanishes. (For a direct proof, see [11, 9], see also §6 of [4].) 4.2. Failure of invariance and divergence issues. In this section, we look more carefully into a subtle issue with instanton corrections and the interpretation of the mirror as a moduli space of Lagrangian submanifolds up to Floer-theoretic equivalence. We return to Example 3.1.1, i.e. (C2 equipped with the standard Kahler form and the holomorphic volume form n = dxl\dY/(XY-E), and use the same notations as above. Consider two special Lagrangian fibers on opposite sides of the wall, Tl = Tr1,o and T2 = Tr2 ,o, where rl < lEI < r2 are chosen in a way such that the points of M corresponding to Tl and T2 (equipped with the trivial local systems) are identified under the instanton-corrected gluing u = Zl + Z2. Namely, the torus Tl corresponds to a point with coordinates w = 1, U = exp( -AI) E lR+, where Al is the symplectic area of a Maslov index 2 disc in (C2, Td, e.g. either of the two portions of the line x = Y where IXY-EI < rl; meanwhile, T2 corresponds to w = 1, Zl = Z2 = exp( -A2) E lR+, where A2 is the symplectic area of a Maslov index 2 disc in (C2, T2), or equivalently half of the area of the portion of the line x = Y where Ixy - EI < r2. The area Ai can be expressed by an explicit formula in terms of ri and E; the actual relation is irrelevant, all that matters to us is that A is a monotonically increasing function of rio Now we choose rl and r2 such that exp(-Al) = 2exp(-A2) and rl < lEI < r2. We will consider the tori Tl and T2 inside XO = X \ D = (C2 \ {xy = E}, where they do not bound any nonconstant holomorphic discs. (Another option would be to instead compactify (C2 to (cIP'2, and choose the parameters of the construction so that exp(-Ad = 2exp(-A2) = exp(-k iClP1 w); then Tl and T2 would be weakly unobstructed and would still have non-vanishing convergent power series Floer homology. The discussion below would carry over with minor modifications.) Working in XO, the convergent power series Floer homologies H F* (T1,T1) and HF*(T2,T2) are isomorphic to each other (and to the cohomology of T2). In fact the same property would hold for any other Tr ,>. due to the absence of holomorphic discs in XO, but in the case of Tl and T2

SPECIAL LAGRANGIAN FIBRATIONS

35

FIGURE 6. Creating intersections between TI and T2 we expect to have a distinguished isomorphism between the Floer homology groups, considering that TI and T2 are in the same instanton-corrected equivalence class and meant to be "isomorphic". However, TI and T2 are disjoint, so C F* (TI' T2) is zero, which does not allow for the existence of the expected isomorphism. (Note that the issue would not arise when working over the Novikov ring: we would then have needed to choose the areas Al and A2 above so that T-Al = 2T- A2, which never holds. In that case, one should instead take Al = A2 and equip TI with a nontrivial local system; but then TI and T2 cannot be made disjoint by Hamiltonian isotopies.) One way to realize the isomorphism between TI and T2 is to deform one of them by a Hamiltonian isotopy (without crossing any walls) in order to create intersections. Namely, projecting C 2 to C by the map f(x, y) = xy, TI and T2 fiber above concentric circles '"'1i = {Iz - EI = ri}, and inside each fiber they consist ofthe "equatorial" 8 1-orbit where Ixl = Iyl. Deform TI by a Hamiltonian isotopy, without crossing E or 0, to a 8 1-invariant Lagrangian torus T{ which fibers above a closed curve '"'1i intersecting '"'12 in two points p and q, and T{ = f-1bDn{lxl = Iyl} (see Figure 6). Then Ti and T2 intersect along two circles, which can be handled either as a degenerate Morse-Bott type intersection (CF*(T{, T2) is then generated by chains on Ti n T2), or by further perturbing T{ to replace each 8 1 by two transverse intersection points. PROPOSITION 4.2. In XO = C 2 \ f- 1(E), the convergent power series Floer homology H F* (T{, T2) is well-defined and isomorphic to H* (T2, q. PROOF. Any holomorphic disc in XO = C 2 \ f-l(E) that contributes to the Floer differential on CF*(T{, T2) is necessarily a section of f over one of the two regions R1 and R2 delimited bY'"'1i and '"'12 (see Figure 6). Recalling that Ixl = Iyl on Ti U T2, the maximum principle applied to x/y implies that, if a disc with boundary in T{ U T2 intersects neither the x axis nor the y axis, then x/y is constant over it. Thus, there is exactly one 8 1-family of such sections of f over R 1 , namely the portions of the lines y = ei'Px which lie in f-l(RI). On the other hand, there are two 8 1-families of sections over R2. Indeed, let g : D2 ~ R2 be a biholomorphism given by the Riemann mapping theorem, chosen so that g(O) = 0, and consider a holomorphic map u: (D2,8D2) ~ (XO,T{UT2), Z H u(z) = (x(z),y(z)) such that fou maps

36

D.AUROUX

D2 biholomorphically onto R2. Up to a reparametrization we can assume that j 0 u = g. Over the image of u, either x or y must vanish transversely once; assume that it is x that vanishes. Then Z r-+ x(z)/y(z) is a holomorphic function on the disc, taking values in the unit circle along the boundary, and vanishing once at the origin, therefore it is of the form z r-+ eicp z for some eicp E 8 1 . Thus u(z) = (eicp/2(zg(z))1/2,e-icp/2(g(z)/z)I/2). This gives an 8 1 _ family of holomorphic sections over R2; the other one is obtained similarly by exchanging x and y. Denote by al (resp. (2) the symplectic area of the holomorphic discs in (XO, T{ U T2) which are sections of j over Rl (resp. R2). By construction, these areas are related to those of the Maslov index 2 discs bounded by T{ and T2 in C 2 : namely, a2 - al = A2 - AI. Thus, the choices made above imply that exp( -al) = 2 exp( -(2). After a careful check of signs, this in turn implies that the contributions of the various holomorphic discs in (XO, T{ UT2) to the Floer differential on CF*(T{, T2) (with C coefficients) cancel out. D

Denote by ep the generator of CFO(T{, T2) which comes from the intersections in j-l(p), and denote by eq the generator of CFO(T2, T{) which comes from the intersections in j-1(q). Then m2(ep , eq ) = e-a:l [T{] is a nonzero multiple of the unit in CF*(T{, T{), and m2(eq , ep ) = e-a: 1 [T2J is a nonzero multiple of the unit in CF*(T2' T2): this makes it reasonable to state that T{ and T2 are isomorphic. This example illustrates the failure of convergent power series Floer homology to be invariant under Hamiltonian isotopies, even without wallcrossing (recall the isotopy from Tl to T{ did not cross j-1(0)); this is of course very different from the situation over the Novikov ring. When we deform T{ back to T 1 , we end up being able to cancel all the intersection points even though they represent nontrivial elements in Floer homology, because the cancellations in the Floer differential occur between families of discs with different symplectic areas (something which wouldn't be possible over Novikov coefficients). At the critical instant in the deformation, the discs with area a1 have shrunk to points, while the discs with area a2 become pinched annuli. At the end of the deformation, the tori T1 and T2 are disjoint, and the discs have become holomorphic annuli with boundary in T1 U T2. It would be tempting to hope that a souped up version of Floer theory that also includes holomorphic annuli would be better behaved. However, in that case we would immediately hit a divergence issue when working with complex coefficients: indeed, there are 2k families of holomorphic annuli with boundary in T1 U T2 which cover k-to-l the annulus bounded by the circles 1'1 and 1'2 in C. Even without considering annuli, divergence issues are already responsible for the bad properties of convergent power series Floer homology exhibited here - first and foremost, the lack of invariance under the Hamiltonian

SPECIAL LAGRANGIAN FIBRATIONS

37

isotopy from Tl to T{. Denote by H : [0, 1] x XO -+ JR. a family of Hamiltonians whose time 1 flow sends Tl to T{, and recall that continuation maps cI> : CP*(Tl, T2) -+ CF*(TL T2) and

~: + J (~~ -

X(S)XH(t, u(s, t)))

= O.

Here u : JR. x [0,1] -+ XO is a map with u(JR. x {O}) C Tl and u(JR. x {I}) C T2 and satisfying suitable asymptotic conditions at infinity, XH is the Hamiltonian vector field associated to H, and X : JR. -+ [0,1] is a suitable smooth cut-off function. In our case, cI> and

0

38

D.AUROUX

a nonzero multiple of ep , whereas m2(ea , m2(eb, ep )) is zero since m2(eb, ep ) E C F* (Tl' T 2 ) = O. Passing to cohomology classes, this contradicts the expected associativity of the product on Floer homology. A closer inspection reveals that this is caused by the divergence of quantities such as m3(e a, fa, ep ) (where fa is the generator of C Fl (Tl' TD corresponding to the intersections in f-l(a)): indeed, this triple product counts discs obtained by cutting open the divergent series of annuli with boundary in Tl U T2 already mentioned above. In conclusion, there are many pitfalls associated to the use of convergent power series Floer homology, even in fairly simple situations (compactifying the above example to CJlD 2 , we would still encounter the same divergence phenomena in a smooth projective Fano variety). A cautious view of the situation would dictate that outside of the very simplest cases it is illusory to even attempt to work over complex coefficients, and that in general mirror symmetry is only a perturbative phenomenon taking place over a formal neighborhood of the large volume limit. Nonetheless, as long as one restricts oneself to consider only certain aspects of Floer theory, the power series obtained by working over the Novikov ring seem to often have good enough convergence properties to allow the construction of a mirror that is an honest complex manifold (rather than a scheme over the Novikov field). Floer theory for a single weakly unobstructed Lagrangian seems to be less prone to divergence than the theory for pairs such as (Ll' L2) in the above example. Also, in the example we have considered, divergence issues can be avoided by equipping all our Lagrangian submanifolds with suitable Hamiltonian perturbation data (Le., "wiggling" Lagrangians so that they intersect sufficiently). However, more sophisticated divergent examples can be built e.g. inside conic bundles over elliptic curves; in some of these examples, Floer products are given by series in Ao for which the radius of convergence is strictly less than 1, i.e. convergence only holds for sufficiently large symplectic forms, regardless of Hamiltonian perturbations.

5. Relative mirror symmetry 5.1. Mirror symmetry for pairs. In this section, we turn to mirror symmetry for a pair (X, D), where X is a Kahler manifold and D is a smooth Calabi-Yau hypersurface in the anticanonicallinear system. Our goal is to clarify the folklore statement that "the fiber of the mirror superpotential W : XV -7 C is mirror to D". The discussion is fairly similar to that in §7 of [4J. Let D c X be a hypersurface in the anticanonical linear system, with defining section (j E HO(X,K)/): then the holomorphic volume form n = ( j - l E nn,O(X\D) (with poles along D) induces a holomorphic volume form nD on D, the residue of n along D, characterized by the property that n = nD 1\ dlog(j + 0(1) in a neighborhood of D. Additionally, the Kahler form w induces a Kahler form on D by restriction.

SPECIAL LAGRANGIAN FIBRATIONS

39

It is reasonable to expect that special Lagrangian torus fibrations on X \ D should have a "nice" boundary behavior. Namely, assuming that the Kahler metric on X is complete, for a reasonable special Lagrangian fibration 7r : X \ D -+ B we expect: CONJECTURE

5.1. Near BB, the fibers of 7r are contained in a neighbor-

hood of D, and the smooth fibers are Sl-bundles over special Lagrangian tori in (D,wID,DD)'

(Here, by BB we mean the part of the boundary of B which lies at finite distance in the symplectic affine structure). In other terms, we expect that near D the special Lagrangian tori in X \ D accumulate onto special Lagrangian tori in D (as observed in the various examples we have discussed). If Conjecture 5.1 holds, then BB is the base of a special Lagrangian fibration on D, and the (uncorrected) SYZ mirror to D, MD, can be identified as a complex hypersurface lying inside the boundary of the (uncorrected) moduli space M of pairs (L, \7) in X \ D. Assume D is smooth, and consider a special Lagrangian torus fiber L = 7r-I(b) near BB: then we expect that L bounds a distinguished family of Maslov index 2 holomorphic discs, namely small meridian discs in the normal direction to D. More precisely, as b approaches the boundary of B, we expect L to collapse onto a special Lagrangian torus A in D, and the meridian discs to be approximated by small discs inside the fibers of the normal bundle of D lying above the points of A. Call 8 the relative homotopy class of the meridian discs, and by Z8 the corresponding holomorphic coordinate on M (which is also the contribution of the family of meridian discs to the superpotential). Then we expect that Z8 is the dominant term in the superpotential near the boundary of M, as the meridian discs have areas tending to zero and all the other holomorphic discs have comparatively much greater areas. The boundary of M corresponds to limiting pairs (L, \7) where the area of the meridian disc becomes 0 (i.e., L is entirely collapsed onto a special Lagrangian torus in D); recalling that IZ81 = exp( w), this corresponds to IZ81 = 1. In fact, the boundary of M fibers above the unit circle, via the map

I8

(5.1) with fiber MD = {Z8 = 1}. The points of MD correspond to pairs (L, \7) where L is entirely collapsed onto a special Lagrangian torus A c D, and the holonomy of \7 around the meridian loop J1 = B8 is trivial, i.e. \7 is pulled back from a U(l) local system on A. Thus MD is precisely the uncorrected SYZ mirror to D. In general, the fibration (5.1) has monodromy. Indeed, a local trivialization is given by fixing a framing, i.e. an (n - 1)-dimensional subspace of HI(L,Z) which under the projection L -+ A maps isomorphically onto HI (A, Z). (Less intrinsically, we can choose a set of longitudes, i.e. lifts to L

D.AUROUX

40

of a collection of n - 1 loops generating HI(A, Z)); the framing data allows us to lift to M a set of local holomorphic coordinates on MD. However, unless the normal bundle to D is trivial there is no consistent global choice of framings: if we move A around a loop in BB and keep track of a longitude ,\ lifting a loop 'Y E A, the monodromy action is of the form ,\ M ,\ + kyj..l, where k-y is the degree of the normal bundle of D over the surface traced out by'Y. A more thorough calculation shows that the monodromy of (5.1) is given by a symplectomorphism of MD which geometrically realizes (as a fiberwise translation in the special Lagrangian fibration MD -t BB dual to the SYZ fibration on D) the mirror to the autoequivalence - ® KXID of DbCoh(D). This is easiest to see if we assume that, in a neighborhood of D, the anticanonical bundle K)/ can be equipped with a semi-flat connection, i.e. a holomorphic connection whose restriction to the fibers of 7r is flat. Then the parallel transport from one fiber of (5.1) to another can be realized geometrically as follows: given a pair (L, \7) where L is almost collapsed onto a special Lagrangian A cD, we can modify the holonomy of \7 around the meridian loop by adding to it a multiple of Im(O"-IBO")IL, where 0" is the defining section of D. The monodromy is then (L, \7) M (L, \7 +Im(O"-IBO")IL), which in the limit where L collapses onto D is exactly the expected transformation. If we can neglect the terms other than Z8 in the superpotential, for instance in the large volume limit, then MD is essentially identified with the fiber of W at 1. In fact, recall from the discussion at the end of §2.2 that changing the Kahler class to [w] + tel (X) "enlarges" the mirror while rescaling the superpotential by a factor of e- t : thus, assuming that X is Fano, or more generally that -Kx is nef, the flow to the large volume limit can be realized simply by rescaling the superpotential. Hence, Conjecture 5.1 implies: CONJECTURE 5.2. If (XV, W) is mirror to X, and if -Kx is nef, then for t -t 00 the family of hypersurfaces {W = et } C XV is asymptotic (up to corrections that decrease exponentially with t) to the family of mirrors to (D, WID + tCI(X)ID)'

For example, considering the mirror to C]P'2 with [w] . [C]P'I] = A, the j-invariant of the elliptic curve {x + y + e- A /xy = et } C (C*)2 can be determined to equal 3

3

e t+A(e t+A - 24)3 __ ~-,--__-'-- = e 9t+3A + ... , e 3t+A - 27

whose leading term matches with the symplectic area of the anticanonical divisor after inflation (observe that ([w] + tCI) . [C]P'I] = 3t + A). There are two reasons why this statement only holds asymptotically for t -t 00. First, the formula for the superpotential includes other terms besides Z8, so the hypersurfaces {W = e t } and {Z8 = e t } are not quite the same.

SPECIAL LAGRANGIAN FIBRATIONS

41

More importantly, the instanton corrections to the mirror of D are not the same as the instanton corrections to the fiber of z§. When constructing the mirror to X, the geometry of M D c M gets corrected by wall-crossing terms that record holomorphic Maslov index 0 discs in X; whereas, when constructing the mirror of D, the corrections only arise from Maslov index o holomorphic discs in D. In other terms: the instanton corrections to the mirror of X arise from walls generated by singularities in the fibration 7r : X \ D -t B (Le., singularities in the affine structure of B), whereas the instant on corrections to the mirror of D arise from the walls generated by singularities in the fibration 7rD : D -t BB (i.e., singularities in the affine structure of BB). Since the singularities of the affine structure on BB are induced by those strata of singularities of B that hit the boundary, the wall-crossing phenomena in D are induced by a subset of the wall-crossing phenomena in X, but there are also walls in X which hit the boundary of B without being induced by singularities at the boundary. On the other hand, the smooth fibers of Ware symplectomorphic to each other and to the hypersurface {z§ = 1}. Moreover, it is generally believed that the Kiihler class of the mirror should not be affected by instanton corrections, so the discrepancy discussed above is no longer an issue. Hence: we expect that the fibers of W, viewed as symplectic manifolds, are mirror to the divisor D viewed as a complex manifold. (Observe that, from this perspective, the parameter t in Conjecture 5.2 no longer plays any role, and accordingly the geometries are expected to match on the nose.) 5.2. Homological mirror symmetry. Assuming Conjectures 5.1 and 5.2, we can try to compare the statements of homological mirror symmetry for X and for the Calabi-Yau hypersurface D. Due to the mismatch between the complex structure on the mirror to D and that on the fibers of W (see Conjecture 5.2), in general we can only hope to achieve this in one direction, namely relating the derived categories of coherent sheaves on X and D with the Fukaya categories of their mirrors. Denote by (XV, W) the mirror to X, and by D V the mirror to D, which we identify symplectically with a fiber of W, say D V = {W = et } C XV for fixed t » O. First we need to briefly describe the Fukaya category of the LandauGinzburg model W : XV -t C. The general idea, which goes back to Kontsevich [27] and Hori-Iqbal-Vafa [22], is to allow as objects admissible Lagrangian submanifolds of Xv; these can be described either as potentially non-compact Lagrangian submanifolds which, outside of a compact subset, are invariant under the gradient flow of -Re(W), or, truncating, as compact Lagrangian submanifolds with (possibly empty) boundary contained inside a fixed reference fiber of W (and satisfying an additional condition). The case of Lefschetz fibrations (Le., when the critical points of Ware nondegenerate) has been studied in great detail by Seidel; in this case, which is

42

D.AUROUX

by far the best understood, the theory can be formulated in terms of the vanishing cycles at the critical points (see e.g. [35]). The formulation which is the most relevant to us is the one discussed by Abouzaid in [1]: in this version, one considers Lagrangian sub manifolds of XV with boundary contained in the given reference fiber D V = W-1(e t ), and which near the reference fiber are mapped by W to an embedded curve C.

,c

DEFINITION 5.3. A Lagrangian submanifold L C XV with (possibly empty) boundary aL c D V = W-1(e t ) is admissible with phase '{) E (-~,~) if IWI < et at every point of int(L) and, near aL, the restriction of W to L takes values in the half-line et - ei'PlR.+. Floer theory is then defined by choosing a specific set of Hamiltonian perturbations, which amounts to deforming the given admissible Lagrangians so that their phases are in increasing order, and ignoring boundary intersections. For instance, to determine H F(L1' L2), one first deforms L2 (reI. its boundary) to an admissible Lagrangian Lt whose phase is greater than that of L1, and one computes Floer homology for the pair of Lagrangians (L1' Lt) inside Xv, ignoring boundary intersections. We denote by F(XV, DV) the Fukaya category constructed in this manner. (In fact, strictly speaking, one should place the reference fiber "at infinity" , i.e. either consider a limit of this construction as t -t +00, or enlarge the symplectic structure on the subset {IWI < et } of Xv so that the symplectic form blows up near the boundary and the Kahler metric becomes complete; for simplicity we ignore this subtlety.) By construction, the boundary of an admissible Lagrangian in Xv is a Lagrangian submanifold of DV (possibly empty, and not necessarily connected). There is a restriction A,xJ-functor P : F(X V, DV) -t F(DV) from the Fukaya category of the Landau-Ginzburg model (XV, W) to the (usual) Fukaya category of DV. At the level of objects, this is simply (L, \7) I-t (aL, \7lad. At the level of morphisms, the Aoo-functor P consists of a collection of maps

P(k) : Hom.r(XV,DV)(L 1 , L 2) ® ... ® Hom.r(XV,DV) (Lk, Lk+l) -t Hom.r(DV)(aLl, aLk+1)'

The first order term P(l) is the easiest to describe: given an intersection point p E int(Ll) n int(Lt), P(l)(P) is a linear combination of intersection points in which the coefficient of q E aLl n aL2 counts holomorphic strips in (XV, L1 U Lt) connecting p to q. Similarly, given k + 1 admissible Lagrangians L1, ... ,Lk+b and perturbing them so that their phases are in increasing order, P(k) counts holomorphic discs in (XV, U Li) with k corners at prescribed interior intersection points and one corner at a boundary intersection point.

SPECIAL LAGRANGIAN FIBRATIONS

43

Homological mirror symmetry for the pair (X, D) can then be summarized by the following conjecture: CONJECTURE

5.4. There is a commutative diagram

DbCoh(X)

~1

~ DbCoh(D)

1~

D 7r F(X V, DV) ~ D7rF(DV) In this diagram, the horizontal arrows are the restriction functors, and the vertical arrows are the equivalences predicted by homological mirror symmetry. The reader is referred to [6] for a verification in the case of Del Pezzo surfaces. Another type of Fukaya category that can be associated to XV is its wrapped Fukaya category Fwr(XV) [3]. The objects of that category are again non-compact Lagrangian submanifolds, but the Hamiltonian perturbations used to define Floer homology now diverge at infinity. Assuming that W is proper, we can e.g. use the Hamiltonian flow generated by a function of IWI that increases sufficiently quickly at infinity; however, the wrapped category can be defined purely in terms of the symplectic geometry of XV at infinity, without reference to the superpotential (see [3]). Homological mirror symmetry for the open Calabi-Yau X \ D then predicts an equivalence between the derived category of coherent sheaves DbCoh(X\D) and the derived wrapped Fukaya category D7r Fwr(X V). Moreover, the restriction functor from DbCoh(X) to DbCoh(X \ D) is expected to correspond to a natural functor 'W from the Fukaya category of the Landau-Ginzburg model (XV, W) to the wrapped Fukaya category of Xv. On objects, 'W is essentially identity (after sending the reference fiber to infinity, or extending admissible Lagrangians to non-compact ones by parallel transport along the gradient flow of Re(W)). On morphisms, 'W is essentially an inclusion map if we set up the Hamiltonian perturbations in the wrapped category to be supported outside of the region where IWI < et ; or, more intrinsically, 'W is the continuation map induced on Floer complexes by the deformation from the small Hamiltonian perturbations used to define the Fukaya category of (XV, W) to the large Hamiltonian perturbations used to define the wrapped category. In fact, the wrapped Fukaya category can alternatively be defined from F(X V , W) as the result of localization with respect to a certain natural transformation from the Serre functor (up to a shift) to the identity, induced by the monodromy of W near infinity (see §4 of [36] and §6 of [37]); this parallels the fact that DbCoh(X \ D) is the localization of DbCoh(X) with respect to the natural transformation from - (9 Kx (Le., the Serre functor up to a shift) to the identity given by the defining section of D.

44

D.AUROUX

Finally, when considering compact closed Lagrangian submanifolds, there is no difference between the Fukaya category of (XV, W) and the wrapped Fulaya category; the full subcategory consisting of these compact objects is expected to be equivalent to the subcategory of DbCoh(X \ D) generated by complexes with compactly supported cohomology. 5.3. Complete intersections. As pointed out to the author by Ludmil Katzarkov, the above ideas can be extended to understand mirror symmetry for complete intersections (remaining in the framework of manifolds with effective anticanonical divisors). Namely, consider divisors DI,"" Dk C X (smooth, or at most with normal crossing singularities), intersecting each other transversely, such that L- Di = - K x. Let (XV, W) be the mirror of X relative to the anticanonical divisor L- Di: then the superpotential on XV splits into a sum W = WI + ... + Wk, where Wi : XV -t C records the contributions to the superpotential of holomorphic Maslov index 2 discs which hit the component Di of the anticanonical divisor. For a subset I ~ {I, ... , k}, consider the complete intersection XI = niEI Di C X, and the divisors DI,j = XI n D j , j ¢ I, whose sum represents the anticanonical class of X I. Then we have: CONJECTURE 5.5. In the large volume limit t -t 00, the mirror to XI equipped with the Kahler form WIX/ +tCI (X)lx/ and the anticanonical divisor L-jr¢I DI,j is approximated (in the sense of Conjecture 5.2) by the complete intersection X'j := niEI Wi-I (e t ) in XV, equipped with the superpotential WI:= L-NI Wj.

As before, if we are only interested in comparing the complex geometry of XI with the symplectic geometry of (X'j, WI), then the construction does not depend on the parameter t, and passage to the large volume limit is not needed. Conjecture 5.5 can be understood geometrically as follows. In this setting, we expect to have a special Lagrangian torus fibration 7r : X \ (U Dd -t B, whose base B has boundary and corners: at the boundary, the special Lagrangian fibers collapse onto one of the hypersurfaces D i , and at the corners they collapse onto the intersection of several Di. (This picture is e.g. obvious in the toric setting, where B is the interior of the moment polytope.) Whenever the fibers of 7r lie sufficiently close to D i , they are expected to bound small meridian discs intersecting Di transversely once, whereas the other families of discs have comparatively larger symplectic area, so that Wi = ZOi + 0(1). Setting ZOi equal to 1 for i E I amounts to considering special Lagrangian tori that are completely collapsed onto XI = niEIDi , equipped with flat connections that have trivial holonomy along the meridian loops, i.e. are pulled back from special Lagrangian tori in XI. Thus, before instanton corrections, niEI{zOi = I} is the (uncorrected) SYZ mirror

SPECIAL LAGRANGIAN FIBRATIONS

45

to XI \ (Uj\lI DI,j). When t ---t 00 the discrepancy between Wi and Z8i and the differences in instanton corrections are expected to become negligible. Moreover, in the limit where LeX \ (U D i ) collapses onto a special Lagrangian A C XI \ (Uj\lI DI,j), for j r:f- I the dominant terms in Wj should correspond to families of holomorphic discs in (X, L) that converge to holomorphic discs in (XI, A) (intersecting DI,j). Hence, '£j\lI Wj should differ from the superpotential for the mirror to XI by terms that become negligible in the large volume limit. As a special case of Conjecture 5.5, taking I = {I, ... ,k}, (in the large volume limit) the fiber of (WI"'" Wk) is mirror to the Calabi-Yau complete intersection X{1, ... ,k} = DI n· .. n D k . (In this case there is no residual superpotential.) This is consistent with standard conjectures. It is also worth noting that, in a degenerate toric limit, Conjecture 5.5 recovers the predictions made by Hori and Vafa [23] for mirrors of Fano complete intersections in toric varieties. To give a simple example, consider X = tClP'3 (with fClP l W = A), and let D I , D2 C X be quadric surfaces intersecting transversely in an elliptic curve E = DI n D 2 . Then the superpotential on XV decomposes as a sum W = WI + W2. In the degenerate limit where DI and D2 are toric quadrics consisting of two coordinate hyperplanes each, and E is a singular elliptic curve with four rational components, we have XV = {ZOZIZ2 Z3 = e- A} C (tC*)4, and W = WI + W2, where WI = Zo + Zl and W2 = Z2 + Z3. Then the mirror to DI is the surface {ZOZIZ2Z3 = e- A, Zo

+ ZI

= et } C

(tC*)4,

equipped with the superpotential W2 = Z2+Z3, and similarly for D 2; and the mirror to E is the curve {ZOZIZ2Z3 = e- A , ZO+ZI = et , Z2+Z3 = et } (a noncompact elliptic curve with four punctures). These formulas are essentially identical to those in Hori-Vafa [23]. To be more precise: viewing Di and E as symplectic manifolds (in which case the degeneration to the toric setting should be essentially irrelevant, i.e. up to a fiberwise compactification of the Landau-Ginzburg models we can think of smooth quadrics and elliptic curves), but taking the large volume limit t ---t 00, these formulas give an approximation to the complex geometry of the mirrors. On the other hand, if we consider the symplectic geometry of the mirrors, then the formulas give exact mirrors to Di and E viewed as singular complex manifolds (torically degenerated quadrics and elliptic curves, i.e. large complex structure limits). Thus Hori and Vafa's formulas for toric complete intersections should be understood as a construction of the mirror at a limit point in both the complex and Kiihler moduli spaces. References [1] M. Abouzaid, Homogeneous coordinate rings and mirror symmetry for toric varieties, Geom. Topol. 10 (2006), 1097-1157. [2] M. Abouzaid, D. Auroux, L. Katzarkov, in preparation.

46

D.AUROUX

[3] M. Abouzaid, P. Seidel, An open string analogue of Viterbo functoriality, arXiv:0712.3177. [4] D. Auroux, Mirror symmetry and T-duality in the complement of an anticanonical divisor, J. Gokova Geom. Topol. 1 (2007), 51-91 (arXiv:0706.3207). [5] D. Auroux, L. Katzarkov, D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. Math. 167 (2008), 867-943. [6] D. Auroux, L. Katzarkov, D. Orlov, Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves, Inventiones Math. 166 (2006), 537-582. [7] A. Bondal, D. Orlov, Derived categories of coherent sheaves, Proc. International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 47-56 (math.AG/0206295). [8] P. Candelas, X. C. De La Ossa, P. S. Green, L. Parkes, A pair of Calabi- Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991), 2l. [9] C.-H. Cho, Products of Floer cohomology of torus fibers in toric Fano manifolds, Comm. Math. Phys. 260 (2005), 613-640 (math.SG/0412414). [10] C.-H. Cho, Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle, arXiv:0710.5454. [11] C.-H. Cho, Y.-G. Oh, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math. 10 (2006), 773-814 (math.SG/0308225). [12] O. Cornea, F. Lalonde, Cluster homology, math.SG/0508345. [13] D. A. Cox, S. Katz, Mirror symmetry and algebraic geometry, Math. Surveys Monographs 68, Amer. Math. Soc., Providence, 1999. [14] K. Fukaya, y'-G. Oh, H. Ohta, K. Ono, Lagrangian intersection Floer theory: anomaly and obstruction, expanded version, 2006. [15] K. Fukaya, y'-G. Oh, H. Ohta, K. Ono, Lagrangian Floer theory on compact toric manifolds I, arXiv:0802.1703. [16] M. Gross, Topological mirror symmetry, Inventiones Math. 144 (2001), 75-137. [17] M. Gross, Special Lagrangian Fibrations II: Geometry, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc., Providence, 2001, pp. 95-150 (math.AG/9809072). [18] M. Gross, B. Siebert, From real affine geometry to complex geometry, math.AG /0703822. [19] M. Gross, B. Siebert, An invitation to toric degenerations, arXiv:0808.2749. [20] N. Hitchin, The moduli space of special Lagrangian submanifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25 (1997), 503-515. [21] K. Hori, Mirror symmetry and quantum geometry, Proc. ICM (Beijing, 2002), Higher Ed. Press, Beijing, 2002, vol. III, 431-443 (hep-th/0207068). [22] K. Hori, A. Iqbal, C. Vafa, D-branes and mirror symmetry, hep-th/0005247. [23] K. Hori, C. Vafa, Mirror symmetry, hep-th/0002222. [24] D. Joyce, Lectures on Calabi- Yau and special Lagrangian geometry, math. DG/OI08088. [25] A. Kapustin, Y. Li, D-branes in Landau-Ginzburg models and algebraic geometry, J. High Energy Phys. 0312 (2003), 005 (hep-th/0210296). [26] M. Kontsevich, Homological algebra of mirror symmetry, Proc. International Congress of Mathematicians (Zurich, 1994), Birkhauser, Basel, 1995, pp. 12(}-139. [27] M. Kontsevich, Lectures at ENS, Paris, Spring 1998, notes taken by J. Bellaiche, J.-F. Dat, I. Marin, G. Racinet and H. Randriambololona. [28] M. Kontsevich, Y. Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publ., 2001, pp. 203-263 (math.SG/0011041).

SPECIAL LAGRANGIAN FIBRATIONS

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[29] M. Kontsevich, Y. Soibelman, Affine structures and non-Ar'chimedean analytic spaces, The unity of mathematics, Progr. Math. 244, Birkhauser Boston, 2006, pp. 321-385 (math.AG/ 0406564). [30] N. C. Leung, Mirror symmetry without corrections, math.DG/0009235. [31] C. C. Liu, Moduli of l-holomorphic curves with Lagrangian boundary conditions and open Gromov- Witten invariants for an Sl-equivariant pair, math.SG/0210257. [32] R. C. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705-747. [33] D. Orlov, Triangulated categories of singularities and D-branes in Landau-Ginzburg models, Proc. Steklov Inst. Math. 246 (2004), 227-248 (math.AG/0302304). [34] P. Seidel, Fukaya categories and deformations, Proc. International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 351-360. [85] P. Seidel, Fukaya categories and Picard-Lefschetz theory, Zurich Lect. in Adv. Math., European Math. Soc., ZUrich, 2008. [86] P. Seidel, Symplect'ic homology as Hochschild homology, math.SG/0609087. [37] P. Seidel, Aoo -subalgebras and natural transformations, Homology, Homotopy Appl. 10 (2008),83-114 (math.KT/0701778). [38] P. Seidel, Homological mirror symmetry for the genus two curve, arXiv:0812.1171. [39] A. Strominger, S.-T. Yau, E. Zaslow, Mirror symmetry is T-duality, Nucl. Phys. B 479 (1996), 243-259 (hep-th/9606040). [40] R. P. Thomas, Moment maps, monodromy and mirror manifolds, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publishing, 2001, pp. 467-498 (math.DG/0104196). [41] R. P. Thomas, S.-T. Yau, Special Lagrangians, stable bundles and mean curvature flow, Comm. Anal. Geom. 10 (2002), 1075-1113 (math.DG/OI04197). DEPARTMENT OF MATHEMATICS, M.I.T., CAMBRIDGE MA 02139, USA E-mail address: auroux~math. mi t . edu

Surveys in Differential Geometry XIII

Sphere theorems in geometry Simon Brendle and Richard Schoen

1. The topological sphere theorem

The sphere theorem in differential geometry has a long history, dating back to a paper by H.E. Rauch in 1951. In that paper [64], Rauch posed the question of whether a compact, simply connected Riemannian manifold M whose sectional curvatures lie in the interval (1,4] is necessarily homeomorphic to the sphere. Around 1960, M. Berger and W. Klingenberg gave an affirmative answer to this question: THEOREM 1.1 (M. Berger [3]; W. Klingenberg [48]). Let M be a compact, simply connected Riemannian manifold whose sectional curvatures lie in the interval (1,4]. Then Ai is homeomorphic to

sn.

More generally, Berger [4] proved that a compact, simply connected Riemannian manifold whose sectional curvatures lie in the interval [1,4] is either homeomorphic to or isometric to a compact symmetric space of rank one. K. Grove and K. Shiohama proved that the upper bound on the sectional curvature can be replaced by a lower bound on the diameter:

sn

THEOREM 1.2 (K. Grove, K. Shiohama [33]). Let M be a compact Riemannian manifold with sectional curvature greater than 1. If the diameter of M is greater than 1f /2, then M is homeomorphic to

sn.

There is an interesting rigidity statement in the diameter sphere theorem. To describe this result, suppose that .A1 is a compact Riemannian manifold with sectional curvature K ~ 1 and diameter diam(M) ~ 1f/2. A theorem of D. Gromoll and K. Grove [27] asserts that M is either homeomorphic to or locally symmetric, or has the cohomology ring of the

sn,

The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS0604960. ©2009 International Press

An

S. BRENDLE AND R. SCHOEN

50

16-dimensional Cayley plane (see also [34]). B. Wilking [74] proved that, in the latter case, M is isometric to the Cayley plane.

2. Manifolds with positive isotropic curvature M. Micallef and J.D. Moore have used harmonic map theory to prove a generalization of Theorem 1.1. In doing so, they introduced a new curvature condition which they called positive isotropic curvature. A Riemannian manifold M is said to have positive isotropic curvature if Rl3l3

+ Rl4l4 + R2323 + R2424 -

2 R l 234

>0

for all points p E M and all orthonormal four-frames {el' e2, e3, e4} C TpM. We say that M has nonnegative isotropic curvature if

for all points p E M and all orthonormal four-frames {el' e2, e3, e4} C TpM. We next describe an alternative characterization of positive isotropic curvature, which involves complex notation. To that end, we consider the complexified tangent space T~ M = TpM ®IR C. A manifold M has nonnegative isotropic curvature if and only if

R(z,w,z,w)

~

0

for all points p EM and all vectors z, WETi'M satisfying g(z, z) = g(z, w) = g(w, w) = 0 (cf. [52]). The main theorem of Micallef and Moore is a lower bound for the index of harmonic two-spheres. Recall that the Morse index of a harmonic two-sphere is defined as the number of negative eigenvalues of the second variation operator (counted according to their mUltiplicities). 2.1 (M. Micallef, J.D. Moore [52]). Let u : 8 2 -+ M be a harmonic map from 8 2 into a Riemannian manifold M. We denote by E = u*TM the pull-back of the tangent bundle of M under u, and by EC = E ®lR C the complexification of E. Moreover, let I be the index form associated with the second variation of energy. Then PROPOSITION

- 1

l(s,8)=4 for all sections on 8 2 .

8

E

82

ID8_81 2 dxdy-4 8z

r( E C ). Here, z = x

1 82

+iy

R

(8U 8U_) dxdy. -8,8'8-,8 Z

Z

denote8 the complex coordinate

SPHERE THEOREMS IN GEOMETRY

51

PROOF OF PROPOSITION 2.1. Let I : r(EC) x r(EC) ~ C denote the complexified index form. Then 1(8, s)

=

r

+ ID JL812) dx dy J82 (ID JL812 ax ay

-h2 (R(~~,8,~~,S) +R(~~,8,~~,S)) for all

8

dxdy

E r(E C). We next define

au = ~ (au _ i au) az 2 ax ay

E

Moreover, for each section

8

r(EC), E r (EC) we define

With this understood, the complexified index form can be written in the form 1(8, s) = 2

r (ID JL812 + ID

J82

-2

for all

8

r

az

h2

(R

a_ az

812) dx dy

(~~'8' ~~,s) +R (~~,8, ~~,s))

E r(EC). Integration by parts yields

r

-ID 81 2) dx dy = 8J82 (ID ~812 dz az J82 g(D JLD Dz az ,D_

8

D_

h2 h2 (

R

(~~, ~~,

=

R

(~~,

8,

D

0

Dz

s) ~~, s)

= -

- R

for all

dxdy

8,

(~~,8, ~~,.s))

D JL8, Dz

s) dx dy

dx dy

dxdy

E r(EC). Putting these facts together, the assertion follows.

THEOREM 2.2 (M. Micallef, J.D. Moore [52]). Let u : 8 2 ~ M be a harmonic map from 8 2 into a Riemannian manifold N!. If M has positive isotropic curvature, then u has Morse index at least [n22]. PROOF OF THEOREM 2.2. We denote by E = u*TM the pull-back of the tangent bundle of M, and by E C the complexification of E. Let z = x+iy the complex coordinate on 8 2 . As above, we define 1 , DJL8 = - (DJL8 +'/,DJL8) c7Z 2 ax ay

52

S. BRENDLE AND R. SCHOEN

for each section D 88 = O.

8

E r(EI(} We say that

8

E r(EiC) is holomorphic if

8'i

Let 1l denote the space of holomorphic sections of EiC. Given two holomorphic sections 81,82 E 1l, the inner product g(81' 82) defines a holomorphic function on 8 2 . Consequently, the function g(81' 82) is constant. This defines a symmetric bilinear form

1l x 1l-+ C, By assumption, the map u : 8 2 -+ M is harmonic. Hence, ~ is a holomorphic section of EiC. Since u is smooth at the north pole on 8 2 , the section ~~ vanishes at the north pole. Thus, we conclude that g( ~~ , 8) = 0 for every holomorphic section 8 E 1l. In particular, we have g(~~, ~~) = O. By the Grothendieck splitting theorem (cf. [30]), the bundle EiC splits as a direct sum of holomorphic line subbundlesj that is,

eC =

L1 EI1 L2 EI1 ••• EI1 Ln.

We assume that the line bundles L 1, L2, ... , Ln are chosen so that

c1(LI) ~ cl(L2) ~ ... ~ cl(Ln). Note that c1(L1 ),C1(L2), ... ,Cl(Ln ) are uniquely determined, but LI, L2,"" Ln are not. By definition, EiC is the complexification of a real bundle. In particular, the bundle EiC is canonically isomorphic to its dual bundle. From this, we deduce that

c1(Lk) + cl(Ln-k+1) = 0 for k = 1, ... ,n (see [52], p. 209). For each k E {I, ... , n}, we denote by F(k) the direct sum of all line bundles L j except Lk and L n- k+1' More precisely, we define

F(k) -

ffi L.

W

J'

jE.7(k)

where :r(k) = {I, ... , n} \ {k, n - k + I}. Note that n~=l F(k) = {O}. Moreover, we have c1(F(k» = 0 and rankF(k) ~ n - 2. Let 1l(k) C 1l denote the space of holomorphic sections of F(k). It follows from the Riemann-Roch theorem that dime 1l(k) ~ n - 2. Fix an integer k E {I, ... , n} such that ~ fj r(F(k». Since dimiC 1l(k) ~ n - 2, there exists a subspace il C 1l(k) such that dimiC il ~ [n22] and g( 8, 8) = 0 for all 8 E il. We claim that the resctriction of I to il is negative definite. To see this, consider a section 8 E il. Since 8 is holomorphic, we have

{ (auaz,8'az'S au)

I(8,s)=-4}S2 R

dxdy

SPHERE THEOREMS

IN

GEOMETRY

53

by Proposition 2.1. Moreover, we have g(s,s) = g(~~,s) = g(~~, ~~) = O. Since M has positive isotropic curvature, it follows that

R

(~~,s, ~~,s) ~ O.

Putting these facts together, we conclude that 1(s, s) ::; O. It remains to analyze the case of equality. If 1(s, s) = 0, then s = f ~~ for some meromorphic function f : 8 2 ---7 C. However, ~~ tJ. r(p(k)) by our choice of k. Since s E r(p(k)), it follows that f vanishes identically. Therefore, the restriction of 1 to il is negative definite. We now complete the proof of Theorem 2.2. Suppose that m < [n22], where m denotes the number of negative eigenvalues of the second variation operator. Then dime il > m. Consequently, there exists a non-vanishing section s E il which is orthogonal to the first m eigenfunctions of the second variation operator. Since s E il, we have 1(s, s) < O. On the other hand, we have 1(s, s) ~ 0 since s is orthogonal to the first m eigenfunctions of the second variation operator. This is a contradiction. Combining their index estimate with the existence theory of Sacks and Uhlenbeck [67], Micallef and Moore obtained the following result: THEOREM 2.3 (M. Micallef, J.D. Moore [52]). Let M be a compact simply connected Riemannian manifold with positive isotropic curvature. Then M is a homotopy sphere. Hence, if n ~ 4, then M is homeomorphic to 8 n . SKETCH OF THE PROOF OF THEOREM 2.3. Suppose that 7rj(M) i: 0 for some integer j ~ 2. By a theorem of Sacks and Uhlenbeck [67], there exists a harmonic map u : 8 2 ---7 M with Morse index less than j - 1. On the other hand, any harmonic map u : 8 2 ---7 M has Morse index at least [Il-1 by Theorem 2.2. Putting these facts together, we obtain j > [Il. Thus, 7rj(M) = 0 for j = 2, ... , [Il. Since M is simply connected, the Hurewicz theorem implies that 7rj(M) = 0 for j = 1, ... , n - 1. Consequently, M is a homotopy sphere. We say that M has pointwise 1/4-pinched sectional curvatures if 0 < K(7rl) < 4K(7r2) for all points p E M and all two-planes 7rl,7r2 C TpM. It follows from Berger's inequality (see e.g. [47]) that every manifold with pointwise 1/4-pinched sectional curvatures has positive isotropic curvature. Hence, Theorem 2.3 generalizes the classical sphere theorem of Berger and Klingenberg. The topology of non-simply connected manifolds with positive isotropic curvature is not fully understood. It has been conjectured that the fundamental group of a compact manifold M with positive isotropic curvature is virtually free in the sense that it contains a free subgroup of finite index (see [23]'[29]). A. Fraser has obtained an important result in this direction:

S. BRENDLE AND R. SCHOEN

54

THEOREM 2.4 (A. Fraser [23]). Let M be a compact Riemannian manifold of dimension n 2: 5 with positive isotropic curvature. Then the fundamental group of M does not contain a subgroup isomorphic to Z EB Z. The proof of Theorem 2.4 relies on the existence theory of Schoen and Yau [68], and a careful study of the second variation of area (see also [22],[69]). The proof also uses the following result due to A. Fraser (see [23], Section 3): PROPOSITION 2.5 (A. Fraser [23]). Let h be a Riemannian metric on T2 with the property that every non-contractible loop in (T2, h) has length at least 1. Moreover, let 8 2 be the two-sphere equipped with its standard metric of constant curvature 1. Then there exists a degree-one map f from (T2, h) to 8 2 such that ID fl ~ C, where C is a numerical constant. PROOF OF PROPOSITION 2.5. Let ~ be the universal cover of (T2, h), and let 7r : ~ -+ (T2, h) denote the covering projection. Note that ~ is diffeomorphic to 1R2 • For each positive integer k, there exists a unit-speed geodesic ,k : [-k, k] -+ ~ such that d(rk(k), ,k( -k)) = 2k. Passing to the limit as k -+ 00, we obtain a unit-speed geodesic, : IR -+ ~ such that d(r(tr), ,(t2)) = It 1 - t21 for all tl, t2 ERIn particular, ,(tl) =1= ,(t2) whenever tl =1= t2· By the Jordan curve theorem, the complement ~ \ {,(t) : t E 1R} has exactly two connected components, which we denote by 0 1 and O2 • We next define functions Dl : ~ -+ IR and D2 : ~ -+ IR by

Dl(p)

=

inf{d(r(t),P) : t E 1R} { ~inf{d(r(t),P): t E 1R}

for p E 0 1 for p E O2 otherwise

and

D2(p) = d(r(O),p) - 1. Clearly, IDj(p) - Dj(q)1 ~ d(p, q) for all points p, q E

Q = {p

E

~: Dl(p)2 +D2(p)2 ~

~.

Let

:4}.

We claim that Q

c B1/ 3 (,(1)) U B 1/ 3 (r( -1)).

To see this, we consider a point p E Q. Then there exists a real number t such that d(r(t),p) = IDl(P)I. This implies

Il t l-ll =

Id(r(O),,(t)) -11

::; Id(r(O), ,(t)) - d(r(O),p)1 ~ d(r(t),p) = ID1(p)1

+ Id(r(O),p) - 11

+ Id(r(O),p) - 11

+ ID 2(P)I·

55

SPHERE THEOREMS IN GEOMETRY

From this, we deduce that min {d(')'(I),p), d(')'( -1),p)}

:s; min {d(')'(I), ')'(t)) , d(')'( -1), ')'(t))} + d(')'(t),p)

= Iltl- 11 + d(')'(t),p) :s; 2I D I(P)1 + ID2(P)1 1

< 3' Thus, Q c B I / 3 (')'(I)) U B I / 3 (,),( -1)). We next define R = QnBI(')'(I)). Clearly, R c B I / 3 (')'(I)). This implies DI(p)2 + D2(P)2 = 6~ for all points p E 8R. Hence, the map (D I ,D2) : R --+ B I/ 8 (0) maps 8R into 8B I/ 8 (0). The map (DI' D2) is smooth in a neighborhood of ')'(1). Moreover, the differential of (Db D2) at the point ')'(1) is non-singular. Since (DI (p), D2(p)) =1= (DI (')'(1)), D2(')'(I))) for all pER \ b(I)}, we conclude that the map (Db D2) : R --+ B I/ 8 (0) has degree one. In the next step, we approximate the functions DI and D2 by smooth functions. Let 8 be an arbitrary positive real number. Using the convolution procedure of Greene and Wu (see [24],[25]), we can construct smooth functions fh : R --+ ]R and D2 : R --+ ]R such that

and for all points p, q E R. Fix a cut-off function 1] : [0, 00) --+ [0, 1] such that 1](8) = 2 for 8 :s; 2 and 1](8) = 0 for 8 ~ 3. We define smooth maps

+ x~), X2 1](X~ + x~), 1 - x~ -

xn

and


= (0,0, -1) whenever xI + x~

F(p) =

~ 3. We now define a

'Ij;(16 DI(p), 16 D2(p)).

There exists a numerical constant C such that d(F(p), F(q)) :s; C d(p, q) for all points p, q E R. Moreover, F maps a neighborhood of the boundary 8R to the south pole on 8 2 • It is easy to see that the map F : R --+ 8 2 has degree one.

56

S. BRENDLE AND R. SCHOEN

By assumption, every non-contractible loop in (T2, h) has length at least 1. Hence, if p, q are two distinct points in ~ satisfying 1f(p) = 1f(q), then d(p, q) 2': 1. Since R C B 1/ 3 b(1)), it follows that the restriction 1fln is injective. We now define a map f : (T2, h) -+ 8 2 by

f(y) = {F(P) (0,0, -1)

if Y = 1f(p) for some point pER otherwise

for y E T2. It is straightforward to verify that f has all the required properties. This completes the proof of Proposition 2.5. We note that the minimal surface arguments in [23] can be extended to the case n = 4 provided that M is orientable. THEOREM 2.6. Let M be a compact orientable four-manifold with positive isotropic curvature. Then the fundamental group of M does not contain a subgroup isomorphic to Z EB Z. PROOF OF THEOREM 2.6. Suppose that 1fl(M) contains a subgroup G which is isomorphic to Z EB Z. For each positive integer k, we denote by G k the subgroup of 1fl (M) corresponding to kZ EB kZ. Moreover, let

Ak = inf{L(a) : a is a non-contractible loop in M with [a] E

Gd.

Note that Ak -+ 00 as k -+ 00. Fix k sufficiently large. By a theorem of Schoen and Yau [68], there exists a branched conformal minimal immersion u : T2 -+ M with the property that u* : 1fl(T2) -+ 1fl(M) is injective and maps 1fl(T2) to Gk. Moreover, the map u minimizes area in its homotopy class. Hence, u is stable. We next consider the normal bundle of the surface u(T 2 ). We denote by E the pull-back, under u, of the normal bundle of u(T 2 ). Note that E is a smooth vector bundle of rank 2, even across branch points. (This follows from the analysis of branch points in [35],[54].) Since M and T2 are orientable, we conclude that E is orientable. Let E iC = E Q9JR C be the complexification of E. Since E is orientable, the complexified bundle E iC splits as a direct sum of two holomorphic line bundles E(l,O) and E(O,I). Here, E(I,O) consists of all vectors of the form a( v - iw) E E iC , where a E C and {v, w} is a positively oriented orthonormal basis of E. Similarly, E(O,l) consists of all vectors of the form a( v + iw) E E iC , where a E C and {v, w} is a positively oriented orthonormal basis of E. Since E iC is the complexification of a real bundle, we have Cl (E(I,O)) +Cl (E(O,I)) = Cl (EiC) = O. Without loss of generality, we may assume that cr(E(I,O)) 2': O. (Otherwise, we choose the opposite orientation on E.) Since u is stable, we have (1)

SPHERE THEOREMS IN GEOMETRY

57

for all sections 8 E f(EC) (see [22],[69]). Every section 8 E f(E(l,O)) is isotropic, i.e. g(8,8) = O. Since M has positive isotropic curvature, there exists a positive constant", such that

R for all sections

8

(~~ ,8, ~~, s) ~ '" ~~ 2 8 2 1

1

1 1

E f(E(l,o)). Putting these facts together, we obtain

(2) for all that

8

E r(E(l,O)). Moreover, we can find a positive constant c = c(k) such

8

E r(E(I,O)). Taking the arithmetic mean of (2) and (3), we obtain

(3) for all

£2ID:/12dXdY~~'"

(4)

£2 (1~~12 +c)

181 2 dxdy

for all 8 E f(E(I,o)). We next define a Riemannian metric h on T2 by

h

= u*g + 2c (dx 12> dx + dy 12> dy) = u*g + c (dz ® dE + dE ® dz).

Every non-contractible loop in (T2, h) has length at least Ak . By Proposition 2.5, there exists a degree-one map I from (T2, h) to the standard sphere 8 2 such that Ak ID II ::; C. This implies

A~ 1~~12 : ; CII:zl: = CI (1~~12 +c),

(5)

CI

where is a positive constant independent of k. Fix a holomorphic line bundle Lover 8 2 with cI(L) > O. We also fix a metric and a connection on L. Finally, we fix sections Wl,W2 E f(L*) such that IWII + IW21 ~ 1 at each point on 8 2. Let ~ = f* L be the pull-back of L under the map f. Since I has degree one, we have CI(~) > O. Since cI(E(I,O)) ~ 0, it follows that cl(E(l,O)®O > O. By the Riemann-Roch theorem, the bundle E(I,O) ®~ admits a non-vanishing holomorphic section, which we denote by (J". For j = 1,2, we define Tj = f*(Wj) E r(C) and 8j = (J" ® Tj E f(E(l,O)). Since (J" is holomorphic, we have

where V' denotes the connection on

C. We next observe that

1V'..ft. Tjl2 = IV' ~Wj12 8:e

8:e

::;

C21 ~I uZ

2

1

,

S. BRENDLE AND R. SCHOEN

58

where C2 is a positive constant independent of k. This implies

for j

= 1,2. Using (5), we obtain

A~ ID:z Sjl2 ~ C C2 (1~:12 +c) lui 1

for j

= 1,2. From this, we deduce that

£2 (ID:z sd + ID:z s212) ~ £2 (I~:12 +c) lul

A~ (6)

2

2C1C2

dxdy

2 dxdy.

Note that

ISll + IS21 = lui (1 7 11 + 17 21) ~ lui at each point on T2. Hence, it follows from (4) that

£2 (ID:z sl1 2+ ID:z s112) ~ ~ ~ £2 (I ~: 12 + c) (l s11 2+ IS212) dxdy

(7)

~ ~~£2 (1~:12 +c)

dx dy

lul 2 dxdy.

Thus, we conclude that ~ A~ ~ 16 C1 C2. This contradicts the fact that Ak -+ 00 as k -+ 00. In the remainder of this section, we describe sufficient conditions for the vanishing of the second Betti number. M. Berger [4] proved that the second Betti number of a manifold with pointwise 1/4-pinched sectional curvatures is equal to O. In even dimensions, the same result holds under the weaker assumption that M has positive isotropic curvature: THEOREM 2.7 (M. Micallef, M. Wang [53]). Let M be a compact Riemannian manifold of dimension n ~ 4. Suppose that n is even and M has positive isotropic curvature. Then the second Betti number of M vanishes.

2.7. Suppose that 'l/J is a non-vanishing harmonic two-form on M. It follows from the Bochner formula that PROOF OF THEOREM

n

f).'l/Jik

=

L Ric{ j=l

n

'l/Jjk

+L j=1

n

Ric{ 'l/Jij

-

2

L j,I=1

Rijkl 'l/Jjl,

SPHERE THEOREMS IN GEOMETRY

59

where b.'ljJ = 'L/;,l=l gj/ DJ,l'ljJ denotes the rough Laplacian of'ljJ. Fix a point p E M where the function 1'ljJ12 attains its maximum. At the point p, we have 1'ljJ12 > 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 < <5(n) IUI 2, where <5(4) =

t, <5(5) = /0'

and 2

<5(n) = (n - 2)(n + 1) for n :2 6. Let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with initial metric go· Then the rescaled metrics 2(n-l)(T-t) g(t) converge to a metric of constant sectional curvature 1 as t -t T. Note that the curvature condition in Theorem 3.5 is preserved by the Ricci flow. Moreover, any manifold (!vI, go) which satisfies the assumptions of Theorem 3.5 necessarily has positive curvature operator (see [45], Corollary 2.5). C. B6hm and B. Wilking proved a convergence result for manifolds with two-positive curvature operator, generalizing Chen's work in dimension 4: THEOREM 3.6 (C. B6hm, B. Wilking [7]). Let (1\1, go) is a compact Riemannian manifold with two-positive curvature operator. Let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with initial metric go. Then the rescaled metrics 2(n-l)(T-t) g(t) converge to a metric of constant sectional curvature 1 as t -t T. C. Margerin [51J used the Ricci flow to show that any compact fourmanifold which has positive scalar curvature and satisfies the pointwise pinching condition IWI 2+ 1V12 < IUI 2 is diffeomorphic to 84 or 1l~.IP,4. By combining Margerin's theorem with a conformal deformation of the metric, A. Chang, M. Gursky, and P. Yang were able to replace the pointwise pinching condition by an integral pinching condition. As a result, they obtained a conformally invariant sphere theorem in dimension 4:

SPHERE THEOREMS IN GEOMETRY

65

THEOREM 3.7 (A. Chang, M. Gursky, P. Yang [16]). Let (M,go) be a compact four-manifold with positive Yamabe constant. 8'uppose that (M, go) satisfies the integral pinching condition (15) Then 1M is diffeomorphic to 8 4 or 1l~JP4 .

Given any compact four-manifold lvI, the Gauss-Bonnet theorem asserts that (IUI 2 -1V12 + IWI2) = 321[2 X(M)

r

1M

(cf. [16]' equation (0.4)). Hence, the condition (15) is equivalent to

1M IWI 2 < 161[2 X(M).

(16)

Here, the norm of W is defined by

IWI 2 = E7,j,k,l=l Wijkl Wijkl.

4. New invariant curvature conditions for the Ricci flow In an important paper [44], R. Hamilton proved that the Ricci flow preserves positive isotropic curvature in dimension 4. Moreover, Hamilton studied solutions to the Ricci flow in dimension 4 with positive isotropic curvature, and analyzed their singularities. Finally, Hamilton [44] devised a sophisticated procedure for extending the flow beyond singularities (see also [19], [59], [60]). In a recent paper [10], we proved that positive isotropic curvature is preserved by the Ricci flow in all dimensions. This was shown independently by H. Nguyen in his doctoral dissertation. Our proof relies on the following algebraic result which is of interest in itself (see [10], Corollary 10): PROPOSITION 4.1. Let R be an algebraic curvature tensor on ]Rn with nonnegative isotropic curvature. Moreover, suppose that {el' e2, e3, e4} is an orthonormal four-frame satisfying

R1313

+ R1414 + R 2323 + R2424 -

2 R 12:{4

= o.

Then

Q(Rh313

+ Q(Rh414 + Q(Rh323 + Q(Rh424 -

2 Q(Rh234 ~ 0,

where Q(R) is given by (14).

SKETCH OF THE PROOF OF PROPOSITION 4.1. Following Hamilton [37], we write Q(R) = R2 + R#, where R2 and R# are defined by n

(R 2 )ijkl

=

L p,q=l

~jpq Rklpq

S. BRENDLE AND R. SCHOEN

66

and

n

(R#)ijkl =

2: ~pkq

2

n

Rjplq -

p,q=l

2: ~plq

2

Rjpkq.

p,q=l

Note that R2 and R# do not satisfy the first Bianchi identity, but does. Since R satisfies the first Bianchi identity, we have (R#h313 + (R#h414 + (R#h323

n

2:

(R1p1 q p,q=l n

- 2

+ R#

+ (R#h424 + 2 (R#h342 + 2 (R#h423

n

=2

R2

2:

+ R2p2q) (R3p3q + R4p4q) -

2:

2

R12pq R34pq

p,q=l

(R1p3q

+ R 2p4q ) (R3p1q + R 4p2q )

p,q=l n

- 2

2:

(R1p4q - R2p3q)

(~p1q

- R3p2q).

p,q=l

We claim that the right hand side is nonnegative. To prove this, we define 4

[(1)

=

4

2:

(R1p1 q

+ R2p2q) (R3p3q + R4p4q) -

p,q=l

2:

R12pq R34pq

p,q=l

4

- 2:

(R1p3q

+ R 2p4q ) (R3p1q + R4p2q)

p,q=l 4

- 2:

(R1p4q - R 2p3q)

(~p1q

- R3p2q),

p,q=l

n

4

[(2) =

2:

4

2:(R1P1q p=lq=5

4

- 2:

2: 2:

+ R2p2q) (R3p3q + ~p4q) -

R12pq R 34pq

p=lq=5

n

2:(R1P3q p=lq=5

4

n

+ R2p4q) (R3p1q + ~p2q)

n

- 2: 2:

(R1p4q - R 2p3q)

(~p1q

- R3p2q),

p=lq=5

n

[(3) =

2:

n

(R1p1q p,q=5 n

- 2:

+ R2p2q) (R3p3q + R4p4q) -

(R1p3q

+ R2p4q) (R3p1q + R 4p2q)

p,q=5

n

- 2: p,q=5

2: p,q=5

(R1p4q - R2p3q) (R4p1q - R3p2q).

R12pq R34pq

SPHERE THEOREMS

IN

GEOMETRY

67

We may view the isotropic curvature as a real-valued function on the space of orthonormal four-frames. This function attains its minimum at {ell e2, e3, e4}. Consequently, the first variation at {el' e2, e3, e4} is zero and the second variation is nonnegative. Using the fact that the first variation is zero, we can show that 1(1) = 1(2) = 0 (see [10]' Propositions 5 and 7). In order to estimate 1(3), we consider the following (n - 4) x (n - 4) matrices:

apq = Rlplq + R 2p2q, pq = R 3p1q + R4p2q, epq = Rl2pq,

bpq = R 3p3q + R4p4q, dpq = R4plq - R3p2q, fpq = R 34pq

C

(5

~ p, q ~

n). Since the second variation is nonnegative, the matrix

-c D

-Dj -C

A

-E

E

A

is positive semi-definite. From this, we deduce that

= tr(AB) + tr(EF) -

1(3)

tr(C 2 )

-

tr(D2) 20

(see [10], Proposition 9). Putting these facts together, we conclude that (17)

(R#h313

+ (R#h414 + (R#h323 + (R#h424

Moreover, we have

(R 2h313 + (R2h414 (18)

+ (R2h323 + (R2h424 + 2 (R 2h342 + 2 (R 2h423

n

=

L (Rl3pq -

p,q=1

n

R24pq)2

+

L (Rl4pq + R23pq)2 20

p,q=l

by definition of R2. Adding (17) and (18), we obtain

+ Q(R)1414 + Q(Rh323 + Q(Rh424 + 2Q(Rh342 + 2Q(Rh423 2 O.

Q(Rh313

(19)

Since Q(R) satisfies the first Bianchi identity, we conclude that

Q(Rh313

+ Q(Rh414 + Q(Rh323 + Q(Rh424 -

2 Q(Rh234 2 0,

as claimed. THEOREM 4.2 (S. Brendle, R. Schoen [10]; H. Nguyen 155]). Let M be a compact manifold of dimension n 2 4, and let get), t E [0, T), be a family of metrics on J\;[ evolving under Ricci flow. If (M,g(O)) has nonnegative isotropic curvature, then (J\;[, g( t)) has nonnegative isotropic curvature for all t E [0, T).

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PROOF OF THEOREM 4.2. It follows from Proposition 4.1 that nonnegative isotropic curvature is preserved by the ODE -9tR = Q(R). Hence, the assertion follows from Hamilton's maximum principle for systems (see [37]). As an application of Proposition 4.1, we are able to generalize a theorem of S. Tachibana [73]: THEOREM 4.3 (S. Brendle [14]). Let (M, g) be a compact Einstein manifold of dimension n ~ 4 with positive isotropic curvature. Then (M, g) has constant sectional curvature. PROOF OF THEOREM 4.3. After rescaling the metric if necessary, we may assume that the scalar curvature of (M,g) equals n(n - 1). Since 9 is an Einstein metric, we have Ric ij = (n - 1) gij. This implies D.R + Q(R)

(20)

= 2(n -1) R.

We define a tensor Sijkl by Sijkl = ~jkl - "i-{gik gjl - gil 9jk),

where /'i, is a positive constant. Note that S satisfies all the algebraic properties of the curvature tensor. Let /'i, be the largest constant with the property that Sijkl has nonnegative isotropic curvature. Then there exists a point p EM and a four-frame {el,e2,e3,ed c TpM such that

S(el, e3, el, e3) + S(el, e4, el, e4) - 2S(el,e2,e3,e4) = O.

+ S(e2, e3, e2, e3) + S(e2, e4, e2, e4)

Therefore, it follows from Proposition 4.1 that (21)

Q(S)(el' e3, el, e3) + Q(S)(el' e4, el, e4) + Q(S)(e2, e3, e2, e3) + Q(S)(e2,e4,e2,e4) - 2Q(S)(el,e2,e3,e4) ~ o.

We next observe that Q(S)ijkl = Q(R)ijkl

+ 2(n -

1) /'i,2 (gik gjl - gil gjk)

- 2/'i, (Ricik gjl - Ricil gjk - Ricjk gil

+ RiCjl gik),

hence Q(S)ijkl

= Q(R)ijkl + 2(n - 1) /'i, (/'i, - 2) (gik gjl -

gil gjk).

Substituting this into (21), we obtain

+ Q(R)(el, e4, el, e4) + Q(R)(e2, e3, e2, e3) + Q(R)(e2, e4, e2, e4) - 2 Q(R)(el' e2, e3, e4) + 8(n - 1) /'i, (/'i, - 2)

Q(R)(el, e3, el, e3)

(22)

~

O.

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Fix a vector wE TpM, and consider the geodesic ,(8) = expp (8w.). Moreover, let Vj(8) be a parallel vector field along, with Vj(O) = ej. The function

8 r-+ R(VI (8), V3(8), Vl(8), V3(8)) + R(Vl(8), V4(8), Vl(8), V4(8)) + R(V2(8), V3(8), V2(S), V3(8)) + R(V2(8), 1)4(8), V2(8), V4(S)) - 2 R( VI (8), V2(8), V3(8), V4(8)) - 41'1: is nonnegative and vanishes for s = O. Hence, the second derivative of that function at 8 = 0 is nonnegative. This implies

(D~),wR)(el' e3, eI, e3) + (D;,wR)(el' e4, el, e4) + (D;,wR)(e2' e3, e2, e3)

+ (D~),wR)(e2' e4, e2, e4) Since (23)

W

2 (D;,wR)(el' e2, e3, e4) 2:: O.

E TpM is arbitrary, we conclude that (~R)(el' e3, el, e3)

+ (~R)(el' e4, el, e4) + (~R)(e2' e3, e2, e3)

+ (~R)( e2, e4, e2, q) -

2 (~R) (el' e2, e3, e4) 2:: O.

Adding (22) and (23) yields

R( el, e3, el, e3) + R( el, e4, el, e4) + R( e2, e3, e2, e3) + R( e2, e4, e2, e4) - 2 R(el' e2, e3, e4)

+ 41'1: (1'1: -

2) 2:: O.

On the other hand, we have

R(el' e3, el, e3) + R(el' e4, el, e4) + R(e2' e3, e2, e3) + R(e2' e4, e2, e4) - 2R(el,e2,e3,e4) - 41'1: = O. Since 1'1: is positive, it follows that 1'1: 2:: l. Therefore, 8 has nonnegative isotropic curvature and nonpositive scalar curvature. Hence, Proposition 2.5 in [53] implies that the Weyl tensor of (M, g) vanishes. In the next step, we apply Theorem 4.2 to the product manifolds (M,g(t)) x]R and (M,g(t)) x ]R2. THEOREM 4.4 (S. Brendle, R. Schoen [10]). Let M be a compact manifold of dimension n 2:: 4, and let g(t), t E [0, T), be a solution to the Ricci flow on M. If (M, g(O)) x]R has nonnegative isotropic curvature. then (M, g(t)) x ]R has nonnegative isotropic curvature for all t E [0, T). THEOREM 4.5 (S. Brendle, R. Schoen [10]). Let M be a compact manifold of dimension n 2:: 4, and let g(t), t E [0, T), be a family of metrics on M evolving under Ricci flow. If (M, g(O)) x ]R2 has nonnegative isotropic curvature, then (M,g(t)) x ]R2 has nonnegative isotropic curvature for all t E [0, T). A similar result holds for products of the form (M,g(t)) x 8 2 (1), where 8 2 (1) denotes a two-dimensional sphere ofradius 1 (see [12], Proposition 10).

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4.6 (S. Brendle [12]). Let M be a compact manifold of dimension n 2:: 4, and let g(t), t E [0, T), be a solution to the Ricci flow on M. If (M, g(O)) x 8 2(1) has nonnegative isotropic curvature, then (M, g(t)) x 8 2 (1) has nonnegative isotropic curvature for all t E [0, T). THEOREM

Theorem 4.6 is quite subtle, as the manifolds (M,g(t)) x 8 2 (1) do not form a solution to the Ricci flow. Theorems 4.4-4.6 provide us with various curvature conditions that are preserved by the Ricci flow. We now discuss these curvature conditions in more detail. Let M be a Riemannian manifold of dimension n 2:: 4. The product M x ~ has nonnegative isotropic curvature if and only if

for all orthonormal four-frames {el,e2,e3,e4} and all'\ E [-1,1] (see [12], Proposition 4). Similarly, the product M x ~2 has nonnegative isotropic curvature if and only if

for all orthonormal four-frames {el, e2, e3, e4} and all '\, /1 E [-1,1] (see [10], Proposition 21). Finally, the product M x 8 2 (1) has nonnegative isotropic curvature if and only if R13l3 +,\2 R14l4 + /1 2

- 2'\/1 R 1234

+ (1 -

R2323 ,\2)

+ ,\2/1 2 R2424

(1 - /1 2 ) 2:: 0

for all orthonormal four-frames {el,e2,e3,e4} and all '\,/1 E [-1,1] (cf. [12], Proposition 7). We can also characterize these curvature conditions using complex notation. The product M x ~ has nonnegative isotropic curvature if and only if R(z,w,z,w) 2:: 0 for all vectors z,w E Ti'M satisfying g(z,z)g(w,w)g(z, w)2 = o. Moreover, the product M x ~2 has nonnegative isotropic curvature if and only if R(z, w, z, w) 2:: 0 for all vectors z, WETi'M (see [53], Remark 3.3). Combining these results with earlier work of Hamilton [31] and of B6hm and Wilking [1], we obtain the following theorem: THEOREM 4.7 (S. Brendle, R. Schoen [10]). Let Riemannian manifold of dimension n 2:: 4 such that

CM, go)

be a compact

for all orthonormal four-frames {el,e2,e3,e4} and all '\,/1 E [-1,1]. Let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with

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initial metric go. Then the rescaled metrics 2(n-l)(T-t) g(t) converge to a metr'ic of constant sectional curvature 1 as t -t T. It follows from Berger's inequality that every manifold with pointwise 1/4-pinched sectional curvatures satisfies (24). Hence, we can draw the following conclusion:

COROLLARY 4.8 (S. Brendle, R. Schoen [10]). Let (M,go) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. 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 2(n-l)(T-t) g(t) converge to a metric of constant sectional curvature 1 as t -t T. Both Theorem 3.6 and Theorem 4.7 are sub cases of a more general convergence theorem for the Ricci flow: THEOREM 4.9 (S. Brendle [12]). Let (M, go) be a compact Riemannian manifold of dimension n 2: 4 such that

for all orthonormal four-frames {el' e2, e3, e4} and all A E [-1,1]. Let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with initial metric go. Then the rescaled metrics 2(n-l)(T-t) g(t) converge to a metric of constant sectional curvature 1 as t -t T. To conclude this section, we provide a diagram showing the logical implications among the following curvature conditions: (C1) AI has 1/4-pinched sectional curvatures (C2) AI has nonnegative sectional curvature (C3) M has two-nonnegative flag curvature; that is, R1313 for all orthonormal three-frames {el' e2, e3} (C4) M has nonnegative scalar curvature (C5) M x ]R2 has nonnegative isotropic curvature (C6) Jovf x 8 2 (1) has nonnegative isotropic curvature (C7) M x ]R has nonnegative isotropic curvature (C8) M has nonnegative isotropic curvature (C9) M has nonnegative curvature operator (ClO) AI has two-nonnegative curvature operator

+ R2323 2: a

Note that conditions (C4)-(C10) are preserved by the Ricci flow, but (C1)-(C3) are not.

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5. Rigidity results and the classification of weakly l/4-pinched manifolds In this section, we describe various rigidity results. The following theorem is based on the strict maximum principle and plays a key role in our analysis: THEOREM 5.1 (S. Brendle, R. Schoen [11]). Let M be a compact manifold of dimension n ~ 4, and let g(t), t E [0, T), be a solution to the Ricci flow on M with nonnegative isotropic curvature. Moreover, we fix a time T E (0, T). Then the set of all four-frames {eI, e2, e3, ed that are orthonormal with respect to g( T) and satisfy Rg(T) (el, e3, el, e3)

+ Rg(T)(el, e4, el, e4) + Rg(T) (e2, e3, e2, e3)

+ R g(T)(e2,e4,e2,e4) -

2Rg(T)(el,e2,e3,e4)

=

°

is invariant under parallel transport.

SKETCH OF THE PROOF OF THEOREM 5.1. Let E denote the vector bundle defined in Section 3, and let P be the orthonormal frame bundle of E; that is, the fiber of P over a point (p, t) E M x (0, T) consists of all n- frames {el, ... , en} C TpM that are orthonormal with respect to the metric g(t). Note that P is a principal O(n)-bundle over M x (0, T). Let 7r denote the projection from P to M x (0, T). For each t E (0, T), we denote by Pt = 7r- l (M x {t}) the time t slice of P. The connection D defines a horizontal distribution on P. For each point ~ = {el, ... ,en} E P, the tangent space T~P splits as a direct sum T~P = 1HI~ EB V~, where 1HI~ and V~ denote the horizontal and vertical subspaces

SPHERE THEOREMS IN GEOMETRY

at

73

~,

respectively. We next define a collection of horizontal vector fields Xl, ... , X n , Y on P. For each j = 1, ... , n, the value of Xj at a point ~ = {el' ... ,en} E P is given by the horizontal lift of the vector ej. Similarly, the value of Y at a point ~ = {el' ... ,en} E P is given by the horizontal lift of the vector Note that the vector fields Xl, ... , Xn are tangential to Pt· We define a function u : P -+ lR by

fk

u :~

= {el, ... , en}

+ R(el' e4, el, e4) + R(e2' e3, e2, e3) + R(e2, e4, e2, e4)

r-+ R(el' e3, el, e3)

- 2 R(el, e2, e3, e4),

where R denotes the Riemann curvature tensor of the evolving metric g(t). By assumption, the function u : P -+ lR is nonnegative. Using (13), we obtain n

Y(u) -

L Xj(Xj(u)) = Q(R)(el, e3, el, e3) + Q(R)(el' e4, el, e4) j=l

(see [11], Lemma 6). Moreover, there exists a positive constant K such that

Q(R)(el, e3, el, e3)

+ Q(R)(el' e4, el, e4) + Q(R)(e2' e3, e2, e3)

+ Q(R)(e2, e4, e2, e4) 2: K

inf

2 Q(R)(el' e2, e3, q)

(D2u)(~,O - K

C:E'V e ,Ic:19

Du(~) - Ku

sup C:E'V£,1c:19

(see [11], Lemma 7). Putting these facts together, we obtain n

Y(u) -

L Xj(Xj(u)) 2: K C:E'V£,1c:19 inf (D2u)(~,~) j=l

- K

sup

Du(~)

- Ku.

C:E'V£,1c:19

Since u satisfies this inequality, we may apply a variant of Bony's strict maximum principle for degenerate elliptic equations (cf. [8]). Hence, if i : [0,1] -+ Pr is a horizontal curve such that i(a) lies in the zero set ofthe function u, then i(l) lies in the zero set of the function u (see [11], Proposition 5). From this, the assertion follows. If we apply Proposition 5.1 to the product manifolds (M, g(t)) x 8 1 , then we obtain the following result: COROLLARY 5.2 (S. Brendle, R. Schoen [11]). Let M be a compact manifold of dimension n 2: 4. Moreover, let g(t), t E [a, T), be a solution to the Ricci flow on M with the property that (M,g(t)) x lR has nonnegative isotropic curvature. Fix real numbers T E (a,T) and A E [-1,1]. Then the

S. BRENDLE AND R. SCHOEN

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set of all four-frames {el' e2, e3, ed that are orthonormal with respect to g(r) and satisfy Rg(r) (el, e3, el, e3)

+ A? Rg(T)(el' e4, el, e4) + Rg(r) (e2' e3, e2, e3)

+ A? Rg(r) (e2' e4, e2, e4) -

2A Rg(T) (el' e2, e3, e4)

=

°

is invariant under parallel transport.

Theorem 5.1 and Corollary 5.2 can be used to prove various rigidity results. For example, we can extend Theorem 4.3 as follows: THEOREM 5.3 (S. Brendle [14]). Let (M, g) be a compact Einstein manifold of dimension n 2: 4 with nonnegative isotropic curvature. Then (M, g) is locally symmetric. In the next step, we classify all Riemannian manifolds (M, go) with the property that (M, go) x lR. has nonnegative isotropic curvature: THEOREM 5.4. Let (M, go) be a compact, locally irreducible Riemannian manifold of dimension n 2: 4. Suppose that (M, go) x lR. has nonnegative isotropic curvature. Moreover, let g(t), t E [0, T), denote the unique maximal solution to the Ricci flow with initial metric go. Then one of the following statements holds: (i) The rescaled metrics 2(n-I)(T-t) g(t) converge to a metric of constant sectional curvature 1 as t -+ T. (ii) n = 2m and the universal cover of (M, go) is a Kahler manifold. (iii) (M, go) is locally symmetric. SKETCH OF THE PROOF OF THEOREM 5.4. By assumption, the manifold (M, go) is locally irreducible and has nonnegative Ricci curvature. By a theorem of Cheeger and Gromoll, the universal cover of M is compact (see [17] or [61], p. 288). If (M, go) is locally symmetric, we are done. Hence, we will assume that (M, go) is not locally symmetric. By continuity, there exists a real number 6 E (O,T) such that (M,g(t)) is locally irreducible and non-symmetric for all t E (0,6). By Berger's holonomy theorem, there are three possibilities: Case 1: Suppose that Holo(M, g(r)) = SO(n) for some r E (0,6). In this case, it follows from Corollary 5.2 that

+ A2 Rg(T)(eI,e4,el,e4) + R g(r)(e2,e3,e2,e3) + A2 R g(r)(e2,e4,e2,e4) - 2A Rg(T) (el, e2, e3, e4) >

R g(T)(el,e3,el,e3)

°

for all orthonormal four-frames {el, e2, e3, e4} and all A E [-1, 1]. By Theorem 4.9, the rescaled metrics 2(n-l)(T-t) g(t) converge to a metric of constant sectional curvature 1 as t -+ T. Case 2: Suppose that n = 2m and Holo(M, g(t)) = U(m) for all t E (0,6). In this case, the universal cover of (M,g(t)) is a Kahler manifold for

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all t E (0,8). Since g(t) ---+ go in Coo, we conclude that the universal cover of (M, go) is a Kahler manifold. Case 3: Suppose that n = 4m 2: 8 and Holo(M,g(r)) = Sp(m)' Sp(l) for some r E (0,8). In this case, the universal cover of (M,g(r)) is a compact quaternionic-Kahler manifold. In particular, (M,g(r)) is an Einstein manifold. Since (M, g( r)) has nonnegative isotropic curvature, Theorem 5.3 implies that (M,g(r)) is locally symmetric. This is a contradiction. In the special case that (M, go) has weakly 1/4-pinched sectional curvatures, we can draw the following conclusion: COROLLARY 5.5 (S. Brendle, R. Schoen [11]). Assume that (M, go) has :S K (7l'1) :S weakly 1/4-pinched sectional curvatures in the sense that 4 K (7l'2) for all points p E M and all two-planes 7l'1, 7l'2 C TpM . Moreover, we assume that (M,go) is not locally symmetric. Finally, letg(t), t E [O,T), denote the unique maximal solution to the Ricci flow with initial metric go. Then the rescaled metrics 2(n-l)(T-t) g(t) converge to a metric of constant sectional curvature 1 as t ---+ T.

°

Finally, we briefly discuss the problem of classifying manifolds with almost 1/4-pinched sectional curvature. This question was first studied by M. Berger [6]. Berger showed that for each even integer n there exists a positive real number c(n) with the following property: if M is a compact, simply connected Riemannian manifold of dimension n whose sectional curvatures lie in the interval (1,4 + c(n)], then M is homeomorphic to 8 n or diffeomorphic to a compact symmetric space of rank one. Building upon earlier work of J.P. Bourguignon [9]' W. Seaman [70] proved that a compact, simply connected four-manifold whose sectional curvatures lie in the interval (0.188,1] is homeomorphic to 8 4 or Cp2. U. Abresch and W. Meyer [1] showed that a compact, simply connected, odd-dimensional Riemannian manifold whose sectional curvatures lie in the interval (1,4(1 + 1O-6?] is homeomorphic to a sphere. Using Theorem 5.4 and Cheeger-Gromov compactness theory, P. Petersen and T. Tao [62] proved that any compact, simply connected Riemannian manifold of dimension n whose sectional curvatures lie in the interval (1,4 + c(n)] is diffeomorphic to a sphere or a compact symmetric space of rank one. Here, c( n) is a positive real number which depends only on n.

6. Hamilton's differential Harnack inequality for the Ricci flow In 1993, R. Hamilton [39] established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator (see [38] for an earlier result in dimension 2). In this section, we describe this inequality, as well as some applications. Let (M,g(t)), t E (O,T), be a family of complete Riemannian manifolds evolving under Ricci flow. Following

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R. Hamilton [36], we define Pijk = DiRicjk - DjRicik

and M"t) =

~Ric"t) - ~2 D2t,),scal + 2 ~kJ'/ Ric kl -

Rickt RicJ'k

+~ 2t Ric"tJ'

Hamilton's matrix Harnack inequality states: THEOREM 6.1 (R. Hamilton [39]). Let (M, g(t)), t E (0, T), be a solution to the Ricci flow with uniformly bounded curvature and nonnegative curvature operator. Then M(w,w)

+ 2P(v,w,w) + R(v,w,v,w)

~

0

for all points (p, t) E M x (0, T) and all vectors v, w E TpM.

Taking the trace over w, Hamilton obtained a gradient estimate for the scalar curvature: COROLLARY 6.2 (R. Hamilton [39]). Assume that (M, g(t)), t E (0, T), is a solution to the Ricci flow with uniformly bounded curvature and nonnegative curvature operator. Then 0 1 , scal + t scal + 2 oi scal vt + 2 Ric(v, v) ~ 0

at

for all points (p, t) E M x (0, T) and all vectors v E TpM.

We note that H.D. Cao [15] has established a differential Harnack inequality for solutions to the Kahler-Ricci flow with nonnegative holomorphic bisectional curvature. In [13], it was shown that Hamilton's Harnack inequality holds under the weaker assumption that (M,g(t)) x JR2 has nonnegative isotropic curvature: THEOREM 6.3 (S. Brendle [13]). Let (M, g(t)), t E (0, T), be a solution to the Ricci flow with uniformly bounded curvature. Moreover, suppose that the product (M,g(t)) x JR2 has nonnegative isotropic curvature for all t E (0, T). Then M(w,w) + 2P(v,w,w) + R(v,w,v,w) ~ 0 for all points (p, t) E M x (0, T) and all vectors v, w E TpM, The Harnack inequality has various applications. For example, it can be used to show that any Type II singularity model with nonnegative curvature operator and strictly positive Ricci curvature must be a steady Ricci soliton (see [40]). A Riemannian manifold (M, g) is called a gradient Ricci soliton if there exists a smooth function f : M -+ JR and a constant p such that Ricij = p gij + Dl,j f· Depending on the sign of p, a gradient Ricci soliton is called shrinking (p > 0), steady (p = 0), or expanding (p < 0). As in Theorem 6.3, we can replace the condition that M has nonnegative curvature

SPHERE THEOREMS IN GEOMETRY

operator by the weaker condition that M curvature:

x ]R2

77

has nonnegative isotropic

PROPOSITION 6.4 (S. Brendle [13]). Let (M,g(t)), t E (-oo,T), be a solution to the Ricci flow which is complete and simply connected. We assume that (M, g(t)) x]R2 has nonnegative isotropic curvature and (M, g(t)) has positive Ricci curvature. Moreover, suppose that there exists a point (po,to) E M x (-00, T) such that scalg(t)(p) :::; scalg(to)(po) for all points (p,t) E M x (-oo,T). Then (M,g(to)) is a steady gradient Ricci soliton.

PROPOSITION 6.5 (S. Brendle [13]). Let (M,g(t)), t E (O,T), be a solution to the Ricci flow which is complete and simply connected. We assume that (M, g(t)) x]R2 has nonnegative isotropic curvature and (M, g(t)) has positive Ricci curvature. Moreover, suppose that there exists a point (po, to) E M x (0, T) such that t· scalg(t)(p) :::; to . scalg(to) (po) for all points (p, t) E M x (0, T). Then (M, g( to)) is an expanding gradient Ricci soliton.

7. Compactness of pointwise pinched manifolds R. Hamilton [41] has shown that a convex hypersurface with pinched second fundamental form is necessarily compact. B. Chen and X. Zhu proved an intrinsic analogue of this result. More precisely, they showed that a complete Riemannian manifold which satisfies a suitable pointwise pinching condition is compact: THEOREM 7.1 (B. Chen, X. Zhu [18]). Let (M, go) be a complete Riemannian manifold of dimension n 2 4. Assume that the scalar curvature of (M, go) is uniformly bounded and positive. Moreover, suppose that (M, go) satisfies the pointwise pinching condition

where c is a positive real number and i5(n) denotes the pinching constant defined in Theorem 3.5. Then M is compact.

Theorem 7.1 was generalized by L. Ni and B. Wu (see [57], Theorem 3.1). In the remainder of this section, we prove another generalization of Theorem 7.1. To that end, we need two results due to L. Ni [56] and 1. Ma and D. Chen [49]:

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78

PROPOSITION 7.2 (L. Ni [56]). Let (lvI,g) be a steady gradient Ricci soliton which is complete and non-compact. Suppose that there exists a point Po E lvI such that 0 < scal(p) ::; scal(po) for all points p E lvI. Moreover, we assume that Ric 2: c scal 9 for some constant c > O. Then there exists a constant a > 0 such that

scal(p) ::;

e-ad(po,p)

scal(po)

fOT d(po, p) 2: 1. PROOF OF PROPOSITION 7.2. Since (lvI,g) is a steady gradient Ricci soliton, there exists a smooth function f : N! ---+ lR. such that Ricij = D;'j f. This implies

o = oiscal- 211 DiRickl + 211 DkRicil = oiscal - 211 Dr,k,zf + 211 D~,i,zf kl . = oi scal + 2 9 Rikjl ff1 f = oiscal + 2 Ricij ai J. By assumption, the scalar curvature attains its maximum at the point Po. This implies oiscal(po) = O. Since NI has positive Ricci curvature, it follows that oi!(po) = O. Hence, the point Po is a critical point of the function f. Since f is strictly convex, we conclude that f has no critical points other than Po. Let a be a positive real number such that Ric 2: a 9 for d(po, p) ::; 1. This implies f(p) - f(po) 2: ~ ad(po,p)2 for d(po,p) ::; 1. Moreover, we have f(p) - f(po) 2: ~ ad(po,p) for d(po,p) 2: 1. Using the inequality Ric 2: cscal g, we obtain

oi(e 2c ! scal) oif

= e2c! (oiscaloif + 2cscalJDfJ2) ::; e2c! (oiscal oi f + 2 Ric ij Oi f ai 1) = O.

Fix a point p E lvI, and let "( : [0, (0) ---+ lvI be the solution of the ODE ,,('(s) = -oif(,,((s)) 8~i with initial condition "((0) = p. It is easy to see that "((s) ---+ Po as s ---+ 00. Moreover, the function s H e2c !(-y(s»scal("((s)) is monotone increasing. Thus, we conclude that scal(p) ::; e -2c(f(p)- !(po» scal(po) for all points p E !vI. This implies scal(p) ::;

e-c(Td(po,p)

scal(po)

for d(po,p) 2: 1. PROPOSITION 7.3 (L. Ma, D. Chen [49]). Let (lvI, g) be an expanding gradient Ricci soliton which is complete and non-compact. Suppose that there exists a point Po E lvI such that 0 < scal(p) ::; scal(po) for all points p E lvI.

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Moreover, we assume that Ric 2: c: scal 9 for some constant c: > O. Then there exists a constant Q > 0 such that scal(p) ::;

e- ad (po,p)2

scal(po)

for all points p EM. PROOF OF PROPOSITION 7.3. Since (M,g) is an expanding gradient Ricci soliton, there exists a smooth function f : M ----r lR and a constant p < 0 such that Ricij = P gij + Dlj f. This implies oiscal + 2 Ricij Qi f = O. By assumption, the scalar curvature attains its maximum at the point po. This implies oisca1(po) = O. Since M has positive Ricci curvature, it follows that od(po) = O. Thus, Po is a critical point of the function f. Since f is strictly convex, Po is the only critical point of f. Since Ric 2: 0, we have f(p) - f(po) 2: pd(pO,p)2 for all p EM. As above, the inequality Ric 2: c: scal 9 implies

-!

oi(e 2c ! scal) oif ::; O. Fix an arbitrary point p E M, and let "( : [0,00) ----r M be the solution of the ODE ,,('(s) = -oi f("((s)) a~t with initial condition "((0) = p. Then "((s) ----r Po as s ----r 00. Moreover, the function s H- e2c !b(s)) scal("((s)) is monotone increasing. Therefore, we have scal(p) ::; e- 2c (f(p)-!(po)) scal(po) ::; ecpd (pQ,p)2 scal(po) for all points p EM. Since p < 0, the assertion follows. The following theorem generalizes Theorem 7.1 above: THEOREM 7.4. Let (M, go) be a complete Riemannian manifold of dimension n 2: 4 with bounded curvature. Suppose that there exists a positive constant c: such that

R1313 + >.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

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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.

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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.

References [1] U. Abresch and W. Meyer, A sphere theorem with a pinching constant below 1/4, J. Diff. Geom. 44, 214-261 (1996) [2] B. Andrews and H. Nguyen, Four-manifolds with 1/4-pinched flag curvatures, to appear in Asian J. Math. [3] M. Berger, Les varietes Riemanniennes 1/4-pincees, Ann. Scuola Norm. Sup. Pisa 14, 161-170 (1960) [4] M. Berger, Sur quelques varietes riemaniennes suffisamment pincees, Bull. Soc. Math. France 88, 57-71 (1960) [5] M. Berger, Sur les varietes 4/23-pincees de dimension 5, C. R. Acad. Sci. Paris 257, 4122-4125 (1963) [6] M. Berger, Sur les varietes riemanniennes pincees juste au-dessous de 1/4, Ann. lust. Fourier (Grenoble) 33, 135-150 (1983) [7] C. B6hm and B. Wilking, Manifolds with positive curvature operator are space forms, Ann. of Math. 167, 1079-1097 (2008) [8] J.M. Bony, Principe du maximum, inegalite de Harnack et unicite du probleme de Cauchy pour les operateurs elliptiques degeneres, Ann. Inst. Fourier (Grenoble) 19, 277-304 (1969) [9] J.P. Bourguignon, La conjecture de Hopf sur S2 x S2, Riemannian geometry in dimension 4 (Paris 1978/1979), 347-355, Textes Math. 3, CEDIC, Paris (1981)

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[10] S. Brendle and R Schoen, Manifolds with 1/4-pinched curvature are space forms, J. Amer. Math. Soc. 22, 287-307 (2009) [11] S. Brendle and R Schoen, Classification of manifolds with weakly 1/4-pinched CU7'Vatures, Acta Math. 200, 1-13 (2008) [12] S. Brendle, A general convergence result for the Ricci flow, Duke Math. J. 145, 585-601 (2008) [13] S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow, J. Diff. Geom. 82, 207-227 (2009) [14] S. Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric, to appear in Duke Math. J. [15] H.D. Cao, On Harnack's inequalities for the Kahler-Ricci flow, Invent. Math. 109, 247-263 (1992) [16] A. Chang, M. Gursky, and P. Yang, A conformally invariant sphere theorem in four dimensions, Pub!. Math. IRES 98, 105-143 (2003) [17] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geom. 6, 119-128 (1971) [18] B. Chen and X. Zhu, Complete Riemannian manifolds with pointwise pinched curvature, Invent. Math. 423-452 (2000) [19] B. Chen and X. Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature, J. Diff. Geom. 74,177-264 (2006) [20] H. Chen, Pointwise 1/4-pinched 4-manifolds, Ann. Global Anal. Geom. 9, 161-176 (1991) [21] D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom. 18, 157-162 (1983) [22] A. Fraser, On the free boundary variational problem for minimal disks, Comm. Pure App!. Math. 53, 931-971 (2000) [23] A. Fraser, Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. 158, 345-354 (2003) [24] RE. Greene and H. Wu, On the subharmonicity and pl'urisubharmonicity of geodesically convex functions, Indiana Univ. Math. J. 22, 641-653 (1972) [25] RE. Greene and H. Wu, Coo convex functions and manifolds of positive cur"UatuT'e, Acta Math. 137, 209-245 (1976) [26] D. Gromoll, Differenzierbare Strukturen und Metriken positiver Krummung auf Sphiiren, Math. Ann. 164, 353-371 (1966) [27] D. Gromoll and K. Grove, A generalization of Berger's rigidity theorem for positively curved manifolds, Ann. Sci. Ecole Norm. Sup. 20, 227-239 (1987) [28] D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. 20, 75-90 (1969) [29] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21st century, Vo!' II (New Brunswick 1993), 1-213, Progr. Math., 132, Birkhauser, Boston (1996) [30] A. Grothendieck, Sur la classification des fibres holomorphes sur la sphere de Riemann, Amer. J. Math. 79, 121-138 (1957) [31] K. Grove, H. Karcher, and E. Ruh, Group actions and curvature, Invent. Math. 23, 31-48 (1974) [32] K. Grove, H. Karcher, and E. Ruh, Jacobi fields and Finsler metrics on compact Lie groups with an appl'ication to differentiable pinching problems, Math. Ann. 211, 7-21 (1974) [33] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. 106, 201-211 (1977) [34] K. Grove, Ramifications of the classical sphere theorem, Actes de la Table Ronde de Geometrie Differentielle (Luminy 1992), 363-376, Semin. Congr. 1, Soc. Math. France (1996)

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[62] P. Petersen and T. Tao, Classification of almost quarter-pinched manifolds, Proc. Amer. Math. Soc. 137, 2437-2440 (2009) [63] A. Petrunin and W. Tuschmann, Asymptotical flatness and cone structure at infinity, Math. Ann. 321, 775-788 (2001) [64] H.E. Rauch, A contribution to differ'ential geometry in the large, Ann. of Math. 54, 38-55 (1951) [65] E. Ruh, Krummung und differenzierbare Struktur auf Spharen II, Math. Ann. 205, 113-129 (1973) [66] E. Ruh, Riemannian manifolds with bounded cu'rvature mtios, J. Diff. Geom. 17, 643-653 (1982) [67] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of2-spheres, Ann. of Math. 113, 124 (1981) [68] R. Schoen and S.T. YaH, Existence of incompr'essible minimal sU1faces and the topology of three dimensional manifolds with non-negative scalar' curvature, Ann. of Math. 110, 127-142 (1979) [69] R. Schoen, Minimal submanifolds in higher codimensions, Mat. Contemp. 30, 169-199 (2006) [70] W. Seaman, A pinching theorem for four manifolds, Geom. Dedicata 31,37-40 (1989) [71] W.X. Shi, Defo'm!ing the metric on complete Riemannian manifolds, J. Diff. Geom. 30, 223-301 (1989) [72] M. Sugimoto and K. Shiohama, and H. Karcher, On the differentiable pinching problem, Math. Ann. 195, 1-16 (1971) [73] S. Tachibana, A theor'em on Riemannian manifolds with posit'ive curvature opemtor, Proc. Japan Acad. 50, 301-302 (1974) [74] B. Wilking, Index parity of closed geodesics and rigidity of Hopf fibmtions, Invent. Math. 144, 281-295 (2001) DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY, STANFORD,

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

85

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R. DONAGI AND T. PANTEV

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..<; the z = 0 incarnation of a twistor-type object that also has a z = 1 interpretation which is identified with the full solution. Non abelian Hodge theory, as developed by Donaldson, Hitchin, Corlette, Simpson [Don87, Hit87a, Cor88, Sim92, Cor93, Sim97], and many others, establishes under appropriate assumptions the equivalence of local systems and Higgs bundles. A richer object (harmonic bundle or twistor structure) is introduced, which specializes to both local systems and Higgs bundles. This is closely related to Deligne's notion of a z-connection: at z = 1 we have ordinary connections (or local systems), while at z = 0 we have Higgs bundles. Depending on the exact context, these specialization maps are shown to be diffeomorphisms or categorical equivalences. The projective (or compact Kahler) case and the one dimensional open case were settled by Simpson twenty years ago - but the open case in higher dimension had to await the breakthroughs by Biquard [Biq97], JostYang-Zuo [JYZ07], Sabbah [Sab05], and especially Mochizuki [Moc06, Moc09, Moc07a, Moc07b]. This higher dimensional theory produces an equivalence of parabolic local systems and parabolic Higgs bundles. This is quite analogous to what is obtained in the compact case, except that the objects involved are required to satisfy three key conditions discovered by Mochizuki. In section 5.1 we review these exciting developments, and outline our proposal for using non-abelian Hodge theory to construct the automorphic sheaves required by the geometric Langlands conjecture. This approach is purely mathematical of course, but it is parallel to physical ideas that have emerged from the recent collaborations of Witten with Kapustin, Gukov and Frenkel [KW06, GW06, FW08], where the geometric Langlands conjecture was placed firmly in the context of quantum field theory. Completion of these ideas depends on verification that Mochizuki's conditions are satisfied in situations arising from the geometric Langlands conjecture. This requires a detailed analysis of instability loci in moduli spaces. Particularly important are the wobbly locus of non-very-stable bundles, and the shaky locus, roughly the Hitchin image of stable Higgs bundles with an unstable underlying bundle. In section 6.1 we announce some results about these loci for rank 2 bundles. These lead in some cases to an explicit construction (modulo solving the differential equations inherent in the nonabelian Hodge theory) of the Hecke eigensheaf demanded by the geometric Langlands correspondence.

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Acknowledgments. We would like to thank Edward Witten for encouragement to write up [DP06], Shing-Tung Yau for inviting us to write this survey, and Dima Arinkin and Dennis Gaitsgory for illuminating conversations. Some of the work surveyed here was completed while the second author was visiting the Aspen Center for Physics, and he wishes to thank the ACP for the excellent and stimulating working environment. The work of Ron Donagi was supported by the NSF grants DMS 0612992 and DMS 0908487. The work of Tony Pantev was supported by NSF grant DMS0700446. Both authors were supported by NSF Research Training Group Grant DMS-0636606. 2. A brief review of the geometric Langlands conjecture

In a nutshell the Geometric Langlands Conjecture predicts the existence of a canonical equivalence of categories (GLC)

~

L

c: Dcoh(.cOC, 0) --+ D coh ( Bun, V),

which is uniquely characterized by the property that c sends the structure sheaves of points V in .coc to Hecke eigen V-modules on LBun:

The characters appearing in the geometric Langlands correspondence depend on certain background geometric data: • a smooth compact Riemann surface C; • a pair of Langlands dual complex reductive groups C, LC.

If we write 9 and Lg for the Lie algebras of G and LG and we fix maximal tori T c G and LT C LG with Cartan subalgebras by t c 9 and Lt c Lg, then group theoretic Langlands duality can be summarized in the relation between character lattices rootg

C

charc

corootg C cocharc

II

II

c

weight g

c

tV

c

coweightg

c

t

c

LtV

II

root[Lg] C Char[LC] C weight[Lg]

II

Here rootg C weight g C tV are the root and weight lattice corresponding to the root system on 9 and charc = Hom(T, eX) denotes the character lattice of G. Similarly corootg = {x E t I (weight g, x) C Z} ~ weight~ coweightg = {x E t I (rootg, x) C Z} ~ root~

R. DONAGI AND T. PANTEV

90

are the coroot and coweight lattices of g, and cochara = Hom(CX, T) = Hom(Sl, TlR) = {x E t I (chara, x) C Z} ~ charb is the cocharacter lattice of G. With this data we can associate various moduli stacks: Bun, LBun: the moduli stacks of principal G, LG bundles Von C, Coc, LCoc: the moduli stacks of G, LG local systems V = (V, V') on C.

The more familiar moduli spaces of stable objects in these stacks are not quite right for (GLC): unstable bundles and local systems must be considered along with the stable ones. For semisimple groups, these stacks are the correct objects. In order to obtain the correct statement also for reductive groups, we need the rigidified versions Bun, LBun, Loc, LLoc, in which the connected component of the generic stabilizer is "removed". In the notation of [AOV08, Appendix A] we have

Bun

= Bun/Zo(G),

LBun

= LBun/Zo(G),

Loc = Coc/Zo(G), LLoc

= LCoc/Zo(LG),

where Zo(G), ZO(LG) denote the connected components of Z(G), Z(LG). A subgroup of Z(G) and Z(LG) will give a normal flat subgroup in the inertia of the moduli stacks and as explained in [AOV08, Appendix A] we can pass to a quotient by such subgroups to obtain rigidified stacks. The rigidified stacks are thus intermediate between the full stacks and the moduli spaces of their stable objects. The appearance of the rigidfied moduli in the Geometric Langlands Conjecture is necessary (see Remark 2.4) to ensure the matching of components of the two categories involved in (GLC). For semi-simple groups this step is unnecessary (see Remark 2.1). REMARK 2.1. To clarify the rigidification process it is useful to introduce the notions of a regularly stable bundle and a regularly simple local system. By definition these are objects whose automorphism group coincides with the generic group of automorphisms, namely the center Z(G), Z(LG) of the structure group G, LG. It is instructive to note that the rigidified stacks often specialize to familiar geometric objects. For instance if G, LG are semi-simple groups, then we are rigidifying by the trivial subgroup and so Bun = Bun, Coc = Loc, etc. Note also that if the center of G is connected, then both Bun and Loc are generically varieties: in this case the open substacks Bunrs C Bun, Loc rs C Loc parameterizing regularly stable bundles or regularly simple local systems coincide with the GIT-moduli spaces of regularly stable bundles and regularly simple local systems respectively.

GEOMETRIC LANGLANDS AND NON-ABELIAN HODGE THEORY

91

To formulate the characteristic property of c we also need the Heeke correspondences LHecke'L C LHecke defined for all dominant cocharacter J-L cochar as follows:

E

fG]

LHecke: the moduli stack of quadruples (V, V', x, ,6), where • V, V' are principal LG-bundles on C, • x E C,

-=+ YI~-{x}·

• ,6 : Vic-{:J:}

the closed substack of LHecke of quadruples (V, V', x,,6) such that if A char is a dominant co character and if p>' is

LHeckelL :

E fG]

the irreducible representation of LG with highest weight A, then ,6 induces an inclusion of locally free sheaves p>'(,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}) .

R. DONAGI AND T. PANTEV

92

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,.

GEOMETRIC LANGLANDS AND NON-ABELIAN HODGE THEORY

93

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

R. DONAGI AND T. PANTEV

94

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,
GEOMETRIC LANGLANDS AND NON-ABELIAN HODGE THEORY

95

Hitchin discovered [Hit87b] that the moduli stack lliggs has a natural symplectic structure and comes equipped with a complete system of commuting Hamiltonians. These are most conveniently organized in a remarkable map h : lliggs -+ B to a vector space B, known as the Hichin map.

The target of this map, also known as the Hitchin base, is the cone

where as before t is our fixed Cartan algebra in 9 = Lie(G), and W is the Weyl group of G. We will see momentarily that the cone B is actually a vector space. To construct the Hitchin map h one considers the adjoint action of G on g. For every principal G-bundle E the quotient map 9 -+ g/iG induces a map between the total spaces of the associated fiber bundles

E

9 --- E

XAd

(gIIG)

XAd

I

I

ad(E)

C x (gIIG)

which induces a (polynomial) map between the fiber bundles ad(E) ® We -+ (g ® wC)IIG.

(3)

The map (3) combines with the canonical identification g/iG = tlW given by Chevalley's restriction theorem [Hum72, Section 23.1] to yield a natural map of fiber bundles l/E : ad(E) ® We -+ (t 0 we)IW

This construction gives rise to the Hitchin map:

h:

lliggs ---- B := HO(C, (t ® welW) (E, y)

-~:>

"y mod W" := l/E(O).

Slightly less canonically if r = dim t = rank 9 we can choose homogeneous G-invariant polynomials h, h, ... , Ir E qg] such that qg]G = qt]W qh, ... , Ir]. With this choice we get an identification

with d s

= deg Is,

and we can rewrite the Hitchin map as h:

lliggs ____ B =

(E, y)

-~)

IT\r

<:17 s =1

0ds ) HO(C 'we

(h(y), ... ,Ir(Y))·

96

R. DONAGI AND T. PANTEV

The points of the Hitchin base admit a natural geometric interpretation as certain Galois covers of C with Galois group W called cameral covers. By d~finition the cameral cover associated with a point b E B is the cover Pb : Cb ~ C obtained as the fiber product Cb ------ tot(t ® we)

Pbt C

~

b

tot(t ®we)/W

Repeating the same construction for the tautological section C x B tot (t ® we) /W we also get the universal cameral cover ~

~

) tot(t ® we)

~

t

C x B -- tot(t ® we)/W which by construction restricts to Cb on the slice C x {b} c C x B. Deformation theory for principal bundles on C together with Serre duality gives a natural identification

lliggs

~

TV Bun

of the stack of Higgs bundles with the cotangent stack TV Bun to the stack of bundles. This gives rise to the symplectic structure on lliggs. The Hitchin map h : lliggs ~ B is a completely integrable system structure on lliggs. Its generic fibers are abelian group stacks which are also Lagrangian for the natural symplectic structure. Concretely the fiber h-l(b) is identi~ed with an appropriately defined Prym stack for the cameral cover Pb : Cb ~ C, i.e. h-l(b) is a special W-isotypic piece for the W-action on the stack of (decorated) line bundles on Cb . The details of this picture were worked out in various situations in [Hit87h, Fa193, Don93, Don95, Sco98, DG02]. The most general result in this direction is [DG02, Theorem 4.4] according to which: • the covering map p : ~ ~ C x B determines an abelian group scheme T over C x B; • if ~ S B is the discriminant divisor parametrizing b E B for which Pb : Cb ~ C does not have simple Galois ramification, then the restriction

h : lliggs IB _t:..

~

B -

~

is a principal homogeneous stack over the commutative group stack TorsT on B - ~ parametrizing T-torsors along C. REMARK 3.1. For every J.l E char( G) we can also consider the associated spectral cover ~tL ~ C x B. It is the quotient of Cj; by the stabilizer of

GEOMETRIC LANG LANDS AND NON-ABELIAN HODGE THEORY

97

JL in W. Very often, e.g. for classical groups and the fundamental weight

[Hit87b, Don93] the fiber of the Hitchin map can also be described as a stack of (decorated) line bundles on the spectral cover. For instance if G = GLnUC) and we use the highest weight of the n-dimensional fundamental representation of G, then the associated spectral cover C b -+ C is of degree n, and the fiber of the Hitchin map h-l(b) can be identified with the stack 1'ic( C b ) of all line bundles on Cb. 3.2. Using abelianization. From the point of view of the Geometric Langlands Conjecture the main utility of the Hitchin map is that it allows us to relate the highly non-linear moduli Bun to an object that is essentially "abelian" . The basic idea is to combine the Hitchin map with the projection L1iiggs -+ LBun, (E, 'P) -+ E. More precisely we have a diagram

in which the fibers of h : 1iiggs IB _b. -+ B - ~ are commutative group stacks and each fiber of h dominates Bun. We can use this diagram to reformulate questions about V-modules or V-modules on LBun to questions about V-modules or V-modules on fibers of h. This process is known as abelianization and has been applied successfully to answer many geometric questions about the moduli of bundles. The fact that each fiber of h : 1iiggs IB _b. -+ B - ~ is an isotypic component of the moduli of line bundles on the corresponding cameral cover, and the fact (see Example 2.5) that the Geometric Langlands Correspondence can be constructed explicitly for rank one local systems, suggests that abelianization can be used to give a construction of the functor c (GLC) in general. A first attempt to reduce the GLC to its abelian case was in the unpublished [Don89]. The Hitchin system was used there to construct abelianized Hecke eigensheaves (M, 6) on the moduli of Higgs bundles. We describe this below, along with one way to push these eigensheaves down to the moduli of bundles. A modern version of the abelianized Hecke eigensheaf construction appeared in [DP06]. Our current approach [DP09, DPS09b] essentially replaces the explicit pushforward (from Higgs to Bun) with recent results from non abelian Hodge theory. We will outline this approach in the remainder of this survey. There are various other ways in which one can employ abelianization to produce a candidate for the functor c. One possibility is to apply a version of the generalizations of the Fourier transform due to Laumon, Rothstein, and Polishchuk [Lau96, Rot96, PROl, Po108] along the fibers of h. The most successful implementation of this approach to date is the recent work of

98

R. DONAGI AND T. PANTEV

Frenkel-Teleman [FT09] who used the generalized Fourier transform to give a construction of the correspondence (GLC) for coherent sheaves on a formal neighborhood of the substack of opers (see [BD05, BD03]) inside .coc. Another idea is to study the deformation quantization of a Fourier-Mukai transform along the fibers of h. This is the main component of Arinkin's approach [Ari02, Ari08] to the quasi-classical geometric Langlands correspondence. This approach was recently utilized by Bezrukavnikov and Braverman [BB07] who proved the geometric Langlands correspondence for curves over finite fields for G = LG = GLn . Last but not least, in the recent work of Kapustin-Witten [KW06] the geometric Langlands correspondence c is interpreted physically in two different ways. On one hand it is argued that the existence of the conjectural map (GLC) is a mirror symmetry statement relating the A and B-type branes on the hyper-Kahler moduli spaces of Higgs bundles. On the other hand Kapustin and Witten use a gauge theory/string duality to show that the functor c can be thought of as an electric-magnetic duality between supersymmetric four-dimensional Gauge theories with structure groups G and LG respectively. This suggests that c can be understood as a conjugation of the Fourier-Mukai transform along the Hitchin fibers with two non-abelian Hodge correspondences. Some non-trivial tests of this proposal were performed in [KW06] and in the work of Frenkel-Witten [FW08] who elaborated further on this conjectural construction. In the rest of the section we construct the construction of [Don89]. This proposal shares many of the same ingredients as the other approaches and highlights the important issues that one has to overcome. It also has the advantage of being manifestly algebraic. In this approach one starts with a local system V on C (taken to be GLn valued for simplicity) and uses it together with some geometry to construct a pair (M,8) where M is a bundle on Higgs, and 8 : M -+ !v! @ niliggs IBis a meromorphic relative Hat connection acting along the fibers of the Hitchin map h : Higgs -+ B. Furthermore by construction the bundle (M,8) is a Hecke eigen V-module with eigenvalue (V, \7) with respect to an abelianized version

of the Hecke functors. These are defined again for i = 1, ... , n - 1 as integral transforms with respect to the trivial local system on the abelianized Hecke correspondences

i

abHeckei

~

Higgs

i

~

Higgs xC

GEOMETRIC LANG LANDS AND NON-ABELIAN HODGE THEORY

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The correspondences ab Hecke i are the "Higgs lifts" of the correspondences Hecke i from Example 2.3, that is abHeckei (V, Vi, /3, x) E Heckei and ((V,
mutative diagram

/3 fits

in a com-

p

----;;.. .. V'

Here the maps abpi, abqi are induced from the maps pi, qi and so the fiber of over ((V',
Suppose now we have V = (V, \7) - a rank n vector bundle with an integrable connection on C. The fiber over V E Bun of the projection Higgs ~ TV Bun -+ Bun is just the fiber T~ Bun = HO(C,ad(V) ® we) of the cotangent bundle to Bun. Restricting the Hitchin map to T~ Bun and pulling back the universal spectral cover we get a cover

Using the spectral correspondence [Hit87b, Don95, DG02] we can find a holomorphic line bundle L on 't'\l such that • 7r\l*L ~ Pc (V ® (@-(n-1)), • \7 induces a (relative over T~ Bun) holomorphic connection D on L.

R. DONAGI AND T. PANTEV

100

TV

Indeed, by definition the vector bundle Pc V on C x Bun comes equipped with a tautological Higgs field i.p E HO(C x Bun,pc(ad(V)@wc)), characterized uniquely by the property that for every E Bun we have i.pICx{O} = e. The cover 'ifv ---+ C x Bun is simply the spectral cover of (Pc V, i.p) and hence comes equipped with a natural line bundle L', such that 7rv*L' = PcV. Notice that for every E Bun the restriction of the line bundle L' to the spectral curve Chv(O) = 'ifvlcx{O} has degree n(n-1)(g-1) and so does not admit a holomorphic connection. To correct this problem we can look instead at L'IG@(0-(n-l)whichhasdegreezeroandso

TV

TV

e TV

e TV

"v(O)

admits holomorphic connections. With this in mind we set L := L'

@

7rvpc(0-(n-l).

To see that V' induces a relative holomorphic connection D on L we will need the following fact. Let E HO( C, ad(V) @ we) be a Higgs field and let

e

(P: C ---+ C, NE 1'icn(n-l)(g-l) (C) ) be the associated spectral data. Suppose that C is smooth and that p : C ---+ C has simple ramification. Let R c C denote the ramification divisor. Then there is a canonical isomorphism of affine spaces

N)

.

holomo~phlc )

T :

( connectIOns on V

-t

(meromorPhic connections on with logarithmic poles along R ( 1) and residue -2

Indeed, since (C, N) is built out from (V, e) via the spectral construction we have that p*N = V. Away from the ramification divisor N is both a subbundle in p*V and a quotient bundle of p*V. Furthermore if V' is a holomorphic connection on V, the pullback p*V' is a holomorphic connection on p*V and so on C - R we get a holomorphic connection on N given by the composition

(4) On all of C the composition (4) can be viewed as a meromorphic connection on N with pole along R. The order of the pole and the residue of this meromorphic connection can be computed locally near the ramification divisor R. The order of the pole is clearly::; 1, since this is true for each step in (4). The residue is clearly locally determined, in particular it is the same at all the (simple) points of R. An appropriate version of the residue theorem then implies that this residue must be (-1/2). Here is an explicit calculation for this. Since p has simple ramification, in an appropriate local (formal or analytic) coordinate centered at a point r E R the map p can be written

GEOMETRIC LANG LANDS AND NON-ABELIAN HODGE THEORY

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as z I---t z2. The image of this local chart in C is, sayan analytic disk JD) c C centered at a branch point. Over JD) the covering C ---+ C splits into n - 1 connected components: p- 1 (JD)) = JD)o Il JD)IIl ... Il JD)n-2 where Po := PI]]))o : JD)o ---+ JD) is the two sheeted ramified cover given by po(z) = z2 and Pi := PI]]))i : JD)i ---+ JD) are one sheeted components for i = 1, ... , n - 2. Over JD) the bundle V will then split into a direct sum of a rank two piece Vo and a rank n - 2 piece W. For the calculation of the polar part of the connection D near this point only the rank two piece Vo of the bundle is relevant since upon restriction to JD)o the natural adjunction morphisms p*V ---+ Nand N ---+ p!V = p*V Q9Oc(R) factor through p(jVo ---+ Nand N ---+ p(jVo Q90]]))o(r) respecti vely. Thus we focus on the covering of disks Po : JD)o ---+ JD), po(z) = z2, where z, ware the coordinates on JD)o and JD) respectively. we denote the covering involution of this map by (j, i.e. (j : JD)o ---+ JD)o, (j(z) = -z. Without a loss of generality we may assume that NI]]))o has been trivialized. This induces a trivialization of Vo = Po*O]]))o: the frame of this trivialization consists of e+, e_ E r(JD), Vo), where e+ is a frame for the subsheaf of (j-invariant sections of Vo, and e_ is a frame for the subsheaf of (j-antiinvariant sections of V. Concretely, if we use the canonical identification r(JD), Vo) = r(JD)o, 0), the section e+ corresponds to 1 E r(JD)o, 0) and the section e_ corresponds to z E r(JD)o, 0). Since V : Vo ---+ Vo Q9 O~ is a holomorphic connection on Vo, we will have that in the trivialization given by the frame (e+, e_) it is given by

V

= d + A(w)dw,

A(w) E Mat2x2(r(JD), 0)).

But the change of frame (e+, e_) ---+ (e+, e_) exp( - J A( w )dw), transforms the connection d + A( w )dw into the trivial connection d and since holomorphic changes of frame do not affect the polar behavior it suffices to check that the connection 7(V) on 0]]))0 induced from d by (4) has a logarithmic pole at z = 0 with residue (-1/2). By (4) we have that the meromorphic connection 7(V) on 0]]))0 is defined as the composition OJ[}x

o

~(P(jVo)I]]))X0 ~(P(jVO)Ij[}x0

Q9

0]]))1 x 0

~0j[}10x

where the first and third maps are induced from the adjunction maps

OJ[}x

o

---+

p!p*Oj[}x = p*p*Oj[}x = p*Vo and p*Vo = p*p*Oj[}x ---+ O]]))x. o 0 0 0 To compute 7(V) choose a small subdisk U C JD)o, s.t. 0 tf. U. Then P01(pO(U)) = UIl (j(U), and the restriction of Po to U and (j(U) induces an identification

(5)

r(u,p*V)

~

r(U, 0) EEl r((j(U) , 0)

so that adjunction morphisms r(U, au) ---+ r(U,p*vo) and r(U,p*Vo) ---+ r(U, au) become simply the inclusion i and projection p for this direct sum decomposition.

R. DONAGI AND T. PANTEV

102

Now the connection Po on the bundle Po Vo has a fiat frame (poe+, poe- ). In terms of the decomposition (5) we have

e G)

Pa + = Now if

J(z)

E

Pa

and

e-= (~z)

r(U, 0) we get

T(Y')(f)

= po PaY' 0 i(f) = po PaY'

(1) + J(z) . (z)] 2z -z [J'(Z) dz. (1) + (J'(z) _ J(z)) dz. ( z )] 2z 2z2 -z

= pop*Y' o

=p

=

p

(J~z))

[J(z) . 2

2

1

1

!¥)

[(J'(Z)1 -J(z)

dZ]

"2-z-

= J'(z)dz - ~ J(z) dz.

2 z Hence Resz=o(T(Y')) = -1/2 as claimed. Next suppose we are given a line bundle 2 on a variety X and a trivalizing open cover {Ua.J for 2 with local frames e a E r(Ua ,2). Let ga(3 E r(Ua(3, OX) be the transition functions for these frames: e a = e(3g(3a' In particular any section of 2 is given by a collection {sa} of locally defined holomorphic functions Sa E qUa, 0) satisfying Sa = ga/3s/3, and a connection Y' : 2 -+ 2 ® ni- is given by connection one forms aa E qUa, ni-) satisfying aa - a/3 = d log g/3a' If g E Q is a fixed rational number and if 5 E qx,.ci81U) is a global section in some rational power of 2, then we can choose a trivializing cover {Ua } for 2 which is also a trivializing cover for .ci81u and such that the transition functions for .ci81u in appropriately chosen local frames are all branches g;/3 of the g-th powers of the transition functions ga/3 for 2 and the section 5 is represented by a collection {sa} of locally defined holomorphic functions satisfying Sa = g;/35/3. Taking d log of both sides of this last identity we get that (

-~dlOg5a) - ( -~dlog5(3)

= dlogg/3a'

In other words the collection {d - (1/ g) d log Sa} gives a meromorphic connection on .c with pole along the divisor 5 = O. Furthermore if the divisor of 5 is smooth this connection has residue -1/ g. Now consider the line bundle N ® p*(i8I-(n-1) on C = C hv ((})' As explained above the connection Y' on V induces a meromorphic connection T(Y') on N with a logarithmic pole along R and residue (-1/2). On

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the other hand we have 0c(R) = p*(02(n-I). Let s be a holomorphic section of p*(02(n-1) that vanishes on R. Then by the discussion in the previous paragraph s induces a meromorphic connection :D on the line bundle p*(0-(n-I): if we trivialize p*(0-(n-1) on an open U C C and if 5 is represented by a holomorphic function Su E r(u, 0) in this trivialization, then in the same trivialization :D := d + !dlog(su). By construction :D has logarithmic poles along R and residue 1/2. Therefore the tensor product connection r(\7) @ id + id @:D is a holomorphic connection on N @ p*(0-(n-1) = Lie x{()}' The whole construction makes sense rela'hV(O)

tively over T~ Bun and so by varying () E T~ Bun we get a relative holomorphic connection D on L, uniquely characterized by the property that its restriction to the slice C hv «()) x {fJ} is equal to r(\7) @id+id@:D. Even though this is not needed in what follows, it is instructive to note here that the relative connection D can be lifted to an absolute (i.e. differentiating in all directions) holomorphic connection on L. This absolute connection is not integrable but has curvature which is a holomorphic two form. Of course this holomorphic two form restricts to zero on each cameral curve which accounts for the integrability of the relative connection D. Now we can use (L, D) as an input for the GLI-version of the geometric Langlands correspondence. More precisely, applying the construction from Example 2.5 to the relative local system (L, D) and along the smooth fibers of 7rv we get a relative rank one local system (L, V) on the part of Pic ('-G'v/C x T~ Bun) = Higgs xBT~ Bun sitting over B - ~. If we push (L, \7) forward to Higgs we get a relative meromorphic local system (!VI, 8), where M is a holomorphic vector bundle on Higgs, and 8 is a meromorphic connection on M differentiating only along the fibers of h : Higgs ---7 B. The bundle M can be described explicitly. Let (E,1/J) E Higgs be any point, then the fiber of M at (E,1/J) is given by M(E,'IjJ)

=

EB

£jJ(L8.N~,),

()ET~ Bun

h( ())=h( 'If!)

where £jJ ---7 'PicO(Ch('IjJ)) x Pic(C h(1/,)) is the standard Poincare line bundle, L() is the restriction of L to the slice C h«()) x {e}, and N'IjJ E Pic(Ch('IjJ)) is the line bundle corresponding to (E,1/J) via the spectral correspondence.

3.2. • The above approach will give rise to a geometric Langlands correspondence if we can find a way to convert the abHi-eigen module (M,8) on Higgs to an Hi-eigen module on Bun. To do this we can take several routes: we can either average the (M,8) over all () E T~ Bun, or use deformation quantization as in [Ari02, Ari05], or

REMARK

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R. DONAGI AND T. PANTEV

use Simpson's non-abelian Hodge theory [Sim91, Sim92, Sim97] as we will do in the remainder of the paper . • To set up the previous construction for an arbitrary group G we need to establish a duality between lliggso and LHiggs. This was done in [DP06] and we will review it in section 4 . • The correspondences Higgs i and ab Higgs i can be related geometrically: ab Higgsi is the total space of the relative conormal bundle of Higgsi C Bun x Bun x Cover C.

4. The classical limit In this section we review the construction of the Fourier-Mukai functor

FM appearing in step (2) of the six step process in section 5.1. 4.1. The classical limit conjecture. Fix a curve C and groups G, LG. The moduli stacks of Higgs bundles arise naturally in an interesting limiting case of conjecture (GLC): the so called classical limit. On the local system side of (GLC) the passage to the limit is based on Deligne's notion of a z-connection [Sim97] which interpolates between the notions of a local system and a Higgs bundle. A z-connection is by definition a triple (V, V', z), where 7f : V -+ C is a principal G-bundle on C, z E C is a complex number, and V' is a differential operator satisfying the Leibnitz rule up to a factor of z. Equivalently, V' is a z-splitting of the Atiyah sequence for V: o~ ad(V) ~ £(V) ~ Tc ~ o. ~

'V

Here ad(V) = V Xad 9 is the adjoint bundle of V, £(V) = (7f*Tv)G is the Atiyah algebra of V, (T : £(V) -+ Tc is the map induced from d7f : Tv -+ 7f*Tc, and V' is a map of vector bundles satisfying (T 0 V' = z . idTcWhen z = 1 a z-connection is just an ordinary connection. More generally, when z i= 0, rescalling a z-connection by z-l gives again an ordinary connection. However for z = 0 a z-connection is a Higgs bundle. In this sense the z-connections give us a way of deforming a connection into a Higgs bundle. In particular the moduli space of z-connections can be viewed as a geometric I-parameter deformation of .coc parametrized by the z-line and such that the fiber over z = 1 is .coc, while the fiber over z = 0 is lliggso the stack of Higgs bundles with trivial first Chern class. Using this picture we can view the derived category Dco h(lliggs o, 0) as the z -+ 0 limit of the category Dcoh(.cOC, 0). On the LBun side the limit comes from an algebraic deformation of the sheaf of rings D of differential operators on LBun. More precisely D is a sheaf of rings which is filtered by the filtration by orders of differential operators. Applying the Rees construction [Ree56, Ger66, Sim91] to this filtration we get a flat deformation of D parametrized by the z-line and such

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that the fiber of this deformation at z = 1 is V and the fiber at z = 0 is the symmetric algebra seT = gr V of the tangent bundle of LBun. Passing to categories of modules we obtain an interpretation of Dcoh(LBun, seT) as the z -+ 0 limit of Dcoh(LBun, V). Since LHiggs is the cotangent stack of LBun the category Dcoh (LBun, seT) will be equivalent to Dcoh (LHiggs, 0) and so we get a limit version of the conecture (GLC) which predicts the existence of a canonical equivalence of categories

(clGLCo) which again sends structure sheaves of points to eigensheaves of a classical limit version of the Hecke functors. The precise construction of the classical limit Hecke functors is discussed in [DP06, Section 2]. Here we will only mention that in a forthcoming work Arinkin and Bezrukavnikov establish an isomorphism between the algebra of classical limit Hecke functors and the algebra of abelianized Hecke functors that we discussed in Section 3.2. We also expect that the equivalence do extends to an equivalence

(clGLC) which again sends structure sheaves of points to eigensheaves of a classical limit version of the Hecke functors. 4.2. Duality of Hitchin systems. The classical limit conjecture

(clGLC) can be viewed as a self duality of Hitchin's integrable system: Hitchin's system for a complex reductive Lie group G is dual to Hitchin's system for the Langlands dual group LG. This statement can be interpreted at several levels: • First, a choice of an invariant bilinear pairing on the Lie algebra g, induces an isomorphism between the bases of the Hitchin systems for G and LG, interchanging the discriminant divisors. • The general fiber of the neutral connected component Higgso of Hitchin's system for G is an abelian variety. We show that it is dual to the corresponding fiber of the neutral connected component LHiggs o of the Hitchin system for LG. • The non-neutral connected components -=~a Higgs form torsors over Higgs o. According to the general philosophy of [DP08], these are dual to certain gerbes. In our case, we identify these duals as natural gerbes over LHiggs o. The gerbe lliggs of G-Higgs bundles was introduced and analyzed in [DG02]. This serves as a universal object: we show that the gerbes involved in the duals of the nonneutral connected components Higgs are induced by lliggs. -==-a

lO6

R. DONAGI AND T. PANTEV

• More generally, we establish a duality over the complement of the discriminant between the gerbe 1iiggs of G-Higgs bundles and the gerbe L1iiggs of LG-Higgs bundles, which incorporates all the previous dualities . • Finally, the duality of the integrable systems lifts to an equivalence of the derived categories of 1iiggs and L1iiggs. As a corollary we obtain a construction of eigensheaves for the abelianized Hecke operators on Higgs bundles. To elaborate on these steps s~mewhat, note that the Hitchin base Band the universal cameral cover C(j' -+ C x B depend on the group G only through its Lie algebra g. The choice of a G-invariant bilinear form on 9 determines an isomorphism I : B -+ L B between the Hitchin bases for the Langlands-dual algebras g, Lg. This isomorphsim lifts to an isomorphism f. of the corresponding universal ;:ameral covers. (These isomorphisms are unique up to automorphisms of C(j' -+ C x B: There is a natural action of (Cx on B which also lifts to an action on 0' -+ C x B. The apparent ambiguity we get in the choice of the isomorphisms I, f. is eliminated by these automorphisms. ) The next step [DP06] is to show that the connected component Il) of the Hitchin fiber h-l(b) over some general b E B is dual (as a polarized abelian variety) to the connected component L Pl(b) of the corresponding fiber for the Langlands-dual system. This is achieved by analyzing the cohomology of three group schemes 7 ~ 7 ~ P over C attached to a group G. The first two of these were introduced in [DG02], where it was shown that h-l(b) is a torsor over HI(C, 7). The third one P is their maximal subgroup scheme all of whose fibers are connected. It was noted in [DG02] that 7 = 7 except when G = SO(2r + 1) for r ~ 1. Dually one finds [DP06] that 7 = P except for G = Sp(r), r ~ 1. In fact, it turns out that the connected components of HI (P) and HI (7) are dual to the connected components of HI (L7), HI (Lp), and we are able to identify the intermediate objects Hl(7),HI(L7) with enough precision to deduce that they are indeed dual to each other. Finally we extend the basic duality to the non-neutral components of the stack of Higgs bundles. The non-canonical isomorphism from non-neutral components of the Hitchin fiber to Pb can result in the absence of a section, i.e. in a non-trivial torsor structure [HT03, DP08]. In general, the duality between a family of abelian varieties A -+ B over a base B and its dual family AV -+ B is given by a Poincare sheaf which induces a Fourier-Mukai equivalence of derived categories. It is well known [DP08, BB07, BB06] that the Fourier-Mukai transform of an A-torsor Au is an O*-gerbe aAv on A v. Assume for concretness that G and LG are semisimple. In this case there is indeed a natural stack mapping to Higgs, namely the moduli stack 1iiggs of semistable G-Higgs bundles on C. Over the locus of stable bundles, the stabilizers of this stack are isomorphic to the center Z (G) of G and so over the stable locus 1iiggs is a gerbe. The stack 1iiggs was analyzed

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in [DG02]. From [DP08] we know that every pair a E 7ro(Higgs) = 7rl (G), j3 E 7rl (LG) = Z(G)/\ defines a U(l)-gerbe ,gHiggs on the connected Higgs and that there is a Fourier-Mukai equivalence of catecomponent -='-
THEOREM [Cor88, Sim92, Cor93, Sim97] Let (X, Ox(l)) be a smooth complex projective variety. Then there is a natural equivalence of dg 0categories:

nahx : (

fi nz't e ran k t X sys ems on

If" IV-

1ol)c a (finite rank Ox (1) -semistable ) , , --t H~ggs bundles on X w~th ChI = 0 d h 0 an c 2 =

REMARK 5.1. (a) Here by a Higgs bundle we mean a pair (E, ()) where E is a vector bundle on X, and () : E ----t E001- is an Ox-linear map satisfying () 1\ () = O. A Higgs bundle (E, ()) is Ox (l)-semistable iffor every (}-invariant subsheaf FeE we have X(F 0 O(n))j rk(F) ~ X(E 0 O(n))j rk(E) for n~O.

(b) We can also consider Higgs sheaves. These are by definition pairs (F, ()) where F is a coherent or quasi-coherent sheaf on X, and () : F ----t F 0 01is Ox-linear map satisfying () 1\ () = O.

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For quasi-projective varieties, the one dimensional analogue of the Corlette-Simpson theorem was settled by Simpson twenty years ago [Sim90j. The open case in higher dimension had to await the recent breakthroughs by Biquard [Biq97], Jost-Yang-Zuo [JYZ07j, Sabbah [Sab05], and especially Mochizuki [Moc06, Moc09, Moc07a, Moc07bj. This higher dimensional theory produces an equivalence of parabolic local systems and parabolic Higgs bundles, quite analogous to what is obtained in the compact case. Mochizuki is able to prove a version of the non-abelian Hodge correspondence which allows for singularities of the objects involved: THEOREM [Moc06, Moc09j Let (X,Ox(l)) be a smooth complex projective variety and let D c X be an effective divisor. Suppose that we have a closed subvariety Z C X of codimension ~ 3, such that X - Z is smooth and D - Z is a normal crossing divisor. Then there is a canonical equivalence of dg ®-categories:

finite rank locally abelian) finite rank tame tame parabolic Higgs bun( dles on (X, D) which are nahX,D : (parabolic C-l.ocal) -+ systems on (X, D) Ox(l)-semistable and satisfy parchl = 0 and parch2 = 0

Mochizuki requires three basic ingredients for this theorem: (1) a good compactification, which is smooth and where the boundary is a divisor with normal crossings away from co dimension 3; (2) a local condition: tameness (the Higgs field is allowed to have at most logarithmic poles along D) and compatibility of filtrations (the parabolic structure is locally isomorphic to a direct sum of rank one objects); and (3) a global condition: vanishing of parabolic Chern classes. A feature of the non-abelian Hodge correspondence that is specific to the open case is captured in another result of Mochizuki: THEOREM [Moc07a, Moc07bj Let U be a quasi-projective variety and suppose U has two compactifications

X

~ut:

Y

where:

• X, Yare projective and irreducible; • X is smooth and X - U is a normal crossing divisor away from codimension 3;

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Then the restriction from X to U followed by the middle perversity extension from U to Y gives an equivalence of abelian categories: irreducible tame) ¢*' 0 'ljJ*: ( parabolic CC-local --+ ( simple V-modules on Y whiCh) . systems on (X, D) are smooth on U

5.2. Using non-abelian Hodge theory. As we mentioned before non-abelian Hodge theory provides a natural approach to constructing the geometric Langlands correspondence c. The relevance of non-abelian Hodge theory to the problem is already implicit in the work of Beilinson-Drinfeld [BD03] on quantization of Hitchin hamiltonians, in the work of Arinkin [Ari02, Ari08] on the quasi-classical version of the geometric Langlands conjecture, and it the work of Bezrukavnikov-Braverman [BB07] on the Fourier-Mukai interpretation of the correspondence in positive characteristic. The non-abelian Hodge theory approach was brought in the spotlight in the mirror symmetry work of Hausel-Thaddeus [HT03] and several key features of the approach were worked out in the ground breaking work of Kapustin-Witten [KW06] on gauge theory/sigma model duality, in the work of Frenkel-Witten [FW08] on endoscopy, and in our own work [DP06] on the classical limit of the Geometric Langlands Conjecture. The possibility suggested by these works is that the known (see [DP06] and the discussion in section 4) eigensheaf of the abelianized Heckes, which is a Higgs-type object (E, 'P), extends by non abelian Hodge theory to a twistor eigensheaf on LBun. The original Higgs sheaf appears at z = 0, while at the opposite end z = 1 we can expect to find precisely the Hecke eigensheaf postulated by the GLC. The situation is essentially non-compact: There is a locus S in the moduli space LBun s of stable bundles along which our Higgs field 'P blows up. This can be traced back, essentially, to the difference between the notions of stability for bundles and Higgs bundles. The cotangent bundle TV (LBun S ) embeds as a Zariski-open in LHiggss. If we ignore stability the two are equal: TV (LBun) = LHiggs. But as moduli of stable objects, there is a locus Un in LHiggs parametrizing stable Higgs bundles with unstable underlying bundle. In order to turn the projection LHiggsS ---+~Buns into a morphism, Un must be blown up to an exceptional divisor Un. Then the Higgs field part '!!..-.of the Hecke eigensheaf (E, 'P) on LBun s blows up along the image S of Un. In current work with C. Simpson [DP09, DPS09b, DPS09a], we are investigating the possibility of applying non-abelian Hodge theory to the GLC. The heart of the matter amounts to verification of the Mochizuki conditions: we need to find where the Higgs field blows up, resolve this locus to obtain a normal crossing divisor, lift the objects to this resolution, and verify that the parabolic chern classes of these lifts vanish upstairs. This

no

R.. DONAGI AND T. PANTEV

would provide the crucial third step in the following six step recipe for producing the candidate automorphic sheaf: I G-Iocal system (V, V) on C

(1)~ I G-Higgs bundle (E, 0) on C (2)

I

I I

II

ab L Hecke-eigensheaf

on LHiggs

(3)n

parabolic Higgs sheaf on LBun s satisfying Mochizuki's conditions

(1)-(3) (4)n

parabolic local system on LBun s satisfying Mochizuki's conditions

(1)-(3) (5)n

ordinary local system on Zariski open in LBun (6)

II

I V-module on LBun

I

Note that all of the other steps in this process are essentially already in place. The functor (1) is given by the Corlette-Simpson non-abelian Hodge correspondence (E,O) = nahc(V, 'V) on the smooth compact curve C. The functor (2) sends (E,O) E lliggs to FM(O(E,lI)) where FM is a FourierMukai transform for coherent sheaves on TVBun = lliggs. In fact FM is the integral transform with kernel the Poincare sheaf constructed (away from the discriminant) in [DP06]. This sheaf is supported on the fiber product of the two Hitchin fibrations h : lliggso ---+ Band Lh : LHiggs ---+ B and we discussed it briefly in section 4.1. The functor (4) is the parabolic non-abelian Hodge correspondence nahLBunss ,5 of Mochizuki. Here LBunss denotes the (rigidified) stack of semistable bundles. Note that here we are applying the first Mochizuki theorem not to a projective variety but to a smooth proper Deligne-Mumford stack with a projective moduli space. In fact Mochizuki's proof [Moc09] works in this generality with no modifications. The functors

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(5) and (6) are the pullback and middle extension functors applied to the two compactifications LBun ss :J LBuns C LBun. In order to conclude that the composition (6) 0 (5) is an equivalence we need a strengthening of Mochizuki's extension theorem which would allow for Y to be an Artin stack which is only locally of finite type [DPS09aj. In the next section we explain some of the issues that one needs to tackle in order to carry out step (3). 6. Parabolic Higgs sheaves on the moduli of bundles To construct the functor (3) we need to convert a translation invariant line bundle 2' on the Hitchin fiber into a stable parabolic Higgs sheaf (£, 'P) on the moduli of bundles. The strategy is: • construct a suitable blow-up of the Hitchin fiber which resolves the rational map to LBuns ; • pull 2' and the taulogical one form on the Hitchin fiber to this blow-up; • twist with an appropriate combination of the exceptional divisors; • push-forward the resulting rank one Higgs bundle on the blow-up to LBun 8 to obtain a quasi-parabolic Higgs sheaf (£, 'P) on (LBunS, 5) • fix parabolic weights for (£, 'P) so that parch1 = 0 and parch2 = O. In [DP09, DPS09b] we work out this ~rategy for G = GL2(C). The first task here is to understand the divisors Un and 5 geometrically. 6.1. Wobbly, shaky, and unstable bundles. A G-bundle E is very stable if it has no nonzero nilpotent Higgs fields () [Lau88j. Very stable bundles are stable [Lau88j. We call a bundle wobbly if it is stable but not very stable, and we call a bundle shaky if it is in S. A major step towards carrying out our program is the identification of shaky bundles: THEOREM [DP09] Let G = LC = GL2(C)' Fix a smooth Hitchin fiber Higgsc' (a) The rational map Higgsc ---+ BunS can be resolved to a morphism mggsc -+ BunS by a canonical sequence of blow-ups with smooth centers. (b) For every translation invariant line bundle 2' on Higgsc ' and for any twist by exceptional divisors of the pullback of 2' to mggsc, the polar divisor of the associated quasi-parabolic Higgs sheaf (£, 'P) is independent of C, 2', and the twist, and is equal to S. (c) The shaky bundles are precisely the wobbly ones. This is in exact agreement with the expected behavior of the Hecke eigensheaf, according to Drinfeld and Laumon [Lau95j.

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In view of this theorem, the key geometric issue needed for a proof of the GLC along these lines is therefore an analysis of the locus of wobbly bundles and of the sequence of blowups needed to convert it into a normal crossing divisor. For G = GL 2 (C) this analysis is carried out in [DP09]. In specific cases it is possible to work out the moduli spaces, wobbly loci, and Hecke correspondences in great detail. One such case is when the curve is JP>I with n marked points, and the group is G = GL2(C)' This is an instance of the tamely ramified Geometric Langlands Conjecture, or the Geometric Langlands Conjecture for parabolic local systems and bundles. This natural extension of the GLC is explained beautifully in [Fre08, GW06], and a simple case (elliptic curve with one marked point) is analyzed in [FW08] from a point of view similar to ours. The six step process outlined above applies equally well to the ramified case: in fact, as explained above, our use of non-abelian Hodge theory has the parabolic structures built in even when the initial objects are defined over a compact curve, so there is every reason to expect that our construction should work just as well when the initial object is itself parabolic. A major surprise is that in the parabolic case, the [DPS09b] characterization of the poles of the parabolic Higgs sheaf (£, 'P) on Bun needs to be modified. Wobbly bundles are still shaky, but new, non-wobbly components of the shaky locus can arise. These seem to be related to the variation of GIT quotients. In this section we illustrate this new phenomenon in the first non trivial case, n = 5. The results will appear in [DPS09b]. There is a large body of work describing the moduli space Mn of semistable GL(2) parabolic bundles (or flat U(2) connections) on JP>I with n marked points as well as its cohomology ring, see e.g. [Bau91, Jef94, BR96, BY96]. In several of these references one can find an identification of !vIs as a del Pezzo surface dP4, the blowup of JP>2 at 4 general points. Actually, lvIn is not a single object: it depends on the choice of parabolic weights at the n points. For instance [Bau91] if we choose all the parabolic weights to be equal to 1/2, then the moduli space Mn can be described explicitly as the blow-up of JP>n-3 at n-points lying on a rational normal curve. The dP4 description of M5 holds for the lowest chamber, when the parabolic weights a are positive but small. By working out the GIT picture, we find [DPS09b] that in the case of balanced weights there are actually four chambers, and the corresponding moduli spaces are: dP4 for 0 < a < t,J dP5 for ~ < a < ~, JP>2 for ~ < a < ~, and empty for ~ < a < 1. The Hecke correspondence essentially relates the space at level a to the corresponding space at level 1 - a. The non-abelian Hodge theory description gives us the flexibility of working in a chamber of our choosing; we choose the self-dual dP5 chamber at ~ < a < ~. We find that in the lowest chamber, the shaky locus does agree with the wobbly locus. It consists of the 10 lines on the dP4 , together with 5 additional rational curves, one from each of the five rulings on the dP4, and all five passing through the same point p E dP4. In particular, this divisor

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fails to have normal crossings at p and so is not suitable for the non-abelian Hodge theory approach. As we move to the next chamber, it is precisely the point p that is blown up to produce the dP5. We check [DPS09b] that the wobbly locus now consists of 15 of the 16 lines on the dP5 - the proper transforms of the 15 previous components. This is where the new phenomenon first shows up: the shaky locus actually consists of all 16 lines on dP5 . In our self-dual chamber, the shaky divisor has normal crossings, the total space of the Hecke correspondence is smooth, the rational map from the Hitchin fiber to M5 ha.<, a natural resolution producing a parabolic Higgs sheaf of on M 5 , and there exist twists and assignments of parabolic weights along the shaky locus that fulfill the Mochizuki conditions from section 5.1. More or less all of this fails on the dP4 or the ]fD2 model; in particular, there is no solution to the Mochizuki conditions involving only 15 of the lines. This gives in this case an explicit construction (modulo solving the differential equations inherent in the non-abelian Hodge theory) of the Hecke eigensheaf demanded by the GLC. 6.2. On functoriality in non-abelian Hodge theory. Showing that the V-module we construct on LBun in step (6) in section 5.2 is indeed a Hecke eigensheaf depends on having good functorial properties of the nonabelian Hodge correspondence and the Mochizuki extension theorem in the parabolic context. The main task is to define direct images of parabolic objects under fairly general circumstances and to establish their basic properties. The aspects of functoriality needed for our construction in examples are relatively easy to establish, basically because the resolved abelianized Hecke correspondences tend to be finite. Nevertheless, it seems natural to try to establish the functorial behavior in general. We are currently pursuing this in a joint project with C.Simpson [DPS09a]. Through the works of Mochizuki [Moc07a, Moc07b] and Jost-YangZuo [JYZ07] we know that the de Rham cohomology of the V-module extensiom (of the restriction to X \ D of) a tame parabolic local system on (X, D) can be calculated directly in terms of L2 sections with respect to the harmonic metric. In the case of a map to a point, the functoriality we need identifies this also with the cohomology of (the Dolbeault complex associated to) the corresponding parabolic Higgs bundle. Our plan is to establish the general case of functoriality by combining this with an appropriate extension of the techniques of Simpson's [Sim93]. References [AOV08] D. Abramovich, M. Olsson, and A. Vistoli. Tame stacks in positive characteristic. Ann. Inst. Fourier (Grenoble), 58(4):1057-1091, 2008. [Ari02] D. Arinkin. Fo'urier tmnsform for quantized completely integmble systems. PhD thesis, Harvard University, 2002. [Ari05] D. Arinkin. On A-connections on a curve where A is a formal parameter. Math. Res. Lett., 12(4):551-565,2005.

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[Ari08] D. Arinkin. Quasi-classical limit of the geometric Langlands correspondence, 2008. talk at KITP, Santa Barbara, available at http:// online.kitp. ucsb.edu/ online/langlandsJn08/ arinkin1 /. [Bau91] S. Bauer. Parabolic bundles, elliptic surfaces and SU(2)-representation spaces of genus zero Fuchsian groups. Math. Ann., 290(3):509-526, 1991. [BB06] O. Ben-Bassat. Twisting derived equivalences, 2006. [BB07] A. Braverman and R. Bezrukavnikov. Geometric Langlands correspondence for V-modules in prime characteristic: the GL(n) case. Pure Appl. Math. Q., 3(1, part 3):153-179, 2007. [BD03] A. Beilinson and V. Drinfeld. Quantization of Hitchin's integrable system and Hecke eigensheaves. Book, in preparation, 2003. [BD05] A. Beilinson and V. Drinfeld. Opers, 2005. arXiv.org:math/0501398. [Biq97] O. Biquard. Fibres de Higgs et connexions integrables: Ie cas logarithmique (diviseur lisse). Ann. Sci. Ecole Norm. Sup. (4),30(1):41-96,1997. [BR96] I. Biswas and N. Raghavendra. Canonical generators of the cohomology of moduli of parabolic bundles on curves. Math. Ann., 306(1):1-14, 1996. [BY96] H. Boden and K. Yokogawa. Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: 1. I. Internat. J. Math, 7:573-598, 1996. [Cor88] K. Corlette. Flat G-bundles with canonical metrics. J. Differential Geom., 28(3):361-382, 1988. [Cor93] K. Corlette. Nonabelian Hodge theory. In Differential geometry: geometry in mathematical physics and related topics (Los Angeles, CA, 1990), volume 54 of Proc. Sympos. Pure Math., pages 125-144. Amer. Math. Soc., Providence, RI, 1993. [DDD+06] D.-E. Diaconescu, R. Dijkgraaf, R. Donagi, C. Hofman, and T. Pantev. Geometric transitions and integrable systems. Nuclear Phys. B, 752(3):329-390, 2006. [DDP07a] D.-E. Diaconescu, R. Donagi, and T. Pantev. Geometric transitions and mixed Hodge structures. Adv. Theor. Math. Phys., 11(1):65-89, 2007. [DDP07b] D.-E. Diaconescu, R. Donagi, and T. Pantev. Intermediate Jacobians and ADE Hitchin systems. Math. Res. Lett., 14(5):745-756, 2007. [DG02] R. Donagi and D. Gaitsgory. The gerbe of Higgs bundles. Transform. Groups, 7(2):109-153,2002. [Don87] S. Donaldson. Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3),55(1):127-131, 1987. [Don89] R. Donagi. The geometric Langlands conjecture and the Hitchin system, 1989. Lecture at the US-USSR Symposium in Algebraic Geometry, Univ. of Chicago, June-July, 1989. [Don93] R. Donagi. Decomposition of spectral covers. Asterisque, 218:145-175,1993. [Don95] R. Donagi. Spectral covers. In Current topics in complex algebraic geometry (Berkeley, CA, 1992/93), volume 28 of Math. Sci. Res. Inst. Publ., pages 65-86. Cambridge Univ. Press, Cambridge, 1995. [DP06] R. Donagi and T. Pantev. Langlands duality for Hitchin systems, 2006. preprint, arXiv.org:math/0604617. [DP08] R. Donagi and T. Pantev. Torus fibrations, gerbes, and duality. Mem. Amer'. Math. Soc., 193(901):vi+90, 2008. With an appendix by Dmitry Arinkin. [DP09] R. Donagi and T. Pantev. Parabolic Hecke eigen V-modules for GL2, 2009. in preparation. [DPS09a] R. Donagi, T. Pantev, and C. Simpson. Functoriality in non-abelian Hodge theory. in preparation, 2009. [DPS09b] R. Donagi, T. Pantev, and C. Simpson. Hodge theory and the tamely ramified langlands correspondence. in preparation, 2009. [Dri80] V. Drinfeld. Langlands' conjecture for GL(2) over functional fields. In Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pages 565-574, Helsinki, 1980. Acad. Sci. Fennica.

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[Lau96] G. Laumon. Transformation de Fourier generalisee, 1996. arXiv.org:alg-geom/ 9603004. [Lau03] G. Laumon. Travaux de Frenkel, Gaitsgory et Vilonen sur la correspondance de Drinfeld-Langlands. Asterisque, 290:Exp. No. 906, ix, 267-284, 2003. Seminaire Bourbaki. Vol. 2001/2002. [Moc06] T. Mochizuki. Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. Asterisque, 309:viii+ 117, 2006. [Moc07a] T. Mochizuki. Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules. I. Mem. Amer. Math. Soc., 185(869):xii+324, 2007. [Moc07b] T. Mochizuki. Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules. II. Mem. Amer. Math. Soc., 185(870):xii+565, 2007. [Moc09] T. Mochizuki. Kobayashi-Hitchin correspondence for tame harmonic bundles. II. Geom. Topol., 13(1):359-455, 2009. [PolO8] A. Polishchuk. Kernel algebras and generalized Fourier-Mukai transforms, 2008. arXiv.org:081O.1542. [PROl] A. Polishchuk and M. Rothstein. Fourier transform for D-algebras. I. Duke Math . .I.,109(1):123-146,200l. [Ree56] D. Rees. Valuations associated with ideals. II. .I. London Math. Soc., 31:221-228, 1956. [Rot96] M. Rothstein. Sheaves with connection on abelian varieties. Duke Math . .I., 84(3):565-598, 1996. [Sab05] C. Sabbah. Polarizable twistor V-modules. Asterisque, 300:vi+208, 2005. [Sco98] R. Scognamillo. An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups. Compositio Math., 110(1):17-37, 1998. [Sim90] C. Simpson. Harmonic bundles on noncompact curves . .I. Amer. Math. Soc., 3(3):713-770, 1990. [Sim91] C. Simpson. Nonabelian Hodge theory. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 747-756, Tokyo, 1991. Math. Soc. Japan. [Sim92] C. Simpson. Higgs bundles and local systems. Inst. Hautes Etudes Sci. Publ. Math., 75:5-95, 1992. [Sim93] C. Simpson. Some families of local systems over smooth projective varieties. Ann. of Math. (2), 138(2):337-425, 1993. [Sim97] C. Simpson. The Hodge filtration on nonabelian cohomology. In Algebraic geometry-Santa Cruz 1995, volume 62 of Proc. Sympos. Pur'e Math., pages 217281. Amer. Math. Soc., Providence, RI, 1997.

Surveys in Differential Geometry XIII

Developments around positive sectional curvature Karsten Grove ABSTRACT. This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are not included, and things are not presented in chronological order.

Spaces of positive curvature have always enjoyed a particular role in Riemannian geometry. Classically, this class of spaces form a natural and vast extension of spherical geometry, and in the last few decades their importance for the study of general manifolds with a lower curvature bound via Alexandrov geometry has become apparent. The importance of Alexandrov geometry to Riemannian geometry stems from the fact that there are several natural geometric operations that are closed in Alexandrov geometry but not in Riemannian geometry. These include taking Gromov Hausdorff limits, taking quotients by isometric group actions, and forming joins of positively curved spaces. In particular, limits (or quotients) of Riemannian manifolds with a lower (sectional) curvature bound are Alexandrov spaces, and only rarely Riemannian manifolds. Analyzing limits frequently involves blow ups leading to spaces with non-negative curvature as, e.g., in Perelman's work on the geometrization conjecture. Also the infinitesimal structure of an Alexandrov space is expressed via its "tangent spaces" , which are cones on positively curved spaces. Hence the collection of all compact positively curved spaces (up to scaling) agrees with the class of all possible so-called spaces of directions, in Alexandrov spaces. So spaces of positive curvature play the same role in Alexandrov geometry as round spheres do in Riemannian geometry. In addition to positively, and nonnegativey curved spaces, yet another class of spaces has emerged in the general context of convergence under a lower curvature bound, namely almost nonnegatively curved spaces. These are spaces allowing metrics with diameter say 1, and lower curvature bound Supported in part by a grant from the National Science Foundation. ©2009 International Press

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arbitrarily close to o. They are expected to playa role among spaces with a lower curvature bound, analogous to that almost flat spaces play for spaces with bounded curvature. In summary, the following classes of spaces play essential roles in the study of spaces with a lower curvature bound:

Pc Po

c P o-

corresponding to positively curved, nonnegatively curved, and almost nonnegatively curved spaces. Here the first and the last class enjoy the useful property of being stable under small perturbations. Among all manifolds, these form "the tip of the iceberg". Yet, aside from being Nilpotent spaces (up to finite covers) and having a priory bounded topology in terms of generators for homology- and fundamental groups, only a few general obstructions are known, and none in the simply connected case. Moreover, so far only obstructions on fundamental groups distinguish the three classes. The study of the two first classes has also played a significant role in the development of comparison theorems, and in this way also influenced the general development of Riemannian geometry. Much of this work even originated in connection with the desire to characterize spheres among spaces with positive curvature, resulting in so-called sphere theorems. It is indeed remarkable how many of the known general tools have been developed during proofs of sphere theorems. This not only applies to a number of so-called comparison theorems, like the Rauch Comparison Theorem for Jacobi fields and the Toponogov Comparison Theorem for geodesic triangles, but also includes critical point theory for distance functions as well as the Ricci flow. Throughout the rest of this article our main focus will be on manifolds with positive (sectional) curvature. Except for the beautifully rich and simple trick provided by Synge, there have been three general approaches used in an attempt to gain understanding of the class, all starting in some sense with the sphere as a uniquely determined extremal object. In short these approaches have been guided by Shape, Size and Symmetry. Here the first two of these have played an important role in the development of the tools alluded to above and hence also to the known obstructions. The latter, on the other hand has provided a natural framework for the discovery and construction of examples, an area of pivotal importance for the subject. We will discuss each of them below. The reader should also consult two recent survey articles on manifolds with positive and nonnegative curvature by Wilking [Wi] and Ziller [ZI]. Most of what we will discuss here can be found in one of these surveys where more details are given as well as references to original papers. Other surveys that cover topics mentioned in this essay are Petersen [Pel and Rong [Ro]. The surveys by Plaut [PI] and Petrunin [Pt] provide good access to Alexandrov spaces. Other related surveys are [Z2] and [GI, G2]. We will use a non-traditional approach and only provide other references if they

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cannot easily be found in one of these sources, or when the reference therein was incomplete. 1. Structure

Without stating it again, we assume throughout that all manifolds considered are complete Riemannian manifolds. The following fundamental result due to Cheeger and Gromoll provides an amazingly simple structure of all non-compact manifolds in Po: THEOREM 1.1 (Soul Theorem). Any non-compact manifold M E Po is diffeomorphic to the total space E of a vector bundle over a compact manifold S E Po. In fact, any soul S is a totally geodesic submanifold of M and E is the normal bundle to S in M. Except for the fundamental: PROBLEM 1. Which vector bundles over a compact S E Po admit a metric with nonnegative curvature? the soul theorem to a large extent reduces the topological study of nonnegatively curved manifolds to compact ones. So far, obstructions in this problem are known only for certain manifolds S with infinite fundamental group by work of Ozaydin-Walschap and of Belegradek-Kapovitch. The problem is difficult even when S is a sphere. Here work of Grove-Ziller shows that all bundles have such metrics when the dimension of the base is at most 5. For positive curvature, one has a complete answer due to Gromoll and Meyer (known prior to the soul theorem): THEOREM 1.2. Any non-compact manifold M E P is diffeomorphic to euclidean space.

In the language of the soul theorem, a soul in this case is simply a point. As shown by Perelman this remains true in the more general case where M has positive curvature on an open set. The key geometric tools in all of this work are the Toponogov comparison theorem and a deep study of convex sets and concavity properties of distance functions in nonnegative curvature. In the remaining part of this essay, we will confine our discussion to compact manifolds. The first global theorems about positive curvature were special applications of the Second Variation Formula for Arc Length, and hence in essence Index Theorems in the context of Morse Theory for suitable path spaces. The particular method pioneered by Synge (variations in directions of parallel fields) applies in general situations where boundary conditions for the relevant

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path spaces are totally geodesic. The original application deals with orientation: THEOREM 1.3 (Orientation). Any odd dimensional manifold in P is orientable, and an orientable even dimensional manifold in P is simply connected. A recent application of the Synge trick due to Wilking illustrates how severely the presence of totally geodesic submanifolds (with small codimension) restricts the structure of manifolds in P: LEMMA 1.4 (Connectivity). The inclusion map of a totally geodesic submanifold V n- k C Mn E P is n - 2k + 1 connected. Recall here that a map f : X ~ Y is i-connected if the induced map fq : ~ 7rq(Y) on homotopy groups is onto for q = t and an isomorphism

7rq(X)

for q < t. For k = 1 a convexity argument as in the soul theorem immediately implies that such a manifold M is homeomorphic to a sphere or is double covered by a manifold homeomorphic to a sphere. The following is a natural open PROBLEM 2. Is any M E P with a codimension one totally geodesic submanifold diffeomorphic to either the standard sphere or the real projective space? Also for k = 2, the following could have a positive answer: PROBLEM 3. Is any (simply connected) ME P with a codimension two totally geodesic submanifold diffeomorphic to either the standard sphere or the complex projective space? From now on we will use the terminology CROSS for a simply connected compact rank one symmetric space, i.e., such a manifold is either the sphere §n, the complex-, or quaternionic- projective space C]p>n, or lHI]p>n, or the Cayley plane Ca]P>2 with their canonical metrics. Each type forms a chain (with fixed codimension), i.e., an infinite sequence of manifolds M i , with Mi c Mi+1 isometrically and in particularly totally geodesically. Positively curved chains are clearly severely restricted by the connectivity lemma. In fact PROBLEM 4. Are the CROSS's the only positively curved chains? It should be pointed out that at present simply connected manifolds M E P not diffeomorphic to a CROSS are known only in dimensions 6 (two), 7 (infinitely many), 12 (one), 13 (infinitely many), and 24 (one).

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2. Symmetry

The round sphere, §n is characterized as the simply connected closed manifold with maximal symmetry degree, i.e., with the largest dimensional Lie group acting on it. This naturally leads to questions of describing manifolds with large transformation groups even without any curvature assumptions as, e.g., in work of W.-Y. Hsiang and many others. It also suggest the following approach to investigate manifolds of positive curvature: Describe the structure and ultimately classify positively curved manifolds with large isometry groups

One of the motivations for this, is that it provides a systematic guide towards the discovery and construction of new examples, a task which is known to be very difficult. In retrospect, the classification of (simply connected) homogeneous manifolds with positive curvature, due to the combined efforts of Berger, Wallach, Aloff-Wallach and Berard-Bergery, is an example of this. Non-CROSS'es occur in dimensions 6 (W 6 ), 7 (B7 and A~,q,p, q E Z), 12 (W 12 ), 13 (B 13 ), and 24 (W24). The driving force in that work, however, was based on the fact that Lie groups with biinvariant metrics have nonnegative curvature and that taking quotients only increases curvature. The same method was behind the construction due to Eschenburg in dimensions 6 (E 6 ) and 7 (Ek i' k, 1 E Z3) and later Bazaikin in dimension 13 (B~3, q E Z5) of so-called 'biquotients with positive curvature. In both cases curvature computations are essentially reduced to Lie algebra computations. The same type of problems are natural for the other classes Po C Poas well, and rich classes in Po, including many exotic spheres have been discovered in this way. There are several natural measures for largeness. The most obvious ones are: symdeg, symrank, cohom, i.e., the degree of symmetry = dimension of Isometry group, the symmetry rank = rank of isometry group, the cohomogeneity = dimension of orbit space. Another very useful measurement, cofix takes into account the fixed point set and essentially measures co homogeneity modulo fixed point sets, called the fixed point cohomogeneity = cohom - dim fix - 1. For each of these invariants there are partial or complete solutions to the program described above. In all cases, orbit spaces play a significant role. These orbit spaces are Alexandrov spaces with positive curvature, and this vaguely prevents "too many", "too singular" points. Typically, the orbit space will have non-empty boundary, and "soul theorems" for these spaces become powerful tools. On the manifold itself, the existence of totally geodesic submanifolds as fixed point sets of groups of isometries are common in this context, often leading to restrictions via the connectivity lemma of section one.

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Torus actions play a particular role for several reasons. On the one hand, it follows from the Cheeger-Fukaya-Gromov theory, that collapse with bounded curvature of simply connected manifolds is given in terms of such actions. Secondly, for manifold in P the method of Synge implies that an isometric action by a torus T has fixed points in even dimensions, and either fixed points or circle orbits in odd dimensions. In particular, there are isotropy groups of maximal rank in even dimension and at most corank 1 in odd dimensions. For the symmetry rank one has the following rigidity and "pinching" theorems due to Grove-Searle and Wilking respectively: THEOREM 2.1 (Rank Rigidity). Any M n E P has symrank(M) ~ [n; 1] with equality if and only if M is diffeomorphic to either §n, Clpm/2, or a lense space §n/71 k . Moreover, the actions are known as well. THEOREM 2.2 (Rank Pinching). Any positively curved simply connected n-manifold M with symrank(M) ~ n/4+1 and n =I- 7 is homotopy equivalent to a CROSS. In this formulation, results have been combined: It is due to Wilking for n ~ 10, to Fang-Rong for 7 < n < 10 and is covered by the rigidity theorem in dimensions n < 7. The conclusion fails spectacularly in dimension 7. In fact all known examples, i.e., all positively curved Eschenburg and Aloff Wallach spaces have isometry group of rank 3. When symrank(M) = [n;I]_ 1, we say that M has almost maximal symmetry rank. The only dimensions not covered by the above result are dimensions 4, 5, 6, and 7 where this is 1, 2, 2, and 3. Here, the flag manifold W 6 = SU(3)/T2 and the Eschenburg "flag" E6 = SU(3)/T2 both have almost maximal symmetry rank. Since dimensions 4 and 5 have been resolved (see below), this suggests the following PROBLEM 5. Classify M n E P with almost maximal symmetry rank when n = 6 and 7. As indicated, all known examples in P in these dimensions have almost maximal symmetry rank. In lower dimensions there is a complete classification: THEOREM 2.3 (Strong Rank Pinching). Any Mn E P with almost maximal symmetry rank and n ~ 5 is diffeomorphic to a CROSS. In dimensions 2 and 3, of course, no symmetry assumptions are needed. In dimension 5, the result is due to Rong and since there are no exotic spheres in this dimension it suffices to prove that such a manifold is a homotopy sphere. The topological classification in dimension 4 is due to Hsiang

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and Kleiner. By Freedman's topological classification of simply connected 4-manifolds, it suffices to prove that the Euler characteristic of such a manifold is at most 3. Since this is also the Euler characteristic of the fixed point set of the circle action, the result follows from a geometric analysis of the fixed point set. When confined to smooth circle actions (cf. [FSS]), the extensive work of Fintushel [Fi] and Pao [Pal gives, modulo the Poincare conjecture, in particular a complete classification of such actions on simply connected 4-manifolds, in terms of so-called weighted orbit space data. Thus, the combination of the work of Hsiang-Kleiner, Fintushel and Perelman provides a proof of the above theorem in dimension 4. Extending the fixed point analysis to non-negative curvature as in Kleiner's thesis and in the work of Searle-Yang, and again combining this with the work of Fintushel and Perelman immediately yields the following classification: THEOREM 2.4. Any non-negatively curved simply connected 4-manifold with an isometric circle action is diffeomorphic to one of §4, CIP'2 ,§2 X §2,

C1P'2#

±

C1P'2.

A classification up to equivariant diffeomorphism is not yet known, since a full understanding of the possible weighted orbit space data in nonnegative and positive curvature has not yet been achieved. In the case of positive curvature, however, we propose the following CONJECTURE. An isometric circle action on a simply connected positively curved 4-manifold is equivariently diffeomorphic to a linear circle action on either §4 or CIP'2 .

The theorem above of course provides support for the classical Hopf conjecture, that there is no metric on §2 x §2 with positive curvature. Indeed if there is one, it can have at most a finite isometry group. One can speculate that the list in the theorem are all simply connected manifolds Po. Topologically this would follow from the ellipticity conjecture (see "Size" section). Replacing the torus Tk by any compact Lie group G, it is also natural to investigate how an isometric action by G on a manifold M E P restricts }vI. For example, in which range of dimensions can one get a complete classification, or an exhaustive list of possibilities, potentially containing new examples. Also PROBLEM 6. Is a simply connected G manifold M n dimension a CROSS with a linear action?

E

P of minimal

In several of the above, as well as in subsequent results, the classification of fixed point homogeneous manifolds of positive curvature by Grove-Searle plays an important role. Here a manifold is called fixed point homogeneous

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if it supports an action where the group acts transitively on the normal sphere to a fixed point component, i.e., cofix = 0 with the terminology provided above. A similar useful classification is also known in fixed point cohomogeneity one by Grove-Kim: THEOREM 2.5 (Almost Minimal Cofix). Any simply connected M E P with cofix(M) ::; 1 is diffeomorphic to a CROSS. There is also a complete classification up to equivariant diffeomorphism in the non simply connected case. The conclusion fails in cofix 2. For the degree of symmetry one has the following satisfactory result due to Wilking: THEOREM 2.6 (Degree of Symmetrys). Any simply connected 1v[n E P with symdeg(.i\I) 2: 2n - 6 is tangentially homotopy equivalent to a CROSS, or isometric to a homogeneous space in P. One of the striking things about the classification of homogeneous manifolds of positive curvature is that apart from the CROSS'es, they occur only in finitely many dimensions. This phenomenon turns out to hold for any cohomogeneity by the following result of Wilking: THEOREM 2.7 (Cohomogeneity Finiteness). Any simply connected 1vf n E P with cohom(M) = k 2: 1 and n 2: 18(1 + k)2 is tangentially homotopy equivalent to a CROSS. One of the remarkable original constructions, which lead to this result, was the existence of a chain of positively curved Gi manifolds Mi (with fixed codimension) and with isometric orbit spaces MdG i once a sufficiently large dimensional positively curved cohomogeneity k manifold is given. A deep analysis of the limit object Moo showed that it was one of §oo, Cpoo or lHIp oo . While the above result illustrates the difficulty in finding new examples of manifolds in P, it also gives hope that it might be possible for low cohomogeneity. In the case of cohomogeneity one, the bound in the above theorem is 72. The optimal number is actually 14. In fact, from the classification in even dimensions due to Verdi ani and the work of Grove-Wilking-Ziller, one has: THEOREM 2.8 (Cohomogeneity One). A simply connected cohomogeneity one n-manifold ME P is diffeomorphic to a CROSS when n -=f 7,13. For n = 13, M is diffeomorphic to a cohomogeneity one Bazaikin space or §13. For n = 7, M is diffeomorphic to a cohomogeneity one Eschenburg space, the Berger space B 7 , or to one of the Konishi-Hitchin manifolds Pk, Qk, k 2: 2, or to an exceptional manifold N7.

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In all cases, the possible actions are known as well (for the CROSS'es they are all "linear"). One of the Bazikin manifolds in the theorem is the Berger space B 13 , and the theorem gives a classification in this dimension. One of the Eshenburg spaces and Ql is the Aloff-Wallach-Wilking normal homogeneous manifold All = W7 = SU(3) SO(3)/ U(2), and PI = §7. The theorem provides a classification of positively curved cohomogeneity one manifolds modulo the now obvious PROBLEM 7. Determine whether Pk , Qk, k ant) metrics of positive curvature.

~

2 and N support (invari-

All the 7-manifolds, M in the above theorem support almost effective actions by S3 x S3 and metrics of nonnegative curvature (Grove-Ziller). Moreover, each of the S3 factors act almost freely on IvI. In particular, there is an orbifold bundle M --+ M / S3 = B, where B supports an almost effective action by the other S3 factor. In fact, B = §4 or C]p>2 and the action is the well known action with singular orbits of co dimension 2. For B7 and N7 the metric on the base is an orbifold metric, singular along both singular orbits. For Pk, Qk, k ~ 2 the metric on the base is an orbifold metric which is singular along one of the singular orbits. In the latter cases Hitchin has constructed a self dual Einstein metric on each base, and the manifolds Pk and Qk are the two-fold covers of the so-called Konishi bundles of self dual two forms of these. By work of Dearricott it is well known that if the Hitchin metrics had positive curvature then the natural 3-Sasakian metric on the Konishi bundle would also have positive curvature once the fiber is shrunk sufficiently. Alas, the Hitchin metrics have curvatures of both signs when k ~ 2. The most interesting among the candidates are the Pk's since they are all 2-connected. In particular, it would follow from the 7r2 finiteness theorem (see Shape section), that if they have positive curvature, their pinching must necessarily approach 0 as k goes to infinity. Although these candidates have been known for several years, some fundamental new ideas are still needed to see if this family as a whole supports positive curvature. However, THEOREM 2.9 (Cohomogeneity One Example). The manifold P2 has an invaTiant metric with positive curvature. This indeed is a new example, since P2 is 2-connected with 7r3(P2) = /Z2, and the only known two connected 7-manifolds with positive curvature are §7 and B7 and 7r3(B7) = /Z1O. It is interesting to note that in fact P2 is homeomorphic to the unit tangent bundle Tl§4 of the 4-sphere, but it is not known if it is diffeomorphic to it. As indicated earlier, any construction of a metric with positive curvature on any of the new candidates must be unlike all previous ones. For connection metrics a necessary and sufficient condition for positive curvature has been derived by Chaves-Derdzinski-Rigas. In a manuscript [De], Dearricott offers

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a proof that this condition can be satisfied by means of a suitable conformal change of the Hitchin metric. A very different construction of an invariant piecewise polynomial C 2 metric with positive curvature has been given by Grove-Verdiani-Ziller in [GVZ]. In fact, that metric is shown to have strongly positive curvature, i.e., there is an (explicitly constructed) 2-form 'f7 such that the modified curvature operator R+i7 (necessarily having the same "sectional curvatures") is positive definite. Checking this amounts to checking that specific polynomials are positive on a given interval, the orbit space of the group action. Another natural attempt for the construction of new examples of positive curvature, is to see if one or more among the large known class of nonnegatively curved manifolds can be deformed to have positive curvature. Although no obstructions are known yet for simply connected manifolds, this is exceedingly difficult. By the orientation theorem of Synge, JR.JP>n x JR.JP>1n provide simple examples where this is impossible. Here JR.JP>3 x JR.JP>2 is particularly striking, since Wilking has constructed a metric with nonnegative curvature, and positive curvature on an open dense set of points. The same property is known for the so-called Gromoll-Meyer sphere ~7 = Sp(2)/ / Sp(1) by work of Wilhelm. In a recent manuscript [PW], Petersen and Wilhelm offer a proof that for this example a deformation is possible: THEOREM 2.10 (Exotic Sphere). ~7 = Sp(2)/ / Sp(1) admits a metric with positive curvature. It is remarkable that during the deformation, metrics with some negative curvatures will arise and the whole proof is both long and involved. The existence of an exotic sphere with positive curvature is of course of pivotal importance for all differentiable sphere theorems. It should also be noted that by work of Weiss and Grove-Wilhelm respectively, ~7 does not support a 1/4 pinched (cf. Shape Section) metric (nor one with radius larger than 1/2 maximal), respectively a positively curved metric with 4 points at distance exceeding 1/2 of the maximal possible diameter (cf. Size Section). We like to mention that at the time of writing, the work on all of the above examples has not yet been through a complete examination by other experts. Note that from the classification theorem above we know in particular that any cohomogeneity one homotopy n-sphere ~ E P is standard. The same conclusion holds in the class Po by work of Grove-Verdiani-WilkingZiller, but not in In fact all cohomogeneity one manifolds belong to by Schawachhofer-Tuschmann, but the Kervaire spheres have cohomogeneity one and some of them are exotic. Since the proposed metric of positive curvature in the Petersen-Wilhelm example has cohomogeneity 4, and a positively curved homotopy 4-sphere with a circle action is standard by the strong rank theorem, the following is natural:

Po

Po.

PROBLEM 8. Is a positively curved cohomogeneity k homotopy sphere standard if k :S 3?

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It is also natural to wonder about whether or not actions are linear. As we have seen they are linear in cohomogeneity one for the class P, and almost for the class Po. The exception here follows from work Grove-Ziller, since there is a non-linear "Kervaire" action, observed by Calabi on §5 where both singular orbits have co dimension 2, and hence the manifold has an invariant metric with non-negative curvature.

3. Size There are several natural metric invariants measuring size, such as, e.g., diameter, radius and volume. This allows for precise ways of saying that positively curved manifolds are "small". The first of these, obtained by a Synge type argument, due to Bonnet (for sectional curvature) and Myers (for Ricci curvature) provides a diameter bound in terms of a positive lower curvature bound: THEOREM 3.1 (Diameter Bound). A manifold with curvature bounded below by 8 > 0 has diameter at most 7r /.J8, the diameter of the sphere with constant curvature 8. In particular, a positively curved manifold has finite fundamental group, and in even dimensions it is either trivial or Z2 by Synge's theorem. No further general restrictions on the structure of the fundamental group of a positively curved manifold are known (cf. Shape section). In particular, the first counter example to the so-called Chern conjecture proposing that an abelian fundamental group would be cyclic, was found by Shankar. It is easy to see that any finite group is a subgroup of some SU(n), and hence the fundamental group of a manifold in Po. In contrast, nothing is known about the natural: PROBLEM 9. Is any finite group the fundamental group of a positively curved manifold? In any given dimension, however, there are restrictions on the size of the topology provided by the following finiteness result due to Gromov: THEOREM 3.2 (Betti Number Theorem). For any n E N there is a C(n) such that the fundamental group and homology groups of any M n E P o- is generated by at most C(n) elements. The main geometric tools used in the proof of this result, is the Toponogov Comparison Theorem, critical point theory for distance functions and the Bishop-Gromov Relative Volume Comparison Theorem. PROBLEM 10. Determine the optimal value for C(n). An extremely strong size and structure restriction is contained in the following proposed extension of a conjecture attributed to Bott:

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CONJECTURE (Ellipticity). The Betti numbers of the loop space (any field of coefficients) of any simply connected M n E P o- grow at most polynomially. A confirmation would have profound consequences. When restricted to the field of rational numbers a lot of restrictions are known for this class of so-called rationally elliptic manifolds. In particular, it would yield the optimal bound C(n) = 2n for the sum of Betti numbers, and also have impact on the classical Hopf conjecture about the Euler characteristic. The extension to the class Po- could be significant in a potential path of a proof of the ellipticity conjecture. The Bonnet-Myers theorem begs the question of what happens in the extreme case. In fact, having maximal diameter characterizes the round sphere as was shown by Toponogov: THEOREM 3.3 (Maximal Diameter Theorem). A manifold with curvature bounded below by 8 > 0 and diameter 7f /../8 is isometric to the sphere with constant curvature 8. The conclusion also holds when sectional curvature is replaced by Ricci curvature. For positively curved manifolds, say with lower curvature bound normalized to 1, the diameter function therefore provides a natural filtration in terms of super level sets, to be thought of as "diameter pinching". In particular, it is natural to ask whether a sufficiently large manifold with positive curvature is a sphere. Rather than relying on classical Morse theory for the loop space, it turned out to be advantageous to work directly on the manifold and develop a "Morse type" theory for (non-smooth) distance functions. In conjunction with Toponogov's comparison theorem, this indeed lead to the following results of Grove-Shiohama, and Gromoll-Grove and Wilking: THEOREM 3.4 (Diameter Sphere Theorem). A manifold with curvature bounded below by 8 and diameter> 7f /2../8 is homeomorphic to a sphere. THEOREM 3.5 (Diameter Rigidity Theorem). A manifold with curvature bounded below by 8 and diameter 7f /2../8 is either a topological sphere, or its universal cover is isometric to a CROSS. In the latter case, there is also an isometric classification when the fundamental group is non-trivial: The Z2 quotient of complex odd dimensional projective spaces, and the space forms where the fundamental group acts irreducibly. These "diameter theorems" raise two natural questions: PROBLEM 11. Is a manifold in P with almost maximal diameter diffeomorphic to the standard sphere?

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or even PROBLEM 12. Is a manifold in P with almost 1/2 maximal diameter diffeomorphic to one of the model spaces? Problems of this type are naturally analyzed using the fact that the class of n-manifolds with given lower bound on Ricci curvature and upper bound on diameter is precompact relative to the so-called Gromov-Hausdorff metric. This is a fairly simple consequence of the relative Bishop-Gromov volume comparison theorem. When a lower (sectional) curvature bound is present, any limit object is a so-called Alexandrov space, i.e., a finite dimensional length/inner metric space with a lower curvature bound expressed by distance comparison equivalent for manifolds to the Toponogov triangle comparison theorem. The dimension of such an Alexandrov space is at most n. One refers to collapse in the general case of dimension less than n, and non-collapse, otherwise. Non-collapse is equivalent to having a lower bound on volume. Relatively little is known in general about collapse. The work so far has culminated in the recent manuscript [KPT] by Kapovitch-PetruninThschman on the class Po-. In low dimensions, much more is known due to work of Shioya-Yamaguchi which has significant impact on the solution of Thurston's geometrization conjectuture due to Perelman. In the noncollapsing case, however, one has the following fundamental result due to Perelmann, a proof of which has recently been published in [Ka]: THEOREM 3.6 (Topological Stability). All n-dimensional Alexandrov spaces with curvature 2: k in a Gromov-Hausdorff neighborhood of an ndimensonal Alexandrov space X with curvature 2: k are homeomorphic to X. This immediately yields topological finiteness in all dimensions (including 3 where only homotopy type was know) for the class of n-manifolds with lower bounds on curvature and volume and upper bound on diameter, originally due to Grove-Petersen-Wu. In the case of positive curvature (where also Hamilton's theorem can be invoked), one gets in particular: THEOREM 3.7 (Topological Finiteness). For any positive 6, v and integer n =1= 4, there are at most finitely many diffeomorphism types of n-manifolds, with curvature and volume bounded below by 6 and v respectively. The same holds for homeomorphism types in dimension 4. Note, that in the diameter problems above no volume bound is present, so collapse will typically happen. Of course limiting objects are Alexandrov spaces with positive curvature and maximal, respectively half maximal diameter. In the case of maximal diameter, the Alexandrov space is not a round sphere as in Toponogov's theorem above, but "only" a so-called spherical suspension of an Alexandrov space with the same positive lower curvature bound.

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The spherical suspension of a manifold in P with more than 1/2 maximal diameter was closely analyzed by Grove-Wilhelm, and used to derived the following result where collapse is avoided: THEOREM 3.8. For any n-manifold M E P with more than 1/2 maximal diameter, there is an n-dimensional Alexandrov space X which is the Gromov-Hausdorff limit of positively curved metrics on !v! as well as on the standard n-sphere. In other words, a differentiable stability version of Perleman's Theorem above would imply that such an M is diffeomorphic to the standard sphere. A curious observation by Wilking based on the suspension construction as well, also shows that if there is an exotic sphere M E P with more than 1/2 maximal diameter, then !vI also carries a metric with almost maximal diameter. In other words, the answer to problem 11 is yes if and only if any positively curved manifold with diameter larger than 1/2 maximal is diffeomorphic to a sphere. The case where M E P has q + 1 > 2 points with individual distances larger than 1/2 maximal diameter was also analyzed via spherical suspensions. It was shown, that 11,1 = IDW x §n- q u§q-1 x lIJJ n - q+ 1 . In particular, Mis diffeomorphic to the standard sphere if q = n-3 invoking a result of Hatcher. All size invariants are difficult to compute, and so filtrations by size do not provide any guide towards the construction of new examples. 4. Shape A third characterization of the round sphere is being simply connected and having constant positive curvature. It is only natural to expect that (simply connected) Riemannian manifolds with almost the same shape are spheres as well. This became known as the pinching problem raised by Hopf. Here, the pinching

8M :=

min sec max sec

~

1

measures the proximity to constant curvature. The first major result in this direction was due to Rauch who in the process derived the so-called Rauch comparison theorem for Jacobi fields. The optimal result in terms of pinching, the so-called 1/4- pinching sphere theorem, was proved by Klingenberg, with rigidity due to Berger: THEOREM 4.1 (Topological Sphere/Rigidity Theorem). A 8 pinched simply connected manifold with 8 2': 1/4 is either homeomorphic to the sphere or isometric to a CROSS. The crucial geometric tools used to prove this was the Toponogov comparison theorem and the so-called long homotopy lemma by Klingenberg.

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The question of diffeomorphism was first succesfully treated independently by Calabi, Gromoll and Shikata each of whom established a diffeomorphism sphere theorem with dimension dependent pinching approaching 1. Each proof also provided the germ for subsequent developments: The Gromoll filtration of the group of homotopy spheres, the idea and notion of distance between manifolds/differentiable structures in the work of Shikata, and the analytic approach used by Calabi. In particular, some of the roots of the far reaching work of Colding and Cheeger on manifolds with a lower Ricci curvature bound have similarities with the work of Calabi. Subsequently, two different methods to achieve a dimension independent diffeomorphism sphere theorem were obtained by Shiohama-Sugimoto, and by Ruh. Moreover, the method of Ruh in conjunction with the general nonlinear center of mass technique developed by Grove-Karcher led to an equivariant sphere theorem, and in particular to a pinching theorem for all space forms. Just recently, following a breakthrough by Bohm and Wilking [BW], the Ricci flow was used by Brendle and Schoen [BSl, BS2] to replace homeomorphism by diffeomorhism in the above classical result, and extend it to pointwise pinching and equivariance. In particular: THEOREM 4.2 (Differentiable Sphere/Rigidty Theorem). A pointwise 1/4-pinched manifold is either diffeomorphic to a space form or locally isometric to a CROSS. A key issue here is which curvature conditions are preserved by the Ricci flow, a very difficult problem. Although 1/4 pinching is not preserved, it turns out that it is in a family which is preserved and which moreover satisfies the general condition set forth by Bohm and Wilking in the situation of strict 1/4 pinching. Already some time ago, Berger using Gromov-Hausdorff limit arguments obtained a below 1/4 pinching theorem in even dimensions (for spheres only deriving homeomorphism of course), and later using non convergence methods, Abresh-Meyer were able to do the same in odd dimensions getting explicit constants. Invoking the Ricci flow, Petersen-Tao [PT] recently derived the satisfactory. THEOREM 4.3 (Below 1/4 pinching). There is an E(n) > 0 such that any simply connected 1/4 - E(n) pinched n-manifold is diffeomorphic to a CROSS. The first c5 < 1/4 where one knows for sure that there is a different c5-pinched manifold, is for c5 = 1/37. In fact Ptitmann showed that each of the three normal homogeneous manifolds, B7 = SO(5)/ SO(3), W7 = SU(3) SO(3)/ U(2), and B13 = SU(5)/ Sp(2) Sl with positive curvature admits a homogeneous 1/37-pinched metric. It is natural to wonder if indeed

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1/37 is the optimal pinching among all positively curved metrics on these examples, and how large E(n) necessarily has to be. An investigation of the class of 8-pinched manifolds equipped with the Gromov-Hausdorff metric does provide some crude insight in terms of finiteness results or even obstructions modulo finiteness. The general Cheeger Finiteness Theorem asserts that the class of closed n-manifolds with bounded curvature and diameter, and with a lower bound for volume contains at most finitely many diffeomorphism types. Since a positive lower curvature bound by the Bonnet-Myers Theorem yields a bound for the diameter, and an upper curvature bound yields a lower bound on the injectivety radius for positively curved manifolds in even dimensions by results of Synge and Klingenberg, one has the following THEOREM 4.4. For any n E N, and 0 < 8 :::; 1, there are only finitely many diffeomorphism types of 8-pinched 2n-man~folds. So far, at most 4 different simply connected examples of positively curved 2n-manifolds are known for any n. In contrast, one easily sees that there exists a sequence of different AloffWallach spaces Apq whose pinching converges to that of Vf,T7 = Au and hence to 1/37. In particular, for any 8 < 1/37 there are infinitely many 8pinched manifolds. Nevertheless, with additional topological restrictions, the following is a corollary of 11"2- Finiteness Theorem due to Petrunin-Tushmann and to Fang-Rong: THEOREM 4.5. For any n' EN, and 0 < 8 :::; 1, there are only .finitely many diffeomorphism types of 8-pinched 2-connected n-manifolds. So far, however, no infinite family of positive curvature is known, where by necessity the pinching constants will approach O. An infinite family in even dimensions, or an infinite 2-connected family in odd dimensions would do according to these finiteness results. The Pk family from the Symmetry section is a natural candidate for this. On the other hand, Fang-Rong have conjectured that the above theorem should hold for positively curved manifolds without a pinching assumption. Utilizing the Cheeger-Fukaya-Gromov theory of collapse with bounded curvature, Rong has obtained he following result about the structure of the fundamental group THEOREM 4.6 (11"1 Structure). For' any n E N, and 0 < 8 :::; 1, there is a constant w( n, 8) so that 11"1 (M) contains a cyclic subgroup of index at most w( n, 8) for any 8-pinched n-manifold M. It is a conjecture of Rong that w(n, 8) can be chosen independently of 8. Although the pinching function provides a natural filtration among positively curved manifolds by superlevel sets, it is typically very difficult to calculate for a particular example and clearly not a guide towards finding new examples.

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References [BW] C. Bi:ihm and B. Wilking, Manifolds with positive C1lrvature operators are space forms, Ann. of Math. 167 (2008), 1079-1097. [BS1] S. Brendle and R. Schoen, Manifolds with 1/4-pinched C1L'T"1Jature are space forms, J. Amer. Math. Soc. 22 (2009), no. 1, 287-307. [BS2] S. Brendle and R. Schoen, Classification of man'ifolds with weakly 1/4-pinched curvatlLres, Acta Math. 200 (2008), no. 1, 1-13. [De] O. Dearricott, A 7-manifold with positive curvature, preprint. [Fi] R. Fintushel, Circle actions on Simply Connected 4-manifolds, Trans. Amer. Math. Soc. 230 (1977), 147-171. Classification of circle actions on 4-manifolds, Trans. Amer. Math. Soc. 242 (1978), 377-390. [FSS] R. Fintushel, R.J. Stern and N. Sunukjian. Exotic group actions on simply connected smooth 4-manifolds, arXiv:0902.0963. [G1] K Grove, Geometry of, and via, Symmetries, Amer. Math. Soc. Univ. Lecture Series 27 (2002), 31-53. [G2] K Grove, Finiteness theorems in Riemannian geometry, Explorations in Complex and Riemannian geometry, 101-120, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003. [GVZ] K Grove, L. Verdiani and W. Ziller, A new type of a positively curved manifold, arXiv0809.2304. [Ka] V. Kapovitch, Perelman's Stability Theorem, Surveys in Differential Geometry, Vol. XI: Metric and Comparison Geometry, ed. KGrove and J.Cheeger, International Press (2007), 103-136. [KPT] V. Kapovitch, A. Petrunin and W. Tuschmann, Nilpotency, almost nonnegative curvature and gradient push, Ann. of Math., to appear. [Pal P.S. Pao, Nonlinear circle actions on the 4-sphere and twisting spun knots, Topology 17 (1978), 291-296. [Pel P. Petersen, Variations on a theme of Synge, Explorations in Complex and Riemannian geometry, 241-251, Contemp. Math., 332, Amer. Math. Soc., Providence, RI,2003. [PT] P. Petersen and T. Tao, Classification of Almost Quarter Pinched Manifolds, Proc. Amer. Math. Soc., to appear. [PW] P. Petersen and F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv.org:0805.0812. [Pt] A. Petrunin, Semiconcave Fnnctions in Alexandrov's Geometry, Surveys in Differential Geometry, Vol. XI: Metric and Comparison Geometry, ed. K Grove and J. Cheeger, International Press (2007), 137-201. [PI] C. Plaut, Metric spaces of curvature 2 k. Handbook of geometric topology, 819-898, North-Holland, Amsterdam, 2002. [Ro] X. Rong, Collapsed Manifolds with Bounded Sectional Curvature and Applications, Surveys in Differential Geometry, Vol. XI: Metric and Comparison Geometry, ed. KGrove and J.Cheeger, International Press (2007), 1-23. [Wi] B. Wilking, Nonnegatively and Positively Curved Manifolds, Surveys in Differential Geometry, Vol. XI: Metric and Comparison Geometry, ed. KGrove and J.Cheeger, International Press (2007), 25-62. [ZI] W. Ziller, Examples of manifolds with nonnegative sectional curvature, in: Metric and Comparison Geometry, ed. KGrove and J.Cheeger, Surv. Diff. Geom. Vol. XI, International Press (2007), 63-102. [Z2] W. Ziller, Geometry of positively curved cohomogeneity one manifolds, in: Topology and Geometric Structures on Manifolds, in honor of Charles P.Boyer's 65th birthday, Progress in Mathematics, Birkhauser (2008), 233-262. UNIVERSITY OF NOTRE DAME

E-mail address: kgrove2COnd. edu

Surveys in Differential Geometry XIII

Einstein metrics, four-manifolds, and conformally Kahler geometry Claude LeBrun The Ricci curvature of a smooth Riemannian n-manifold (M,g) is the function on the unit tangent bundle UT M = {v E T M Ig( v, v) = I} given by vl---t1'(v,v)

where l' is the Ricci tensor of g. This function gives a precise measure of the volume distortion of the exponential map, since in geodesic normal coordinates the metric volume element becomes

If the metric 9 has constant RicCi curvature, we call it an Einstein metric, and (M, g) is then said [4] to be an Einstein manifold. This of course happens precisely when 9 satisfies the so-called Einstein equation

(1)

1'=)..g

for some real constant )... The number ).., which then represents the constant value of the Ricci curvature, is often called the Einstein constant of (M, g). It is related to the scalar curvature s = 1'i i = Rij ij by s = n).., so the Einstein constant and the scalar curvature in particular have the same sign. All of this terminology is now completely standard among mathematicians, but Einstein himself would probably have been deeply uncomfortable with it. After all, mathematicians are primarily interested in equation (1) as a plausible avenue for geometrizing smooth compact manifolds. The fact that we are interested in Riemannian rather than Lorentzian solutions of (1) is simply not an issue for us, but it most certainly would have puzzled Einstein - especially insofar as we have adopted his notation 9 for the metric, while forgetting that he intended this as an abbreviation for gravitational field. In any case, the supreme historical irony is perhaps that Einstein later [14] called equation (1) the "greatest mistake of his life," since the introduction Supported in part by NSF grant DMS-0604735. ©2009 International Press

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of the so-called cosmological constant A into the gravitational field equations prevented him [31] from predicting the observed Hubble expansion of the universe. Of course, we mathematicians have a long history of latching onto good ideas, without worrying terribly much about where they came from. This is nothing new. Indeed, it has been nearly two centuries since Goethe [15] complained that, "Mathematicians are like Frenchmen; you tell them something, they translate it into their own language, and, before you know, it's something entirely different." In any case, one of the central problems of modern differential geometry is to determine precisely which smooth compact n-manifolds admit Einstein metrics. When n = 2 or 3, the Einstein metrics are just the metrics of constant sectional curvature, so when such a metric exists, it geometrizes the manifold in the extremely strong sense of displaying it as a quotient of a standard, homogeneous model by a discrete group of isometries. In fact, the existence of such metrics on any 2-manifold is guaranteed by the classical uniformization theorem. By contrast, not every compact 3-manifold admits an Einstein metric; but Perelman's successful attack [20, 34, 35, 36] on the Thurston geometrization program via the Hamilton Ricci flow [18] has still taught us that every 3-manifold can at least be broken up into Einstein and collapsed pieces. Dimension four represents an important transition for equation (1); when n = 4, Einstein metrics are usually no longer locally homogeneous, but special low-dimensional phenomena, discussed below, nonetheless provide powerful links between their geometry and the differential topology of the underlying manifold. On the other hand, when n 2: 5, Einstein metrics do not seem to offer a plausible geometrization of manifolds, because [5, 6] even familiar manifolds like high-dimensional spheres typically admit unitvolume Einstein metrics for many different values of the Einstein constant A. The case of n = 4 thus seems particularly interesting and important. But unfortunately, we are still far from being able to determine precisely which smooth compact 4-manifolds !v14 admit Einstein metrics. Nevertheless, Kahler geometry provides a rich source of examples of Einstein metrics on compact 4-manifolds, and Seiberg-Witten theory allows one to mimic Kahler geometry when treating even non-Kahler metrics on compact complex surfaces. This article will therefore focus on the following restricted version of the problem: QUESTION 1. If M4 is the underlying smooth manifold of a compact complex surface (M, J), when does M carry an Einstein metric?

It turns out that there is a powerful analogy between complex surfaces and 4-manifolds that carry symplectic forms (closed, non-degenerate 2-forms). It is therefore natural to also ask QUESTION 2. If M4 is a smooth compact 4-manifold that admits a symplectic form w, when does M carry an Einstein metric?

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Without some restriction on the Einstein metric, a full answer to even these restricted questions remains tantalizingly out of reach. However, if we are willing to also make an assumption about the sign of the Einstein constant A, definitive answers like the following [9] can now be given. 1. Suppose that M is a smooth compact oriented 4-manifold which admits a complex structure J. Then M also admits a (possibly unrelated) Einstein metric 9 with A > 0 if and only if M appears on the following list of diffeotypes: THEOREM

O:S k :S 8,

Here CJP>2 denotes the smooth oriented manifold obtained by giving CJP>2 the non-standard orientation, and

CJP>2#kCJP>2 = CJP>2# CJP>2# ... #CJP>2, ,

v

#

k

where # indicates the connected sum operation, which glues two oriented 4-manifolds together by first removing a standard ball from each, and then identifying the resulting boundary spheres via a reflection. The relevance of this operation to complex geometry arises from the fact that if N is a complex surface, we may may replace any point pEN with a CJP>1 of selfintersection -1 to obtain a new complex surface N, called the blow-up of N at p, which is diffeomorphic to N#CJP>2. Thus the diffeotypes listed above can be realized by OP,! x CJP>1 and of CJP>2 blown up at k points in general position, O:S k :S 8. In other words [11, 27], this list describes the diffeotypes of the Del Pezzo surfaces, which are by definition the compact complex surfaces which are Fano, in the sense that E H2(M, JR) is a Kahler class. If we broaden the question by merely requiring that the Einstein constant be non-negative, more diffeotypes are allowed, but a complete classification [24] can still be given.

cf

THEOREM 2. Suppose that M is a smooth compact oriented 4-manifold which admits an integrable complex structure J. Then M also admits an Einstein metric 9 with A 2: 0 if and only if M appears on the following list of diffeotypes:

CJP>2#kCJP>2, O:S k :S 8, S2 X S2, K3, M~

K3/Z2, T4 , T4 /Z2, T4 /Z3, T4 /Z4, T 4/Z6, T 4/(Z2 EEl Z2), T 4/(Z3 EEl Z3),or T 4/(Z2 EEl Z4).

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The diffeotypes we have added in passing from Theorem 1 to Theorem 2 are exactly those represented by the compact complex surfaces of Kahler type for which Cl E H2(M, Z) is a torsion class. These are traditionally sorted into four baskets [3, 16]. The first basket holds the K3 surfaces, named in honor of Kummer, Kahler, and Kodaira, and defined to be the simply connected compact complex surfaces with Cl = 0; they are all deformation equivalent [21], and so, for example, are all diffeomorphic to the smooth quartic t 4 + u 4 + v 4 + w 4 = 0 in CP3' Next, there are the Enriques surfaces, which are Z2-quotients of K3 surfaces; again, there is only one diffeotype. Then there are the Abelian surfaces, which are diffeomorphic to the 4-torus T4. And finally, there are the hyper-elliptic surfaces, which are quotients of T4 by one of seven finite groups of affine motions, each of which acts in a uniquely specified way. Symplectic analogs of these results are also true: THEOREM 3. Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure w. Then M also admits an Einstein metric g with A > 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

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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

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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) <

00,

Q(u) = N},

NEZ.

EXISTENCE OF FADDEEV KNOTS

151

In [LY1, LY4], it is shown that EN is attainable at N = ±1 and that there is an infinite subset of Z, say §, such that EN is attainable for any N E §. The purpose of the present work is to extend this existence theory for the Faddeev knot problem to arbitrary settings beyond 3 dimensions. Our motivation of engaging in a study of the Faddeev knot problem beyond 3 dimensions comes from several considerations: (i) Theoretical physics, especially quantum field theory, not only thrives in higher dimensions but although requires higher dimensions [GSW, P, Z]. (ii) The 3dimensional Faddeev model may be viewed naturally as a special case of an elegant class of knot energies stratified by the Hopf invariant in general dimensions (see our formulation below). (iii) Progress in general dimensions helps us achieve an elevated level of understanding [LY3, LY5] of the intriguing relations between knot energy and knot topology and the mathematical mechanism for the formation of knotted structures. (iv) Knot theory in higher dimensions [H, K, R] is an actively pursued subject, and hence, it will be important to carry out a study of "ideal" knots for the Faddeev model in higher dimensions. Note that minimization of knot energies subject to knot invariants based on diagrammatic considerations has been studied considerably in literature. For example, knot energies designed for measuring knotted/tangled space curves include the Gromov distortion energy [G1, G2], the Mobius energy [BFHW, FHW, 01, 02], and the ropelength energy [B, CKS1, CKS2, GM, Na]. See [JvR] for a rather comprehensive survey of these and other knot energies and related interesting works. See also [KBMSDS, Kf, M,

S, SKK]. Although there are various available formulations when one tries to generalize the Faddeev energy (1.3), the core consideration is still to maintain an appropriate conformal structure for the energy functional which works to prevent the energy to collapse to zero. The simplest energy is the conformally invariant n-harmonic map energy, where n is the dimension of the domain space, which is also known as the Nicole model [Ni] when specialized to govern maps from ~3 into 8 2 • Another type of energy functionals is of the Skyrme type [MRS, Sl, S2, S3, S4, ZB] whose energy densities contain terms with opposite scaling properties and jointly prevent energy collapse. In fact, these terms interact to reach a suitable balance to ensure solitons of minimum energy to exist. The Faddeev model (1.3) belongs to this latter category for which the solitons of minimum energy are realized as knotted energy concentration configurations [BS1, BS2, FN1, FN2, Su]. In this paper, our main interest is to develop an existence theory for the energy minimizers of these two types of knotted soliton energies. Specifically, we will study both the Nicole-Faddeev-Skyrme (NFS) type and Faddeev type knot energy (see (2.4), (2.5) and (2.6) for definitions). The two energy functionals have very different analytical properties. In

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F. HANG, F. LIN, AND Y. YANG

particular, the conformally invariant term

(1.6)

r

J~4n-l

IV'u1 4n - 1 dx

in the NFS model enables us to carry out a straightforward argument which shows that the Hopf-Whitehead invariant Q (u) (see (2.3)) must be an integer for any map u with finite NFS energy. More importantly, it allows us to get an annulus lemma (Lemma 3.1) which permits us to freely cut and paste maps under appropriate energy control. In this way, as in [LY2], the minimization problem fits well in the classical framework of the concentrationcompactness principle [EI, E2, LI, L2]. Along this line, we shall arrive at the main result, Theorem 7.1, which guarantees the existence of extremal maps for an infinite set of integer values of the Hopf-Whitehead invariant. The situation is different for the Faddeev energy (see (2.6)). In this case, it seems difficult to know whether a map with finite energy can be approximated by smooth maps with similar energy control. In particular, it is not clear anymore why the Hopf-Whitehead invariant (see (2.3)), which is given by an integral expression, should always be an integer. Based on some recent observations of Hardt-Riviere [HR] in the study of the behavior of weak limits of smooth maps between manifolds in the Sobolev spaces, and some earlier approach of Esteban-Muller-Sverak [Sv, EM], we are able to show that the Hopf-Whitehead invariant of a map with finite Faddeev energy must be an integer (see Theorem 10.1). Such a statement is not only useful for a reasonable formulation of the Faddeev model but also plays a crucial role in understanding the behavior of minimizing sequence and the existence of extremal maps. One of the main difficulties in understanding the Faddeev model is that it is still not known whether an annulus lemma similar to Lemma 3.1 exists or not. In particular, we are not able to freely cut and paste maps with finite energy and it is not clear whether the minimizing problem would break into a finite region one and another at the infinity. That is, in this situation, the minimizing problem does not fit in the framework of the classical concentration-compactness principle anymore. This difficulty will be bypassed by a decomposition lemma (Lemma 12.1) for an arbitrary map with finite Faddeev energy (in the same spirit as in [LYI] for maps from ]R3 to S2). Roughly speaking, the lemma says we may break the domain spaces into infinitely many blocks, each of which can be designated with some "degree". By collecting those nonzero "degree" blocks suitably we may have a reasonable understanding of the minimizing sequence of maps for the Faddeev energy (Theorem 13.1). Based on this and the sublinear growth law for the Faddeev energy, we will obtain several existence results of extremal maps for the Faddeev energy (see Section 13.1). We point out that the method to bypass the breakdown of the concentration-compactness principle is along the same line as [LYI]. However, due to the fact that we do not have the tool of lifting through the classical Hopf map S3 ----t S2 in higher dimensions, we have to resort to different approaches to deal with the

EXISTENCE OF FADDEEV KNOTS

153

nonlocally defined Hopf-Whitehead invariant. When reduced to the Faddeev model from ]R3 to 8 2 , this method gives a different route towards the main results in [LYI]. Moreover, by establishing the subaddivity of the Faddeev energy spectrum (see Corollary 13.3), we are able to strengthen the Substantial Inequality in [LYI] to an equality. That is, we are actually able to establish an additivity property for the Faddeev knot energy spectrum. We will also use the same approach to improve the Substantial Inequality for the Skyrme model to an equality (see Theorem 14.3). Here is a sketch of the plan for the rest of the paper. The first part, consisting of Sections 2-7, is about the NFS model. In Section 2, we introduce the generalized knot energies of the Nicole type [AS, ASVW, Ni, We], the NFS type extending the two-dimensional Skyrme model [Co, dW, GP, KPZ, LY2, PMTZ, PSZI, PSZ2, PZ, SB, Wei], and the Faddeev type [FI, F2], all in light of the integral representation of the Hopf invariant in the general (4n - 1) dimensions (referred to as the Hopf dimensions). We will also obtain some growth estimates of the knot energies with respect to the Hopf number in view of the earlier work [LY3, LY5]. In Section 3, we establish a technical (annulus) lemma for the NFS model which allows truncation of a finite-energy map and plays a crucial role in proving the integer-valuedness of the Hopf-Whitehead integral and the validity of an energy-splitting relation called the "Substantial Inequality" [LY4]. We shall see that the conformal structure of the leading term in the energy density is essential. In Section 4, we show that the Hopf-Whitehead integral takes integer value for a finite-energy map in the NFS model. In Section 5, we consider the minimization process in view of the concentrationcompactness principle of Lions [LI, L2] and we rule out the "vanishing" alternative for the nontrivial situation. We also show that the "compactness" alternative is needed for the solvability of the Faddeev knot problem stated in Section 2 for the NFS energy. In Section 6, we show that the "dichotomy" alternative implies the energy splitting relation or the Substantial Inequality. These results, combined with the energy growth law stated in Section 2, lead to the existence of the NFS energy minimizers stratified by infinitely many Hopf charges, as recognized in [LYI]. We state these results as the first existence theorem in Section 7. We then establish a simple but general existence theorem for both the generalized NFS model and the generalized Faddeev model in the compact case. For the Nicole model over ]R3 or 8 3 , we prove the existence of a finite-energy critical point among the topological class whose Hopf number is arbitrarily given. The second part, consisting of Sections 8-13, is about the Faddeev model. In Section 8, we briefly describe the formulation of Faddeev model. In Section 9, various basic tools necessary for the study of Faddeev model are discussed. Section 10 is devoted to showing that for a map with finite Faddeev energy, the Hopf-Whitehead invariant is well defined and takes only integer values. We also derive a similar result for maps with mixed differentiability (see Section 10.1). Such kind of results are needed in proving the crucial

F. HANG, F. LIN, AND Y. YANG

154

decomposition lemma (Lemma 12.1). In Section 11, we describe some basic rules concerning the Hopf-Whitehead invariant for maps with finite Faddeev energy and the sublinear energy growth rate. Note that such kind of sublinear growth is a special case of results derived in [LY5]. The arguments are presented here to facilitate the discussions in Section 11, Section 12 and Section 13. In Section 12, we prove a crucial technical fact: the validity of a certain decomposition lemma for a map with finite Faddeev energy. The proof of this lemma shares the same spirit as that in [LYl] but is technically different due to the lack of lifting arguments. In Section 13, we prove the main result of the second part, namely, Theorem 13.1, which describes the behavior of a minimizing sequence of maps. Based on this description and the sublinear growth law, we discuss some facts about the existence of minimizers in Section 13.1. In Section 14, we apply our approach in the second part to the standard Skyrme model to derive the subadditivity of the Skyrme energy spectrum and strengthen the substantial inequality to an equality. Finally, we conclude with Section 15. 2. Knot energies in general Hopf dimensions Recall that the integral representation of the Hopf invariant by Whitehead [Wh] of the classical fibration 8 3 -> 8 2 can be extended to the general case of the fibration 8 471 - 1 -> 8 271 . More precisely, let u : 8 471 - 1 -> 8 2n (n :::: 1) be a differentiable map. Then there is an integer representation of u in the homotopy group 7l"4n_1(8 2n ), say Q(u), called the generalized Hopf index of u, which has a similar integral representation as (1.4) as follows. Let Ws2n be a volume element of 8 271 so that (2.1 ) is the total volume of 8 2n and u* the pullback map A(8 2n ) --t A(84n - 1) (a homomorphism between the rings of differential forms). Since u* commutes with d, we see that du*(wS2n) = 0; since the de-Rham cohomology H2n(84n-l, JR) is trivial, there is a (2n - I)-form v on 8 471 - 1 so that dv = U*(WS2n) (sometimes we also write U*(WS2n) simply as U*WS2n when there is no risk of confusion). Of course, the normalized volume form WS2n = 18 2n l- 1wS2n gives the unit volume and f; = 182n l- 1 v satisfies df; = u* (WS2n). Since WS2n can be viewed also as an orientation class, Q(u) may be represented as [GHV, Hu] (2.2)

Q(u) = ( } S4n-l

f; 1\ U*(WS2n)

= I LI2 ( 8

} S411-1

v

1\

U*(WS2n).

The conformal invariance of (2.2) enables us to come up with the Hopf invariant, or the Hopf-Whitehead invariant, Q(u), for a map u from JR 4n - 1

155

EXISTENCE OF FADDEEV KNOTS

to s2n which approaches a fixed direction at infinity, as (2.3)

Q(u)

=

Is;nl2

f

JR4n-l

dv

v 1\ U*(WS2n),

= U*(WS2n).

With the above preparation, we introduce the generalized Faddeev knot energies, subclassified as the Nicole, NFS, Faddeev energies over jR4n-l, respectively, as (2.4)

ENicole(u) =

(2.5)

ENFS(u) =

(2.6)

EFaddeev(u) =

f

lV'uI 4n - 1 ,

f

{1V'uI 4n - 1 + IU*(WS2n) 12 +

f

{1V'u I4n - 2 + -21Iu*(WS2n)12},

JR4n-l JR4n-l JR4n-l

In -

uI 2},

where and in the sequel, we omit the Lebesgue volume element dx in various integrals whenever there is no risk of confusion, we use the notation lV'ul, Idul, and IDul interchangeably wherever appropriate, and we use n to denote a fixed unit vector in jR2n+l or a point on s2n. Besides, we use CO to denote the best constant in the Sobolev inequality (2.7)

over jR4n-l with q satisfying 1/q = 1/2 - 1/(4n - 1) = (4n - 3)/2(4n -1), given by the expression

(2.8)

Co

=

1 (

([4n - 1][4n - 3])2 W4n-l

!

)r(2n + r(4n -1)

r(2n -

!)) (4n~1) ,

with Wm being the volume of the unit ball in jRm. 2.1. Let E be the energy functional defined by one of the energy functionals given by the expressions {2.4}, {2.5}, and {2.6}. Then there is a universal constant C = C( n) > 0 such that THEOREM

4n-l

E(u) ~ ClQ(u)I4n".

(2.9)

In the case when E is given by {2.6}, the constant C has the explicit form

2n-l C( n-2co ) _ n( IS2nI2)4n-l 4nn2.

(2.10) PROOF.

(2.11)

Recall the Sobolev inequality over jR4n-l of the form

C(n,p)lIfll q

~ IIV' flip,

From the pointwise bound (2.12)

1
< 4n-1,

q=

(4n - 1)p . 4n -1- p

156

F. HANG, F. LIN, AND Y. YANG

and assuming dv = u* (WS2n) and I5v = 0, where 15 is the codifferential of d which is often denoted by d* as well, we have

where we have used an V-version of the Gaffney type inequality [ISS, Sc] for differential forms (we thank Tom Otway for pointing out these references). Choose p = (4n - 1)/2n so that q = (4n - 1)/(2n - 1) in (2.11). The conjugate exponent q' with respect to q is q' = q/(q - 1) = (4n - 1)/2n. Thus the Holder inequality and (2.13) lead us to

Is 2n121Q( u) 1~ Il v ll q llu*(WS2n) Il q, ~

(2.14)

CIIV'vll (4n-1)/2nll'u* (WS2n) 11(4n-1)/2n

~ C1

(r

4n

lV'uI

4n - 1 )

4n-l ,

JJR4n-l

which establishes (2.9) for the energy functional given by (2.4) or (2.5). Consider now the energy functional (2.15 ) In [LY5], we have shown that, when the exponent p in (2.15) lies in the interval (2.16)

1

4n(4n - 1) 4n+ 1 '

there holds the universal fractionally-powered topological lower bound 411.-1

(2.17)

Ep(U) 2: C(n,p)IQ(u)I"4n,

where the positive constant C(n,p) may be explicitly expressed as

(2.18) X

4n ) ( (4n - 1)(8n - p) - p(4n + 1)

(471-1) (8" -p) -p( 4,,+ 1) 87/.(4n-,,)

It is seen that our stated lower bound for the energy defined in (2.6) corresponds to p = 4n - 2 so that C(n, 4n - 2) is given by (2.10) as claimed. 0 For the earlier work in the classical situation, n Note that the energy (2.19)

= 1, see [KR, Sh, VK].

EXISTENCE OF FADDEEV KNOTS

157

is also of interest and referred to as the AFZ model [AFZ] when n = 1. Combining (2.13) and (2.14), we have ClQ(u)1 ~ Ilu*(WS2n)II(4n_1)/2n'

(2.20)

which implies that the energy E AFZ defined in (2.19) satisfies the general fractionally-powered topological lower bound (2.9) as well. We next show that the lower bound (2.9) is sharp. THEOREM 2.2. Let E be defined by one of the expressions stated in (2.4), (2.5), (2.6), and (2.19). Then for any given integer N which may be realized as the value of the Hopf- Whitehead invariant, i.e., Q( u) = N for some differentiable map u : jR4n-1 --+ s2n, and for the positive number EN defined as

(2.21)

EN

= inf{E(u)IE(u) < 00, Q(u) = N},

we have the universal topological upper bound 4n-l

EN ~ CINI----:rn,

(2.22) where C

> 0 is a constant independent of N.

PROOF. In [LY5], we have proved the theorem for the general energy functional

E(u) =

l4n-l 1-£('\1u) dx,

where the energy density function 1-£ is assumed to be continuous with respect to its arguments and satisfies the natural condition 1-£(0) = O. Hence the theorem is valid for the energy functionals (2.4) and (2.6). For the energy functional (2.5), there is an extra potential term lu - n12. However, this term does not cause problem in our proof because the crucial step is to work on 1 a ball in jR4n-1 of radius INI4n and u = n outside the ball. Therefore, the potential term upon integration contributes a quantity proportional to the volume of the ball, which is of the form CINI----:rn. 0

4n-l

In the following first few sections, we will concentrate on the energy functional (2.5).

3. Technical lemma Let B be a sub domain in restricted to B,

(3.1)

E(u; B)

=

jR4n-1

and consider the knot energy (2.5)

L

{1'\1uI 4n - 1 + IU*(WS2n )1 2 + lu - nI2}.

We use BR to denote the ball in jR4n-l centered at the origin and of radius R > O. The following technical lemma plays an important part in our investigation of the first part of this paper.

F. HANG, F. LIN, AND Y. YANG

158

LEMMA 3.1. For any small e > 0 and R 2 1, let u : B2R \ BR --t s2n satisfy E(u;B2R \ BR) < e. Then there is a map u : B2R \ BR --t s2n such that (i) u = u on VBR, (ii) u = n on VB2R, (iii) E(u; B2R \ BR) < Ge, where C > 0 is an absolute constant independent of R, e, and u. The same statement is also valid when u is modified to satisfy u = n on vBRand u = u on VB2R'

To obtain a proof, it will be convenient to work on a standard small domain. First, for the map stated in the lemma, define (3.2) Hence y E B2 \ Bl and

(3.3)

e> 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 <

1

such that for any

C.

(b) Vanishing: For any R > 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 <

E,

r

I JIR4n-l \B Rj fj(X)dX-(1-t)E(Uj)1 <E.

For convenience, we assume Rj > 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}, ~,
(7.11) where q = cosh'f} -

cos~

and 0

< 'f} < 00,0 :::;

~,

27T. The AFZ ansatz

[AFZ, ASVW, HS] reads (7.12)

where the undetermined function condition (7.13)

1(0) = lim 1('f}) = 0, ry-O

1

satisfies the "normalized" boundary

1(00) = lim 1('f}) = 00, ry_oo

EXISTENCE OF FADDEEV KNOTS

171

so that the Hopf map is given by the choice f( 'r/) = sinh'r/ with m = n = 1, or (7.14) After some calculation, it can be shown [AFZ, HS] that the Hopf invariant of u designated by (7.10)-(7.13) is given as

Q(u) = mn.

(7.15) Besides, with the new variable (7.16)

the function f becomes a function of t, which is still denoted by f(t) for simplicity, so that the Nicole energy (7.9) takes the form [ASVW] (7.17)

rX) {t(1+t2) (P 1 [m2 ] p ) ~} dt, (1 +tJ2)2 + 1 +t2 i2+n2 (1 + J2)2

E(f) = 3211"2 io

and the boundary condition (7.13) is reinterpreted in terms of t given in (7.16). The Euler-Lagrange equation of (7.17) is [ASVW] (7.18)

t 2(1

+ t 2)(1 + f2)(2t 2[1 + t 2]f? + [m 2 + n 2t 2lf2)ftt - 4t4(1 + t 2)2 f ft + t 3(1 + 3t 2)(1 + t 2)(1 + f2)Jf - 2t2(1 + t 2)(m2 + n 2t 2)f3 f? + t 3(m 2 + n 2[1 + 2t2])(1 + f2)f2 ft - (m 2 + n 2t 2)2 f3(1- f2) = o.

It is important to note that the advantage of using the AFZ ansatz (7.12) is that it is a compatible ansatz [ASVW], meaning that (7.18) gives rise to the critical points of the original Nicole energy (7.9). More precisely, the critical points of (7.17) subject to the boundary condition f(t) ----+ 0 as t ----+ 0, f (t) ----+ 00 as t ----+ 00, give rise to the critical points of the Hopf number (7.15) for the Nicole energy through (7.10)-(7.12) and (7.16). Although (7.18) looks complicated, it has a nontrivial solution f(t) = t when m = n = 1, which implies that the Hopf map is an explicit critical point [ASVW]. Our purpose below is to show that, for any m, n, the equation (7.18) has a finite-energy solution satisfying the stated boundary condition at t = 0 and t = 00. In fact, such a solution also minimizes the energy (7.17). To proceed, we introduce another new variable

(7.19)

9

Then the boundary condition for (7.20)

= arctanf.

f

g(O) = 0,

becomes

g(oo) =

11"

2'

171

EXISTENCE OF FADDEEV KNOTS

so that the Hopf map is given by the choice f ('T7)

= sinh'T7 with m = n = 1, or

(7.14) After some calculation, it can be shown [AFZ, HS] that the Hopf invariant of u designated by (7.10)-(7.13) is given as

Q(u)

(7.15)

= mn.

Besides, with the new variable (7.16) the function f becomes a function of t, which is still denoted by f(t) for simplicity, so that the Nicole energy (7.9) takes the form [ASVW] (7.17)

E(f)

= 3211"

2 {'X! {

io

2 (Jl

t(l +t) (1 + j2)2

1

+ 1 + t2

2] (1 +f2j2)2 ) ~ } dt,

[m2 (2 +n

and the boundary condition (7.13) is reinterpreted in terms of t given in (7.16). The Euler-Lagrange equation of (7.17) is [ASVW] (7.18) t 2(1

+ t 2)(1 + f2)(2t 2[1 + t 2]fl + [m 2 + n 2t 2lJ2)ftt - 4t4(1 + t 2)2 f ft + t 3(1 + 3t2)(1 + t 2)(1 + f2)ff - 2t2(1 + t 2)(m2 + n 2t 2)f3 f? + t 3(m 2 + n 2[1 + 2t2])(1 + f2)f2 ft - (m 2 + n 2t 2)2 f3(1 - f2) = O.

It is important to note that the advantage of using the AFZ ansatz (7.12) is that it is a compatible ansatz [ASVW], meaning that (7.18) gives rise to the critical points of the original Nicole energy (7.9). More precisely, the critical points of (7.17) subject to the boundary condition f(t) --t 0 as t --t 0, f(t) --t 00 as t --t 00, give rise to the critical points of the Hopf number (7.15) for the Nicole energy through (7.10)-(7.12) and (7.16). Although (7.18) looks complicated, it has a nontrivial solution f(t) = t when m = n = 1, which implies that the Hopf map is an explicit critical point [ASVW]. Our purpose below is to show that, for any m, n, the equation (7.18) has a finite-energy solution satisfying the stated boundary condition at t = 0 and t = 00. In fact, such a solution also minimizes the energy (7.17). To proceed, we introduce another new variable

(7.19)

9

Then the boundary condition for (7.20)

= arctanf.

f

g(O) = 0,

becomes

g(oo) =

11"

2'

F. HANG, F. LIN, AND Y. YANG

172

and the energy (7.17) is converted into the simplified form given as

(7.21) I(g)

=

f

{t(l + t')

(gl + 1 ~ t' [r~' + n'L:~::~ g)')~} dt,

where we have suppressed an irrelevant constant factor. It is seen that the Hopf map, defined by g( t) = arctan t, is of finite energy for any integers m, n. We now define the admissible space as

A (7.22)

=

{g(t) I g(t) is absolutely continuous over the interval (0,00), satisfies the boundary condition (7.20), and I (g) < oo},

and consider the associated minimization problem

10 == inf{I(g) I g E A}.

(7.23)

Let {gj} be a minimizing sequence of (7.23). We may assume that I(gj) ::; 10+1 (say) for all j = 1,2, .... We will show that {gj} contains a subsequence which converges in a well-defined way to an element in A, go (say), and I(go) = 10· In fact, collectively writing

P

(7.24)

_ tan 2 g (g) - (1 + tan2 g)2'

we see that P(·) is a periodic even function of period 7r, whose singularities at odd-integer multiples of 7r /2 are removed if we understand P( ~) = limg->~ P(g) = 0, etc. In the sequel, we always observe such a convention for P(·). Therefore, for any g E A, the modified function (7.25) lies in A and satisfies 0 ::; 9 ::; ~ and I (g) ::; I (g). Hence, with suitable modifications if necessary, we may assume that our minimizing sequence {gj} satisfies the same boundedness condition 0 ::; gj ::; ~, j = 1,2, .... On the other hand, near t = 0 and t = 00, we have, respectively,

(7.26)

and (7.27)

2' -

7r 1

1 gj(t)::;

((Xl it 3ds ) ~ (fOO it S-2

2

1

s3 1dg d:'1

3

ds )

t

1

::; 23 C 3 (I(gj))3,

which indicates in particular that {gj} satisfies the boundary condition (7.20) uniformly.

173

EXISTENCE OF FADDEEV KNOTS

The structure of the energy I given in (7.21) shows that for any numbers 0< a < b < 00, the sequence {gj} is bounded in W 1,3(a, b). Using a diagonal subsequence argument, we may assume without loss of generality that {gj} is weakly convergent in W 1,3(a, b) for any 0 < a < b < 00. We use go to denote the so-obtained weak limit of {gj} over the entire interval (0,00). We need to prove that go E A and I(go) = 10· For convenience, we set

(7.28)

J(g,h;a,b)=

lb {t(1+t2)(g;+1~t2[~: +n2]p(h))~}dt,

where g,h are absolultely continuous over (0,00) and P(·) is defined by (7.24). We note that

(7.29)

pi (h) = 2 tan h(1 - tan2 h) (1 + tan2 h)2 ' pl(h) = 0,

h i= odd-integer multiple of

h = odd-integer multiple of

~2 ;

7r

2"'

Hence, pi is bounded. Besides, we may check that J(., h; a, b) is convex for fixed h, a, b. Therefore, we have (7.30) and the weakly lower semicontinuity of J(., go; a, b) implies that (7.31)

J(go,go;a,b):S liminfJ(gj,go;a,b). )-->00

Consequently, we get

2: lim inf J(gj, gj; a, b) (7.32)

)-->00

= )-->00 lim (J(gj, gj; a, b) - J(gj; go; a, b)) + lim inf J(gj, go; a, b) )-->00 2: J(gO, go; a, b).

Letting a ---> 0 and b ---> 00 in (7.32), we see that I(go) = J(90, go; 0, 00) :S 10 as claimed. The fact that go satisfies the boundary condition (7.20) follows from the uniform bounds (7.26) and (7.27). Thus, go EA. The Euler-Lagrange equation of (7.21) is

(7.33)

174

F. HANG, F. LIN, AND Y. YANG

With the help of this equation, we may show that go satisfies 7r (7.34) 0 < go(t) < 2' 0 < t < 00. In fact, if there is a point to > 0 such that go(to) = 0 or g(to) = 7r/2, then the property 0 ~ go(t) ~ 7r /2 implies that gb(to) = O. In view of the uniqueness theorem for the initial value problem of an ordinary differential equation, we infer that go(t) == 0 or go(t) == 7r /2 since g = 0 and g = 7r /2 are two trivial solutions of (7.33). This conclusion contradicts the boundary condition (7.20) enjoyed by the function go obtained earlier. The property (7.34) ensures the invertibilityofthe transformation (7.19) so that we obtain a critical point for the original energy (7.17). We may summarize our study above in the form of the following existence theorem. THEOREM 7.3. For any NEZ, the Nicole energy {7.9} has a finiteenergy critical point n in the topological class Q = N. More precisely, for any m, nEZ, the energy functional (7. 9) has a finite-energy critical point 1/. represented in terms of the toroidal coordinates through the expressions {7.10}-{7.13} so that its Hopf invariant satisfies Q = mn, its associated configuration function f defined in {7.12} is positive-valued with range equal to the full interval (0,00) and minimizes the reduced one-dimensional energy {7.17} in the variable t = sinh 1'}.

As mentioned already, since (7.9) is conform ally invariant, it covers the spherical energy (7.8) when n = 1. Therefore, Theorem 7.3 establishes the existence of a critical point of the energy (7.8) at n = 1 among the topological class Q = N for each NEZ.

8. Generalized Faddeev knot energy In the subsequent sections, we shall study the topologically constrained minimization problem of the generalized Faddeev knot energy in arbitrary (4n - 1) dimensions. The generalization we will be focused on is defined by the energy (8.1 ) where, for convenience, we have absorbed the unimportant coefficient ~ in (2.6) to unity. One may argue that a more natural generalization of the Faddeev knot energy should take the original "quadratic" form so that (8.2) However, at this moment, the energy (8.2) seems to be too hard to approach. Indeed for n 2:: 2 and a map n : 1R4n - 1 ~ s2n with J1R4n-1 {ldnl 2+ In*wS2n 12} < 00, it is not necessary that n*wS2n is a closed form. On the other hand,

EXISTENCE OF FADDEEV KNOTS

175

it is worth mentioning that (8.1) may be viewed as a "natural" extension of the Faddeev energy as well because (i) both energy density terms are quadratic when n = 1, and (ii) with respect to the rescaling of coordinates, x f---+ AX (A > 0), the two energy terms respond with A-1 and A, respectively, as in the classical Faddeev model. As mentioned in the introduction, one of the main difficulties in understanding the Faddeev model is that it is still not known whether an annulus lemma similar to Lemma 3.1 exists or not. In particular we are not able to freely cut and paste maps with finite energy and it is not clear the minimizing problem would break into a finite region one and a problem at the infinity, that is, the minimizing problem does not fit in the frame of classical concentration compactness principle anymore. This difficulty will be bypassed by a decomposition lemma (Lemma 12.1) for an arbitrary map with finite Faddeev energy (in the same spirit as in [LYl] for maps from IR3 to 8 2 ).

9. Some general facts and useful properties and relations In this section we collect some basic facts which will be used frequently later. We will use the algebraic notations in [F, Chapter 1]. Assume n is an integer and 1 :::; k :::; n. Then we denote

(9.1) A (n, k) = {A = (AI, ... , Ak) I Ai'S are integers and 1 :::; Al < ... < Ak :::; n}. If A = (AI, ... , Ak) E A (n, k), k < n, then XE A (n, n - k) is obtained from the complement {l, ... ,n} \ {A1, ... ,Ak}. If X1"",Xn are the coordinates on IRrt, then we write

dX A = dX A1 If, for every A E A (n, k),

WA

1\

dX A2

1\ ... /\

dx Ak .

is a distribution on an open subset of IRrt, then

we call W

=

L

wAdx A

AEA(n,k)

a (k-form) distribution. Occasionally, we will need to verify some weak differential identities. It is convenient to have the following basic rule. LEMMA

IRn, a E Lf~c (0),

9.1. Assume that 1 :::; PI,, P2, P3 :::;

00,

0 is an open subset of

Lf~c (0) is a k-form, j3 E Lf~c (0) is another form such that da E

j3 ELf! (0), a E distribution, we have

Lf;c (0)

d (a 1\ (3)

and dj3 E

Lf~c (0). Then, in sense of

= da 1\ j3 + (_l)k a 1\ dj3.

Here p~ = P~-=-l is the conjugate power of Pl· Similar for P2 and Ps·

F. HANG, F. LIN, AND Y. YANG

176

First assume 1 ::; Pl,P2,P3 < 00. By mollifying arguments we may find a sequence of smooth k- forms ai E Coo (0) such that ai -> a in Lf~c (0) and Lf;c (0), dai -> da in Lf;c (0). Taking a limit in the equation PROOF.

d (ai A ,8) = dai A ,8 + (-1 ) k ai A d,6, the lemma follows. For the remaining cases, without loss of generality, we assume PI = 00, 1 ::; P2,P3 < 00. Then we may find a sequence of smooth k-forms ai E Coo (0) such that ai ~ a in L~c (0), ai -> a in Lf;c (0) and dai ----) da in Lf;c (0). Then ai A,6 -> a A ,6 in sense of distribution and dai A,6 ----) da A,6, ai A d,6 -> a A d,6 in Lfuc(O). The same limit process as above implies the lemma. 0 For a smooth map u, the exterior differential d commutes with the pullback operator u*. It remains true under suitable integrability condition on the derivatives of u when it is only weakly differentiable. 9.2. Assume that 0 C ~n is open, a is a smooth k-form on ~l with compact support, u E Wl~,:+l (0, ~l). Then in sense of distribution LEMMA

du*a = u*da. PROOF.

we may find Ui E Coo (0, ~l) such that Ui

->

U in Wl~+l (0, ~l)

.ti.!.

and Ui -> U a.e. It follows that uia -> u*a in Llo~ (0) and uida -> u*da in Lfoc (0). Taking limit in duia = uida, we arrive at the conclusion. 0 The conclusion of the above lemma can be strengthened when we know that the map is bi-Lipschitz.

9.3. Assume that 0 1 , O2 are open subsets in ~n, ¢ : 0 1 -> O2 is a bi-Lipschitz map, and a E Lfoc (0 2 ) is a k-form such that da E Lfoc (0 2 ). LEMMA

Then d¢*a

= ¢*da.

PROOF. We may find a sequence of smooth k-forms ai E ergo (0 2 ) such that ai ----) a in Lfuc (0 2 ) and dai -> da in Lfuc (0 2 ), Hence ¢*ai ----) ¢*a in L[oc (0 1 ) and ¢*dai -> ¢*da in Lfoc (Od. It follows from Lemma 9.2 that d¢*ai = ¢*dai. Letting i -> 00, we obtain d¢*a = ¢*da. 0

Later on we will need to verify weak differential identities for maps with mixed differentiability on different domains. For that purpose we state the following smoothing lemma. Bi1 - 1 x (-1, 1), f : B~-l -> (-1,1) function. Assume that 1 ::; PI, ql < 00, 1 ::; P2, q2 < 00, a k-form such that da E Lf;c (0). For x E 0, we write x = (x', Denote 0_ = {x EO: xn < f (x')}. If LEMMA

9.4. Let 0

=

alrL E Li~c

({ x E 0 I

Xn ::;

f (x')})

be a continuous E Lf~c (0) is a

x n ), x' E ~n-l.

177

EXISTENCE OF FADDEEV KNOTS

and OOIB-1 E Li;c ({ x EO: xn :::; f (X')}) , then there exists a sequence of smooth k-forms ai on 0 such that ai

--+

a in Lf~c (0) ,

OOi

--+

00 in Lf! (0) ,

ailr2-

--+

alrL

OOi lr2-

--+

OOl fL

in Lf~c ({x E 0 I Xn :::; f (x')}) , in Li;c ({ x E 0 I Xn :::; f (x')}) .

If anyone of the Pl,P2, q1, q2 is infinite, then the conclusion remains true if we replace the strong convergence by the weak * convergence in L/;c' PROOF. For 6 > 0 small we denote io : x t--t (x', Xn - 6), then i;5a is 1 x (-1 + 6,1). We may choose 0 < c < 6 small enough such defined on that for y' E B~.:::-io' we have f(x ' ) + 6 > f(y') + c for all x' E B~-1 (y'). For P E COO (IRn, 1R), p(x) = 0 for x E lR n\B1 and Int n p(x)dx = 1, write Pc(x) = c~ p( ~). Let f30 = Pc * i;5a be defined on B~.:::-io x (-1 + 36, 1 - 36). Choose a ¢>o E crgo(B~.:::-io x (-1 +36,1- 36)) with ¢>o = 1 on B~.:::-lo x (-1 +46,1-46). Then ao = ¢>o . f30 satisfies all the requirements as 6 --+ 0+. 0

Br-

Based on the above smoothing lemma, we can derive another differential identity.

9.5. Assume that 0 is an open subset in lR n , 1: C 0 is a continuous hypersurface which separates 0 into 0 1 and O2 i.e. 0\1: = 0 1 U02, 1 :::; Pl,P2,P3,Ql,q2,q3:::; 00, a E Lf~c(O) is a k-form with 00 E Lf!(O), a E Lf;c (0) and LEMMA

aln2 E Li~c (02 U 1:) ,

001 02

al 02 E Li~c (02 U 1:) .

ELi! (02 U 1:) ,

Let f3 E L[oc (0) be another form with df3 E L[oc (0) and p'

1

E Lz~c (0 1 U 1:) ,

f310 2

E Lz~c (02 U 1:) ,

f310

q'

f310

p'

1

E Lz;c (0 1 U 1:) ,

f310 2

E Lz;c (02 U 1:) ,

q'

p'

df31 01 E Lz;c (0 1 U 1:) . q'

df310 2 E Lz;c (02 U 1:) .

Then, in sense of distribution, d (a 1\ (3) = 00 1\ f3 + (_1)k a PROOF.

Q2, Q3

1\

df3.

Without loss of generality, we may assume that 1 :::; PI, P2, P3, Ql,

< 00. By localization and rotation, we may assume that 0 is a cylinder

and 1: is the graph of a continuous function. It follows from Lemma 9.3 that we can find a sequence of smooth k-forms ai on 0 such that

ai

--+

a in Lf~c (0) ,

ai

--+

a in Li~c (0 2 U 1:) ,

ai

--+

a in Li~c (02 U 1:) .

dai

--+

da in Lf;c (0) ,

dai

--+

ai

--+

a in Lf! (0) ,

da in Li! (02 U 1:) ,

F. HANG, F. LIN, AND Y. YANG

178

Taking limit in d (O:i 1\ (3)

= dO:i 1\(3+( _l)k O:i I\d(3, we obtain the conclusion. D

For later purposes, we review a little bit of the Hodge theory on domains ([T, Section 9 of Chapter 5]). Let 0 c lRn be a bounded open subset with smooth boundary E = a~, v be the outer normal direction and i : E ----+ 0 be the natural put-in map. For 1 < p < 00,

w;iP (0) =

{o: E W 1,p (0) 10: is a form with i*o: = O},

HR (0) = {o:

E

Coo (0)

10: is a form with do:

= 0,

d*o:

= 0,

i*o:

= O}.

Here the subscript R refers to the imposed relative boundary condition: i*o: = O. That is, the tangential part of 0: on the boundary E is zero. Then we have

LP (0) = dwh'P (0) EB d*wh'P (0) EB HR (IT) and HR (0) ~ H* (0, a~, lR), the real singular cohomology group. More precisely, if w E LP (0), then w

= do: + d* (3 + 1',

with 0:, (3 E wh'P (0), I' E HR (0) and 11001Iwl.P(0), 11(3llw1,p(0) :::; c (p, 0) Ilwllv)(o). If we know (w, d*
10

in Hence T

E

10 (d* (3, d
V' (0),

hence

T

=

(w, d*
=

(d* (3, d
1 ' 0 for every

d0: 1 + d(31

in

in

+ 1'1

(d* (3, T) dx

for 0:1,(31 E

=

in

wh'P' (0)

(d* (3, d(31) dx

=

P~1' For every

and 1'1 E HR (0),

= O.

This implies d* (3 = O. One of the ingredients in proving the crucial decomposition lemma (Lemma 12.1) is the construction of suitable functions on annulus which connects the original map to constant maps. The next two lemmas are about the existence of such auxiliary functions. First, we derive some basic inequalities for the harmonic extension of a function on the boundary of a domain.

9.6. Let 0 c lRn be a bounded open subset with smooth boundary E, 1 < p < 00, f E W 1 ,p (E), and u the harmonic extension of f to O. Then LEMMA

Ilull W 12!L < c (p, 0) IlfIIWl,P(:E)' 'n=T(0) -

EXISTENCE OF FADDEEV KNOTS

179

We need the following basic fact (compare with [HWYl, Proposition 2.1]): Assume that p E C~ (IRn-l), 9 is a function on IR n- 1 , and PROOF.

(Tg) (x) =

1n-l

for x E IR+., x = (x', x n), 1 < p <

00.

p (~) 9 (x' -

xn~) d~

Then

IITgl1 L1f~!T (JR+) :S c (n, p, p) IIgIILP(JRn-l) ,

IIV'Tgll L~ (JR+) :S c (n,p, p) IIV'gIILP(JRn-l). To prove the two inequalities, we claim that

IITgIIL~ (JR+) :S c (n, p) Ilgll£1(JRn-l). If the claim is true, then the first inequality follows from the Marcinkiewicz interpolation theorem (see [SW, p197]) and the basic fact that IIT91ILoo(JR+)

:S Ilpll£1(JRn-l)lIgIlV:>O(JRn-l). To prove the claim, assume that IlglI£1(JRn-l) = l. Then ITg (x) I :S c(~~) and Xn

f

} XEJR+ ,O<Xn
ITg (x)1 dx :S c (n, p) a

for a > O. Hence, for t > 0,

IITgl > tl =

I{x E IR+: 0 <

Xn

< c(n,p)Cn~l, ITg(x)1 > t}1

:S~ f l

t }o<xn
ITgl(x)dx:Sc(n,p)Cn~l.

Thus the claim follows. Next we observe that, for 1 :S i :S n - 1,

and

n-l On (Tg) (x) = -

t; 1n-l

p (~) ~jOjg (x' -

Hence it follows that, for 1 < p <

00,

xn~) df

IIV'Tgll ~ ( ) :S c (n,p, p) Ln- JR+

IIV'g IILP(JRn-l). By decomposition of unity, flattening the boundary and applying the 1 ....!!L above fact, we may find some v E W 'n-l (0) with Ivllwl'~(n) :S c (p, 0)

IIfllw1,p(L:) and vlL: = f. Then ~ (u - v) follows from elliptic estimate that

=

-~v and (u -

v)lan = O. It

F. HANG, F. LIN, AND Y. YANG

180

Hence c (p, 0) Ilvll W1..!!£... c (p, 0) IlfIIW 1 ,p(E)' Il ull W1..!!£... 'n-1 (0) < 'n-l (0) < -

o The next lemma gives us the existence of suitable auxiliary functions with energy control. LEMMA 9.7. Assume that n ;:::: 3, f : 8B? - t SI-l C jRI such that JaB 1 Idfl n - 1 dS ::; c(l, n) small,then there exists au E w 1 ,n (B2\Bl' SI-l) such that ula B1 = f, ulaB2 = const and

IIVUIILn(B2\Bd ::; c (l, n) IIdfllu.-1(aBI). PROOF. Set faBl = 1811 1 JaBl fdS. By the Poincare inequality, we have

[ If - faBll dS ::; c (l, n) IldfIILn-l(aB]) ::; c (l, n) cn~l. JaBl 1

Hence IlfaBll-ll::; c(l,n)c n - l . We can solve the Dirichlet problem

b..v = 0 on B2\Bl, { vlaB1 = f, vlaB2 = faB!' Then b.. (v - faB l ) = 0 on B2\Bl, (v - faB1)laBl It follows from Lemma 9.6 that

= f, (v - fa Bl)l aB2 = O. 1

IIV - faBJWl,n(B2\BJ)::; c(l,n) IIf - fa BlIIW1.n-l(aBd::; c(l,n)c n- 1 • It follows that, for 8 > 0 small, 1

IIV - faBJLOO(B2\B 1+6)

::;

c (n, l, 8)c n -

l .

For x E B;l \B1 " E 8B1 U8B2, we let P (x, ') be the Poisson kernel. For 2

8B2, define f (') = faB]. Then v (x) = J~BluaB2 P (x, ') f (') dS (,). Set = I~I' r = lxi-I. Then classical estimate for the Poisson kernel gives (see [HWY2, lemma 2.2 and section 5])

, E

'0

For k ;:::: I with kr ::;

!, we write

EXISTENCE OF FADDEEV KNOTS

181

Using the Poincare inequality, we see that

laB

1

n ~ (e )1 kr

0

r

J aBlnBkr(~O)

fkr,~ol dS ~ c (l, n) IldfIILn-l(aBlnBkr(~o))

If -

~

1

c(l,n)En-l.

1

Hence II!kr,~ol-11 ~ c(l,n)En-l. On the other hand,

Iv(x)-!kr,~ol=1 JaBIUaB r p(x,e)(f(e)-!kr,~o)dS(e)1 ~

2

!

(oBI \Bkr(~O))UaB2

+

r

J aBlnBkr(~O)

P (x, e) If (e) -

P (x, e) If (e) -

!kr,~o IdS (e)

fkr,~ol dS (e)

~ c (I, n) (r + ~) + cr ~~~ JraBlnBkr(~O) If (e) - !kr,~ol dS (e) c(l,n)

~ -k-

+ c (l, n ) k n- 1 1df ILn-l (aBlnBkr(~O))

~ c (I, n) (~ + kn - 1E n~ Hence Ilv (x)l- 11

= l~t~~l'

.

~ c (l, n) (~+ kn-1En~1) .

By fixing k large, r small, and then Let u (x)

1)

E

small, we have Illvl-lIILoo(B2\BIl ~

Then u satisfies all the requirements of the lemma.

!. 0

To prove that the Hopf-Whitehead invariant Q (u) must be an integer for any map u with finite Faddeev energy, we need to show that the invariant of a suitable weakly differentiable map must be an integer. For this purpose, we recall some ideas from [Sv, EM]. PROPOSITION 9.8. ([Sv, Section 2])Assume that M n and N n are both smoothly oriented Riemannian manifolds, u E Wl~; (M n , Nn) such that and Ju = Idet dul E £1 (Mn). Then there exists a measure zero subset E of M n such that the function

L

dCu,y) =

sgn(detdu(x))

xEu- 1 (y)\E

is integrable on N n and for every f E £00 (Nn),

r

JMn

u* (fWNn)

= =

r

JMn

f (u (x)) det du (x) dpMn (x)

r f (y) . d(u, y) dPNn (y).

JNn

F. HANG, F. LIN, AND

182

Here WN" is the volume form on N n , with the Riemannian metric.

/-lMn

Y.

YANG

is the measure on M n associated

Proposition 9.8 follows from the Lusin type theorems and the usual coarea formula for Lipschitz functions. The idea of [Sv, EM] to show that d( u, y) is independent of y is to show f M" u* (fw N") = 0 whenever fNn fWNl1 = O. To achieve that, the following basic fact is useful. LEMMA 9.9. Assume that n ~ 2, 1 ::::: p ::::: n~l' or n = 1 but 1 ::::: p < and a E LP (l~n) is a (n - I)-form with da ELI (JR n ). Then fIR" cia = O.

00,

PROOF. By a mollifying function argument, we may assume that a E Coo (JR n ). Fix some ¢ E C~ (JRr!) such that ¢IB 1 / 2 = 1 and ¢IIRTI\B 1 = O. For

R> 0, we write ¢R (x) = ¢ (-n)' Then 0= { d (¢Ra) = { d¢R!\ a i IR" iIR"

+ {

iIR n

¢Rda.

Note that

1

d¢R!\ al :::::

{

iIR

c(n) R

TI

lal : : : c(n,p)

{ iBR\BR/2

as R ---+ 00. Hence, by letting R fIR" da = O.

lalP)

( (

*Rn-1-~

---+

0

iBR\BR/2 ---+

00

in the first equation, we get D

In Lemma 9.9, the requirement p ::::: n~l is crucial. Indeed, for n ~ 2, let r be the fundamental solution of the Laplacian, ¢ E C~ (JRr!) with fIR n ¢ (x) dx = 1, and let

Then for any q fIR n da = 1.

>

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

)
,y -

I

x-y 14n-

2

2n-1 4n-l (:JR4n- 1) . It follows from the Because Ju E L-nn L 2 , we see Ju E L ~ classical Hardy-Littlewood-Sobolev inequality ([St]) that Ju E L1 (:JR8n - 2 ), that is U*WS2n xS2n xS4n-2 ELI. 0

To continue, note that for x, y satisfies

c~ =~I) *

Wsm-l

=

E :JR m ,

Ix -\Im f

the map I~::::~I

L

t

::JRm X :JRm ---+ sm-1

(_l)m-k sgn

k=O 'xEA(m,k) i=O

X (X'xi -

y,X;) (dx,X)

la

XAi

1\

dyX'

(A,

~

EXISTENCE OF FADDEEV KNOTS

185

Indeed, under the spherical coordinate, the metric and volume forms of ]Rm and sm-l are given by

l~i,j~m-l

respectively, where B (0)

= det (bii (0)).

Hence

It follows that

Developing the product out we get the needed formula. PROOF CONTINUED.

We may write U*WS2n

=

L 1>. (x) dxA. A

Here A runs over elements in A (4n - 1, 2n), and the same for IL, v we use below. Then U*WS2nxS2nxS4n-2

= -

L

fA (x) dX A /\

A

L ip. (y) dyp. /\ Ix - yl1

4n _ 1

p.

2n

X

L L sgn (v, v) (XII; II

1

=-

YIIJ . (dx lI ) lax,,; /\ dyv

i=O

Ix - yl

2n

4n-l

L L L iA (x) ip. (y) dxA/\ dyp. A

p.

i=O

/\ sgn (IL, J1) (XP.i - yp.J . (dxp.) lax!'i /\ dY/i

186

F. HANG, F. LIN, AND Y. YANG

4n-1

1 = -Ix _

4n - 1

1

Y

L). L L I>. (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} <

00,

and that a is a smooth 2n-Jorm on s2n. Then du*a = O. If 2 ::; p < (2n~!~~~)I), (3 E V (JR 4n - 1 ) is a (2n - I)-form such that d{3 = u*a, then

r a

J~4n-l

u * 1\ {3

= Q (u)

(r a)

2

JS2n

2n-l

2

PROOF. Claim 10.2 implies that du*a = O. Since u*a E L-n- n L , it follows that dd*(r * u*a) = u*a and d*(r * u*a) E LP(JR4n-l). Hence we may find (3 E V with d{3 = u*a. Using (2n-l)(4n-l) < 2n-l we get n(4n-3) n-l ' u*a 1\ (3 E L 1(JR4n-l). We claim that flR 4n - 1 u*a 1\ {3 does not depend on the choice of (3. Indeed, if 13 E V satisfies d13 = u*a, then d({3 - 13) = O. Hence {3 - 13 = d'Y for some (2n - 1 )-form 'Y E LP· (JR4n - 1 ), where ;. = ~ - 4n1_l' Indeed we may choose 'Y = d*(r* ({3 - 13)). Note that u*al\'Y ELI. It follows from Lemma 9.1 that d (u*a 1\ 'Y)

= u*a 1\ ((3 - 13) .

Using Lemma 9.9 we see

r

JlR 4n - 1

u*al\({3-13) =0.

EXISTENCE OF FADDEEV KNOTS

191

The claim follows. Next we look at the case J8 2n 0: = O. In this case we may find a smooth (2n - l)-form "( on s2n such that 0: = d"(. It follows from Lemma 9.2 and the fact u E w 1,4n-2 that u*o: = u*d"( = du*"(. Note that u*"( E L2. Hence we may choose f3 = u*"(. It follows that

r

J~4n-l

u*O:I\f3=

r

J~4n-l

u*o:l\u*"(=O=Q(u)

(rJ82n 0:)2

Finally, if J82n 0: f= 0, by rescaling we may assume J82n 0: = W82n + d"( for some smooth (2n - l)-form "(. Hence

with T Hence

= d* (f * U*W82n). Let f3 = T + u*"(. Then f3

r

J~4n-l

0:

=

E L2 and df3

r U*o:I\T+U*o:I\U*"( = r U*W82n T + r du*"( J~4n-l J~4n-l

u*O:I\f3=

1s2n I. Then

= u*o:.

J~4n-l

1\

1\

Note that because u*"( E L2, T E L2, du*"( = u*d"( E L2, dT 4n-l 2 L 2;;:- n L , we see that d (u*"( 1\ T)

= du*"( 1\ T -

u*"( 1\ dT

= du*"( 1\ T -

u*"( 1\ U*W82n

T.

= U*W82n

E

= du*"( 1\ T. Hence J~4n-l du*"( 1\ T = J~4n-l d (u*"( 1\ T)

r

J~4n-l

u*o: 1\ f3

=

r

J~4n-l

U*W82n

1\ T =

= o. It follows that

Is 2nl 2 Q (u)

= Q (u)

(rJ 82n 0:)2 o

Using Proposition 10.6 we easily derive the following expected corollary. 10.7. For every v E Coo (s4n-1, S2n), let u = V01l"~1, where 11"0 : s4n-1\ {n} ~ ffi.4n-1 is the stereographic projection with respect to the north pole n. Then J~4n-l Idul 4n - 2 + IU*W82n 12 < 00 and COROLLARY

Q (u)

=

Q (v).

Here Q (v) is defined as in [BT, p228] as follows: Ifv*w82n smooth (2n - l)-form 'fl on s4n-1, then Q (v) =

r

~ V*W82n 1\ 'fl. IS2nl J8 4n - 1

= d'fl

for some

F. HANG, F. LIN, AND Y. YANG

192

PROOF. Indeed since l'Vu (x)1 ~ (lxl:1)2' we see that

flR

4n - 1

Idul 4n - 2 +

IU*WS2n 12 < 00. On the other hand, V*WS2n = d7J implies U*Ws2n

= (-1)* 7l"n V*Ws2n = d( 7l"n-1)* 7J = d~ T.

Here T = (7l"~1)*7J. Then ITI ~ (lxl+~)4n Using Proposition 10.6, we see that

Q (v)

= _1-2 { IS 2nl

J

V*WS2n /\ 7J

2'

It follows that T E L2 (lR 4n - 1).

= _1-2 {

JlR

IS 2nl

S4n-l

U*WS2n /\ T = Q (u).

4n - 1

o When n i- 1, 2, 4, v E Coo (S4n-1, S2n), classical algebraic topology tells us Q (v) can only be an even integer (see [Hu, Corollary 3.6 on p214 and Theorem 4.3 on p215]). It is natural to make the following CONJECTURE 1. Under the assumption of Theorem 10.1, Q (u) must be an even integer when n i- 1,2,4.

10.1. Further discussions on the Hopf-Whitehead invariant. In the proof of the crucial decomposition lemma (Lemma 12.1), we will see that some maps to be constructed have finite Faddeev energy on one piece of the domain and finite conformal dimensional energy on other piece of the domain. It is necessary to show such kind of maps still have integer Hopf invariant. Indeed we have the following analogue of Theorem 10.1.

Wl!;

THEOREM 10.8. Assume that u E (lR4n - 1, s2n) and that n c lR 4n - 1 is a bounded open subset with continuous boundary such that

{ Idul 4n - 2 + IU*WS2n 12 +

In

( JlR

4n - 1

\0.

Idul 4n - 1 <

00,

where WS2n is the volume form on s2n. Then du*wS2n = O. Let 1

r(x)- (4n-3) IS4n- 21IxI 4n - 3 '

T=d*(r*U*WS2n),

where d* is the L2-dual of d, Is4n- 2 1is the area of s4n-2. Then T E 4n-l L2n-l (lR 4n - 1), dT = U*WS2n, d*T = O. The generalized Hopfinvariant Q(u)

=~ {

IS 2nl A~.4n-l

U*WS2n /\T

is well defined and equal to an integer.

Again the first step is to show that du*wS2n = O. CLAIM 10.9. For any smooth 2n-form a on s2n, we have du*a = O.

EXISTENCE OF FADDEEV KNOTS

PROOF.

fo, ... , hn that

E

193

By linearity we may assume a = fodb /\ ... /\ dhn, where C~ (lR2n+l, lR). Because u E Wz::n , it follows from Lemma 9.2

du* (bdh /\ ... /\ dhn) = u* (db /\ ... /\ dhn). Hence

du* (db /\ ... /\ dhn) = o. Note that foou E L oo (lR 4n - 1 ), d(foou) E L~-2(lR4n-l), d(foou) L 4n-l(lR4n - 1\0), u*(db/\···/\dhn) E L2(0), u*(db/\···/\dhn) L 4~;;1 (lR 4n - 1 \0), and

E E

du* (db /\ ... /\ dhn) E L OO (lR 4n - 1 ). It follows from Lemma 9.5 that

du*a

= d (fo 0 U· u* (db /\ ... /\ dhn)) = d

(fo

0

u) /\ u* (db /\ ... /\ dhn)

=u*da=O.

o To continue we observe that

(U*WS2n )10 E L 2n,,-1 (0) n L2 (0) ,

(U*wS2n)llRn\0 E L 4~;;1 (lRn\o).

4n-1 ( 4 Hence U*WS2n E L""2n lR n-l). Let T = d* (r * U*WS2n). Then 4n-1 4 1 T E L2n-1 (lR n- ), dT = u* (WS2n). In particular,

4n-1 is well defined. Because Ju = IU*WS2n 1 E L2n-1, the proofs of Claim 10.3 and 10.4 remain valid with minor modifications (e.g., replacing Lemma 9.1 by Lemma 9.5 when necessary). Similar to Proposition 10.6, we have 1 ,1 (lR4n - 1 s2n) 0 C lR4n - 1 zs 10 . 10 . Assume that u E w:Z oe' , a bounded open subset with continuous boundary such that

PROPOSITION

f Idul 4n - 2 + IU*WS2nI2

Jo

+

f

JlR4n - \0

Idul 4n - 1 <

00,

1

and that a is a smooth 2n-form on s2n. Then du*a 4n-1 L2n-1 (lR4n-l) such that d(3 = u*a, then for n 2: 2 we have f

O. If (3 E

u*a/\(3=Q(u)(f a)2 JS2n For n = 1, the equality remains true if, in addition, u is constant near infinity.

JlR4n- 1

F. HANG, F. LIN, AND Y. YANG

194

This follows from a similar argument as that in the proof of Proposition 10.6.

11. Energy growth estimate In this section we will describe some basic rules concerning the Hopf invariant for maps with finite Faddeev energy and the sublinear energy growth law. Note that such kind of sublinear growth is a special case of results derived in [LY5]. We provide the arguments here to facilitate the further discussions in Section 12 and Section 13. Recall for U E Wl~; (lR 4n - 1 , S2n) , we denote

Let

LEMMA

11.1. For any

U

E

X, 4n

IQ(u)l::; c(n)E(U)4n-1. PROOF. Indeed,

Q(u) =

~ r U*WS2n IS 2nl JlR4n - 1

I\T

with T = d* (f * U*WS2n). It follows that

IQ (u)1

::; c (n) l4n-l lu*wS2n I·ITI ::; c (n) IIU*WS2n 11£2 IITIIL2 ::; c (n) IIU*WS2n 11£2 IIU*WS2n II 2~4Yl) L

n

4n-2

1

::; c(n) IIU*WS2nllL211U*WS2nllL4~-1 IIU*WS2nll4~~~1 L-n~ 2n(4n-2) ::; C (n) IIU*WS2n 111~-1 IIVuII L4'!."-21 ::; C (n)

4n

E (U) 4n-l

.

o

For NEil, denote

EN = inf{E(u): U E X,Q(u)

= N}.

The above lemma gives a lower bound for EN. The upper bound may be derived by choosing suitable test functions.

EXISTENCE OF FADDEEV KNOTS

LEMMA

11.2. For n

= 1,2,4, we have 4n-l

EN ~ c(n) INI~

For n

=1=

195

for all integers N.

1,2,4, we have 4n-l

EN ~ c (n) INI~

for all even integers N.

We start with some basic facts. • If U E X, ¢ : lR4n - 1 -+ lR4n - 1 is an orthogonal transformation, then U 0 ¢ E X and Q (u 0 ¢) = sgn (det ¢) . Q (u). Indeed, we have (u

Here T

0

= ¢*U*WS2n = ¢*dT = d¢*T.

¢)* WS2n

= d* (r * U*WS2n) Q (u

0

¢)

E L2. Hence

r _1-2 r 18

=~ 182n l JlR =

2n l

=

¢* (U*WS2n

l

JlR 4n -

sgn (det ¢)

18

¢*U*WS2n 1\ ¢*T

4n - 1

2n l

2

lR4n -

*

1

.

• If u E X, 'ljJ E Coo (8 211.,8 211.), then 'ljJ (deg'ljJ? Q (u) .. Indeed, denote Q: = 'ljJ*WS2n. Then 0

T)

U WS2n 1\ T

= sgn (det ¢) . Q (u)

('ljJ

1\

1

0

u E X and Q('ljJou)

=

u)* Ws2n = u*Q: = dT

for some T E L2. It follows from Proposition 10.6 that Q('ljJou) =

r

~ U*'ljJ*WS2n I\T 18 2n JS2n l

=

C8;nl fs2n 'ljJ*WS2n)

= (deg'ljJ)2 Q

e

2

Q (u)

(u).

• Assume X1,X2 E lR 4n - 1, E 8 211., rl,r2 > 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" <

(m + 1)2.

Let k = I¥- - m 2. Then 0 :S k :S 2m. By scaling and packing we can find a 'l/J E Coo (S2n, S2n) such that 'l/J (n) = n, deg'l/J = m and Id'l/JI :S c (n) m2~. Let

'l/J (uo (m-2~x)) , for Ixl :S m2~ + 1,

U (x) =

Uo (x n,

(m2~

+ 1+

1:Sj:Sk otherwise,

4j) el) , for Ix - (m2~ + 1 + 4j) ell :S 1,

197

EXISTENCE OF FADDEEV KNOTS

where e1 = (1,0, ... ,0) E IR 4n - 1 . Then Q (v) since Idul ::; c (n), we see that

= 2m2 + 2k = N.

Moreover

4n-l

E(u)::; c(n)m2n +c(n)k 4n-l

::; c(n)m2n +c(n)m 4n-l

::; C

(n) m 2 n

4n-l

::;

c (n) N--:rn.

o

12. The decomposition lemma In this section, we prove the crucial decomposition lemma. Roughly speaking, the lemma says that we may break the domain space into infinitely many blocks, on the boundary of each block the map is almost constant, and hence, we can assign a Hopf-Whitehead invariant for it. By collecting nonzero "degree" blocks suitably, we may achieve a good understanding of the minimizing sequence of maps for the Faddeev energy (Theorem 13.1). Note that such a decomposition lemma for maps from IR3 to 8 2 was proven in [LY1] using the lifting through the Hopf fibration 8 3 --t 8 2 . In higher dimensions, we will use the Hodge decomposition of differential forms in place of the lifting. Let us introduce some notation. For x E IRm we write

For R > 0, Y E IRm

,

QR(Y) = {x E IRm : Ix -Yloo::; R}. QR

= QR (0).

Denote Zm = {x E IRm :

Xi

E Z for 1 ::; i ::; m}

as the lattice of all integer points. Then IRm

U QR(e)·

=

~E2RZm

Here 2RZm means the scaling of the lattice boundaries of these cubes is given by ~R

= {x

LEMMA

E IRm

: Xi

= (2j + 1) R

= {

(lduI4n-2

E

Wl~; (IR4n -1, 8 2n ) with

+ IU*WS2n 12) dx ::; A< 00.

J1R4n-1

Let T

by factor 2R. The union of

for some 1 ::; i ::; m and integer j}.

12.1. Assume u EX. That is, u E (u)

zm

= d:" (f

* U*WS2n) .

F. HANG, F. LIN, AND Y. YANG

198

Here r is the fundamental solution of -Do on ]R411,-1. Then for every c > 0, there exists R = R (n, c, A) > 0, y E QR/4 and "'t; E Z for every ~ E 2RZ411, - 1 such that

1

~ 22 1 6 U *WS2n t;E2RZ4n-1 18 11,1 QR(t;)+y In particular, except for finitely many ~ 's, "'t;

L

"'t;

1\

T - "'t;

:s: c.

= 0 and, when c <

1,

= Q (u).

t;E2RZ4n-1 PROOF. Since Ilu*WS2nIIL2nn-l(lR4n_l)

:s: c(n) Ildulli~n-2(lR4n-l)'

it follows

from Holder's inequality that

Ilu*WS2n II L""""2n 4n-1 (lR4n-l) -< c (n, A). Hence

4n-1 (lR4n- 1) + IIDTII L""""2n 4n-1 (lR4n-l) -< c (n , A). II TII L 2n=I It follows from the Fubini type estimate (Section 3 of [HL]) that we may find some y E QR/4 such that U

II;R+Y

1,411,-2 ( " ) E W loe LJR + y,

1 4n-l TII;R+Y E Wlo~ 2n (~R

+ y) ,

and

f

(lduI411,-2

}I;R+Y

:s: :s:

c(n)

f

R }lR4n-1

+ ITI ~~=t + IDTI4~;;1 ) (lduI411,-2

d8

+ ITI~~=~ + IDTI4~;;I) dx

c(n,A)

R

.

By translation we may assume y = O. Pick up a cube QR (~) with ~ E 2Rz 4n-1. Without loss of generality, we may assume ~ = O. We have

f

}aQR

(lduI4n-2

+ ITI ~~=t + IDTI4~;;I) d8 :s:

12.2. There exists U1 E u11aQR = ulaQR' u11aQ2R = const and CLAIM

IlduIilL4n-l(Q2R\QR) Here we set

c (n, A) .

R

w 1,4n-1 (Q2R\QR' 8 211,)

:s: c(n) Il du IIL4n-2(aQR)'

such that

199

EXISTENCE OF FADDEEV KNOTS

Indeed, consider the map
=

IxllxL.

Then
r

Idvl4n-2 dS ::; c (n, A).

R

J8BR

It follows from Lemma 9.7 and scaling that, when R = R (n, A) is large enough, there exists a VI E w I ,4n-I (B2R\BR' S2n) with vII8BR = v, vl18B2R = canst such that

IldvIIIL4n-1(B 2R \BR) ::; c (n) IldvIIL4n-2(8BR) ::; c (n) IlduIIL4n-2(8QR)· Let U1

= VI 0
satisfies all the requirements in Claim 12.2. 4n-l

CLAIM 12.3. There exists some 7"1 E L2n-l (lR4n-I) with 7"II QR = 7",

7"IllR4n-1\Q2R = 0, d7"l = UiWS2n and

Indeed we may write

f>. (x) dx,\.

7"= '\EA( 4n-1,2n-1)

Then we define

!,\(X),

f-,\ (x ) -- {

2R-lxl oo f R

,\

(JJ:£) Ixl"" '

0, and 7"2

=

L

x E QR, x E Q2R\QR, x E lR 4n - I \Q2R,

]; (x) dx,\.

'\EA( 4n-1 ,2n-1)

It follows that 7"2 E L ~~=~ (lR 4n - 1) , D7"2 E L 4~;:1 (lR 4n - I ), and

F. HANG, F. LIN, AND Y. YANG

200

O. Let v (x)

Note that (dT2)IQR = U*WS2n and (dT2)11R4n-1\Q2R Ul (¢(x)). Then v E Wl~; (lR4n-I,S2n),

r

(ldvI4n-2

+ IV*WS2n 12) dx +

lBR

r

Idvl 4n - 1 dx <

00,

lB2R\BR

and VI1R4n-I\B2R = canst. It follows from Theorem 10.8 that dV*WS2n = O. 4n-l Let Tf2 = ¢*T2' It follows from Lemma 9.3 that Tf2 E L2n-l (lR4n-l) and dTf2 = ¢*dT2' In particular, (dTf2)IB R = V*WS2n, (dTf2)11R4n-I\B 2R = 0, and

Note that d (V*WS2n - dTf2) = O. Hence for any forms

r

1B2R\BR

(V*WS2n - dTf2' d*
r

l1R 4n - 1 =0.

C~

(lRn),

(V*WS2n - dTf2, d*
U sing the fact

H 2n (B2R\BR, aBR u aB2R, lR) ~ H 2n - 1 (B 2R\BR, lR) ~ H 2n - 1 (s4n-2, lR)

= 0,

it follows from the Hodge theory that we may find some Tf E W (B2R\BR) such that

1

4n-l

'""2n

and

Here iR : aBR ~ lR 4n - 1 and i2R : aB2R ~ lR 4n - 1 are the identity maps. Let Tf = { Tf, 0,

on B 2R\BR, on BR U (lR 4n - 1\BR ) .

201

EXISTENCE OF FADDEEV KNOTS

Then it follows that for any form cp E C~ (lRn ),

r

"I 1\ dcp =

r r r

"I 1\ dcp

J B2R\BR

JJR4n-l

=

-d ("I 1\ cp)

+ drJ 1\ cp

J B2R\BR

=

(V*WS2n

-drJ2)l\cp-

JB2R\BR

+

r

JaBR

=

r

r

JaB2R

i 2R {rJI\CP)

i'R ("I 1\ cp) (V*WS2n -

JJR4n-l

drJ2) 1\ cp.

Hence

d"I -- v *WS2n Let "II 4n-l

L2n-l

= "I + "12· (lR4n -

1 ).

Then drJl

=

-

d"12

V*WS2n.

on

lIll4n-l lA. •

Denote Tl

By Lemma 9.3, we have

It follows from Proposition 10.10 that we have

Hence

= (¢-1)* "II.

Then Tl E

202

F. HANG, F. LIN, AND Y. YANG

::; c (n) Ildullir.ln-2(8QR) (1Idullir.ln-2(8QR) +

+ R;~=t Ilrll ::; C

X

4n-l

L2n=I(8QR)

(n) IldUlli4n-2(8QR)

(r (r

X

r R r

J8QR

Idul 4n - 2 dS +

(~)

IlduIII~~~2(8QR)

Idul 4n - 2 dS + c (n)

Idul 4n - 2 dS + R

J8QR

::; C

)

foQR

J8QR

R4~~1 IIDrl1 L 42~1 (8QR)

IDrI4~;;-1 dS)

+

C

(n)

IIduI111~~2(8QR)

Irl ~~=t dS)

J8QR

•

R

r

(lduI4n-2 + Irl ~~=i +

IDrI4~;;-1) dS,

J8QR

R4n-l

if R 2: 1. We may set

/\'0

= Q (Ul)

E Z. Then we get

J(r

'~ " - -12 U *WS2n A r - /\,f" f"E2RZ4n-l IS 2n l QR(f,,)

::;

c(~)

.R

R4n-l

c (n, A) ::;

2n::;

r (lduI 4n - 2 + Irl~~=t + IDrI4~;;-1) dS IER

c:,

R4n-l

when R is large enough. As a consequence,

L

1/\'f"I::;

f"E2RZ4n-l This implies /\,f"

Q (u) -

nl

= 0 except

L

f"E2RZ4n-l

+, 1 IS

/\,f,,::;

IU*WS2n

Arldx+c:

<

00.

QR(f,,)

for finitely many es. On the other hand,

L

f"E2RZ4n-l

+, 1 IS nl

u* WS2n A r - /\,f" ::;

C:.

QR(f.)

Using the fact that Q (u) - 2:::f"E2RZ4n-l /\,f" is an integer, we see that, when c: < 1,

Q(U) =

L

/\,f".

f"E2RZ4n-l

D

13. Existences of minimizers After the fore-going preparation, we are ready to prove the main result of the second part of this article, Theorem 13.1 below. This theorem describes the behavior of a minimizing sequence of maps for the Faddeev model. Based on this result and the sublinear growth law, we will obtain several existence

203

EXISTENCE OF FADDEEV KNOTS

statements in Section 13.1. It is worth pointing out that even for the Faddeev model for maps from IR3 to 8 2 , Theorem 13.1 improves the substantial inequality in [LYl] to an equality. Such a result is based on some special operations on maps with finite Faddeev energy given in Lemma 13.2 and establishes a subadditivity property for the Faddeev knot energy spectrum. Recall that

X = { u E Wl~; (IR 4n -

=

r

JJR4n-l

(lduI 4n -

1,

8 2n )

IE (u)

2+ IU*WS2n 12) dx < oo} .

For NEil, we set EN

= inf{E (u)

1U

E XN}

where X N

= {u E X

1

Q (u)

= N} .

13.1. Assume that N is an nonzero integer such that XN i=
{ Ui} C

• N=N1+···+Nm . • IYij - Yikl ~ 00 as i ~ 00 for 1 ~ j, k ~ m, j i= k. • If we set Vij (x) = Ui (x - Yij) for 1 ~ j ~ m, then there exists a Vj E X such that

as i

~ 00

and

for all j.

•

m

EN

= LENj' j=1

In particular, if EN < EN'

+ EN"

for N

= N' + Nil, N', Nil i= 0, then

EN is achieved.

Before carrying out the proof of this theorem, we make some general discussion. Assume Ui E X with E (Ui) ~ A < 00. Then, after passing to a subsequence, we may find a Uoo E X such that Ui ~ Uoo a.e., dUi --->. 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 <

e)

F (r,~, () = Fr, (r, = (r, ¢ (~) + () E BI and ( E B.l.. It follows from the discussion in [HL,

00, ~

2

16

Section 3] that for a.e. ( E B.l., U 0 Fr, E 16

r (Id (u lv,

0

WI!; (VI).

Fd1 4n- 2 + I(u 0 Fr,)* wS2nI2)

:S c(n);; {O
2

Moreover

dx

(lduI4n-2 + IU*WS2n 12) (r, ¢ (~) + () . r4n-2drd~.

Hence

r

lBrt;

d(

r (ld(uoFdI4n-2+I(uoFd*wS2nI2)dx

lv,

(lduI4n-2 + IU*WS2n 12) (r, () . r 4n - 2drd(

:S c(n);; {O
:S c(n)

r

lVl

(lduI 4n- 2+ IU*WS2n 12) dx.

It follows that we may find a ( E B.l. such that 16

Let

Then

VI

r

l~

EX,

(ldvlI4n-2 + IviwS2nl2) dx:S c(n)

kr

(lduI 4n- 2+ IU*WS2nI2) dx

EXISTENCE OF FADDEEV KNOTS

209

and VI (r,~)

vIIlR. 4n -

l \ Vl

= u (r, () = U.

for

~

E BL

,

16

Let

We have

V2 E

~ E

B16,

~ E

B32\B16,

~ rJ.

B32.

X,

r (ldv214n-2 + I 2WS2n 12) dx ::; V

r (lduI 4n - 2 + IU*WS2n 12) dx

c (n)

JV32

JV32

and V2 (r,~)

v2llR 4n -

l\

= U (r, ()

for ~ E B 16 ,

u.

V32 =

Let

f (r) = U (r, ()

for 0

< r < 00.

Then

roo If' (r) 4n- 2 r 4n- 2dr ::; c (n) r 1

h

J~

(lduI 4n - 2 + IU*WS2n 12) dx < 00.

Hence If' (r)1 = If' (r) I r· ~ E £1 ([1,00)). It follows that limr -+ oo f (r) exists. Without loss of generality we may assume that lim

r-+oo

f (r)

= -no

Here n is the north pole of s2n. We may find R > 1 large enough such that for r ~ R, f (r) lies in lower half sphere. Let 7fn : s2n\ {n} ~ ~2n be the stereographic projection with respect to n. Define 9 (r) =

Then 9 (r)

~

0 as r

~

7f n

(f (r))

for r

~

R.

00, 19 (r)1 ::; 1, and

{'Xl 19' (r) 14n - 2 r 4n - 2dr ::; c (n)

JR

r (lduI 4n - 2 + IU*WS2n 12) dx. J~

It follows from Hardy's inequality that

loo

19 (r)1 4n - 2 dr::; c (n) ::; c (n)

loo 19'

(r)1 4n - 2 r 4n - 2dr

r (lduI 4n- 2 + IU*WS2n 12) dx.

lVl

F. HANG, F. LIN, AND Y. YANG

210

Let

1' { rJ (x) = xi;/T~i2, 0,

if X4n-l ~ Ix'l- 1,

~f

Ix'i - 1 ~ X4n-l ~ -2,

If x4n-1

~

-2.

Note that

c (n) IdrJ (x)1 ~ Ixl + l' Denote

w (x) = rJ

(2~) g (Ixl)

(ldwI4n-2

+ IA2n (dw)1 2) dx

> R.

for Ixl

Then

r

lJR4n-l\BR

~ c (n)

loo Ig

~ c (n)

r

(r)1 4n - 2 dr

+ c (n)

loo Ig'

(r)1 4n - 2 r 4n - 2 dr

(lduI 4n - 2 + IU*WS2n 12) dx.

lVl

Finally, we set ( ) _ {V2

(x),

if X4n-l

~

Ix'l- 2R,

7r~1 (w (x)), if X4n-l ~ Ix'l- 2R.

v x -

Then it follows from the construction that v EX,

r

1~2

(ldvI 4n - 2 + IV*W S 2n I2 ) dx

~ c(n)

r

(lduI4n-2

+ IU*WS 2n I2 )

dx,

lV32

and VIJR4n-l\V32 For every c

= u,

V

(x) = -n for

X4n-l ~

-4R.

> 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 <

00,

I~I < a}

C ]R3

214

F. HANG, F. LIN, AND Y. YANG

as the corresponding cone with origin as the vertex. Note that VI = {X E ]R3 : X3

< O}.

To continue, we define a function,

a,

¢ (e) =

{ 2 (lei - l) ftr'

e,

We also write for

°<

F (r,

< 00, e

r

e, () = F( (r, e) = (r, ¢ (e) + ()

E BI and ( E B.l... It follows from the discussion in [HL, 2

16

section 3] that for a.e. (E B.l.., uoF( E 16

f

WI!;; (VI)' Moreover 2

(Id (u 0 Fdl 2 + Id (u 0 Fe.) /\ d (u 0 F(1 2 ) dx

lv!

~c f

(ldul2 + Idu /\ dul2) (r, ¢ (e)

+ () . r 2drde.

l{o
Hence

f

f

d(

lB-fG

(Id (U 0 Fdl 2 + Id (u 0 Fe.) /\ d (u 0 F(1 2 ) dx

lv,

~c f

l{O
~c f lVl

(ldul2 + Idu /\ dul2) (r, (). r 2drd(

(ldul2 + Idu /\ dul2) dx.

It follows that we may find some ( E B.l.. such that 16

Let VI

(r, e) = {u (r, ¢ (e u (r, e),

Then

VI

EX,

()

+ (),

eE

B~ (() ,

e~ BI (() . 2

EXISTENCE OF FADDEEV KNOTS

215

and (r,~)

VI

= u (r, ()

for ~ E B.l, 16

Vl\lR3\V1 = u. Let

I Vl

V2 (r,~) =

(r, zk) ,

~ E

VI (r, (~~~ (\~\ - 16) +

VI

l6) ~) ,

~ E B 32 \B 16 ,

~

(r,~),

B l6 ,

rt.

B 32 ·

We have V2 E X,

r (\ dV2\2 + \dV2

1\

lV32

dV2\2) dx

~c

r (\du\2 + \du

lV32

1\

du\2) dx,

and

V2

(r,~) =

u (r, ()

for ~ E B16,

V2\lR3\V32 = u. Let

f(r)=u(r,() Then

roo If' (r)1

~c

forO
r

(\du\2 + \du 1\ du\2) dx < 00. lV1 If' (r)\ r· ~ E Ll ([1,00)). It follows that limr--+oo f (r) exists.

lo Hence If' (r)\ =

2

r 2dr

Without loss of generality we may assume lim

r--+oo

f (r)

=

(0,0,0, -1).

We may find R > 1 large enough such that for r 2: R, f (r) lies in lower half sphere. Let n = (0,0,0,1) and ?Tn : 8 3 \ {n} ---t ]R3 be the stereographic projection with respect to n, define

Then 9 (r)

---t

°

as r

9 (r) = ---t

?Tn

00, \9 (r)\

roo 19' (r)1

2

lR

r 2dr

(f (r))

for r 2: R.

~ 1 and

~c

r

lV1

(\du\2

+ \du 1\ du\2) dx.

It follows from Hardy's inequality that

roo \9 (r)\2 dr ~ c roo 19' (r)1 2 r 2dr ~ c r

lR Let

lR

l~

(\du\2

+ \du 1\ du\2) dx.

F. HANG, F. LIN, AND Y. YANG

216

Note that

c

Id1J (X)I ::; Ixl Denote w (x) = 1J

+1

(2~) 9 (Ix!)

for Ixl > 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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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 01<:,e and oi,e to the abelian groups of one-dimensional subfields of K and L, respectively. It is shown in [2] that in the case of functional fields of transcendence degree 2 over k = iFp and t' =1= p, any isomorphism Wi defines an isomorphism between K and some finite purely inseparable extension of L. In this paper we treat the first problem which arises when we try to extend the result to general homomorphisms. By the description above, it suffices to treat the corresponding homomorphisms of abelian groups W~ :

OK,e -+ OL,e·

By Kummer theory, these can be identified with homomorphisms

'II; : Hom(K*jk*,Ze) -+ Hom(L*jl*,Ze). that wi commutes with projections onto Galois

The condition groups of one-dimensional fields is the same as commuting with projections

Hom(K* jk*, Ze(l)) -+ Hom(E*, Ze(l)). If it were possible to dualize the picture we would have a homomorphism

'11* : L* jl* -+ K* jk*, mapping multiplicative groups of one-dimensional subfields in L to multiplicative groups of one-dimensional subfields of K. This is the problem that we consider in the paper. In order to solve the problem for Galois groups we need to consider the maps ~; : L* -+ K*, between t'-completions of the dual spaces (as in [2]) and to· find conditions which would allow to reconstruct '11* from ~;. This problem will be addressed in a future publication.

228

F. BOGOMOLOV AND Y. TSCHINKEL

3. Functional equations LEMMA 6 .. Let

x, y E K be algebraically independent elements and z E k(x, y) a nonconstant rational function. Let f, g E k(t)* be nonconstant functions such that f(x)/g(y) E k(z). Then there exist j,g E k(t)* such that k(z) PROOF.

Write z

= k(j(x)jg(y)).

= p(x, y)/q(x, y), with coprime p, q E k[x, y]. Then

f(x)/g(y)

= II(p/q-ci)ni = q-Li ni II(p-ciq)n i , i

i

modulo k*, for pairwise distinct Ci E k and some ni E Z. The factors on the right are pairwise coprime, Le., their divisors have no common components. Thus the divisors of q(x, y) and p(x, y) - Ciq(X, y) are either "vertical" or "horizontal", Le.,

q(x, y)

= t(x)u(y)

and

p(x, y) - ciq(X, y)

= Vi(X)Wi(Y),

for some t, u, Vi, Wi E k(t). It follows that

z(x, y) - Ci and we can put

9=

= Vi(X)Wi(y)/t(x)u(y)

Vi(X)/t(x) and

j

=

Z(Y)/Wi(Y).

D

A rational function f E k(x, y)* is called homogeneous of degree r if

>,Tf(x,y)

(3.1)

A function

= f(>.x,>.y),

for all >. E k*.

f is homogeneous of degree 0 iff f

E

k( x / y) * .

Let PI,P2 E k(x, y)* be rational functions with disjoint divisors. Assume that PI(X,y), P2(X,y) is homogeneous of degree r. Then PI is homogeneous of degree rl, P2 is homogeneous of degree r2 and rl + r2 = r. LEMMA 7.

8. Let f, g E k[t] be nonzero polynomials. Assume that p(x, y) := g(x)f(y) is homogeneous of degree dEN. Then COROLLARY

= axn f(y) = byd-n,

g(x)

for some n E Nand a,b E k*. LEMMA

(3.2)

9. Let f, g

E

k[t] be polynomials such that

p(x, y) = ax r f(y) - c!lg(x) E k[x, y]

MILNOR K2 AND FIELD HOMOMORPHISMS

is homogeneous of degree

l'

229

EN. Then g(x) = adxT + ao, f(y) = cdyT + CO,

and aCd - cad

= O.

Write g(x) = L:iaixi and f(y) = L:jCjyj, substitute into the equation (3.2), and use homogeneity. D PROOF.

LEMMA

10. Let /I,h,gl,g2

E

k[t] be polynomials such that

gcd(gl,g2) = gcd(fl, h) = 1 E k[tl/k*. Let p(x,y) = gl(x)h(y) - g2(X)/I(y) E k[x,y] be a polynomial, homogeneous of degree l' EN. Then gi(X) = aixT + bi , Ji(y) = ciyT + di , for some ai, bi, Ci, di E k, for i = 1,2, with bld2 - b2dl = 0, alC2 - a2Cl = 0., PROOF.

By homogeneity, p(O, 0)

= 0,

i.e.,

gl (0)12(0) - g2(0)/I (0) = O. Rescaling, using the symmetry and coprimality of /I, 12, resp. gl, g2, we may assume that

/I (0) 12(0)) _ ( gl (0) g2 (0) -

(11 11)

or

(11 0)0 .

In the first case, restricting to x = 0, resp. y = 0, we find

gl (x) - g2 (x) = axT, /I(y) - h(Y) = cyT, for some constants a, C E k*. Solving for 12, g2 and substituting we obtain

In the second case, we have directly

gl(X) /I(y)

= ax T, = cyT,

F. BOGOMOLOV AND Y. TSCHINKEL ;

230

for some a, c E k*, and p(x, y)

= axTJ2(y)

- cyT g2(X).

o

It suffices to apply Lemma 9.

PROPOSITION 11. Let x, y E K* be algebraically independent elements. Fix nonzero integers rand s and consider the equation

(3.3) with

R E k(x/y), p E k(x), q E k(y), S E k(p/q), where p E k(x) and q E k(y) are nonconstant rational functions. Assume that (i) x, y, p, q are multiplicatively independent; (ii) R, S are nonconstant. Then or with rI

EN,

We have

with dI = PI,I/qI,I and r = -rIs in the second case. Conversely, every pair (p, q) as above leads to a solution of (3.3).

Equation (3.3) gives, modulo constants,

PROOF.

J

I

(3.4)

yT

II (x/y - Ci)n

i

i=O

= qS

II (p/q - dj)mj, j=O

for pairwise distinct constants Ci, dj E k, and some ni, mj E Z. We assume that CO = do = 0 and that Ci, dj E k*, for i, j 2: 1. Expanding, we obtain XnOyT-L-i?,On i II(x - ciy)n; i>O -

mo - L-j?,omj mo-s S-L-n~:omj

- PI P2

q2

qI

II(PIq2 - djP2qI )mj , j>O

MILNOR

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AND FIELD HOMOMORPHISMS

231

where p = pI/P2 and q = qI/q2, with Pl.P2 and ql, q2 coprime polynomials in x, resp. y. It follows that:

(AI)

xno

= p~O(x)P2-mo- 2:::.p Om'] (x),

LEMMA 12. If nl all i,j ;::: 1.

=1=

0 then the exponents ni, mj have the same sign, for

PROOF. Assume otherwise. Collecting terms in (A3) with exponent of the same sign we obtain:

II (x i>O,ni>O

Ciy)ni

=

II (Plq2 j>O,mj>O

djP2QI)m j ,

(x II i>O,ni
Ciyt i

=

(PIQ2 II j>O,mj
djP2Ql)m j

Thus there are a, bEN such that

is a nontrivial rational function of x/y with trivial divisor at infinity in pI x pI, with standard coordinates x, y. The same holds for

a nontrivial rational function of p/Q. Thus k(p/Q) n k(x/y) =1= k, which contradicts the assumption that p/q and x/yare multiplicatively independent. Indeed, the functions p/q and x/y generate a subgroup of rank 2 in K* /k* and hence belong to fields intersecting by constants only. 0 By Lemma 12, if Li>O 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

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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.

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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

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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

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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

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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).

E. VIEHWEG

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.
5. Milnor-Wood inequalities Before we discuss families of abelian varieties over a higher dimensional base, let us mention a numerical condition, which applies to a different class of Higgs bundles over curves Y, the Milnor-Wood inequality for the Toledo invariant. We refere to [BGG06] for an introduction and for a guide to the literature. Let 'JI' be a local system, induced by a representation of 7rl (Y, *) in a connected non-compact semi-simple real Lie group G. Since the representations of the fundamental group of Yare not semi-simple we can not apply

E. VIEHWEG

258

Simpson's correspondence stated in Theorem 3.1. As in [BGG06, Section 2J one has to add on the representation side the condition "reductive" and on the Higgs bundle side the condition "polystable". As explained in [BGG06, Section 3.2J, for G = SU(p, q) (or for G = Sp(2n, lR)) the corresponding Higgs bundle (F, 'lj;) is given by two locally free sheaves V and W on Y of rank p and q, respectively, and the Higgs field 'lj; is the direct sum of two morphisms

13 : W

-----+ V

@

n}

and

-y: V -----+ W

For G = Sp(2n, lR) one has p = q = nand W dual to each other. The Toledo invariant of 'JI' or of (F, r) is

@

n}.

= VV. Moreover 13 and -yare

r('JI') = r((F, 'lj;)) = deg(V) = - deg(W), and the classical Milnor-Wood inequality says that

Ir('JI') I :S Min{p, q} . (g - 1). In fact, on page 194 of [BGG06J one finds a more precise inequality, and again equality has strong implications on the structure of the Higgs field: PROPOSITION 5.1. Let 'JI' be a local system on Yo induced by a representation of 1fl(Y, *) in SU(p,q) (or in Sp(2n,lR)). Assume that the Higgs bundle

is stable, where -y : V -----+ W

@

n} (log S)

and

13: W

-----+ V

@

n} (log S).

Then

(5.1)

-rk(f3) . deg(n}) :S -2· deg(W)

= 2 . deg(V) :S rkb) . deg(n}).

The inequality 2· deg(V) :S rkb) . deg(n}) is strict, except if -y is an isomorphism (and hence rk(V) = rk(W)). PROOF. It is sufficient to prove the inequality on the right hand side. The other one follows by interchanging the role of p and q, hence of V and W. Since this inequality is compatible with exact sequences of Higgs bundles, the Jordan-Holder filtration for Higgs bundles allows to assume that (F = V EB W, 'lj; = -y + (3) is stable. The subbundle 9 = V EB -y(V) @ (n} )-1 of (F,O) and the kernel K of V --+ -y(V) are compatible with the Higgs field, hence

2· deg(V) - rk(-y(V)) . deg(n})

:S deg(V) + degb(V)) - rkb(V)) . deg(n}) :S O. If equality holds, the stability implies that K is zero and that 9 particular -y is an isomorphism.

= F.

In 0

259

ARAKELOV INEQUALITIES

EXAMPLE 5.2. Let (E, ()) be the Higgs bundle of a polarized variation of Hodge structures of weight one. Then one could choose V = El,o and W = EO,l. Since (3 = 0 and () = '"Y the inequality (5.1) says that

o ::; 2· deg(V) ::; rk('"'() . deg(O}), hence it coincides with the inequality (3.6) in Variant 3.5. (Kang Zuo). Let (E, ()) be the Higgs field of a variation of Hodge structures of weight k, with k odd. Choose EXAMPLE 5.3

k-l

k-l -2-

2

V=

E9 E

k - 2m ,2m

and

=

W

m=O

E9 E

k-

2m - 1,2m+l,

m=O

and for '"Y and (3 the restriction of the Higgs field. The Milnor-Wood inequality says that k-l -2-

-L

(5.2)

k-l -2-

deg(E k - 2m - l ,2m+l)

m=O

=

L

1 deg(E k - 2m ,2m) ::;

"2 . deg(O}).

17L=O

The Arakelov equality in Lemma 3.3, c) implies that (5.2) is an equality. On the other hand, having equality in (5.2) just implies that the morphisms ()k-m,m : Ek-m,m --7 Ek-m-1,m+1 Q9

O}

are isomorphisms for m even, but it says nothing about the other components of the Higgs field. So the two equalities are not equivalent. To get an explicit example, consider Y = pI and S = {O, 1,00}. Choose E 3 ,o

= 0lP'1(1),

E 2,1 = OlP'l E 1,2

= OlP'l

EO,3 =

OlP'l(-l).

SO ()3,O and ()1,2 are isomorphisms, whereas ()2,1 : OlP'l -+ O~l (log(O + 1 + (0)) is injective and has a zero in some point, say 2. The degree of E is zero, and obviously the Higgs bundle is stable. Hence by Theorem 3.1 (E, ()) is the Higgs bundle of a local system, and by construction the local system underlies a polarized variation of Hodge structures. This is one example for which (5.1) is an equality, without V or W being stable, a quite common effect which will be studied in a follow-up of [BGG06] in more details.

6. Arakelov inequalities for variations of Hodge structures of weight one over a higher dimensional base From now on Y denotes a projective manifold and S a normal crossing divisor in Y with Yo = Y \ S. We will need some positivity properties of the sheaf of differential forms on the compactification Y of Yo. 6.1. We suppose that O}(logS) is nef and wy(S) is ample with respect to Yo.

ASSUMPTIONS

(*)

260

E. VIEHWEG

As a motivation, assume for the moment that Yo C Ag is a Shimura variety, or that there is a Kuga fibre space fo : Xo -+ Yo. In both cases Yo is the quotient of a bounded symmetric domain and replacing Yo by an etale finite cover, we may choose for Y a Mumford compactijication, i.e. a toroidal compactification, as studied in [Mu77]. There it is shown that n~ (log S) is nef, and that Y maps to the Baily-Borel compactification. Then the finiteness of Yo -+ Ag implies that wy(S) is ample with respect to Yo. The main reason why we need an extra condition is Yau's Uniformization Theorem ([Ya93], discussed in [VZ07, Theorem 1.4]), saying that (*) forces the sheaf n~(log S) to be p,-polystable. Here, as in Definition 2.1, we will consider for coherent sheaves F the slope p,(F) with respect to wy(S). Usually to define stability and semistability on higher dimensional projective schemes, one considers slopes with respect to polarizations. Replacing "ample" by "nef and big", hence considering semi-polarizations there might exist effective boundary divisors D not recognized by the slope. So one has to identify p,-equivalent subsheaves. 6.2. 1. A subsheaf 0 of F is p,-equivalent to F, if F /0 is a torsion sheaf and if Cl (F) - Cl (Q) is the class of an effective divisor D with p,(Oy(D)) = O. 2. 0 C F is saturated, if F /0 is torsion free. 3. F is weakly p,-polystable, if it is p,-equivalent to a p,-polystable sub sheaf.

DEFINITION

The way it is stated, Theorem 3.1 only generalizes to a higher dimensional base Yo if Yo = Y is compact. For variations of Hodge structures however the polystability of the induced Higgs bundle remains true and, as recalled in [VZ07, Proposition 2.4], there is no harm in working with the "semi-polarization" Wy (S). PROPOSITION 6.3. Let E be the logarithmic Higgs bundle of a C-variation of Hodge structures W on Yo with unipotent local monodromy around the components of S. If GeE is a sub-Higgs sheaf then for all dim(Y) - 1 2: l/ 2: 0 and for all ample invertible sheaves 1£ on Y one has

(6.1)

Cl(G).Cl(Wy(s))dim(y)-v-l. C1 (llt ~

O.

Moreover, if GeE is saturated the following conditions are equivalent: (1) For some v 2: 0 and for all ample invertible sheaves 1£ the equality holds in {6.1}. (2) For all v and for all ample invertible sheaves 1£ the equality holds in {6.1}. (3) G is induced by a local sub-system ofW. Most of the standard properties of stable and semistable sheaves carry over to the case of a semi-polarization, hence to the slope with respect to

ARAKELOV INEQUALITIES

261

the nef and big sheaf wy (S). In particular one can show the existence of maximal {t-semistable destabilizing subsheaves, hence the existence of the Harder-Narasimhan filtration. In addition the tensor product of {t-semistable sheaves is again {t-semistable. The starting point of [VZ07] was a generalization of the Arakelov inequality (3.5) for k = 1 to a higher dimensional base. It is a direct consequence of the Simpson correspondence, as stated in Proposition 6.3, using quite annoying calculations of slopes and degrees. Following a suggestion of Martin Moller we present below a simplified version of those calculations. THEOREM 6.4. Under the Assumption (*) consider a polarized rr:-variation of Hodge structures V on Yo with logarithmic Higgs bundle (E, ()). Assume that V is non-unitary, irreducible and that the local monodromy in s E S is unipotent. Then

(6.2) The equality {t(V) {t-semistable.

=

{t(n}(log S)) implies that El,o and EO,l are both

PROOF. Consider the Harder-Narasimhan filtrations 0= Go C Gl C ... C Go = El,o 7= 7= 7=~

0 = G'OC G'l C ... C Ge', = Eo,l . 7= 7= 7=

and

Next choose two sequences of maximal length

o= (6.3)

jo < jl < ... < jr = f

and

0=

()(GjJ C Gji ®n~(logS),

and hence

()(GjJ

jb < j~

< ... < j~ = f',

()(Gj'_1+l)

rt.

rt.

with

Gji-l ®n~(logS)

Gji-l ® n~(log S).

Starting with jo = jb = 0 this can be done in the following way. Assume one has defined j~-l and j:-l' Then j: is the minimal number with

()(G j,_l+1) C Gji 0 n~(log S), and

j~

is the maximum of all j with

()(Gj ) C Gj[ 0 n}(logS). Writing E;'o = Gj, and E?,l = Gj[ we obtained two filtrations

o = El,o C El,o C ... C El,o = El,o 07=17= 7=r

and

0= EO,l C EO,l C ... CEO'! = EO,l 07=17= 7=r . Let us define F!',q = Ef,q / Ef~ql' Remark that F!',q is not necessarily {tsemistable. For (p, q) = (1,0) for example, the Harder Narasimhan filtration is given by

O-G· /G·),-1 cG· ),-1 7= ),-1+ l/G·),-1 C 7= ... CG·/G· 7=)' ),-1 -_Fl,o ~ .

E. VIEHWEG

262

So J.t(Gj'_l+l/Gj'_l) ~ J.t(FL1,o) ~ J.t(GjjGj,-l)

(6.4)

and, replacing G by G' and j by j',

J.t(Gj~_l+l/Gj~_) ~ J.t(FLO,l) ~ J.t(Gj/Gj~_l)·

(6.5)

6.5. A. (Cl (EZ'o) + Cl (E?,l)) . Cl (wy(s))dim(Y)-l ::; 0 for all " E {I, ... ,r}.

CLAIM

> J.t(Fi'o) > ... > 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 <

K,

S

e

K, sp,q = "'" ri!,q , K, L.J Pi

y1,0 K,

i=l

and 8K,

= rl'o _

'o~1' J-Li

"K,

1,0

1,0

= L..ti=l J-Li . Pi

10'

SK,'

yO,l K,

"K,

0,1

0,1

= L..ti=l J-Li . Pi

01'

SK,'

r~,l.

Remark that rl,o is the slope of the sheaf E~'o, whereas r~,l is just a virtual slope without any geometric meaning. By the choice of p?,l one finds

P;'o . J-Li'o

+ p?,l . J-L?,1 = rk( F/'o) . J-Li'o + rk( FiO,l) . J-L?,1 -

Ii

=0

and we can state:

(5) pi'o. J-Li'o = _p?,l . J-L?,1 and hence p?,l > 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.

E. VIEHWEG

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)

<;(V) ~

rk(El,O) . rk(EO,l) . (ni rk(E) . ni

+ 1) .

Moreover (7.4) is an equality if and only if the kernel of the morphism

1l0m(EO,1, E1,o)

~

n}(log S),

induced by 0, is a direct factor of 1l0m(EO,1, E1,O).

Here again one uses Simpson's polystability, applied to the variation of f

f

Hodge structures /\ V with Higgs bundle /\ (E, 0). THEOREM 7.7. Suppose in Theorem 7.4 thatfori = i(V) the sheafni is of type A or B. Assume that one has the length equality (7.5)

<;(V) =

rk(El,O) . rk(EO,l) . (ni rk(E) . ni

+ 1) .

Then

a. Mi is the complex ball SU(l, ni)/ K, and V is the tensor product of a unitary representation with a wedge product of the standard representation of SU(l, ni). b. Let M' denote the period domain for V. Then the period map factors as the projection Yo ~ Mi and a totally geodesic embedding Mi ~ M'. In Theorem 7.7, a) the Higgs field of the standard representation of SU(l, ni) (or of its dual) is given by 1

1

1

1

E1,O = w~ ni+ 1 iO. n. EO,l = w~ ni+ 1 and () = id . w~ ni+ 1 iO. n. --+ W~ niH iO. n. ~ '
w;n

where det(n i ).

1 i +l

stands for an invertible sheaf, whose (ni

+ l)-st

power is

REMARK 7.S. We do not know, whether the Arakelov equality implies the condition (7.5). In [MVZ07] this implication has been verified for rk(V) ~ 7. Nevertheless, the necessity of the Yau-equality in the characterization of complex ball quotients indicates that besides of the Arakelov equality one needs a second condition, presumably one using second Chern classes. Although the second Chern class does not occur in Theorem 7.7 it seem to be hidden in the condition on the length of the Higgs field stated there.

ARAKELOV INEQUALITIES

269

As an illustration of the latter, let us consider a second numerical invariant, the discriminant. Recall that for a torsion free coherent sheaf F on Y

6(F) = [2. rk(F) . c2(F) - (rk(F) -1) . cl(F)2] . Cl(wy(s))dim(Y)-2. For the Higgs bundle (E,O) of V we define 6(V) = Min{ 6(E 1,O), 6(EO,1)}. The Bogomolov inequality for semi-stable locally free sheaves allows to state as a corollary of Theorem 6.4: COROLLARY 7.9. Keeping the assumptions and notations from Theorem 6.4 the Arakelov equality p,(V) = p,(OHlogS)) implies that 6(V) ~ O. THEOREM

i

7.10. Suppose in Theorem 7.4 that wy(S) is ample, that for is of type A or B and that 6(V) = O. Then

= i(V) the sheaf Oi

a. Mi is the complex ball SU(1, ni)/ K, and V is the tensor product of a unitary representation with the standard representation of SU(1, ni)' b. Let M' denote the period domain for V. Then the period map factors as the projection Yo -+ Mi and a totally geodesic embedding Mi -+ M'. rk(E1,O) . rk(EO,l) . (ni + 1) c. <;(V) = rk(E) . ni . Note that in a) we have to exclude the wedge products of the standard representations. For those 6(V) is larger than O. In [MVZ07] we are mainly interested in subvarieties of A g • If one assumes the conditions in 7.1 to hold for all non-unitary local C-subvariations of Hodge structures of the induced family then one can deduce the following numerically characterization of Kuga fibre spaces: THEOREM

7.11. Let f : A -+ Yo be a family of polarized abelian varieties

such that Rl fo*CA has unipotent local monodromies at infinity, and such that the induced morphism Yo -+ Ag is generically finite. Assume that Yo has a projective compactification Y satisfying the Assumption (*). Then the following two conditions are equivalent: 1. There exists an etale covering Y~ -+ Yo such that the pullback family f' : A' = A x Yo Y~ -+ Y~ is a K uga fibre space. II. For each irreducible subvariation of Hodge structures V of Rl f*oCA

with Higgs bundle (E,O) one has: 1. Either V is unitary or the Arakelov equality p,(V) = p,(O~ (log S)) holds. 2. If for a p,-stable direct factor OJ of O~ (log S) of type B the composition

E. VIEHWEG

270

is non-zero, then

(( E e.)) <; REMARK

, J

=

rk(El,O). rk(EO,l). (nj rk(E) . nj

+ 1) .

7.12.

(1) Theorem 7.11 partly answers the question on the JL-stability of the Hodge bundles El,o and EO,l, at least for subvariations of Hodge structures in Rl fo*Cxo for a family fo : Xo ~ Yo of abelian varieties. In fact, choosing in part I) a Mumford compactification Y' of Y~ one can show that the Hodge sheaves E,l,O and E,O,l of the pullback V' of the irreducible subvariation of Hodge structures V are JL-stable. So up to replacing Yo by an etale cover and Y by a suitable compactification, the JL-stability of the Hodge sheaves follows from the Arakelov equality if V is of type A or C, whereas for type B we need an additional numerical condition. (2) If one knows the JL-stability of Efl,o and E'0,l on some compactification of an etale covering Y~ of Yo, and if wy(S) is ample, then llom(EO,l, El,O) is JL-polystable and the Arakelov equality implies that the morphism

is surjective and splits. So by Lemma 7.6 the numerical condition, saying that (7.4) is an equality, holds and by Theorem 7.7 Mi must be a complex ball. As remarked in 7.8 we think it is unlikely to have a characterization of a complex ball, which is only using first Chern classes. (3) The condition "wy(S) ample" appears in (2) since one uses that the tensor product of polystable sheaves is polystable. The same is used in the proof of Theorem 7.10. There however the ampleness is needed for a second reason. One uses the characterization of unitary bundles as those polystable bundles with vanishing first and second Chern class. S.T. Yau conjectures that for both statements "wy(S) nef and big" is sufficient. He and Sun promised to work out a proof of those results. 8. Open ends

I. As mentioned already, under the assumptions made in 7.1 for variations of Hodge structures V of weight one and of small rank, the Arakelov equality implies that the length inequality 7.4 is an equality. Let us write in 7.1 q = rk(El,O), p = rk(EO,l) and assume that q ~ p. Since

ARAKELOV INEQUALITIES

deg(E 1,O) (6.2) as

(8.1)

+

deg(E O,l)

= 0 one can rewrite the Arakelov inequality

c (E 1 ,o) . c (w (s))dim(Y)-l 1

1

271

Y

< -

p. q

(p + q) . dim(Y)

. c (w (s))dim(Y). 1

Y

8.1. Assume that V satisfies the Arakelov equality and that for i = i(V) the sheaf Oi is of type A or B. Then LEMMA

ni . q ~ p

~

q,

for

ni

= rk(Oi),

and if p = ni . q the numerical condition (7.5) in Theorem 7.7 holds. In particular Mi is a complex ball in this case. PROOF. This follows from the definition of <;((E,fJ)) and Lemma 7.6, implying that

(8.2)

o

Assume the Arakelov equality. If q = 1 E1,o is invertible. Moreover, E 1 ,o ® Ty( -log S) and E O,l have to be J.L-equivalent. So p = m and (8.2) must be an equality, as required in (7.5). Hence Mi is a complex ball of dimension ni (see [MVZ07, Example 8.5]). If q = 2, assuming that wy(S) is ample, one can apply [MVZ07, Lemma 8.6 and Example 8.7J, and again one finds that the Arakelov equality implies the length equality (7.5). COROLLARY

and that for i

8.2. Assume in 7.1 that V satisfies the Arakelov equality,

= i(V) the sheaf Oi is of type A or B. Assume that Min{ rk(E 1,o), rk(Eo,l)} S 2,

Then Mi is the complex ball SU(l, ni)/ K, and V is the tensor product of a unitary representation with a wedge product of the standard representation ofSU(l,ni).

II. In [KM08aJ Koziarz and Maubon define a Toledo invariant for representations p of the fundamental group of a projective variety Y with values in certain groups, in particular in SU(q,p). They assume that X is of general type, and they use the existence of the canonical model Xcan of X, shown in [BCHMJ. Let us assume here for simplicity, that X is the canonical model, hence that Wy is ample. As in Section 5 the Higgs bundle corresponding to p is of the form V EB W and the Higgs field has two components

f3 : W --+ V ® O} and

'Y: V --+ W ® O}.

E. VIEHWEG

272

In [KM08b, Section 4.1J the Toledo invariant is identified with deg(V) = - deg(W), and for 1 ::; q ::; 2 ::; p the generalized Milnor-Wood inequalities in [KM08a, Theorem 3.3J and [KM08b, Proposition 4.3J say that

(8.3)

Ic (V) . c (w )dim(Y)-ll < q 1 1 Y - dim(Y)

+1

. c (w )dim(Y). 1 Y

For q = 2 one finds in [KM08a, Proposition 1.2J a second inequality, saying

(8.4)

2p . c (w )dim(Y) Ic 1 (V) . c1 (w Y)dim(y)-ll < - (p + 2) . dim(Y) 1 Y .

In [KM08b, Theorem 4.1 J the authors also study the case that the MilnorWood inequality (8.3) is an equality. They show that this can only happen if p 2': m . q, and that the universal covering Y is a complex ball. As in Example 5.2 one can apply (8.3) and (8.4) to a polarized variation of Hodge structures of weight one over Y = Yo. So we will assume that q = rk(El,O) is smaller than or equal to p = rk(EO,l) and we will write n = dim(Y). Here the second inequality (8.4) coincides with the Arakelov inequality (8.1). As pointed out in [KM08a, Section 3.3.1J, for variations of Hodge structures of weight one (8.3) also holds for q > 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
Surveys in Differential Geometry XIII

A survey of Calabi-Yau manifolds Shing-Tung Yau

CONTENTS 1. 2.

Introduction General constructions of complete Ricci-flat metrics in Kahler geometry 2.1. The Ricci tensor of Calabi-Yau manifolds 2.2. The Calabi conjecture 2.3. Yau's theorem 2.4. Calabi-Yau manifolds and Calabi-Yau metrics 2.5. Examples of compact Calabi-Yau manifolds 2.6. Noncompact Calabi-Yau manifolds 2.7. Calabi-Yau cones: Sasaki-Einstein manifolds 2.8. The balanced condition on Calabi-Yau metrics 3. Moduli and arithmetic of Calabi-Yau manifolds 3.1. Moduli of K3 surfaces 3.2. Moduli of high dimensional Calabi-Yau manifolds 3.3. The modularity of Calabi-Yau threefolds over Q 4. Calabi-Yau manifolds in physics 4.1. Calabi-Yau manifolds in string theory 4.2. Calabi-Yau manifolds and mirror symmetry 4.3. Mathematics inspired by mirror symmetry 5. Invariants of Calabi-Yau manifolds 5.1. Gromov-Witten invariants 5.2. Counting formulas 5.3. Proofs of counting formulas for Calabi-Yau threefolds 5.4. Integrability of mirror map and arithmetic applications 5.5. Donaldson-Thomas invariants 5.6. Stable bundles and sheaves 5.7. Yau-Zaslow formula for K3 surfaces 5.8. Chern-Simons knot invariants, open strings and string dualities

278 278 278 279 279 280 281 282 283 284 285 285 286 287 288 288 289 291 291 291 292 293 293 294 296 296 297

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6. 7.

Homological mirror symmetry SYZ geometric interpretation of mirror symmetry 7.1. Special Lagrangian submanifolds in Calabi-Yau manifolds 7.2. The SYZ conjecture ~ SYZ transformation 7.3. Special Lagrangian geometry 7.4. Special Lagrangian fibrations 7.5. The SYZ transformation 7.6. The SYZ conjecture and tropical geometry 8. Geometries related to Calabi-Yau manifolds 8.1. Non-Kahler Calabi-Yau manifolds 8.2. Symplectic Calabi-Yau manifolds References

299 300

300 301 301 302 303 303 304 304 305 306

1. Introduction

Calabi-Yau manifolds are compact, complex Kahler manifolds that have trivial first Chern classes (over JR). In most cases, we assume that they have finite fundamental groups. By the conjecture of Calabi [45] proved by Yau [293, 295], there exists on every Calabi-Yau manifold a Kahler metric with vanishing Ricci curvature. Currently, research on Calabi-Yau manifolds is a central focus in both mathematics and mathematical physics. It is partially propelled by the prominent role the Calabi-Yau threefolds play in superstring theories. While many beautiful properties of Calabi-Yau manifolds have been discovered, more questions have been raised and probed. The landscape of various constructions, theories, conjectures, and above all the fast pace progress in this subject, have made the research of Calabi-Yau manifolds an extremely active research field both in mathematics and in mathematical physics. Note: In writing an overview of such a broad subject area, the need to be inclusive was recognized and many experts were consulted. But unfortunately, in the citing of original references and the topics covered, omissions inevitably always occur, and for this, sincere apology is offered.

2. General constructions of complete Ricci-flat metrics in Kahler geometry 2.1. The Ricci tensor of Calabi-Yau manifolds. A complex manifold is a topological space covered by complex coordinate charts such that the transition between overlapping charts are holomorphic; a Hermitian metric on a complex manifold is a smooth assignment of Hermitian inner product structures on the holomorphic tangent spaces of the manifold; a Hermitian metric is called a Kahler metric if near every point the Hermitian metric is approximated by a fiat metric up to second order. In a holomorphic coordinate chart with coordinate variables (Zl,'" ,zn), a Hermitian metric has its

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associated Hermitian form

A Hermitian metric is Kahler if its Hermitian form is closed. In such a case, we call the Hermitian form the Kahler form of the Kahler metric. Given any Kahler metric, one defines its full curvature tensor by certain expressions of covariant derivatives of the metric; the Ricci curvature is a partial contraction of the full curvature tensor. The indices of the Ricci curvature is identical to that of the Kahler form. In local coordinates, Ric =

R

L Ri]dzi

1\

dz j •

We call a Kahler metric Ricci-flat if its Ricci tensor vanishes identically. 2.2. The Calabi conjecture. According to a well-known theorem of Chern, the Ricci form divided by 27r is a (1, I)-form that represents the first Chern class of a compact complex manifold. Rooted in his attempt to find canonical Kahler metrics for a Kahler manifold, in 1954, E. Calabi [45] proposed his celebrated conjecture. Conjecture. To every closed (1, I)-form 2~Cl(X) representing the first Chern class q (X) of a Kahler manifold X, there is a unique Kahler metric in the same Kahler class whose Ricci tensor (form) is the closed (1,1)form C 1 (X). In case the complex manifold has vanishing first Chern class, the zero form represents the first Chern class of the manifold. The Calabi conjecture implies the existence of a unique Ricci-flat Kahler metric in every Kahler class. Early on, Calabi realized that his conjecture can be reduced to a complex Monge-Ampere equation. 2.3. Yau's theorem. By the late 1960s, many were doubtful of the Calabi conjecture. Some attempted to use a reduction theorem of CheegerGromoll [64] (in 1971) to construct counterexamples to the Conjecture. Using the reduction theorem and assuming the conjecture, Yau announced the following splitting theorem in his 1973 lecture at the Stanford geometry conference: Every compact Kahler manifold with non-negative Ricci curvature can be covered by a metric product of a torus and a simply connected manifold with a Ricci-fiat Kahler metric. He then used this theorem to produce a "counterexample" to the conjecture. The "counterexample" was soon discovered to be flawed; Yau withdrew his Stanford lecture. (The flaw was due to the mistaken assumption that manifolds with numerically nonnegative anti-canonical divisor admits a first Chern form which is pointwise non-negative. )

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In 1976, Yau [293, 295] proved the Calabi conjecture by solving the complex Monge-Ampere equation for a real valued function ¢ det ( 9i)

()2¢ )

+ {)zi{)zj = ef

det (9i)),

where ef is any average 1 smooth function and 9{j + ()/Jj¢ is required to be positive definite. The solution ¢ of the above equation ensures that the new Kahler metric

w+ H{)8¢ can attain Ricci (curvature) form in any form referred to in the Calabi conjecture. 2.4. Calabi-Yau manifolds and Calabi-Yau metrics. The first application to Yau's proof of Calabi conjecture is the existence of Ricciflat Kahler metric on every compact complex Kahler manifold with trivial canonical class. (Trivial canonical class is equivalent to the existence of a nowhere vanishing holomorphic volume form, which is equivalent to that the top wedge power of the holomorphic cotangent bundle is the trivial line bundle.) The converse is also true: any Ricci-flat simply connected Kahler manifold has trivial canonical line class. This proves the existence and provides a criterion for Kahler Calabi-Yau manifolds. By convention, Calabi-Yau manifolds exclude those with infinite fundamental groups. The Ricci-flat metrics on Calabi-Yau manifolds are called Calabi- Yau metrics. The existence of Calabi-Yau metrics has other important consequences. In his paper [293, 295], Yau demonstrated that for a Calabi-Yau manifolds (X,w), using the Chern-Wei! form representing the second Chern form C2 (X) in terms of the curvature tensor Rm of a Calabi-Yau Kahler metric of X, one gets

Ix

C2(X)

A wn -

2= C

Ix

IRml2 vol

:c 0

for some positive constant C. Thus the Chern number C2(C) n [w]n-2 is nonnegative. Moreover, when it is zero, we have Rm = 0, and therefore X is covered by the Euclidean space Another application is the reduction of holonomy groups of Calabi-Yau manifolds. One important consequence of a Calabi-Yau metric is that the parallel transports along contractible closed loops preserve the metric and the holomorphic volume form. This implies that the restricted holonomy group of a Calabi-Yau manifold is a subgroup of SU(n), the group of special unitary transformations. Sometimes, this group can be strictly smaller than SU(n). Following the Bochner technique on Calabi-Yau manifolds, every holomorphic (p,O)-form is parallel. Such a form then reduces the holonomy group from SU(n) to a smaller subgroup. Thus if the holonomy group of X is the full SU(n), then

en.

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the Dolbeault cohomology group HP,O(X) = 0 for 1 ::; p::; n-1 and Hn,O(X) is one dimensional, spanned by the holomorphic volume form n of X. The statement of the structure theorem was known to many people, including the announcement made by Yau in 1973. It was also announced by Kobayashi, by Michelsohn [218], and appeared in a subsequent survey paper of Beauville [19]. It states that any compact, complex Kahler manifold with trivial canonical class has a cover that is a metric product of a complex Euclidean space with copies of manifolds with holonomy groups SU(m) and copies of manifolds with holonomy groups Sp(m/2); here m's are the dimensions of the corresponding manifolds. The proof is based on the above-mentioned splitting theorem of CheegerGromoll and also an argument of Calabi who drew some consequences on the first Betti number; this construction of Calabi is referred to as the Calabi construction in the sixties. 2.5. Examples of compact Calabi-Yau manifolds. By Yau's solution to the Calabi conjecture, finding (non-hyperkahler) Calabi-Yau manifolds is equivalent to finding smooth projective varieties of trivial canonical class. The first example of a Calabi-Yau threefold is the smooth quintic in the complex projective space JID4. Due to a condition imposed by superstrings theories, Calabi-Yau threefolds having Euler characteristic X = ±6 and non-trivial fundamental group playa special role. Such examples were first discovered and announced by Yau in a lecture given at the 1985 Argonne conference [296] as the Z3 quotient of an intersection of two cubics and a hypersurface of bi-degree (1,1) in the product JID3 x JID3. More examples were found later by Tian and Yau; due to an observation of Greene and Kirklin [115], these examples are deformation-equivalent to the one found by Yau. A systematic search for Calabi-Yau threefolds with X = ±6 turned up no essentially new example among complete intersections in toric varieties [59]. After Yau's examples of complete intersection Calabi-Yau threefolds in product of projective spaces, various groups, notably the group in University of Texas, employed computer algorithm to search for new examples [147, 117, 48]. Soon after, about 8000 constructions, with 256 distinct Hodge diamonds [120] were found. This pool grew by about 3 orders of magnitude by embedding in products of weighted complex projective spaces [60]. Such threefolds typically have finite quotient singularities inherited from the weighted projective spaces, and a minimal blow-up following Roan and Yau's 1987 construction [245] is understood to provide smooth models. Roan and Yau also proposed to use toric method to construct more examples. This was carried out by Batyrev and Borisov for complete intersections in toric varieties [14]. Yau conjectured that there are finitely many topological types of CalabiYau manifolds in each dimension. This conjecture is still open. By a rough count, by 2002, over 473 million toric embeddings of Calabi-Yau threefolds

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were constructed, with over 30,000 distinct Hodge diamonds [169, 170]. Based on Wall's theorem [281] one sees that complete intersections of hypersurfaces in products of projective spaces produce at least 2590 distinct diffeomorphism classes [57]. It is worth noticing that so far all Calabi-Yau threefolds can be constructed as deformations or small resolutions of complete intersections of toric varieties. The non-trivial check of this for CalabiYau complete intersections in Grassmannians and flag varieties was done by Batyrev, Ciocan-Fontanine, Kim and van Straten [15, 16]. 2.6. Noncompact Calabi-Yau manifolds. Immediately after his proof of the Calabi conjecture, Yau generalized the construction of CalabiYau manifolds to non-compact Kahler manifolds. He presented this result in his plenary lecture in the 1978 Helsinki International Congress of Mathematics. The construction is that in the complement M \ S of a compact Kahler manifold M with an effective anticanonical divisor 8 removed, suppose the first Chern class of M is either positive or trivial in a neighborhood of 8, then there is a complete Ricci-flat metric on M \ 8 if 8 is connected and geometrically stable. In case 8 is nonsingular and connected and assuming 8 admits a Kahler Einstein metric with either positive or zero scalar curvature, the detail of the generalization was presented in a joint paper of Tian-Yau [271, 272]. Around the same time, Bando-Kobayashi [11, 12] worked out some more restrictive cases. The condition that 8 should be geometrically stable was added on after the Helsinki Congress. When 8 is singular, the definition need to be clarified. If the complement M \ S admits a complete Ricci-flat metric, S has to be connected unless the complement is a product of the complex line with other manifolds. Based on his result on the volume growth of complete manifolds with non-negative Ricci curvature and compactification of complete Kahler manifolds, Yau conjectured that his construction gives all examples of noncompact complete Kahler Ricci-flat metrics with connected end. There are counterexample to this conjecture, Taub-NUT metric is one such example was pointed out by Anderson-Kronheimer-LeBrun [3]. The conjecture remains open assuming the manifolds are of finite topological type. Explicit Calabi-Yau metrics have been constructed in many cases when symmetries are present. First there is the explicit Eguchi-Hanson metric [89] on the cotangent bundle T* 8 2 of the two sphere. Viewing this as the canonical line bundle over pI, Calabi [46] constructed a complete Calabi-Yau metric on the total space X = K B of (a fraction of) the canonical line bundle of a positive Kahler-Einstein manifold B. Later, FUtaki [104] generalized to that B is any toric Fano manifold. Candelas and de la Ossa [49] constructed a one-parameter family of explicit Calabi-Yau metrics on the cotangent bundle T* 8 3 of the three sphere, by reducing the Ricci-flat equation to an ODE. The parameter of this

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family corresponds to the size of S3. This was generalized to all dimensions by Stenzel [263]. Candelas and de la Ossa [49] also constructed a one-parameter family of explicit Calabi-Yau metrics on the total space of the vector bundle X = O( -1) EJj O( -1) over pl. As the size of pI shrinks to zero, the total space X degenerates to a cone threefold Xo with rational double point singularity

Xo = {z5

+ zr + z~ + z~ = o} C ((:4.

This singular space has two different small resolutions X+ and X_, both have total space 0(-1) EJj O( -1). The pair (X+, X_) is a local model of a simple flop, a birational change of threefolds. If one varies the quadratic equation z5 + zf + z§ + z~ = t by adding a small nonzero t, then it defines a smooth hypersurface X t in ((:4 that is diffeomorphic to T* S3. The change from X± to X t is called an extremal transition; it is the basic building block for topological changes of Calabi-Yau threefolds. 2.7. Calabi-Yau cones: Sasaki-Einstein manifolds. An important class of non-compact and possibly incomplete Calabi-Yau manifolds are Calabi- Yau cones. The metric cone over a compact Riemannian manifold (8, g) is defined to be (C(S) = IR.+ x S, 9 = dr 2 + r2g), where r > 0 is a coordinate on IR.+. If the dimension of this cone is 2n and the (restricted) holonomy group of (C(S),g) is contained in SU(n), then the manifold (S,g) is called Sasaki-Einstein. In particular, since the cone is Ricci-flat it follows that (8, g) is a (2n-1 )-dimensional Einstein manifold of positive Ricci curvature, Ricg = 2(n - 1)g. In fact, many of the complete non-compact CalabiYau manifolds referred to above are asymptotic to such a cone, meaning that they are modelled at infinity by the large r (complete) end of the cone. Sasaki-Einstein manifolds in low dimensions are also important in string theory, and in particular in the AdS/CFT correspondence. For example, the latter conjectures that to every Sasaki-Einstein 5-manifold there is an associated super conformal field theory on 1R.4. Much work has gone into understanding this correspondence, and the relationship between Sasaki-Einstein geometry and superconformal field theory. The definition above is easily generalized: if the metric cone has (restricted) holonomy contained in U (n), so that the cone is Kahler, the manifold (S, g) is said to be a Sasakian manifold [252]. These manifolds should be viewed as odd-dimensional analogs of Kahler manifolds. A Sasakian manifold inherits a strictly pseudo-convex hypersurface-type CR structure from the complex structure of the cone. Sasakian manifolds are also equipped with a unit norm Killing vector field ~, called the Reeb vector fi~ld! dellned"as the restriction of J(r8/8r) to {r = 1} ~ S c C(S), where J is the complex structure tensor of the Kahler cone. The dual one-form l1{X):::; g(~, X) is a contact form on S. The flow of ~ defines a one-dimensifi)naJ. foliation of S, and it turns out that the transverse leaf space is Kihlet. Indeed, (S, g) is SasakiEinstein if and only if this transverse Kibler structure, with transverse

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metric gT, is Kahler-Einstein with positive Ricci curvature, RicgT = 2ngT. Sasakian manifolds may be classified according to the global properties of this foliation. If the orbits of ~ all close, thus defining a locally free circle action on 8, the Sasakian manifold is said to be quasi-regular, and the leaf space is naturally a Kahler orbifold. In the special case that the circle action is free, the Sasakian manifold is said to be regular, and the leaf space is a Kahler manifold. If there is a non-closed orbit of ~ the Sasakian manifold is said to be irregular. The simplest example of a Sasaki-Einstein manifold is the round sphere, viewed as the unit sphere in C n equipped with its flat Kahler metric. This is regular, with the Kahler-Einstein leaf space being CJIDn - 1 equipped with its Fubini-Study metric. The study of regular Sasaki-Einstein manifolds is in fact essentially equivalent to the study of positive (Fano) Kahler-Einstein manifolds. Boyer-Galicki [33, 34] and their collaborators have constructed large classes of quasi-regular Sasaki-Einstein manifolds by constructing appropriate Kahler-Einstein orbifold leaf spaces. These are typically realized as weighted projective varieties, and the continuity method is used to prove existence, see [283, 83] for the existence of Kahler-Einstein metrics on toric varieties. Boyer-Galicki-Kollar [35] have also shown the existence of numerous Sasaki-Einstein metrics on standard and exotic spheres using the Kahler-Einstein metrics on certain types of Fano orbifolds constructed by Demailly-Kollar [72]. The first examples of irregular Sasaki-Einstein manifolds were constructed by Gauntlett-Martelli-Sparks-Waldram [108]; these authors constructed infinitely many explicit quasi-regular and irregular Sasaki-Einstein metrics on 8 2 x 8 3 . Recently, Futaki-Ono-Wang [105] have proven the existence of toric Sasaki-Einstein metrics, following earlier work of Martelli-Sparks-Yau [214]. In this case the Kahler cone is the smooth part of an affine toric variety. Finally, Gauntlett-Martelli-Sparks-Yau [108] have described some simple obstructions to the existence of Sasaki-Einstein metrics, which also give new obstructions to the existence of Kahler-Einstein metrics on Fano orbifolds.

2.8. The balanced condition on Calabi-Yau metrics. After observing that the tangent bundles of Calabi-Yau manifolds are stable with respect to any polarization, Yau conjectured that the Calabi-Yau manifolds are also stable in the sense of geometric invariant theory. This was first openly discussed in the problem session in the UCLA geometry conference in 1990, where he proposed to approximate the Ricci-flat metric of a CalabiYau manifold X by the induced metric from embedding X into complex projective space by powers of an ample line bundle, and suggested that the action of the projective linear group on the embedding would link the stability of the manifold with the existence of the Ricci-flat metric. Yau [297] initiated the program of approximating Calabi-Yau metrics (and more generally KE-metrics) by embeddings. Under his guidance, Tian

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[270] wrote his thesis on the C 2 convergence of the pullback Fubini-Study metric via projective embeddings. (The Coo convergence was proved in Ruan's Harvard thesis [246]. See also [70].) The refined structure of this embedding was investigated by Catlin [61], Zelditch [300] and Lu [208]. For the problem of finding a canonical position in embedding a Calabi-Yau manifold in projective space, the balanced condition introduced by BourguignonLi-Yau in [32] played a crucial role. (The idea of Bourguignon-Li-Yau was based on the concept of the conformal area introduced by Li-Yau [191] in 1982.) Following their idea, Luo [210] in his MIT thesis (under the guidance of Yau) generalized the notion of balanced embeddings to all projective manifolds and related the concept to questions of geometric stability. The relation between geometric stability and the balanced condition was also studied by Zhang [301]. This program of Yau was carried out by Donaldson [80, 81] in more precise manner. Donaldson showed that for the sequence of balanced embeddings of a Calabi-Yau manifold into projective spaces via increasing powers of an ample line bundle, the sequence of normalized induced metrics converges to the Ricci-flat metric of X [80]. (See [202] for some clarifications.) Based on the balanced embeddings, he then developed an algorithm to numerically approximate the Ricci-flat metrics of K3 surfaces [81]. This algorithm was generalized by Douglas-Karp-Lukic-Reinbacher [87]' and subsequently Braun-Brelidze-Douglas-Ovrut [37, 38] to approximate the Ricciflat metrics on various projective Calabi-Yau threefolds. The work of Donaldson also showed that every Calabi-Yau manifold is asymptotically Chow stable, proving partially Yau's conjecture on the stability of Kibler Einstein manifolds [80].

3. Moduli and arithmetic of Calabi-Yau manifolds 3.1. Moduli of K3 surfaces. Two dimensional Calabi-Yau manifolds are K3 surfaces. Moduli of K3 surfaces are classically known to be smooth. It has a modular description based on the Hodge structures on the K3 surfaces. On any K3 surface X, the middle cohomology group H2(X,Z) is a free Abelian group of rank 22 and coupled with the (intersection) quadratic form (.,.) the lattice (H2(X,Z) (-,.)) is isometric to the lattice

L:= (Zam, -E8 EEl -E8 EEl U EEl U EEl U), where E8 is the Cartan matrix of the corresponding root system and U is the rank two hyperbolic matrix. The holomorphic 2-form 0 on X, which is unique up to scalars, spans a ray in H2(X, Z) ®z C and satisfies the well-known Riemann bilinear relation

(0,0) = 0, and (0, 0) > O.

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Thus after fixing a marking of X that is an isomorphism (H2(X, Z) (-, .)) ~ L, the period [0] lies in D

= {v E JIDLc I (v, v) = 0, (v, v) > o}.

This is the period domain for K3 surfaces, and the assignment X H [0] for marked Kahler K3 surface X is called the period map. The Torelli problem is on how the element [0] ED determines the complex structure of a marked Kahler K3 surface. The global Torelli theorem for algebraic surfaces was proved by Pjateckii-Sapiro and Safarevic [241]; the Torelli theorem for Kahler K3 surfaces, proved by Burns-Rapoport [43] and Looijenga-Peters [207]' states that the loci of the holomorphic two-form [0] in D uniquely determines the marked K3 surface X up to an isomorphism of L generated by Picard-Lefschetz transformation. The surjectivity of the period map was proved Kulikov [171] followed by Persson-Pinkham [238]. The approach based on the Calabi-Yau metric was pioneered by Todorov [273] and completed by Siu [257] and Looijenga [206]. The proof relies heavily on Yau's solution of the Calabi conjecture. Later, based on Yau's solution of the Calabi conjecture, the main lemma of Burns-Rapoport [43], and with the surjectivity of the period map of Kahler K3 surfaces, Todorov and Siu [258] proved that every K3 surface is Kahler. 3.2. Moduli of high dimensional Calabi-Yau manifolds. The existence of the moduli of polarized Calabi-Yau manifolds was settled by the work of Viehweg [279]. The next question is the regularity of the moduli space. The first theorem was due to Bogomolov [30] who proved that the universal deformation space of a compact Kahler-Hamiltonian manifold is unobstructed. In one of his unpublished manuscript, he also claimed that the same is true for any projective manifold with trivial first Chern class. Todorov [274] and Tian [269] each confirmed this claim by proving that every Calabi-Yau manifold has unobstructed deformations. Both proofs used essentially the Calabi-Yau metric of the manifold to derive a differentialgeometric computational equality that allows them to solve the Kuranishi equation in analytic deformation theory. This theorem is now referred to as the Bogomolov-Tian-Todorov unobstructedness theorem. Ran [243] and Kawamata [156, 157] gave new proofs of this unobstructedness result, based on the notion of Tl-lifting property. Their method was later applied to non-Kahlerian Calabi-Yau manifolds and to some singular projective Calabi-Yau varieties. On the moduli space, there is a natural Kahler metric obtained from the variation of the Ricci-fiat metric called the Weil-Petersson metric. The volume of the moduli space of polarized Calabi-Yau manifolds with respect to this metric was proved by Lu-Sun [209] and Todorov[275] to be finite. This follows from finding a suitable metric that bounds the Weil-Petersson metric from above and satisfies the conditions of the Scwharz lemma [294],

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which implies generally that for any Hermitian metric defined on a quasiprojective manifold whose Ricci curvature has a strongly negative upper bound, the total volume is finite. The study of the moduli space of complex structures on a Calabi-Yau threefold led Hitchin to study invariant functionals on differential forms [134]. This approach is also useful when studying the associated flow equations that describes the geometry in terms of an evolving hypersurface. This approach has also led Hitchin [135] to develop the geometry based on open orbits of GL (n,JR) on k-forms, especially when k = 3 [134]. As observed later, many examples of pairs of topologically distinct Calabi-Yau threefolds can be connected by flops or by small contractions followed by smoothing. For instance, Kawamata [158] has recently proved that any two birational smooth Calabi-Yau manifolds can be connected by a sequence of flops. (Kawamata's general result is valid in any complex dimension. The proof for threefolds was given earlier by Kollar [162] and fourfolds by Burns-Hu-Luo [42].) One might speculate that the collection of all Calabi-Yau threefolds can be connected by such process. Reid raised this as his fantasy [244]. For high dimensional Calabi-Yau manifolds, a major question is the Torelli problem. For hyperkahler manifolds, Huybrechts [149] proved that the period map from the moduli space of marked hyperkahler manifolds to the period domain is surjective. The case of general Calabi-Yau manifolds has been recently studied by Liu-Sun-Todorov-Yau.

3.3. The modularity of Calabi-Yau threefolds over Q. In search of Calabi-Yau manifolds that distinguish themselves from the rest, their modularity become the focus of some researchers. This is interesting from the perspective of arithmetic geometry. For a d-dimensional projective Calabi-Yau manifolds X defined over Q, it is said to be modular if the L-function of the Galois representation on the middle £-adic etale cohomology group H1t(XQ, Ql) is equal to the product of L-functions of modular forms up to factors associated to bad primes. Part of Langlands philosophy is the conjecture that all motives, in particular our X, are modular. When d is even, H1t(XQ, Ql) contains d/2-dimensional algebraic cycles, and the interesting part is the modularity of the sub-representation on the orthogonal complement of the images of algebraic cycles. Calabi-Yau varieties of dimension 1 are elliptic curves. The modularity of elliptic curves over Q has been established by Wiles [284], and TaylorWiles [266]. They proved that the two-dimensional Galois representation associated to an elliptic curve over Q does come from a weight 2 = d + 1 modular form. Dimension 2 Calabi-Yau varieties are K3 surfaces. For a K3 surface X defined over Q, H;t(XQ, Qf), which has dimension 22, factors into a direct

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sum (NS(X) EB T(X)) ® Q£ of the Neron-Severi group of algebraic cycles NS(X) and the group of transcendental cycles T(X). There are partial result for small rank T(X). The lattice NS(X) has rank at most 20; when it is 20, X is called a singular K3 surface. (The term "attractive" K3 is sometimes also used in physics [223J.) In this case, T(X) has rank 2 and defines a two-dimensional Galois sub-representation and the associated L-function L(T(X), s). The modularity of L(T(X), s) has been established by Livne [204J that L(T(X), s) does come from a weight 3 = d+l modular form of CM type. When T(X) has rank 3, its modularity follows from the modularity of elliptic curves, because T(X) is endowed with an orthogonal pairing, so that it is essentially the symmetric square of a GL(2) representation. For Calabi-Yau threefolds defined over Q, much is known for the rigid case, where there is no complex structure deformation. More specifically, we say that a Calabi-Yau threefold X is rigid if the H~t(XQ' Q£) has dimension 2. In this case, there is a two-dimensional Galois representation associated to X; the modularity has been established, under some mild conditions, that L( X, s) is determined by some weight 4 = d + 1 modular forms. It is worth noticing that currently more than 50 modular rigid CalabiYau threefolds over Q have been constructed, and expanding. The modularity question for a non-rigid Calabi-Yau threefold X over Q poses more serious challenge as the dimension of H~t(XQ' Q£) gets larger. Much less is known. An attractor flow equation on the complex structure moduli space of Calabi-Yau threefolds was found by Ferrara-Kallosh-Strominger [93J in their study of BPS black holes solutions in string theory. Moore [223J has shown that Calabi-Yau manifolds with complex structure located at an attractor fixed point on the moduli space exhibit interesting arithmetic properties. 4. Calabi-Yau manifolds in physics Calabi-Yau manifolds admit Kahler metrics with vanishing Ricci curvatures. They are solutions of the Einstein field equation with no matter. The theory of motions of circles inside of a Calabi-Yau manifold provide a model of a conformal field theory. (It is called a O'-model in physics.) Because of this, Calabi-Yau manifolds are pivotal in superstring theory. 4.1. Calabi-Yau manifolds in string theory. Superstring theory is a unified theory for all the forces of nature including quantum gravity. In superstring theory, the fundamental building block is an extended object, namely a string, whose vibrations would give rise to the particles encountered in nature. The constraints for the consistency of such a theory are extremely stringent. They require in particular that the theory takes place in a 10dimensional space-time. To make contact with our 4-dimensional world, it is expected that the lO-dimensional space-time of string theory is locally

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the product M4 x X of a 4-dimensional Minkowski space M3,1 with a 6dimensional space X. The 6-dimensional space X would be tiny, which would explain why it has not been detected so far at the existing experimental energy levels. Each choice of the internal space X leads to a different effective theory on the 4-dimensional Minkowski space M3,1, which should be the theory describing our world. It has long been argued that, in order to solve certain classic problems of unified gauge theories such as the gauge hierarchy problem, the 4-dimensional effective theory should admit an N = 1 supersymmetry. In a fundamental paper, Candelas-Horowitz-Strominger-Witten [58] analyzed what the constraint of that N = 1 supersymmetry would mean for the geometry of the internal space X. They found that, for the most basic product models with N = 1 supersymmetry, the space X must be a Calabi-Yau manifold of complex dimension 3. Shortly afterwards, Strominger [264] considered slightly more general models, allowing warped products. For these models, the N = 1 supersymmetry constraint results in a modification of the Ricci-flat equation of the earlier model. 4.2. Calabi-Yau manifolds and mirror symmetry. Around 19871988, physicists including Dixon [76], and Lerche, Vafa, and Warner [181] observed that in mapping an abstract N = 2 superconformal field theory to a possible geometrical realization as a Calabi-Yau sigma model, an ambiguity arose. A superconformal field theory has two natural rings (called (c,c) and (a,c) rings)) as does a Calabi-Yau sigma model (the Dolbeault cohomology naturally splits into even and odd dimensional forms). The question which came to light was which of the two possible pairings of the conformal field theory and geometrical rings is induced by the map between the superconformal theory and the Calabi-Yau sigma model. Lerche, Vafa, and Warner conjectured that maybe both pairings are realized because, they suggested, Calabi-Yau threefolds come in pairs in which the even and odd cohomologies are interchanged. (To be precise, the interchange is between the Dolbeault cohomology H(p,q) with H(3-p,q) for Calabi-Yau threefolds.) At the time, the evidence in support of this conjecture was thin. Some suggested that a less radical solution to the observed ambiguity might be to keep the base Calabi-Yau manifold fixed and merely consider completing the geometrical model in two ways: by including its tangent bundle or its co-tangent bundle. Nevertheless, in 1989, Greene and Plesser [116] , using the methods of conformal field theory as applied to Calabi-Yau sigma models realized as twisted products of N = 2 minimal models, were able to establish that certain pairs of Calabi-Yau manifolds come in pairs in which their Hodge diamonds are mirror reflections (through a diagonal) of one another. Moreover, Greene and Plesser were able to establish that these pairs of CalabiYau manifolds, even though topologically distinct, when used as the basis for Calabi-Yau sigma models, give the same physical string theory. They named such pairs of Calabi-Yau manifolds mirror manifolds. The existence

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of such pairs of Calabi-Yau manifolds with specified properties are known to mathematicians as the Mirror Symmetry conjecture. Of the few hundred mirror manifold pairs which Greene and Plesser's approach explicitly generated, the most famous example is the Fermat quintic X in defined by the vanishing of

cr

f (zo, Zl,·· ., Z4) = z8 + zf + ... + zg + 'lj! (ZOZI .•• Z4) with its mirror being a crepant resolution of X/(715)3, known as the mirror quintic. Beyond constructing such pairs of mirror manifolds, Greene and Plesser noted that one implication of having a mirror pair yielding identical physical models is the existence of a highly nontrivial identity involving the so-called Yukawa couplings of each-quantities determined by the (quantum deformations of the) even cohomology ring of one manifold and the odd cohomology ring of its mirror. A short time later, Candelas, de la Ossa, Green and Parkes (CDGP) [52], studied the example of the mirror quintic pair by undertaking a detailed examination of the variation of Hodge structures of the mirror quintic. This work, interpreted mathematically by Morrison [221] and AspinwallMorrison [6]' produced a beautiful solution to a long-standing problem in enumerative geometry - "counting" rational curves on a general quintic. It is important to note that this work relied on another conjecture - the mirror map conjecture - purporting to give the explicit map between the moduli spaces of this pair of Calabi-Yau mirror manifolds. The foundational discovery of Greene and Plesser, and of Candelas-de la Ossa-Green-Parkes, helped set in motion one of the most spectacular developments in modern mathematics. A far-reaching generalization of the (genus zero) variation of Hodge structure - the so-called Kodaira-Spencer theory of gravity of Bershadsky, Cecotti, Ooguri and Vafa (BCOV) [21] - later led to conjectural counting formulas for GW invariants of all genera for many Calabi-Yaus. BCOV generalizes the variation of Hodge structure incorporating a natural hermitian structure which comes from the special Kahler geometry on the moduli space of Calabi-Yau manifolds. The generalized theory of Hodge structure is regarded as a special case of "t-t* geometry" of two dimensional N = 2 supersymmetric QFT due to Cecotti-Vafa [62]. At genus zero, t-t* geometry includes the successful associative relation, called WittenDijkgraaf-Verlinde-Verlinde [286, 15] (WDVV) equation, in quantum cohomology of projective manifolds. In 1993, BCOV [26] conjectured that one point function on a torus in the t-t* geometry of a Calabi-Yau manifold provides a non-trivial extension of the CDGP counting formula to genus one, called BCOV genus-one formula. Soon after, BCOV [21] introduced a certain recursion formula for higher genus (g ~ 2) GW invariants, which is called the holomorphic anomaly equation. This recursion formula due to BCOV is still under extensive study for its mathematical ground.

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A physical proof of mirror symmetry has been given by Hori-Vafa [137]. They demonstrated the equivalence at the level of two-dimensional gauged linear sigma model [289] which in the low-energy limit leads to the conformal field theory with Calabi-Yau manifold target space. (See also MorrisonPlesser [222] for an earlier attempt along the same lines.) 4.3. Mathematics inspired by mirror symmetry. One area inspired by the mirror symmetry conjecture is the construction of various enumerative invariants of Calabi-Yau manifolds. They appeared in the counting formula of CDGP. This development has led to a proof of the mirror symmetry conjecture by independent works of Givental [110, 29, 232] and of Lian-Liu-Yau [195]. In 1994, Kontsevich [164] expanded and formulated his version of mirror symmetry as an equivalence between complex and symplectic geometry of Calabi-Yau manifolds in all dimension. The geometric approach to mirror symmetry was finally unveiled by Strominger, Yau and Zaslow [265] in their 1997 paper in which they proposed that mirror symmetry is a geometric version of the Fourier transformation along dual special Lagrangian tori fibrations on mirror Calabi-Yau manifolds. This SYZ proposal has guided many research works. 5. Invariants of Calabi-Yau manifolds 5.1. Gromov-Witten invariants. GW invariants are enumerative invariants that play an integral part of the Mirror Symmetry conjecture. The GW invariants were introduced by physicists for counting the holomorphic curves in Calabi-Yau threefolds which are needed to calculate worldsheet instanton corrections to the sigma model part it on function. In their paper, CDGP proposed a formula that counts the number of rational curves of fixed degree on a general quintic Calabi-Yau. (For Calabi-Yau manifolds, Mori theory of rational curves does not apply and it has only be shown by Heath-Brown and Wilson that Calabi-Yau manifolds with Picard number p> 13 must have rational curves. See also [285, 239].) Interpreting the CDGP work mathematically, Aspinwall-Morrison [6] realized that the content of the CDGP formula were related to the work of Gromov [119]' who first introduced pseudoholomorphic curves to study symplectic geometry, and Witten's work on two-dimensional topological topological field theory [287, 288]. Since then, many mathematicians have contributed to the mathematical foundation of this invariants. There are two mathematical approaches to the problem - one based on symplectic geometry via pseudo-holomorphic maps to symplectic almost complex manifolds and the other based on algebraic geometry and the notion of stable maps. Both are dependent on understanding the gluing formula of WDVV [88], which has been interpreted to be the associative law of quantum cohomology. The rigorous approach to pseudo-holomorphic maps

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which proved Gromov's compactness theorem in full generality is due to Parker-Wolfson [236] and R. Ye [299] based on techniques developed by Sacks-Uhlenbeck [251] and Siu-Yau [260]. Around 1994, Ruan-Tian [250] used pseudo-holomorphic maps to define the symplectic GW invariants for all semipositive manifolds, which include Calabi-Yau manifolds. (The genus zero GW definition was also given by McDuff and Salamon [211].) From the algebro-geometric perspective, Kontsevich-Manin [165] also in 1994 gave an axiomatic treatment of GW classes and their properties for Fano varieties. Kontsevich, then introduced the notion of stable maps in algebraic geometry [163]. Using the moduli of stable maps, Li-Tian [185] and BehrendFantechi [24, 23] constructed the virtual cycles of DM-stack with perfect obstruction theories, thus constructing the GW invariants for all smooth projective varieties. The analytical framework for GW invariants was subsequently developed in full generality by Fukaya-Ono, Li-Tian, Ruan, Siebert [103, 186, 249, 255]; some details were clarified later by Zinger [303]. The two approaches give identical invariants, confirmed by Li-Tian and Siebert [187, 256].

5.2. Counting formulas. In 1993-1995, Hosono-Klemm-Theisen-Yau [138, 139] and Hosono-Lian-Yau [144] made an interesting observation that was crucial to the understanding of the CDGP counting formula (g = 0 mirror symmetry) and its generalizations. They observed that the Picard-Fuchs PDEs that compute the periods of a Calabi-Yau manifold have a "motivic" interpretation. Namely, in order for the classical Frobenius method to yield the periods, the Frobenius parameters must satisfy the cohomological relations on the mirror manifold exactly. Using a combinatorial recipe of Batyrev and Borisov [13, 14] for constructing Calabi-Yau manifolds in toric varieties, Hosono-Klemm-TheisenYau [138,139] and Hosono-Lian-Yau [144] showed that this motivic relation holds true for all such Calabi-Yau manifolds, first for Calabi-Yau complete intersections in weighted projective spaces, then in toric varieties, and finally for noncompact Calabi-Yau manifolds which are sums of line bundles over toric varieties. This motivic relation allowed them to write down the counting formula for a Calabi-Yau manifold easily: it expresses the genus zero GW invariants of a Calabi-Yau manifolds explicitly in terms of the special geometry prepotential of the mirror manifold. Independently, Candelas-de la Ossa-Font-Katz-Morrison [50, 51] also generalized the CDGP work and gave detail analyses of models of CalabiYau hypersurfaces in weighted projected space with two Kahler parameters. For positive genus, the Kodaira-Spencer theory of gravity of BershadskyCecotti-Ooguri-Vafa (BCOV) [27] has led to counting formulas for GW invariants of positive genera for many Calabi-Yau manifolds. Hosono-LianYau [144] generalized the BCOV genus-one formula to arbitrary Calabi-Yau complete intersections in toric varieties. Inspired by F-theory and M-theory,

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Klemm-Lian-Roan-Yau [161] have also found a counting formula for GW invariants of Calabi-Yau manifolds of dimension 4 or higher. More recently, based on the theory of BCOV and the geometry of the moduli of Calabi-Yau threefolds, Yamaguchi and Yau [292] have shown that for the quintic threefold the topological partition functions of all genera can be expressed explicitly as polynomials of five known holomorphic functions. They conjectured that similar polynomials exist for all Calabi-Yau threefolds. As shown by Dijkgraaf [74] in 1995, the BCOV theory applied to an elliptic curve has a close similarity to the theory of quasi-modular forms of Kaneko-Zagier [155]. It has also allowed Huang-Klemm-Quackenbush [146] to calculate the partition function up to genus 51 for the quintic. The discovery by Yamaguchi and Yau has led to renewed interest on quasi-modular forms. 5.3. Proofs of counting formulas for Calabi-Yau threefolds. In 1994, Kontsevich demonstrated [163] that one can approach the GW-invariants of quintics by applying the Atiyah-Bott localization formula to the top Chern classes of vector bundles on the stable map moduli spaces of cJrl. Though his method in principle can determine the genus zero GW-invariants of all degrees, more insights are required to settle the mirror conjecture for quintics. Two independent proofs of the CDGP formula used localization techniques in different ways. One approach based on quantum differential equation in the case of the quintics was due to Givental [110] in 1996, and was later expanded and clarified by others [29, 232] in 1998. An independent approach based on functorial localization was given by Lian-Liu-Yau [195] in 1997; they later generalized their work to complete intersections in toric varieties in 1999. (See [194] for a comparison of the two approaches.) The theory developed in [195, 196, 197] - which is called the mirror principle - have been applied to many other generalizations of the CDGP formula. By varying the possible K-classes and evaluating their Chern classes, their approach has also led to a number of new counting formulas for noncompact Calabi-Yau manifolds [195]. To prove the BCOV counting formula for higher genus GW invariants of quintics, a new localization formula for virtual fundamental classes had to be developed. This localization formula was worked out by Li and Zinger [190] for complete intersection of projective spaces; the genus-one formtila of BCOV for quintic Calabi-Yau was subsequently proved by Zinger [304J. 5.4. Integrability of mirror map and arithmetic applications. Mirror symmetry has many interesting and often unexpected connections and applications to mathematics. For instance, it is conjectured that near a certain large complex structure limit, the moduli space of a Calabi-Yau manifold admits certain special coordinates. Lian-Yau [198] conjectured that the power series expansion of the mirror map in these coordinates always have

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integer coefficients. These expansions depend on some choices, but the integrality seems to be independent of such choices. This integrality conjecture has been proved for quintics and several other Calabi-Yau threefolds with hi,i = 1, and a number of isolated examples with hi,i > 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

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[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

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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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