An Introduction to Symplectic Topology through Sheaf theory Princeton, Fall 2010-New York, Spring 2011 C. Viterbo April 30, 2011 15:35 Contents Chapter 1. Introduction 5 Part 1. Elementary symplectic geometry 7 Chapter 2. Symplectic linear algebra 9 1. Basic facts 9 2. Complex structure 13 Chapter 3. Symplectic differential geometry 17 1. Moser’s lemma and local triviality of symplectic differential geometry 17 2. The groups Ham and Di f f! 21 Chapter 4. More Symplectic differential Geometry: Reduction and Generating functions 25 1. Symplectic Reduction 25 2. Generating functions 28 3. The Maslov class 32 4. Contact and homogeneous symplectic geometry 32 Chapter 5. Generating functions for general Hamiltonians. 37 1. Applications 38 2. The calculus of critical values and first proof of the Arnold Conjecture 41 Part 2. Sheaf theory and derived categories 47 Chapter 6. Categories and Sheaves 49 1. The language of categories 49 2. Additive and Abelian categories 52 3. The category of Chain complexes 57 4. Presheaves and sheaves 60 5. Appendix: Freyd-Mitchell without Freyd-Mitchell 66 Chapter 7. More on categories and sheaves. 69 1. Injective objects and resolutions 69 2. Operations on sheaves. Sheaves in mathematics. 74 3. Injective and acyclic resolutions 79 3 4 CONTENTS 4. Appendix: More on injective objects 83 Chapter 8. Derived categories of Sheaves, and spectral sequences 87 1. The categories of chain complexes 87 2. Spectral sequences of a bicomplex. Grothendieck and Leray-Serre spectral sequences 94 3. Complements on functors and useful properties on the Derived category 102 Part 3. Applications of sheaf theory to symplectic topology 105 Chapter 9. Singular support in the Derived category of Sheaves. 107 1. Singular support 107 2. The sheaf theoretic Morse lemma and applications 114 3. Quantization of symplectic maps 116 4. Appendix: More on sheafs and singular support 121 Chapter 10. The proof of Arnold’s conjecture using sheafs. 125 1. Statement of the Main theorem 125 2. The proof 126 Index 129 Bibliography 133 CHAPTER 1 Introduction This are the notes of graduate lectures given in the fall semester 2010 at Prince- ton University, and then as the Eilenberg lectures at Columbia in the spring 2011. The first part of the symplectic part of the course (chapter 2 to 4) corresponds to a course given at Beijing Unversity on 2007 and 2009, with notes by Hao Yin (Shanghai Jiao- tong University). The aim of this course is to present the recent work connecting sheaf theory and symplectic topology, due to several authors, Nadler ([Nad, Nad-Z], [Tam], Guillermou-Kashiwara-Schapira [G-K-S]. This is completed by the approach of [F-S-S], and the paper [F-S-S2] really helped us to understand the content of these works. Even though the goal of the paper is to present the proof of the classical Arnold con- jecture on intersection of Lagrangians, and the more recent work of [F-S-S] and [Nad] on the topology of exact Lagrangians in T ¤X , we tried to explore new connections be- tween objects. We also tried to keep to the minium the requirements in category theory and sheaf theory necessary for proving our result. Even though the appendices contain some material that will be useful for those interested in pursuing the sheaf theoretical approach, much more has been omitted, or restricted to the setting we actually use1 The experts will certainly find that our approach is “not the right one”, as we take ad- vantage of many special features of the category of sheafs, and base our approach of derived categories on the Cartan-Eilenberg resolution. We can only refer to the papers and books in the bibliography for a much more complete account of the theory. The starting point is the idea of Kontsevich, about the homological interpreta- tion of Mirror symmetry. This should be an equivalence between the derived cate- gory of the Db(Fuk(M,!)), the derived cateogory of the category having objects the (exact) Lagrangians in (M,!) and morphisms the elements in the Floer cohomology (i.e. Mor(L1,L2) FH ¤(L1,L2)) the derived category of coherent sheafs on the Mirror, b Æ D (Coh(Mˇ ,J)). Our situation is a toy model, in which (M,!) (T ¤X ,d(pdq)), and Æ Db(Coh(Mˇ ,J)) is then replaced by Db(Sheaf (X R)) the category of constructible cstr £ sheafs (with possibly more restrictions) on X R. £ There is a functor b b SS : D (Sheaf (X R)) D (Fuk(T¤X,!)) cstr £ ¡! 1For example since the spaces on which our sheafs are defined are manifolds, we only rarely discuss assumptions of finite cohomological dimension. 5 6 1. INTRODUCTION determined by the singular support functor. The image does not really fall in b D (Fuk(T¤X,!)), since we must add the singular Lagrangians, but this a more a fea- ture than a bug. Moreover we show that there is an inverse map, called “ Quantiza- tion” obtained by associating to a smooth Lagrangian L, a sheaf over X , FL with fiber (FL)x (CF (L,Vx ),@x ) where Vx is the Lagrangian fiber over x and CF (L,Vx ),@x ) is Æ ¤ ¤ the Floer complex of the intersection of L and Vx . This is the Floer quantization of L. This proves in particular that the functor SS is essentially an equivalence of cat- egories. We are also able to explain the condition for the Floer quantization of L to be an actual quantization (i.e. to be well defined and provide an inverse to SS). Due to this equivalence, for complexes of sheafs F ²,G ² on X , we are able to define H ¤(F ²,G ²) H ¤(F ² (G ²)¤) as well as FH ¤(SS(F ),SS(G )) and these two objects co- Æ ­ incide. We may also define FH ¤(L,G ) as H ¤(FL,G ). I thank Hao Yin for allowing me to use his lecture notes from Beijing. I am very grateful to the authors of [Tam], [Nad], [F-S-S] and [F-S-S2] and [G-K-S] from where theses notes drew much inspiration, and in particular to Stéphane Guillermou for a talk he gave at Symplect’X seminar, which led me to presomptuously believe I could understand this beautiful theory, and to Pierre Schapira for patiently explaining me many ideas of his theory and dispelling some naive preconceptions, to Paul Seidel and Mohammed Abouzaid for discussions relevant to the General quantization the- orem. Finally I thank the University of Princeton, the Institute for Advanced Study and Columbia University for hospitality during the preparation of this course. A warm thanks to Helmut Hofer for many discussions and for encouraging me to turn these notes into book form. New-York, Spring 2011 This material is based upon work supported by the National Science Foundation under agreement DMS-0635607 and DMS-0603957. Any opinion, findings and con- clusions or recommendation expressed in this material are those of the author and do not necessariy reflect the views of the National Foundation. The author hereby certifies that this material has not been tested on animals. Part 1 Elementary symplectic geometry CHAPTER 2 Symplectic linear algebra 1. Basic facts Let V be a finite dimensional real vector space. DEFINITION 2.1. A symplectic form on V is a skew-symmetric bilinear nondegen- erate form, i.e. a two-form satisfying: (1) x, y V !(x, y) !(y,x) 8 2 Æ¡ ( x V !(x,x) 0); Æ) 8 2 Æ (2) x, y such that !(x, y) 0. 8 9 6Æ For a general 2-form ! on a vector space, V , we denote by Ker(!) the subspace given by Ker(!) {v V w V !(v,w) 0} Æ 2 j 8 2 Æ The second condition implies that Ker(!) reduces to zero, so when ! is symplectic, there are no “preferred directions” in V . There are special types of subspaces in symplectic manifolds. For a vector subspace F , we denote by F ! {v V w F , !(v,w) 0} Æ 2 j 8 2 Æ the symplectic orthogonal From Grassmann’s formula applied to the surjective map ! ' : V F ¤ given by ' (v) !(v, ), it follows that dim(F ) dim(Ker(' )) codim(F) F ! F Æ ² Æ F Æ Æ dim(V) dim(F). Moreover the proof of the following is left to the reader ¡ PROPOSITION 2.2. (F !)! F Æ (F F )! F ! F ! 1 Å 2 Æ 1 \ 2 DEFINITION 2.3. A map ' :(V ,! ) (V ,! ) is a symplectic map if '¤(! ) ! ) 1 1 ! 2 2 2 Æ 1 that is x, y V ,! ('(x),'(y)) ! (x, y). It is a symplectomorphism if and only if it 8 2 1 2 Æ 1 is invertible- its inverse is then necessarily symplectic. A subspace F of (V,!) is isotropic if F F ! ( ! 0); ² ½ () jF Æ coisotropic if F ! F ² ½ Lagrangian if F ! F . ² Æ PROPOSITION 2.4. (1) Any symplectic vector space has even dimension. 9 10 2. SYMPLECTIC LINEAR ALGEBRA (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space. (3) If (V1,!1), (V2,!2) are symplectic vector spaces with L1,L2 Lagrangian sub- spaces, and if dim(V ) dim(V ), then there is a linear isomorphism ' : V V 1 Æ 2 1 ! 2 such that '¤! ! and '(L ) L . As a consequence, any two symplectic vec- 2 Æ 1 1 Æ 2 tor spaces of the same dimension are symplectomorphic. PROOF. We first prove that if I is an isotropic subspace it is contained in a La- grangian subspace. Indeed, I is contained in a maximal isotropic subspace.