Fukaya categories and Picard-Lefschetz theory

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Zurich Lectures in Advanced Mathematics Edited by Erwin Bolthausen (Managing Editor), Freddy Delbaen, Thomas Kappeler (Managing Editor), Christoph Schwab, Michael Struwe, Gisbert Wüstholz Mathematics in Zurich has a long and distinguished tradition, in which the writing of lecture notes volumes and research monographs play a prominent part. The Zurich Lectures in Advanced Mathematics series aims to make some of these publications better known to a wider audience. The series has three main constituents: lecture notes on advanced topics given by internationally renowned experts, graduate text books designed for the joint graduate program in Mathematics of the ETH and the University of Zurich, as well as contributions from researchers in residence at the mathematics research institute, FIM-ETH. Moderately priced, concise and lively in style, the volumes of this series will appeal to researchers and students alike, who seek an informed introduction to important areas of current research. Previously published in this series: Yakov B. Pesin, Lectures on partial hyperbolicity and stable ergodicity Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry Sergei B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Pavel Etingof, Calogero-Moser systems and representation theory Guus Balkema and Paul Embrechts, High Risk Scenarios and Extremes – A geometric approach Demetrios Christodoulou, Mathematical Problems of General Relativity I Camillo De Lellis, Rectifiable Sets, Densities and Tangent Measures Michael Farber, Invitation to topological robotics Alexander Barvinok, Integer points in polyhedra Published with the support of the Huber-Kudlich-Stiftung, Zürich

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Paul Seidel

Fukaya Categories and Picard–Lefschetz Theory

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Author: Prof. Paul Seidel MIT Room 2-270 77 Massachusetts Avenue Cambridge, MA 02139 USA E-mail: [email protected]

2000 Mathematics Subject Classification (primary; secondary): 53D40; 32Q65, 53D12, 16E45 Key words: Floer homology, Fukaya category, homological algebra, Lagrangian submanifolds, Lefschetz fibrations, mirror symmetry, pseudo-holomorphic curve, symplectic geometry

ISBN 978-3-03719-063-0 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

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© 2008 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed on acid-free paper produced from chlorine-free pulp. TCF °° Printed in Germany 987654321

Preface

The subject of Fukaya categories has a reputation for being hard to approach. This is due to the amount of background knowledge required (taken from homological algebra, symplectic geometry, and geometric analysis), and equally to the rather complicated nature of the basic definitions. The present book is intended as a resource for graduate students and researchers who would like to learn about Fukaya categories, and possibly use them in their own work. I have tried to focus on a rather basic subset of topics, and to describe these as precisely as I could, filling in gaps found in some of the early references. This makes for a rather austere style (for that reason, a thorough study of this book should probably be complemented by reading some of the papers dealing with applications). A second aim was to give an account of some previously unpublished results, which connect Fukaya categories to the theory of Lefschetz fibrations. This becomes predominant in the last sections, where the text gradually turns into a research monograph. I have borrowed liberally from the work of many people, first and foremost among them Fukaya, Kontsevich, and Donaldson. Fukaya’s foundational contribution, of course, was to introduce A1 -structures into symplectic geometry. On the algebraic side, he pioneered the use of the A1 -version of the Yoneda embedding, which we adopt systematically. Besides that, several geometric ideas, such as the role of Pin structures, and the construction of A1 -homomorphisms in terms of parametrized moduli spaces, are taken from the work of Fukaya, Oh, Ohta and Ono. Kontsevich introduced derived categories of A1 -categories, and is responsible for much of their theory, in particular the intrinsic characterization of exact triangles. He also conjectured the relation between Dehn twist and twist functors, which is one of our main results. Finally, in joint work with Barannikov, he suggested a construction of Fukaya categories for Lefschetz fibrations; we use a superficially different, but presumably equivalent, definition. Donaldson’s influence is equally pervasive. Besides his groundbreaking work on Lefschetz pencils, he introduced matching cycles, and proposed them as the starting point for a combinatorial formula for Floer cohomology, which is indeed partly realized here. Other mathematicians have also made important contributions. For instance, parts of our presentation of Picard–Lefschetz theory reflect Auroux’ point of view. A result of Smith, namely that the vanishing cycles in a four-dimensional Lefschetz pencil necessarily fill out the fibre, was crucial in suggesting that such cycles might “split-generate” the Fukaya category. Besides that, work of Fukaya–Smith on cotangent bundles provided a good testing-ground for some of the more adventurous ideas about Lefschetz fibrations. Our approach to transversality issues is the result of several conversations with Lazzarini. Finally, Abouzaid’s suggestions greatly improved the discussion of symplectic embeddings.

vi

Preface

Among organizations, I would like to thank ETH Zürich for inviting me to teach the course on which this book is based; the NSF for financial support (under grants DMS-9729992 and DMS-0405516); and the EMS Publishing House for accepting the manuscript, and shepherding it to publication.

Contents

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

v

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I

A1 -categories . . . . . . . . . . . . . . . 1 Definitions . . . . . . . . . . . . . . 2 Identity morphisms and equivalences 3 Exact triangles . . . . . . . . . . . . 4 Idempotents . . . . . . . . . . . . . 5 Twisting . . . . . . . . . . . . . . . 6 Z=2-actions . . . . . . . . . . . . .

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II Fukaya categories . . . . . . . . . . . . . . . . 7 A little symplectic geometry . . . . . . . . 8 Classical Floer theory . . . . . . . . . . . 9 The Fukaya category (preliminary version) 10 Some basic properties . . . . . . . . . . . 11 Indices and determinant lines . . . . . . . 12 The Fukaya category (complete version) . 13 Polygons on surfaces . . . . . . . . . . . . 14 Symplectic involutions . . . . . . . . . . .

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III Picard–Lefschetz theory . . . . . . . . . . . . . 15 First notions . . . . . . . . . . . . . . . . . 16 Vanishing cycles and matching cycles . . . . 17 Pseudo-holomorphic sections . . . . . . . . 18 The Fukaya category of a Lefschetz fibration 19 Algebraic varieties . . . . . . . . . . . . . . 20 .Am / type Milnor fibres . . . . . . . . . . .

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211 212 220 235 265 291 304

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Introduction

First impressions. A Fukaya category is an algebraic structure associated to a symplectic manifold. It encodes information about Lagrangian submanifolds, the way in which any two of them intersect, and also about pseudo-holomorphic discs (or polygons). The amount of data packed into a single object makes Fukaya categories fascinating, but also somewhat intimidating, especially from a symplectic topology viewpoint. Before going on, we would like to elaborate a little further on this observation. This will be an informal discussion, addressed to non-specialist readers, and intended to convey just a general idea of what the situation looks like; it contains many statements that are not rigorous, and it is also somewhat more wide-ranging than the rest of book. Fix a symplectic manifold M 2n . Roughly speaking, objects of the Fukaya category are Lagrangian submanifolds Ln  M . The space of such submanifolds is infinite-dimensional (locally modelled on the space of closed one-forms on L). However, Lagrangian submanifolds which are Hamiltonian isotopic give rise to isomorphic objects in the Fukaya category, which cuts down the local degrees of freedom to the first Betti number b1 .L/. Indeed, the classification up to Hamiltonian isotopy is completely understood in some simple cases: curves on surfaces, where the issue is essentially one of topology; less obviously, Lagrangian spheres in S 2  S 2 [75], and a few related examples. There are other (general) effects which work in one’s favour: not all Lagrangian submanifolds actually appear in the Fukaya category, due to obstructions in the sense of [60]; others may occur but be trivial (the zero object); and isomorphism in the Fukaya category may be weaker than Lagrangian isotopy. For an example of the first point, one can simply look at circles in S 2 , where the necessary cancellation of obstructions only happens when the two hemispheres in the complement have equal areas; for more interesting cases see [31], [32]. The second phenomenon occurs for the real locus RPn  CPn for n > 1, assuming that that is Spin (n D 4k  1), and that the Fukaya category is taken with Q-coefficients. For the third point, there are some candidate examples in the form of Lagrangian tori in CP2 , but as far as this author knows, none of them have been rigorously verified. With all this at hand, it becomes feasible in principle to compute the entire Fukaya category for manifolds whose symplectic topology is comparatively simple; the papers quoted above (and others, for instance [132], [102], [101], [59]) can be viewed as part of a wider ongoing effort in that direction. Still, for any reasonably complicated symplectic manifold (for instance, a complex hypersurface in CP3 of degree  4), a direct classification of the objects in the Fukaya category appears to be out of reach. Maybe we have just been setting the goalposts too high. After all, even if one considers a purely algebraic context, such as modules over a finite-dimensional alge-

2

Introduction

bra, the cases where a complete classification can be achieved form a tiny minority [41]. There is, however, one significant difference: for two explicitly given modules, there is an obvious algorithm which decides whether they are isomorphic. More generally, given a finite collection of modules, it is straightforward to determine the morphisms between them, and the composition maps of morphisms. In contrast, in a Fukaya category, the morphisms are Floer cochain groups, and the composition maps operations on such groups, both defined in terms of suitable moduli spaces of pseudo-holomorphic maps. There are some situations where these spaces can be dealt with geometrically: when M is a cotangent bundle and the Lagrangian submanifolds L D graph.df / are sections, an “adiabatic limit” method reduces the question to one in finite-dimensional Morse theory [58] (variations of this method have been successful in several related cases); when M is an algebraic variety defined over the real numbers, and one considers only the real locus L D MR , algebro-geometric tools can be applied; finally, in the case where M is a surface, the study of pseudo-holomorphic curves becomes combinatorial to a large extent. However, beyond these situations, Floer theory computations traditionally rely on case-by-case methods, which offer no a priori guarantee of success. This book is an attempt to address that problem. Together with other developments, it will hopefully help to make Fukaya categories into a more manageable tool. The strategy. In view of the difficulties described above, it is clear that if one wants to make any headway, some drastic simplifications are needed. To begin with, we consider only exact symplectic manifolds M , namely those where the symplectic form ø is the exterior derivative of some one-form . This of course implies that M cannot be closed, but we impose standard convexity conditions, which make this deficiency invisible to finite area pseudo-holomorphic curves. Affine algebraic varieties are a good source of examples. The simplest class of Lagrangian submanifolds L  M are the exact ones, which means that jL is the derivative of some function; we will only deal with these, and exclude all others a priori. This restriction has many useful technical consequences: for instance, exact Lagrangian submanifolds cannot be obstructed, and each of them is a nonzero object of the Fukaya category, whose endomorphism ring is the classical cohomology ring (no quantum corrections). Beyond that, the more fundamental advantage of exactness is that it eliminates the dependence of the Fukaya category on the symplectic area (or Novikov ring) parameter. Mirror symmetry suggests that this dependence is often given by nontrivial transcendental functions, and this is confirmed by sample computations, such as [108], [57]. In our case, the parameter is set equal to zero (or infinity, depending on one’s interpretation), which makes the situation less interesting from an enumerative point of view, but more amenable to computation. We should mention that ultimately, one can try using the Fukaya categories of affine varieties as a stepping-stone towards those of their projective completions [129].

Introduction

3

To bring things closer to classical homological algebra, we will also impose a Calabi–Yau type condition (2c1 .M / D 0, to be precise), together with a corresponding assumption on the Lagrangian submanifolds, which allows us to equip all Floer groups with Z-gradings. The resulting setup can be summarized as follows. Objects of the Fukaya category F .M /, which we call exact Lagrangian branes, are triples L# D .L; ˛ # ; P # /, where L is a closed exact Lagrangian submanifold of M , ˛ # is a real-valued function on L called the grading, and P # is a Pin structure on L. The morphisms spaces are Floer cochain groups CF  .L#0 ; L#1 /, which are finite-dimensional graded vector spaces (over some fixed, but arbitrary, coefficient field K; the Pin structures are really only necessary if char.K/ ¤ 2). These carry multilinear composition maps d , d  1, which form an A1 -structure. The definition requires auxiliary choices, but the outcome is independent of those choices up to quasi-isomorphism. Our second main idea, which comes from Kontsevich’s work, is to take formal enlargements of the Fukaya category. These are purely algebraic constructions, which can be applied to any A1 -category A. The first step is to introduce analogues of chain complexes, so-called twisted complexes, which form another A1 -category Tw A containing A. This contains mapping cones of morphisms, hence is what we call a triangulated A1 -category; the resulting cohomology level category H 0 .Tw A/, also called the derived category D.A/, is a triangulated category in the classical sense. As a second useful step, one can take the split-closure (or Karoubi completion) D  .A/ of the derived category, which introduces formal direct summands of all idempotent endomorphisms; there is also a corresponding construction on the A1 level, denoted by ….Tw A/. An apparently paradoxical statement, but one whose truth will be evident to many readers, is that each enlargement makes the category easier to describe. As an illustration, consider the notion of split-generators: we say that a subset of objects split-generates ….Tw A/ if one can construct all objects out of them by repeatedly applying mapping cones and splitting off direct summands. Whenever B  ….Tw A/ is the full A1 -subcategory formed by such a set of splitgenerators, ….Tw B/ is quasi-equivalent to ….Tw A/. Hence, if one is only interested in ….Tw A/, it is actually sufficient to determine one such B. Let us keep in mind that split-generation is a very weak property; it does not, for instance, imply that the Grothendieck group K0 .D  .A// is spanned by the classes of split-generators. Hence, it is not unreasonable to hope that a finite set of Lagrangian submanifolds can be found which split-generate ….Tw F .M //, and we will see that this is indeed the case in favourable circumstances. The use of the word “computation” in connection with Fukaya categories means that the underlying geometric structures (symplectic manifolds, Lagrangian submanifolds) have to be somehow encoded in a combinatorial way. For that, we will make systematic use of Picard–Lefschetz theory. In general, the importance of this approach in symplectic geometry largely comes from Donaldson’s work [38]; here, we only use the most elementary aspects of the theory, namely the symplectic nature of

4

Introduction

monodromy maps, as well as the Lagrangian nature of Lefschetz thimbles and vanishing cycles. To take advantage of these properties, we will introduce (following an idea of Kontsevich) Fukaya categories of Lefschetz fibrations, which in a sense stand between the ordinary Fukaya category of the total space and that of a smooth fibre. These will then be used to build a machine, first proposed conjecturally in [126], for doing computations by dimensional induction. Computing Fukaya categories. To simplify formulations, we will limit ourselves to the case of algebraic varieties. Namely, let X  CN be an .nC1/-dimensional affine algebraic variety which is smooth, as well as smooth at infinity. The latter condition means that its closure Xx  CPN is again smooth, and intersects the hyperplane at infinity CPN 1 D CPN n CN transversally. We equip X with the restriction of the Fubini–Study Kähler form, which turns it into an exact symplectic manifold. Additionally, we will assume that the canonical bundle KXx is isomorphic to OXx .d / for some d 2 Z, which makes X Calabi–Yau. All these properties are also inherited by a generic affine hyperplane section X 1 D X \ CN 1 . Take the Fukaya categories F .X/ and F .X 1 /, with coefficients in some field K. We begin by discussing X 1 , since our results there are somewhat better: Theorem A. Suppose that d ¤ 2, n  1, and char.K/ ¤ 2. Then ….Tw F .X 1 // is computable from the combinational data obtained by applying Picard–Lefschetz theory to X . The statement requires some explanation. First of all, by saying that ….Tw A/ is computable for some A1 -category A, we mean the following. One can explicitly write down another A1 -category B, which is finite in a strong sense (finitely many objects, the morphisms form finite-dimensional vector spaces, and composition maps dB of sufficiently high order d  0 all vanish), such that ….Tw B/ and ….Tw A/ are quasi-equivalent A1 -categories. In particular, this means that D  .B/ Š D  .A/ as (classical) triangulated categories. Note that for a reasonable choice of coefficient field K (a finite field, or Q, for instance), B really contains a finite amount of information. One can take that information and enter it into a computer program which, by explicitly constructing twisted complexes and their idempotent endomorphisms, will generate the full (infinite) list of isomorphism classes of objects in D  .B/, together with morphisms, composition maps, and exact triangles. This, of course, does not by itself solve any of the big structural questions about D  .B/. As for the meaning of “combinational data produced by Picard–Lefschetz theory”, we leave the precise explanation to Section 19, but the rough idea is to look at linear maps from X to C and C2 , and encode the branch data of those maps in terms of braid monodromy. The same process can be applied to X 1 , and repeated until the dimension is reduced to zero. Given explicit equations defining X, these data can be extracted algorith-

Introduction

5

mically using elimination theory (even though in all but the simplest examples, the complexity tends to be prohibitive). We also need to look briefly at the limitations of the theorem. n  1 excludes the lowest nontrivial dimension, where X 1 is a Riemann surface. This is really for convenience only: the Riemann surface case can be treated with similar (in fact, somewhat simpler) methods, but requires some particular adjustments, which would complicate the discussion. The char.K/ ¤ 2 condition is a technical artifact of our proof, which uses double branched covers and Z=2-actions; quite probably, other methods exist which would allow one to lift this restriction. The other requirement d ¤ 2 is somewhat more essential, and we postpone its discussion to a later point (see Remark 19.8). As mentioned before, the results for F .X/ itself are somewhat weaker. This is not unexpected: for instance, there is no known method for deciding whether X contains any closed exact Lagrangian submanifold, hence whether F .X/ (and its derived categories) are nontrivial. This is in contrast to the case of X 1 , where we could use vanishing cycles as a natural source of objects. The best we can do is this: Theorem B. Assume that char.K/ ¤ 2. Then, from the combinatorial data obtained by applying Picard–Lefschetz theory to X, one can construct an A1 -category which contains a full subcategory quasi-equivalent to F .X/. The A1 -category in question is of the form Tw B, where B is finite in the same sense as before. In fact, by construction B will be directed, and this has some noteworthy consequences. First of all, Tw B is already split-closed, so it will contain not only F .X/, but the whole of ….Tw F .X//. Secondly, directed A1 -categories are quasi-isomorphic to dg categories with finite-dimensional morphism spaces. Since that property is inherited by the category of twisted complexes, we have: Corollary C. In the situation of Theorem B, F .X/ is quasi-isomorphic to a dg category with finite-dimensional hom spaces. The practical usefulness of Theorem B depends on how well one understands the geometric meaning of the map F .X/ ! Tw B. For the simplest class of Lagrangian submanifolds in X, namely Lagrangian spheres which can be represented as matching cycles for a Lefschetz pencil of hyperplane sections, these issues can be addressed easily, so that one indeed gets an algorithm for computing Floer cohomology groups (strictly speaking, the case n D 1 ought to be excluded here, for the same reason as in Theorem A). For more general Lagrangian submanifolds, the statement remains more of a theoretical nature, at least for now. The proofs of the results stated above will be given in Section 19; they involve most of the material introduced throughout the book. As a natural by-product, we establish a general relation between Hurwitz moves of vanishing cycles, and mutations

6

Introduction

of directed Fukaya categories. In fact, we will give two proofs of this (Theorem 17.20 and Theorem 18.24; the proof originally envisaged in [127] is different from either one, and remains unpublished at present). Another interesting consequence is the existence of a spectral sequence for the Floer cohomology of Lagrangian submanifolds lying in the total space of a Lefschetz fibration (Corollary 18.27). Of course, like any spectral sequence, this merely decomposes the problem into a visible part (the starting term) and hidden higher order information (the differentials). In this particular case, the differentials express the way in which the Lagrangian submanifolds are built up from Lefschetz thimbles, where the word “built” is interpreted algebraically, as a Postnikov decomposition in the Fukaya category. With one exception at the end, this book does not deal with examples or applications. Luckily, the existing literature makes up for that shortcoming (for some pointers, see the beginning of Chapter III). Beyond that, any reader seriously interested in Fukaya categories will eventually have to tackle the groundbreaking [60], which discusses many geometric and algebraic issues far outstripping the framework set up here.

I A1 -categories

This chapter describes the homological algebra underlying Fukaya categories. We begin with a quick review of A1 -structures. This is by no means a complete exposition of the theory. Basically, the only aspect which is discussed at length is that of identity morphisms, since it is potentially tricky for Fukaya categories. Also, the formalism is deliberately kept down-to-earth. This means that we do not use operads or noncommutative geometry, even though either language could shed much light into the tangle of formulae. There are alternative references in various flavours for this material. Here are a few suggestions: some of the foundational older papers, for instance [70], [62], still make good reading, even though the emphasis is somewhat different from the one here. Turning to more recent sources, Keller’s work [82], [83], [84] contains accessible explanations of the basic material, as well as applications to questions in algebra and representation theory. Lefevre’s thesis [95] is a comprehensive account from a similar point of view, in the language of homotopical algebra. Fukaya and his collaborators have written several expositions geared specifically towards symplectic geometry [51], [52], [53], [58], [60], [56], [55]. The level of generality in the last three references far exceeds our needs, but the reader who is willing to take that into account will find that many arguments can be scaled down to our more modest framework. Finally, there is a wealth of insights in Kontsevich’s papers and lecture notes [90], [88], [89]. The overall theme of the rest of the chapter is the construction and manipulation of objects in A1 -categories. We discuss mapping cones and exact triangles, as well as the splitting of objects into direct summands. We will also look at one more specific construction, the so-called twist of one object along another. This has previously appeared in the literature in two distinct contexts. One is the notion of exceptional collections and mutations, which arose as a means of constructing rigid vector bundles in algebraic geometry: see [67] and the collection [117]. The theory was gradually extended to general triangulated categories (see [65], [26], [27], [28] for various aspects of this); a useful up-to-date survey is [66]. The second and more recent occurrence of the twisting construction is in connection with spherical objects, where it gives rise to automorphisms of derived categories [134]. The generalization to A1 -categories was pioneered by Kontsevich, whose ideas we borrow freely. Besides setting up the general formalism, we also derive some consequences, notably Corollary 5.8 and Proposition 5.17, which will come in useful later on, when making the connection with symplectic geometry and Lefschetz fibrations. The chapter ends on a somewhat technical note, with a short discussion of A1 -categories with Z=2-symmetry.

8

I A1 -categories

1 Definitions (1a) Categories. Fix a coefficient field K. All our categories will be assumed to be linear over K and small. A non-unital A1 -category A consists of a set of objects ObA, a graded vector space homA .X0 ; X1 / for any pair of objects, and composition maps of every order d  1, dA W homA .Xd 1 ; Xd / ˝    ˝ homA .X0 ; X1 / ! homA .X0 ; Xd /Œ2  d : (1.1) As usual, Œk means shifting the grading of a vector space down by k 2 Z. The maps (1.1) must satisfy the (quadratic) A1 -associativity equations X d mC1 .1/zn A .ad ; : : : ; anCmC1 ; m A .anCm ; : : : ; anC1 /; an ; : : : ; a1 / D 0; m;n

(1.2) where the sum is over all possible terms .1  m  d , 0  n  d  m), and the sign is determined by zn D ja1 j C    C jan j  n. The same sum will appear in many other sign expressions, so we declare it to be standing notation from now on. One can associate to any non-unital A1 -category A its cohomological category H.A/. This has the same objects as A; morphism spaces are the cohomology groups H.homA .X0 ; X1 /; 1A /; and composition is defined by Œa2   Œa1  D .1/ja1 j Œ2A .a2 ; a1 /:

(1.3)

Except for the possible lack of identity morphisms, this is an ordinary linear graded category. There is also a variant of this, namely the subcategory H 0 .A/  H.A/ which retains only the cohomology in degree zero. Along the same lines, a nonunital A1 -category A with dA D 0 for all d > 2 is the same thing as a non-unital differential graded (dg) category. More precisely, the associated dg category has @a D .1/jaj 1A .a/, and a2 a1 D .1/ja1 j 2A .a2 ; a1 / as in (1.3). The opposite of a non-unital A1 -category is defined by setting ObAopp D ObA, homAopp .X0 ; X1 / D homA .X1 ; X0 /, and reversing order in the composition maps: dAopp .ad ; : : : ; a1 / D .1/zd dA .a1 ; : : : ; ad /: On the cohomological level, this yields the ordinary opposite category (taken, however, without the Koszul signs) H.Aopp / D H.A/opp . Remark 1.1. Some footnotes about terminology and conventions: (i) “non-unital” should really be “not necessarily unital”. (ii) The fact that our categories are supposed to be small causes occasional formal trouble, for instance when defining twisted complexes; we will leave it to the reader to take the necessary set-theoretic precautions. (iii) When using categories without identity morphisms, one needs to proceed with

9

1 Definitions

care: for instance, there is no sensible notion of isomorphic objects; full and faithful functors make sense, and so do isomorphisms of categories, but equivalences do not, or at least not in a straightforward way. (iv) Our signs follow the “bar-convention”, which is the one most commonly used by algebraists; the decreasing numbering in (1.1) is an attempt to reconcile the standard convention for composition of morphisms with the equally standard cyclic ordering of marked points on the circle, see Section (9c). Remark 1.2. The compositions of order d > 2 are not chain maps. Nevertheless, they have partial incarnations on the cohomological level, in the form of Massey products. Suppose that we have morphisms Œak  2 HomH.A/ .Xk1 ; Xk / (k D 1; 2; 3) such that both products Œa3   Œa2  and Œa2   Œa1  are zero. We choose chain representatives ak , and additionally hk 2 homA .Xk1 ; XkC1 / (k D 1; 2) such that 1A .hk / D 2A .akC1 ; ak /. Then c D 3A .a3 ; a2 ; a1 /  2A .h2 ; a1 /  2A .a3 ; h1 / 2 homA .X0 ; X3 / is a cocycle of degree ja1 j C ja2 j C ja3 j  1, and the class Œc 2

HomH.A/ .X0 ; X3 / Œa3   HomH.A/ .X0 ; X2 / C HomH.A/ .X1 ; X3 /  Œa1 

(1.4)

depends only on the original Œak . One usually writes hŒa3 ; Œa2 ; Œa1 i D .1/ja2 j Œc. (1b) Functors. A non-unital A1 -functor between two non-unital A1 -categories A, B consists of a map F W ObA ! ObB and multilinear maps of all orders d  1, F d W homA .Xd 1 ; Xd / ˝    ˝ homA .X0 ; X1 / ! homB .F X0 ; F Xd /Œ1  d  (1.5) for X0 ; : : : ; Xd 2 ObA, satisfying the following (polynomial) equations: X X rB .F sr .ad ; : : : ; ad sr C1 /; : : : ; F s1 .as1 ; : : : ; a1 // (1.6) r

s1 ;:::;sr

D

X .1/zn F d mC1 .ad ; : : : ; anCmC1 ; m A .anCm ; : : : ; anC1 /; an ; : : : ; a1 /; m;n

where the left-hand sum is over all r  1 and partitions s1 C    C sr D d . There is an induced ordinary (linear, graded, non-unital) functor H.F / W H.A/ ! H.B/, whose action on morphisms is Œa 7! ŒF 1 .a/. We say that F is a quasiisomorphism if H.F / is an isomorphism of categories. We say that F is cohomologically full and faithful if H.F / is full and faithful. Non-unital A1 -functors can be composed as follows: .G ı F /d .ad ; : : : ; a1 / X X D G r .F sr .ad ; : : : ; ad sr C1 /; : : : ; F s1 .as1 ; : : : ; a1 //; r

s1 ;:::;sr

(1.7)

10

I A1 -categories

where the sum is as in (1.6). Composition is strictly associative, and the identity functor (F X D X, F 1 D id as an endomorphism of homA .X; X/, all higher order terms zero) is a neutral element. (1c) Formal diffeomorphisms. Here is a slightly different viewpoint on (a special class of) A1 -functors. Suppose that we have a set ObA of objects, and for each pair of objects a graded vector space homA .X0 ; X1 /, but for now without any composition maps. A formal diffeomorphism ˆ from A to itself is an arbitrary sequence of maps (1.5), with the only condition that each ˆ1 should be a linear automorphism of homA .X0 ; X1 /. The formula (1.7) turns formal diffeomorphism into a group. Now suppose that we have maps A which make A into a non-unital A1 -category. Given ˆ, one can show by recursively solving (1.6) that there is a unique other A1 -structure Q (with the same underlying vector spaces) such that ˆ W A ! AQ is a non-unital A A1 -functor. This defines an action of the group of formal diffeomorphism on the set of all possible A ; we write AQ D ˆ A. Remark 1.3. The name “formal diffeomorphism” comes from noncommutative geometry, where one can think of A1 -structures as vector fields of degree 1 on graded noncommutative affine space, and the action is just a change of coordinates. (1d) Natural transformations. Non-unital A1 -functors F W A ! B are themselves the objects of a non-unital A1 -category Q D nu-fun.A; B/. An element T 2 homgQ .F0 ; F1 / of the chain space of morphisms in this category is a sequence .T 0 ; T 1 ; : : : / where each T d is a family of multilinear maps homA .Xd 1 ; Xd / ˝    ˝ homA .X0 ; X1 / ! homB .F0 X0 ; F1 Xd /Œg  d ; for all .X0 ; : : : ; Xd /. In particular, T 0 is a family of elements in homgB .F0 X; F1 X/, one for each object X of A. Following Fukaya, we call such T a pre-natural transformation of degree g D jT j from F0 to F1 . The boundary operator is (1.8) 1Q .T /d .ad ; : : : ; a1 / X X  s s iC1 D .1/ rB F1 r .ad ; : : : ; ad sr C1 /; : : : ; F1 .: : : ; as1 CCsi C1 /; r;i s1 ;:::;sr

T si .as1 CCsi ; : : : ; as1 CCsi1 C1 /; s



X m;n



F0 i1 .as1 CCsi1 ; : : : /; : : : ; F0s1 .as1 ; : : : ; a1 / .1/zn CjT j1 T d mC1 .ad ; : : : ; anCmC1 ; m A .anCm ; : : : ; anC1 /; an : : : ; a1 /:

The first sum is over 1  i  r and partitions s1 C  Csr D d , where si may be zero; and  D .jT j1/.ja1 jC  Cjas1 CCsi1 js1   si1 /. The cocycles 1Q .T / D 0

11

1 Definitions

are the actual natural transformations, and coboundaries provide a notion of chain homotopy between them. The formula for the composition of T1 2 homQ .F0 ; F1 /, T2 2 homQ .F1 ; F2 / is much simpler: 2Q .T2 ; T1 /d .ad ; : : : ; a1 / X X s D .1/ı rB .F2sr .ad ; : : : ; ad sr C1 /; : : : ; F2 j C1 .: : : /; r;i;j s1 ;:::;sr s

s

s

T2 j .as1 CCsj ; : : : ; as1 CCsj 1 C1 /; F1 j 1 .: : : /; : : : ; F1 iC1 .: : : /; s

(1.9)

s

T1 i .as1 CCsi ; : : : ; as1 CCsi1 C1 /; F0 i1 .: : : /; : : : ; F0s1 .as1 ; : : : ; a1 //; Ps1 CCsi1 Ps1 CCsj 1 .jT2 j  1/.jak j  1/ C kD1 .jT1 j  1/.jak j  1/. Note with ı D kD1  d that A no longer appears. The formulae for Q , d > 2, follow exactly the same pattern as (1.9). Given a natural transformation T between A1 -functors F0 ; F1 W A ! B, consider the elements ŒTX0  2 HomH.B/ .F0 X; F1 X/. These satisfy the naturality condition ŒTX01   ŒF01 .a/ D .1/jajjT j ŒF11 .a/  ŒTX00  for all Œa 2 HomH.A/ .X 0 ; X 1 /. Hence, they constitute a natural transformation, which we denote by H.T /, between the cohomological functors H.F0 /; H.F1 /. More formally, denote by Nu-Fun.H.A/; H.B// the category consisting of nonunital (linear, graded) functors H.A/ ! H.B/ and their natural transformations. Then, the assignment F 7! H.F /, T 7! H.T / defines a non-unital functor H.nu-fun.A; B// ! Nu-Fun.H.A/; H.B//:

(1.10)

(1e) Composition functors. Left and right composition with a fixed G W A ! B defines, for any non-unital A1 -category C , two non-unital A1 -functors LG W nu-fun.C; A/ ! nu-fun.C; B/; RG W nu-fun.B; C/ ! nu-fun.A; C/: We have already described their action on objects. The first order map on pre-natural transformations is, for T 2 homnu-fun.C;A/ .F0 ; F1 /, .L1G T /d .ad ; : : : ; a1 / X X s D .1/ G r .F1sr .ad ; : : : ; ad sr C1 /; : : : ; F1 iC1 .: : : /; r;i s1 ;:::;sr

T si .as1 CCsi ; : : : ; as1 CCsi1 C1 /; s

F0 i1 .: : : /; : : : ; F0s1 .as1 ; : : : ; a1 //

(1.11)

12

I A1 -categories

with  as in (1.8); and for T 2 homnu-fun.B;C/ .F0 ; F1 /, .RG1 T /d .ad ; : : : ; a1 / X X D T r .G sr .ad ; : : : ; ad sr C1 /; : : : ; G s1 .as1 ; : : : ; a1 //: r

s1 ;:::;sr

The higher order terms RGd , d  2, all vanish. This is not the case for LG , but the relevant formula is a straightforward generalization of (1.11), see for instance [55, Equation (8.42)]. In particular, the structure maps  of the A1 -categories A, B, C do not appear in it. Composition functors themselves can be composed, and this behaves as one would expect: RG1 ıRG2 D RG2 ıG1 ;

LG1 ıLG2 D LG1 ıG2 ;

LG1 ıRG2 D RG2 ıLG1 : (1.12)

A slightly less obvious relation, which generalizes Gerstenhaber’s observation that the Yoneda product on Hochschild cohomology is graded commutative, is the following: Lemma 1.4. Let Q D nu-fun.A; A/. Suppose that we have functors G1 ; G2 2 ObQ and natural transformations T1 2 homQ .G1 ; IdA /, T2 2 homQ .G2 ; IdA /. Then Œ2Q .T1 ; L1G1 .T2 // D .1/$ Œ2Q .T2 ; RG1 2 .T1 // 2 H.homQ .G1 ı G2 ; IdA //; where $ D .jT1 j  1/.jT2 j  1/ C 1. 1 jCjT2 j1 .G1 ı G2 ; IdA / by Proof. Define U 2 homjT Q

U d .ad ; : : : ; a1 / X X D .1/.jT2 j1/zn T1rCd nmC1 .ad ; : : : ; anCmC1 ; m;n;r s1 ;:::;sr

T2m .anCm ; : : : ; anC1 /; G2sr .an ; : : : ; ansr C1 /; : : : ; G2s1 .as1 ; : : : ; a1 //: A straightforward computation shows that 1Q .U / is the difference between the two sides of (1.12).  Remark 1.5. Composition with a fixed functor is, of course, only part of the story. We stop here for the sake of simplicity, even though that puts us at a slight disadvantage when discussing quasi-equivalences later on. (1f) The length filtration. Let G0 ; G1 be objects of Q D nu-fun.A; B/. The length filtration is a complete decreasing filtration F  of homQ .G0 ; G1 /: the F r term consists

1 Definitions

13

of those pre-natural transformations T such that T 0 D    D T r1 D 0. Setting A D H.A/, B D H.B/, Gk D H.Gk /, the starting term of the associated spectral sequence is Y  HomsK HomA .Xr1 ; Xr / ˝    ˝ HomA .X0 ; X1 /; E1rs D (1.13)  X0 ;:::;Xr HomB .G0 X0 ; G1 Xr / : Even though the spectral sequence may not converge in general, one can still use it for comparison arguments. At the most elementary level, the fact that (1.13) only involves cohomological data has the following consequences: Lemma 1.6. Take G0 ; G1 ; G2 2 ObQ, and let T be a natural transformation from G0 to G1 . Suppose that for each X 2 ObA, right composition with ŒT 0  yields an isomorphism HomH.B/ .G1 X; G2 X/ ! HomH.B/ .G0 X; G2 X/. Then, right composition with T in H.Q/ is an isomorphism HomH.Q/ .G1 ; G2 / ! HomH.Q/ .G0 ; G2 / (a parallel result holds for left composition).  Lemma 1.7. If G W A ! B is a cohomologically full and faithful non-unital A1 functor, then so is LG W nu-fun.C; A/ ! nu-fun.C; B/ for any C (the corresponding statement for RG W nu-fun.B; C / ! nu-fun.A; C/ is false in general, but holds e.g. if G is a quasi-isomorphism).  From (1.8), one sees that the differential @ D @rs on (1.13) is .@T /.arC1 ; : : : ; a1 / D .1/.rCs/.ja1 j1/C1 T .arC1 ; : : : ; a2 /G0 .a1 / C .1/zr C.rCs/ G1 .arC1 /T .ar ; : : : ; a1 / X C .1/znC1 C.rCs1/ T .arC1 ; : : : ; anC2 anC1 ; : : : ; a1 /: n

(The conventions here are slightly different from our usual ones: T is an element of E1rs , hence acts on the cohomological level, and similarly the ai are morphisms in A). We will denote this bigraded complex by CC rCs .A; B/s D E1rs , its @-cohomology by HH rCs .A; B/s D E2rs , and call them the Hochschild complex and Hochschild cohomology of A with coefficients in B, respectively (this depends on the functors G0 , G1 , even though the notation does not show that). The first nontrivial column is E20s D HH s .A; B/s D HomsNu-Fun.A;B/ .G0 ; G1 /; and the edge homomorphism H s .homQ .G0 ; G1 // ! E20s is T 7! H.T /. Hence, the Hochschild cohomology groups in the other columns can be viewed as obstructions to (1.10) being full and faithful.

14

I A1 -categories

Remark 1.8. The classical definition of Hochschild cohomology is for a K-algebra A together with a bimodule M , and yields a graded group HH.A; M /. This generalizes in a straightforward way to graded algebras and bimodules, and in that case the Hochschild cohomology becomes bigraded. As a special case, suppose that we have two algebra homomorphisms G0 ; G1 W A ! B. Then B becomes an A-bimodule (by acting through G1 on the left, and G0 on the right), so we have a Hochschild cohomology group HH.A; B/. The group considered above is the obvious extension of this notion from algebras (which can be viewed as categories with one object) to general categories, except that our labeling of the bigraded pieces is somewhat unconventional (we will continue this discussion, from a more abstract homological algebra point of view, in Remark 2.7 below). To prevent terminological confusion, we point out that there are also Hochschild cohomology theories in the A1 -world, namely a graded group HH.A; M/ associated to an A1 -algebra together with an A1 -bimodule over it. Again, if G0 ; G1 W A ! B are homomorphisms of A1 -algebras, one can use them to turn B into an A1 -bimodule over A, and thereby to define HH.A; B/. The generalization of this to A1 -categories is precisely the cohomology H.homQ .G0 ; G1 // which is the target of our spectral sequence. (1g) Constructing functors. The next couple of paragraphs are devoted to constructions of A1 -categories and A1 -functors, which belong to homotopical rather than homological algebra. As our first problem, suppose that we have non-unital A1 -categories A; B with their underlying cohomological categories A; B, and a non-unital functor G W A ! B. Consider the Hochschild cohomology HH.A; B/ associated to the pair .G; G/. Lemma 1.9. If HH 2 .A; B/2r D 0 for all r  3, G can be realized by a non-unital A1 -functor G W A ! B, in the sense that H.G / D G. Proof. This is an obstruction theory argument, which can be regarded as a nonlinear analogue of a spectral sequence. Suppose that for some r  3, G 1 ; : : : ; G r1 have been defined in such a way that the A1 -functor equations (1.5) hold up to order r  1 (for r D 3, this is always possible since G is a functor). Take an arbitrary provisional G r . The resulting difference between the two sides of the A1 -functor equation of order r is a family of maps  W homA .Xr1 ; Xr / ˝    ˝ homA .X0 ; X1 / ! homB .G .X0 /; G .Xr //Œ2  r: (1.14) A straightforward computation, which we omit, shows that  induces a map T on cohomology. Moreover, T is a cocycle in CC 2 .A; B/2r . By assumption, this can be written as @S in the Hochschild complex. Choose a family of maps  on the chain

1 Definitions

15

level, similar to (1.14), which represent S , and change the given G r1 to G r1  . This does not affect the A1 -functor equation of order r  1, but it changes T to T  @S , which is zero by assumption. Hence, the A1 -functor equation at order r now holds on the cohomological level. By choosing a new G r suitably, one can make it hold on the chain level, which completes the induction step.  We will also need a relative version: Lemma 1.10. Let A, B be non-unital A1 -categories, and AQ  A a full subcateQ Let G W A ! B be gory, with corresponding cohomological categories A, B, A. z its restriction to A. Q Suppose that the obvious an ordinary non-unital functor, and G restriction map on the resulting Hochschild cohomology groups, Q B/s ; HH rCs .A; B/s ! HH rCs .A; is injective for r C s D 2, r  3, and surjective for r C s D 1, r  2. Then, given z one can extend it to a any non-unital A1 -functor Gz W AQ ! B such that H.Gz / D G, G W A ! B such that H.G / D G. For that, one uses the subcomplex formed by chains in CC.A; B/ whose restriction Q B/ vanishes. The assumption ensures that this is acyclic in bidegrees to CC.A; .2; 2  r/ for r  3. With that modification, the proof goes through as before. (1h) Homotopy. Consider two non-unital A1 -functors F0 and F1 in ObQ, Q D nu-fun.A; B/, acting in the same way on objects. Let D D F0  F1 2 hom1Q .F0 ; F1 / be the pre-natural transformation defined by D 0 D 0, and D d D F0d  F1d for d > 0. This is an actual natural transformation, 1Q .D/ D 0, and one says that F0 ; F1 are homotopic if D D 1Q .T / for some T 2 hom0Q .F0 ; F1 / which also satisfies T 0 D 0. At least superficially, the name is justified by the fact that two homotopic functors induce the same functor on cohomology, H.F0 / D H.F1 /. Homotopy is preserved by left and right composition with a fixed functor. This means that if T is a homotopy from F0 to F1 , then RG1 .T / is a homotopy between F0 ı G and F1 ı G ; and L1G .T / a homotopy between G ı F0 and G ı F1 . Homotopy is an equivalence relation. In principle, this is not hard to check, but the computations may look a little confusing, for the following reason. Since we are considering functors Fk which act in the same way on objects, the graded vector spaces homQ .Fk ; Fl / are the same for all k, l, and we shall want to move pre-natural transformations from one to the other. To prevent mixups, we will use the notation TŒkl to indicate that T is being thought of as lying in homQ .Fk ; Fl /. In this terminology, the basic observation is that for any three functors Fj ; Fk ; Fl , 1Q .TŒj l / D 1Q .TŒj k /  2Q .Fk  Fl ; TŒj k / D 1Q .TŒkl / C 2Q .TŒkl ; Fj  Fk /:

(1.15)

16

I A1 -categories

To show transitivity, suppose that T1 is a homotopy from F0 to F1 , and T2 one from F1 to F2 . Using (1.15) one sees that T D .T1 /Œ02 C .T2 /Œ02 C 2Q .T2 ; T1 /

(1.16)

satisfies 1Q .T / D F1  F2  2Q .F1  F2 ; T1 / C F0  F1 C 2Q .T2 ; F0  F1 / C 1Q 2Q .T2 ; T1 / D F0  F2 ; hence is a homotopy from F0 to F2 . Symmetry goes along the same lines: let T1 be a homotopy from F0 to F1 . Consider the maps  W homQ .F1 ; F0 / ! homQ .F1 ; F1 /; .T / D .1/jT j 2Q .T1 ; T / C TŒ11 ; W homQ .F1 ; F1 / ! homQ .F0 ; F1 /;

.T / D 2Q .T; T1 / C TŒ01 :

(1.17) Due to the last term in each formula, these are isomorphisms, and by applying (1.15) one sees that they are chain maps. Finally, again using (1.15) one gets ..F0  F1 /Œ10 / D 1Q ..T1 /Œ11 /; .1Q ..T1 /Œ11 // D 1Q 2Q ..T1 /Œ11 ; T1 / C .F0  F1 /: The conclusion is that the preimage of T1  2Q ..T1 /Œ11 ; T1 / under homotopy from F1 to F0 .

ı  is a chain

Remark 1.11. The meaning of the computations above, and of homotopies in general, becomes somewhat clearer when B has strict identity morphisms; see Section (2c) below. Meanwhile, here is a different interpretation, which comes from rational homotopy theory [68, Chapter X]. Consider the following dg algebra: I D Ku0 ˚ Ku1 ˚ Kh, with ju0 j D ju1 j D 0, jhj D 1; the differential and multiplication (omitting the expressions which vanish for degree reasons) are @u0 D h D @u1 and u0 u0 D u0 ; u1 u1 D u1 ; u0 u1 D u1 u0 D 0; u1 h D h; hu1 D u0 h D 0; hu0 D h: H.I / Š K, with generator u0 Cu1 , so the maps fk W I ! K (k D 0; 1) which project to the subspace generated by uk are both quasi-isomorphisms. If B is a non-unital A1 -algebra, it is straightforward to define the tensor product I ˝ B: on the tensor product of underlying graded vector spaces, one sets 1 .i ˝ b/ D i ˝ 1B .b/ C .1/jijCjbj @.i/ ˝ b and for dP 2, d .id ˝ bd ; : : : ; i1 ˝ b1 / D .1/G id    i1 ˝ dB .bd ; : : : ; b1 /, where G D j >k .jbj j  1/jik j. A simple computation shows that T is a homotopy between two A1 -homomorphisms F0 ; F1 W A ! B iff G .ad ; : : : ; a1 / D u0 ˝ F0 .ad ; : : : ; a1 / C u1 ˝ F1 .ad ; : : : ; a1 / C .1/zd h ˝ T .ad ; : : : ; a1 /;

17

1 Definitions

defines an A1 -homomorphism G W A ! I ˝ B. The situation is summarized by the commutative diagram 2B w; F0

A

G

w ww ww w ww f0 ˝ id /I ˝B GG GGf1 ˝ id GG GG G# F1 ,

B:

The same applies to A1 -categories and functors, with the obvious modifications. (1i) Homological perturbation theory. Let B be a non-unital A1 -category. Suppose that for each pair of objects .X0 ; X1 / we have a diagram F1

T1

%

homA .X0 ;e X1 /



homB .X0 ; X1 / G1

where .homA .X0 ; X1 /; 1A / is some chain complex of vector spaces; F 1 ; G 1 are chain maps; and T 1 is of degree 1, satisfying 1B T 1 C T 1 1B D F 1 G 1  id. The Perturbation Lemma is the following statement: Proposition 1.12. One can construct a non-unital A1 -category A with ObA D ObB, morphism spaces homA .X0 ; X1 /, and first order structure map 1A ; this comes with non-unital A1 -functors F W A ! B, G W B ! A which are the identity on objects, and whose first order pieces are the given maps F 1 , G 1 ; as well as a homotopy T between F ı G and IdB , which starts with the given T 1 .  All these structures are defined by explicit formulae, typically written in recursive form. For instance, X X F d .ad ; : : : ; a1 / D T 1 .rB .F sr .ad ; : : : ; ad sr C1 /; : : : ; r

dA .ad ; : : : ; a1 / D

s1 ;:::;sr

X X r

s1 ;:::;sr

F s1 .as1 ; : : : ; a1 ///; G 1 .rB .F sr .ad ; : : : ; ad sr C1 /; : : : ;

(1.18)

F s1 .as1 ; : : : ; a1 ///;

where both sums are over partitions s1 C    C sr D d with r  2. It is an easy verification that A is indeed an A1 -structure, and F an A1 -functor. To make the

18

I A1 -categories

computations more geometric, one can “unroll” the recursive definitions into direct expressions involving sums over trees (more precisely, stable trees in the sense of Section (9d)). The same applies to the construction of G and T , but the formulae in those cases are slightly more involved. Remark 1.13. A typical application goes as follows. Given a non-unital A1 -category B, split each complex .homB .X0 ; X1 /; 1B / into a summand where the differential is zero plus an acyclic complement. Take homA .X0 ; X1 / to be the first summand, with F 1 the inclusion and G 1 the projection. A choice of contracting homotopy for the acyclic complement provides the necessary T 1 . Proposition 1.12 then constructs an A with 1A D 0, which is quasi-isomorphic to B. In this particular case, it is also true that G ı F is homotopic to IdA . To see that, note first that since F ı G is homotopic to IdB , G ı F ı G is homotopic to G . Lemma 1.7 says that RG1 W homnu-fun.A;A/ .IdA ; G ı F / ! homnu-fun.B;A/ .G ; G ı F ı G / is a quasi-isomorphism of chain complexes. Moreover, that still applies if we consider instead the subcomplexes of those pre-natural transformation which have vanishing order zero term. Since RG1 clearly maps IdA  G ı F to G  G ı F ı G , this proves the existence of the desired second homotopy. Corollary 1.14. Any quasi-isomorphism between non-unital A1 -categories has an inverse up to homotopy. z We apply the PerProof. Suppose that our quasi-isomorphism is K W B ! B. turbation Lemma to construct a diagram of non-unital A1 -categories and quasiisomorphisms A O  o  

H 1

F G

/

B

 

 H      AQ o

K Fz Gz

 / z B

where 1A D 1Q D 0. The composition H D Gz ıK ıF , being a quasi-isomorphism A between two A1 -categories with vanishing first order composition, is necessarily a formal diffeomorphism, hence has a strict inverse H 1 with respect to composition. Consider I D F ı H 1 ı Gz . By construction, K is homotopic to Fz ı H ı G D Fz ı Gz ı K ı F ı G . Hence K ı I is homotopic to Fz ı H ı G ı F ı H 1 ı Gz , which in turn is homotopic to the identity. The other homotopy is constructed in the same way. 

1 Definitions

19

Remark 1.15. Proposition 1.12 has a long history, even if one excludes topological precursors [137]. Early special cases can be found in [80] and [69], and through various subsequent generalizations and simplifications, these led to the version stated here, taken from [98]; we refer to that paper for the proof, further references, and a picture of a carp. At least for char.K/ D 0, one can derive the existence of an abstract Perturbation Lemma from the “cofibrant” property of the operad governing A1 structures, see [99]. The first references for Corollary 1.14 seem to be [81], [111]. The analogous result for L1 -algebras, in a slightly weaker form, appears in [88, Section 4.5], with essentially the same proof as above. For an obstruction-theoretic approach in the spirit of Lemma 1.9, see [60, Appendix A5]. (1j) A1 -modules. Let Ch be the dg category of chain complexes of K-vector spaces, considered as an A1 -category. For a fixed non-unital A1 -category A, define the non-unital A1 -category of non-unital (right) A1 -modules over A to be nu-mod.A/ D nu-fun.Aopp ; Ch/: It is worth while spelling out the details concretely. An object M of Q D nu-mod.A/ consists of a graded vector space M.X/ for each X 2 ObA, together with maps of all orders d  1, dM W M.Xd 1 / ˝ homA .Xd 2 ; Xd 1 / ˝    ˝ homA .X0 ; X1 / ! M.X0 /Œ2  d : If one thinks in terms of A1 -functors, then 1M is the differential on the image object M.X0 / 2 ObCh, while for d  2, dM is the action of the functor on morphisms, more precisely Md 1 with the ordering of the arguments reversed. Equation (1.6) becomes X d n .1/zn nC1 (1.19) M .M .b; ad 1 ; : : : ; anC1 /; : : : ; a1 / n

C

X d mC1 .1/zn M .b; ad 1 ; : : : ; anCmC1 ; m A .anCm ; : : : ; anC1 /; : : : ; a1 / D 0; m;n

where the second sum is over n C m < d . Next, let M0 ; M1 be two non-unital A1 -modules. Morphisms in homQ .M0 ; M1 /, which are called pre-module homomorphisms, are written as sequences of maps t d W M0 .Xd 1 / ˝ homA .Xd 2 ; Xd 1 / ˝    ˝ homA .X0 ; X1 / ! M1 .X0 /Œjt j  d C 1; d  1. This is related to the general language of pre-natural transformations T by the obvious switch in arguments, accompanied by an opportunistic sign change: t d .b; ad 1 ; : : : ; a1 / D .1/ T d 1 .a1 ; : : : ; ad 1 /.b/;

(1.20)

20

I A1 -categories

where T d 1 .a1 ; : : : ; ad 1 / 2 homCh .M0 .Xd 1 /; M1 .X0 // and  D .jT j  1/jbj C jT j.jT j  1/=2. The actual module homomorphisms are those t such that 1Q .t/ D 0. To see the concrete meaning of this, we write down the composition maps in Q, which are: .1Q t/d .b; ad 1 ; : : : ; a1 / X d n D .1/ nC1 .b; ad 1 ; : : : ; anC1 /; an ; : : : ; a1 / M1 .t n

C

X

.1/ t nC1 .dMn .b; ad 1 ; : : : ; anC1 /; an ; : : : ; a1 / 0

n

C

X

.1/ t d mC1 .b; ad 1 ; : : : ; nA .anCm ; : : : ; anC1 /; an ; : : : ; a1 /I (1.21)

m;n

.2Q .t2 ; t1 //d .b; ad 1 ; : : : ; a1 / D

X .1/ t2nC1 .t1d n .b; ad 1 ; : : : ; anC1 /; an ; : : : ; a1 /I n

dQ

D 0 for all d  3;

where D janC1 j C    C jad 1 j C jbj  d C n C 1. The vanishing of 3Q ; 4Q ; : : : is an instance of an informal but useful general principle, which says that nu-fun.A; B/ inherits the properties of B rather than those of A. On the cohomology level, any M gives rise to a non-unital H.A/-module H.M/, which more formally is a non-unital contravariant functor from H.A/ to the category of graded vector spaces. This consists of the cohomology of M.X/ with respect to @.b/ D .1/jbj 1M .b/, and the module structure induced by b  a D .1/jaj 2M .b; a/. Similarly, given an A1 -module homomorphism t , we get induced maps H.t/ W H.M0 .X// ! H.M1 .X//, Œb 7! Œ.1/jbj t 1 .b/, which form a module homomorphism in the ordinary sense. The following is a special case of Lemma 1.6: Lemma 1.16. Let t W M0 ! M1 be an A1 -module homomorphism such that for each X 2 ObA, the map H.t/ W H.M0 .X// ! H.M1 .X// is an isomorphism. Then, left and right composition with t induces quasi-isomorphisms homQ .M1 ; N / ! homQ .M0 ; N /, respectively homQ .N ; M0 / ! homQ .N ; M1 /, for any N .  In particular, if M is a non-unital A1 -module such that H.M.X/; 1M / D 0 for all X , then H.homQ .M; N // D H.homQ .N ; M// D 0 for all N . Borrowing the terminology of [136], [21], one could say that in the world of A1 -modules, everything is both K-projective and K-injective. (1k) Pullback. Any non-unital A1 -functor G W A ! B induces a pullback functor G  D RG opp W nu-mod.B/ ! nu-mod.A/:

(1.22)

21

1 Definitions

Like any right composition, this has vanishing higher order terms, hence is a nonunital dg functor between dg categories. Explicitly, .G  M/.X/ D M.G .X//; dG M .b; ad 1 ; : : : ; a1 / D

X X r

.G  t/d .b; ad 1 ; : : : ; a1 / D

rM .b; G sr .ad 1 ; : : : ; ad sr /;

s1 ;:::;sr

X X r

: : : ; G s1 .as1 ; : : : ; a1 //; t r .b; G sr .ad 1 ; : : : ; ad sr /;

s1 ;:::;sr

: : : ; G s1 .as1 ; : : : ; a1 //:

(1l) The Yoneda embedding. For every object Y of A there is an associated nonunital A1 -module Y, defined by Y.X/ D homA .X; Y / and dY D dA . This is part of a canonical non-unital A1 -functor I D IA W A ! Q D nu-mod.A/, called the Yoneda embedding. For c 2 homA .Y0 ; Y1 /, I 1 .c/ 2 homA .Y0 ; Y1 / is the premodule homomorphism Y0 .Xd 1 / ˝ homA .Xd 2 ; Xd 1 / ˝    ˝ homA .X0 ; X1 / ! Y1 .X0 /; .b; ad 1 ; : : : ; a1 / 7! dAC1 .c; b; ad 1 ; : : : ; a1 /I the higher order terms I d , d  2, are defined by the obvious analogues of this formula. The Yoneda embedding is compatible with pullback, in the following sense. Let F W A ! B be a non-unital A1 -functor. Then there is a canonical natural transformation T W IA ! F  ı IB ı F : (1.23) The first order term T 1 consists of a module homomorphism for each Y , which in turn is a sequence of maps .IA .Y //.Xd 1 / ˝ homA .Xd 2 ; Xd 1 / ˝    ˝ homA .X0 ; X1 / ! .F  ı IB ı F .Y //.X0 /:

(1.24)

By definition, .IA .Y //.Xd 1 / D homA .Xd 1 ; Y / and .F  ı IB ı F /.Y /.X0 / D homB .F .X0 /; F .Y //; with this in mind, (1.24) is given by .b; ad 1 ; : : : ; a1 / 7! .1/zd 1 Cjbj F d .b; ad 1 ; : : : ; a1 /. Similar expressions define T r , r  2. In a different direction, one can generalize I 1 to a chain map

D M W M.Y / ! homQ .Y; M/;

.c/d .b; ad 1 ; : : : ; a1 / D dMC1 .c; b; ad 1 ; : : : ; a1 /

(1.25)

for any non-unital A1 -module M. On the cohomological level, this has the following easily verified properties with respect to multiplication on either side: Œ M .2M .c; b// D Œ2Q . M .c/; I 1 .b// 2 HomH.Q/ .Y0 ; M/

(1.26)

22

I A1 -categories

for Œc 2 H.M.Y1 //, Œb 2 HomH.A/ .Y0 ; Y1 /; and Œ2Q .t; M0 .c// D Œ M1 .t 1 .c// 2 HomH.Q/ .Y; M1 /

(1.27)

for Œt 2 HomH.Q/ .M0 ; M1 /, Œc 2 H.M0 .Y //. Note that if we restrict to modules which come from the Yoneda embedding, and correspondingly take t to be the image of some morphism in A, then both equations reduce to special cases of the A1 -functor equation for I. Remark 1.17. The map can be interpreted as part of a wider picture, which also explains the properties listed above. Define the linearization functor to be the nonunital dg functor obtained by combining the Yoneda embedding for Q with pullback by that for A, IQ

 IA

L W Q ! nu-mod.Q/ ! Q (the name comes from the fact that the image of each module homomorphism under L is a homomorphism whose only nonzero term is the first order, linear, one). This functor comes with a canonical natural transformation L W IdQ ! L. At first order, this is given by a module homomorphism l D L1M W M ! Mlin D L.M/ for each A1 -module M, and the starting term of that consists of linear maps l 1 W M.Y / ! Mlin .Y / for all Y . By definition Mlin .Y / D homQ .Y; M/, where Y is the image of Y under the Yoneda embedding, and the linear map is our up to a sign .1/jcj . The fact that l is a homomorphism of A1 -modules implies (1.26); and the fact that L is a natural transformation implies (1.27).

2 Identity morphisms and equivalences (2a) The definitions. Let A be a non-unital A1 -category. There are several versions of what it means for A to have identity morphisms. • A is strictly unital if for each object X there is a (necessarily unique) eX 2 hom0A .X; X/ which satisfies 1A .eX / D 0I .1/jaj 2A .eX1 ; a/ D a D 2A .a; eX0 / for a 2 homA .X0 ; X1 /; dA .ad 1 ; : : : ; anC1 ; eXn ; an ; : : : ; a1 / D 0 for d > 2, ak 2 homA .Xk1 ; Xk / and any 0  n < d .

(2.1)

• A is cohomologically unital or c-unital if H.A/ is unital (has identity morphisms for any object X, which means that it is a graded linear category in the ordinary sense).

2 Identity morphisms and equivalences

23

• A homotopy unital A1 -category is a set ObA together with graded vector spaces homA .X0 ; X1 / and multilinear maps d;.id ;:::;i0 /

A

W homA .Xd 1 ; Xd / ˝    ˝ homA .X0 ; X1 / P ! homA .X0 ; Xd /Œ2  d  2 k ik 

(2.2)

for all d Ci0 C  Cid > 0. These must satisfy certain generalized associativity equations, which reduce to (1.2) for i0 D    D id D 0. The first few new equations are 1A .0;.1/ A / D 0; 1;.1;0/ 1 .1/jaj1 2A .0;.1/ .a// C 1;.1;0/ .1A .a// D a; A ; a/ C A .A A 1;.0;1/ 1 2A .a; 0;.1/ .a// C .1/jaj1 1;.0;1/ .1A .a// D a; A / C A .A A

(2.3)

1;.0;1/ 0;.2/ 1 1;.1;0/ .0;.1/ .0;.1/ A A / C A A / C A .A / D 0:

Modulo the usual signs, the first two parts of this say that multiplication with the cocycle eX D 0;.1/ A , on either side, is chain homotopic to the identity. Using either of the two homotopies, one sees that 2A .eX ; eX / is equal to eX up to a coboundary. The difference of these two coboundaries is a cocycle, but the last part of (2.3) shows that this is itself trivial in cohomology. The full set of equations can be thought of as continuing this hierarchy of higher and higher homotopies; see [55, Section 5] for the details. D eX , and takes Any strictly unital A1 -category is homotopy unital (one sets 0;.1/ A all other maps (2.2) with i0 C    C id > 0 to be zero). Moreover, as the preceding discussion shows, homotopy unitality implies c-unitality. We should also mention the corresponding notions for functors. If A and B are strictly unital, a strictly unital A1 -functor F W A ! B is one that satisfies F 1 .eX / D eF .X / for all objects X, as well as F d .ad 1 ; : : : ; anC1 ; eXn ; an ; : : : ; a1 / D 0 for any d  2 and any n. If A and B are c-unital, a c-unital A1 -functor F W A ! B is simply one such that H.F / is unital. For the homotopy unital version, see [55]. Strict unitality is convenient in many ways, and widely used in the literature for that reason, even though it may appear to be out of sync with the general philosophy of homotopy algebraic structures. C-unitality may also look a priori suspicious, as it falls into the opposite extreme of relegating the question entirely to the cohomology level. As far as our intended application is concerned, Fukaya categories do not naturally come with strict units, but they are easily seen to be c-unital. In fact, there is a geometric construction of homotopy units for them, which defines the necessary maps (2.2) with i0 C    C id > 0 in terms of supplementary moduli spaces, see [60, §20]. While all this seems to point towards homotopy unitality as the most appropriate notion, the discussion will now take a possibly unexpected turn: inspection of the

24

I A1 -categories

relevant obstructions reveals that there is a general algebraic procedure which equips any c-unital A1 -category with homotopy units. Moreover, from the derivation in [55] one sees that any homotopy unital A1 -category is quasi-isomorphic to a strictly unital one. This means that the difference between the three notions is one of taste and convenience, rather than substance. Our choice is to work mainly in the c-unital world. However, as a technical tool for proving some basic properties, we will provide a direct way of turning a c-unital A1 -category into a strictly unital one. Lemma 2.1. Let A be a cohomologically unital A1 -category. Then there is a formal diffeomorphism ˆ with ˆ1 D Id, such that the modified A1 -structure ˆ A is strictly unital. Proof. The initial step goes as follows. Choose for each object X a cocycle eX 2 hom0A .X; X/ representing the identity in H.A/. For any triple of objects one can find a chain map 2Q W homA .X1 ; X2 / ˝ homA .X0 ; X1 / ! homA .X0 ; X2 / representing A the composition in H.A/, and which satisfies the second condition in (2.1). Take a formal diffeomorphism with ˆ1 D Id, such that each ˆ2 is a chain homotopy between 2A and 2Q . The first two composition maps of AQ D ˆ A are 1Q D 1A A A and the chosen 2Q . A

From here on the construction becomes recursive. Suppose that we have a nonunital A1 -category A together with elements eX 2 hom0A .X; X/ which satisfy the first two equations in (2.1). Consider the following family of conditions .Ud;n / on such an A, indexed by d > 2 and 0  n  d : .Ud;n / iA .ai 1 ; : : : ; aj C1 ; eXj ; aj ; : : : ; a1 / vanishes in the following two cases: for i < d and arbitrary j , as well as for i D d and j < n. The case n D 0 is included for purely notational reasons, since clearly .Ud;0 / D .Ud 1;d 1 /. Suppose that A already satisfies .Ud;n / for some n < d . Take a formal diffeomorphism ˆ such that ˆ1 D Id, ˆk D 0 for 2  k < d  1, and ˆd 1 .ad 1 ; : : : ; a1 / D .1/zn dA .ad 1 ; : : : ; anC1 ; eXn ; an ; : : : ; a1 /; ˆd .ad ; : : : ; a1 / D .1/zn dAC1 .ad ; : : : ; anC1 ; eXn ; an ; : : : ; a1 /:

(2.4)

It is obvious from the definition that the composition maps in AQ D ˆ A agree with those of A for orders < d  1. A slightly more involved computation, using the A1 -associativity equations and the inductive assumption, shows that the same holds at order d  1. We now look at order d , writing .b1 ; : : : ; bd C1 / D

2 Identity morphisms and equivalences

25

.a1 ; : : : ; an ; eXn ; anC1 ; : : : ; ad / to simplify the formulae. One finds that dAQ .ad ; : : : ; a1 / D dA .ad ; : : : ; a1 /  .1/ 2A .dA .ad ; : : : ; eXnC1 ; anC1 ; : : : ; a2 /; a1 / X  .1/znC1 .1/ dA .ad ; : : : ; eXnC1 ; anC1 ; : : : j n; i D d and j D 0; finally, i D d C 1. Note that the expression in the last-mentioned sum also vanishes if i D 1 and j D n, or if 3  i  d  1, by induction assumption. Suppose first that n D 0, in which case the third expression on the right-hand side of (2.5) is an empty sum. Removing that, and applying the A1 -associativity equation to the fourth expression, we end up with dAQ .ad ; : : : ; a1 / D dA .ad ; : : : ; a1 /  2A .dA .ad ; : : : ; a2 ; eX1 /; a1 /: This clearly vanishes if a1 is also the identity, showing that AQ has property .Ud;1 /. Next, consider the case when n > 0. Then, the summand in the last expression in (2.5) also vanishes if one sets i D 2 and takes the sum over all j  n; or for i D d and j D 1. Using the A1 -associativity equation as before, one shows that the sum vanishes. Having killed that, one checks easily that AQ has property .Ud;nC1 /. By repeating this process, one gets a sequence of formal diffeomorphisms ˆ1 ; ˆ2 ; : : : such that as k ! 1, .ˆk ı    ı ˆ1 / A satisfies more and more of the strict unitality condition. The ˆk themselves are increasingly close to the identity, meaning that the first nontrivial term of ˆk will be of increasingly high order as k ! 1. This implies that the infinite composition ˆ D    ı ˆ3 ı ˆ2 ı ˆ1 makes sense, and that ˆ A is strictly unital, as claimed.  Remark 2.2. Lemma 2.1 is due independently to Lefevre [95, Theorem 3.2.1.1] and the present author. There is a more conceptual proof, based on homological perturbation theory and the fact that the full and reduced Hochschild complex of a unital algebra are quasi-isomorphic (the expressions (2.4) were suggested by the classical homotopy equivalence between these complexes). Analogous results hold for A1 -functors [95, Theorem 3.2.2.1], A1 -modules [95, Theorem 3.3.1.2], and so on.

26

I A1 -categories

(2b) Functor categories. If A is a strictly unital A1 -category, then nu-fun.C; A/ is again strictly unital for any non-unital A1 -category C. Explicitly, the strict identity morphisms are the natural transformations EF defined by EF0 D eF .X/ 2 hom0B .F .X/; F .X//, EFd D 0 for all d > 0. A little less obviously, Lemma 2.3. If A is c-unital, then so is Q D nu-fun.C; A/. Moreover, if EF 2 hom0Q .F ; F / is any chain representing the identity morphism in H.Q/, then for each X 2 ObC, EF0 2 hom0Q .F .X/; F .X// represents the identity in H.A/. Proof. This can be reduced to the strictly unital case, as follows. Let ˆ be a formal z D diffeomorphism with ˆ1 D id, such that AQ D ˆ A is strictly unital, and set Q Q nu-fun.C ; A/. Consider the A1 -functor of left composition with ˆ, and write Fz D ˆ ı F . On the level of natural transformations, this yields a commutative diagram HomH.Q/ .F ; F /

H.L1 ˆ/

PPP PPP PPP PPP P(

/ Hom z .Fz ; Fz / H.Q/ nnn n n n nnn nv nn

(2.6)

HomNu-Fun.C;A/ .F; F / .

Q F D H.F / D H.Fz / are the cohomology Here C D H.C /, A D H.A/ D H.A/, level structures, and the descending arrows are the maps (1.10) which take natural transformations of A1 -functors to their cohomological counterparts. Lemma 1.7 shows that H.L1ˆ / is an isomorphism. Choose some chain EF such that ŒL1ˆ .EF / is the class of the strict unit ŒEFz . By functoriality of H.L1ˆ /, this is an identity morphism in H.Q/. Diagram chasing in (2.6) shows that H.EFz / D H.EF / is the identity natural transformation in Nu-Fun.C; A/. This implies that each ŒE 0z  D ŒEF0  F is an identity morphism in A, as claimed.  Lemma 2.4. Let G W A ! B be a c-unital A1 -functor. Then for any non-unital A1 -category C, the composition functor LG W nu-fun.C; A/ ! nu-fun.C ; B/ is cunital. (A similar result holds for right composition: let G W A ! B be a non-unital A1 -functor between non-unital A1 -categories. Then, for any c-unital A1 -category C , the composition functor RG W nu-fun.B; C/ ! nu-fun.A; C/ is c-unital). Proof. Let EF 2 hom0nu-fun.C;A/ .F ; F / be a chain representing the identity morphism in H.nu-fun.C ; A//. Take its image under composition, 0 ŒL1G .EF / 2 HomH.nu-fun.C ;B// .G ı F ; G ı F /:

(2.7)

We have seen that for each X, EF0 2 hom0A .F .X/; F .X// represents the cohomological identity, hence so does L1G .EF /0 D G 1 .EF0 / 2 hom0B .G .F .X//; G .F .X///.

2 Identity morphisms and equivalences

27

Using Lemma 1.6 and the c-unitality of nu-fun.C ; B/, it follows that (2.7) is an invertible morphism. Because of the functoriality of LG , it is also idempotent, which implies that it is equal to the identity (this kind of argument will be used repeatedly later on). The same works for RG .  (2c) Homotopy and isomorphism. Let A be a strictly unital A1 -category, C a nonunital one, and F0 ; F1 W C ! A non-unital A1 -functors, acting in the same way on objects. Let T be a pre-natural transformation of degree zero from F0 to F1 , such that T 0 D 0. Denote by S the modified pre-natural transformation defined by S 0 D eF0 .X / for all X, S d D T d for d > 0. A straightforward computation shows that T is a homotopy between F0 and F1 iff S is an actual natural transformation. In this context, the otherwise mysterious expression (1.16) for composition of homotopies can be seen as the ordinary composition of the associated natural transformations, and similarly for (1.17). Using Lemma 1.6 and the strict unitality of the A1 -functor category, it follows that two homotopic functors are isomorphic as objects of H 0 .nu-fun.C ; A//. We will need the analogous result in the c-unital context: Lemma 2.5. Let A be a c-unital A1 -category, and C a non-unital one. If two nonunital A1 -functors F0 ; F1 W C ! A are homotopic, then they are also isomorphic as objects of H 0 .nu-fun.C ; A//. Proof. We use the notation from the proof of Lemma 2.3. Since left composition preserves homotopy, the Fz k D ˆ ı Fk are homotopic, hence by our previous obserz On the other hand, Lˆ is cohomologically vation isomorphic as objects of H 0 .Q/. full and faithful by Lemma 1.7, hence the Fk themselves must be isomorphic.  (2d) Morita invariance. Let A and B be (linear, graded, unital) categories, and let F0 , F1 W A ! B (linear, graded, unital) functors. Let AQ  A be a subcategory such that the inclusion is an equivalence. Lemma 2.6. The restriction map yields an isomorphism of Hochschild cohomology Q B/. groups, HH.A; B/ Š HH.A; Q B/ be the restriction on the chain level. Choose, Proof. Let  W CC.A; B/ ! CC.A; z For for any X 2 ObA, an Xz 2 Ob AQ together with an isomorphism uX W X ! X. Q z those objects X which already lie in A, we may take X D X and set uX to be the identity morphism. The Dennis cotrace is the chain map Q B/s ! CC rCs .A; B/s ; W CC rCs .A; 1 1 . T /r .ar ; : : : ; a1 / D G1 .uXr /1 T d .uXr ar uX ; : : : ; uX1 a1 uX /G0 .uX0 /: r1 0

28

I A1 -categories

This is a homotopy equivalence, since  ı D id, and id  ı  D @ C @ with . T /.ar1 ; : : : ; a1 / X 1 D .1/zn C.rCs/ G1 .uXr1 /1 T .uXr1 ar1 uX ;:::; r1 n

1 ; uXn ; an ; : : : ; a1 /:  uXnC1 anC1 uX n

Remark 2.7. Alternatively, one could take the following more abstract route. Take the abelian category Bimod 0 .A/ of graded A-bimodules, or more precisely bifunctors from .Aopp ; A/ to the category of graded vector spaces (the bimodule homomorphisms are natural transformations of degree zero). Any pair of functors .G0 ; G1 / gives rise to a bimodule, consisting of the vector spaces HomB .G0 .X0 /; G1 .X1 //. In an abuse of notation, we denote this bimodule simply by B, and the corresponding object obtained by looking at the pair of functors .IdA ; IdA / by A. The Hochschild complex can be interpreted as coming from a projective resolution of the last-mentioned bimodule in Bimod 0 .A/, which means that HH.A; B/ Š Ext Bimod 0 .A/ .A; B/:

(2.8)

Here we are considering the Ext groups of any order, and also taking B with its given grading shifted in all possible ways, so that the right-hand side is bigraded. For elementary reasons, restriction of bifunctors from A to AQ yields an equivalence of Q hence we get an isomorphism of Ext groups. categories Bimod 0 .A/ Š Bimod 0 .A/, The relationship with the classical Morita invariance of Hochschild (co)homology for algebras, see for instance [143, Theorem 9.5.6], should now be clear. In fact, our situation is akin to the basic example of Morita invariance provided by matrix algebras, and the elementary argument above copies [96, Theorem 1.2.4]. (2e) Quasi-equivalences. We need a little more notation and terminology: given two c-unital A1 -categories A and B, write fun.A; B/ for the full subcategory of nu-fun.A; B/ consisting of c-unital A1 -functors. A c-unital A1 -functor F W A ! B is called a quasi-equivalence if the underlying cohomology level functor is an equivalence. Suppose that A is a c-unital A1 -category, and AQ  A a full A1 -subcategory such that the inclusion is a quasi-equivalence. Lemma 2.8. For any c-unital A1 -category B, restriction of c-unital A1 -functors and their pre-natural transformations yields a quasi-equivalence Q D fun.A; B/ ! z D fun.A; Q B/. Q Proof. Consider any two c-unital A1 -functors Gk (k D 0; 1) and their restrictions Q The restriction map on pre-natural transformations, Gz k to A. homQ .G0 ; G1 / ! homQz .Gz 0 ; Gz 1 /

(2.9)

2 Identity morphisms and equivalences

29

is obviously compatible with the length filtrations, hence induces a map of the associated spectral sequences. At the E2 level, this reduces to the restriction map on Hochschild cohomology for the associated cohomological functors, which we have just shown to be an isomorphism. Hence (2.9) is a quasi-isomorphism, which means that the restriction is cohomologically full and faithful. Finally, given any c-unital A1 -functor Gz W AQ ! B, one can extend the underlying cohomological functor uniquely to a G W H.A/ ! H.B/; then Lemma 1.10 allows one to construct a G W A ! B which extends Gz and satisfies H.G / D G, hence is c-unital. This proves that restriction is onto on objects, hence a quasi-equivalence.  In particular, the extension procedure shows that there is a c-unital A1 -functor P W A ! AQ such that P jAQ D IdAQ . If we denote by K W AQ ! A the inclusion, then P ı K D IdAQ . P is a quasi-equivalence, in particular cohomologically full and faithful, hence so is LP by Lemma 1.7. Since LP .K ı P / is equal to LP .IdA /, it follows that K ı P is isomorphic to IdA as objects of H 0 . fun.A; A//. Theorem 2.9. Let F W A ! B be a quasi-equivalence. Then there is a quasiequivalence G W B ! A such that G ıF Š IdA in H 0 . fun.A; A//, and F ıG Š IdB in H 0 . fun.B; B//. z  B such that the inclusions Proof. There are full A1 -subcategories AQ  A, B are quasi-equivalences, and such that F restricts to a quasi-isomorphism Fz from AQ z As we have already determined, both inclusions can be inverted up to isomorto B. phism in the respective functor categories. Fz itself has an inverse up to homotopy by Corollary 1.14, but homotopy implies isomorphism in the functor category by Lemma 2.5.  Lemma 2.10. If G W A ! B is a quasi-equivalence, then so is RG W fun.B; C/ ! fun.A; C / for any c-unital A1 -category C (the parallel statement for LG also holds; in fact, in that case there is also a version for non-unital C and the categories nu-fun). Proof. We first need to know that RG is cohomologically full and faithful. If G is a quasi-isomorphism, this is part of Lemma 1.7; and if G is the inclusion of a full subcategory, it is part of Lemma 2.8. In parallel with the discussion above, these two special cases imply the general result. Next, Theorem 2.9 provides an F W B ! A such that F ı G Š IdB . By Lemma 2.4, for any H 2 Obfun.A; C / one gets RG .H ı F / D LH .F ı G / Š LH .IdB / D H , which implies that RG is a surjective map on isomorphism classes of objects. The argument for LG is similar, but a little simpler.  Remark 2.11. The strictly unital version of Theorem 2.9 was apparently first stated by Kontsevich (unpublished). It appears in [55, Theorem 8.6] and [95, Theorem 9.2.04]. The c-unital version discussed here can also be found in [97, Theorem 8.8].

30

I A1 -categories

(2f) A1 -modules. Let A be a c-unital A1 -category. Since Ch is strictly unital, so is Q D nu-mod.A/ D nu-fun.Aopp ; Ch/. Note that, due to the signs introduced in (1.20), the strict identity for M is the module homomorphism t D eM given by t 1 .b/ D .1/jbj b;

t d D 0 for all d  2.

(2.10)

An A1 -module M is called c-unital if the underlying cohomological module is unital, which means that if eX 2 hom0A .X; X/ is a representative for the cohomology unit, then the chain map 2M .; eX / W M.X/ ! M.X/ induces the identity on cohomology. We denote by mod.A/ the full subcategory of c-unital A1 -modules. If G W A ! B is a c-unital A1 -functor, we can restrict the pullback (1.22) to these subcategories, and get G  W mod.B/ ! mod.A/ (2.11) which is a strictly unital dg functor. As a special case of Lemma 2.10, if G is a quasi-equivalence, then so is (2.11). (2g) The Yoneda embedding. By definition, the image of the Yoneda embedding I D IA for c-unital A lies in the subcategory mod.A/  nu-mod.A/. Lemma 2.12. For any c-unital A1 -module M and any object Y of A, the map

from (1.25) is a quasi-isomorphism. Proof. By definition, the mapping cone (shifted by 1) of is the chain complex    1 M 0 M.Y / ˚ homQ .Y; M/Œ1; : (2.12)

1Q Filter this by first taking the subcomplex homQ .Y; M/Œ1, followed by its usual length filtration. Writing A D H.A/ and M D H.M/, the resulting spectral sequence has

„ M .Y /;

r D 0;

s

E1rs D

Y

 HomsK HomA .Xr1 ; Y / ˝ HomA .Xr2 ; Xr1 / ˝     X0 ;:::;Xr1 r > 0:    ˝ HomA .X0 ; X1 /; M.X0 / ;

From (1.25) and (1.21), one sees that the differential @ D @rs W E1rs ! E1rC1;s is given by .@c/.b/ D .1/jbj cb for r D 0, and .@t/.b; ar ; : : : ; a1 / D .1/zr Cjbj t .b; ar ; : : : ; a2 /a1 C .1/jbj t.bar ; ar1 ; : : : ; a1 / X C .1/4 t .b; ar ; : : : ; anC2 anC1 ; : : : ; a1 / n

2 Identity morphisms and equivalences

31

for r > 1, where 4 D janC2 j C    C jar j C jbj  r C n C 1. This is a standard bar resolution, which in the presence of identity morphisms in A, acting as the identity on M , admits a contracting homotopy . t/.b; ar1 ; : : : ; a1 / D t.eY ; b; ar1 ; : : : ; a1 /. Hence (2.12) is acyclic, which implies the desired statement.  Specializing to A1 -modules M which come from the Yoneda embedding, we have D I 1 , hence: Corollary 2.13. I is cohomologically full and faithful (hence also c-unital).



By looking at the image of I, one also sees the following: Corollary 2.14. Any c-unital A1 -category is quasi-isomorphic to a strictly unital dg category, in a canonical way.  We now consider the naturality of the Yoneda embedding with respect to cohomologically full and faithful A1 -functors F W A ! B between c-unital A1 -categories. For any such functor F , the first order term of the natural transformation T from (1.23) is an isomorphism of c-unital A1 -modules, by Lemma 1.16. In view of Lemma 1.6, this implies that T itself is an isomorphism in the underlying cohomological category of A1 -functors. In other words, the diagram A 

F

IA

mod.A/ o

F

/B 

IB

(2.13)

mod.B/

commutes up to canonical isomorphism in H 0 . fun.A; mod.A///. Remark 2.15. Here is a more abstract reformulation of our proof of Lemma 2.12, in the spirit of Remark 2.7. Let Mod 0 .A/ be the abelian category of contravariant (unital) functors from A to graded vector spaces. If M0 ; M1 are c-unital A1 -modules, the underlying cohomological modules M0 ; M1 are objects of Mod 0 .A/. Via bar resolutions, the E2 term of the spectral sequence coming from the length filtration of homQ .M0 ; M1 / can be interpreted as E2 D ExtMod 0 .A/ .M0 ; M1 /;

(2.14)

where the right-hand side is bigraded in the same way as in (2.8). The cohomology modules associated to Yoneda A1 -modules are projective objects of Mod 0 .A/. Hence, If M0 D Y, there is only nonvanishing column in (2.14), and that column can be identified with M1 .Y /. Hence, M1 .Y / is quasi-isomorphic to homQ .Y; M1 /.

32

I A1 -categories

(2h) Terminology. Having completed the discussion of the issue of identity morphisms, we will now drop it from our terminology, by adopting the following convention: in future, all A1 -categories, A1 -functors, and A1 -modules are assumed to be c-unital unless otherwise specified.

3 Exact triangles (3a) Philosophy. When constructing objects in any kind of category, a widespread strategy is to first characterize the desired object in terms of its putative representing functor, and then subsequently to address the existence (representability) question. In our case, the appropriate formalism is that of A1 -modules and theYoneda embedding. Fix an A1 -category A, and write Q D mod.A/. Let M be an A1 -module. Suppose that there is a Y 2 ObA whose image Y under the Yoneda embedding is isomorphic to M in H 0 .Q/, and fix an isomorphism Œt W Y ! M in that category. We say that the pair .Y; Œt/ quasi-represents M. Corollary 2.13 implies that if .Yz ; ŒtQ / is another such pair, then there is a preferred isomorphism Œa W Y ! Yz in H 0 .A/ satisfying ŒtQ   ŒI 1 .a/ D Œt. In an abuse of terminology, we will sometimes just refer to Y as the object quasi-representing M, but the choice of Œt is still supposed to be part of the data. One can give a more concrete characterization in terms of the chain map

D M W M.Y / ! homQ .Y; M/ from (1.25): Lemma 3.1. .Y; Œt/ quasi-represents M iff the following holds. There is a c 2 M0 .Y /, satisfying 1M .c/ D 0 and Œt D Œ .c/, such that for all X 2 ObA, the map homA .X; Y / ! M.X/;

b 7! .1/jbj 2M .c; b/

(3.1)

is a quasi-isomorphism. Proof. Since 1M .c/ D 0, .c/ is an A1 -module homomorphism Y ! M, whose first order term is (3.1) except for the sign. Lemma 1.16 implies that Œt D Œ .c/ is an isomorphism in H 0 .Q/, hence .Y; Œt/ quasi-represents M. Conversely, if we have a quasi-representing pair .Y; Œt/, then by Lemma 2.12 there is a c such that 1M .c/ D 0 and Œt D Œ .c/. Since Œ .c/ is an isomorphism in H 0 .Q/, it must in particular induce an isomorphism between the underlying cohomology modules, which implies that (3.1) is a quasi-isomorphism for all X.  (3b) Direct sum. Here is an elementary example of this approach. Given A1 -modules M0 and M1 , one defines their direct sum M0 ˚ M1 by taking the sums of the respective graded vector spaces M0 .X/ ˚ M1 .X/, X 2 ObA, together with the structural maps dM0 ˚ dM1 . Now suppose that Y0 ; Y1 are objects of A, and Y0 ; Y1

33

3 Exact triangles

their images under the Yoneda embedding. We write Y0 ˚ Y1 for any object which quasi-represents the A1 -module Y0 ˚ Y1 . By definition, this comes with a canonical isomorphism for any X, HomH.A/ .X; Y0 ˚ Y1 / Š HomH.A/ .X; Y0 / ˚ HomH.A/ .X; Y1 /: (3c) Tensor product. On the same basic level, let Z D .Z; @Z / be a chain complex of K-vector spaces, and M an A1 -module. The tensor product module Z ˝ M is defined by .Z ˝ M/.X/ D Z ˝ M.X/; 1Z˝M .z ˝ b/ D .1/jbj1 @Z .z/ ˝ b C z ˝ 1M .b/; dZ˝M .z ˝ b; ad 1 ; : : : ; a1 / D z ˝ dM .b; ad 1 ; : : : ; a1 /

(3.2) for d  2:

One can make Z ˝  into an A1 -functor from Q to itself. At first order, it takes t 2 homQ .M0 ; M1 / to idZ ˝ t 2 homQ .Z ˝ M0 ; Z ˝ M1 /, which by definition is .idZ ˝ t/d .z ˝ b; ad 1 ; : : : ; a1 / D .1/jzj.jtj1/ z ˝ t d .b; ad 1 ; : : : ; a1 /I and the higher order terms are all set to zero. Given an object Y of A, with Y its Yoneda image, we write Z ˝ Y for any object which quasi-represents Z ˝ Y. On the cohomology level, one has HomH.A/ .X; Z ˝ Y / Š HomH.Q/ .X; Z ˝ Y/ Š H..Z ˝ Y/.X// Š H.Z/ ˝ H.Y.X// Š H.Z/ ˝ HomH.A/ .X; Y / where the isomorphisms are given by the Yoneda embedding, the map , and the ordinary Künneth formula. z D H.Z/ with @ z D 0, and Remark 3.2. Given a chain complex .Z; @Z /, set Z Z z ! Z which selects a chain representative for each cohochoose a linear map W Z z ˝ M ,! Z ˝ M induces mology class. By Lemma 1.16, the resulting inclusion Z 0 an isomorphism in H .Q/ (moreover, this isomorphism is independent of the choice of ). This shows that, without essential loss of generality, one can restrict the tensor product operation to cases where Z is simply a graded K-vector space, with vanishing differential. (3d) Shift. The most frequently used example of tensor product is where Z is the one-dimensional vector space KŒ placed in some degree . The outcome is denoted by KŒ ˝ M D SS  M respectively KŒ ˝ Y D S  Y , and called the

34

I A1 -categories

-fold shift of M or Y . In the case  D 1, we omit the index and write simply SSM, S Y . These shifts, if they exist, come with canonical isomorphisms HomH.A/ .Y0 ; S  Y1 / Š HomH.A/ .Y0 ; Y1 /Œ; HomH.A/ .Y0 ; Y1 / Š HomH.A/ .S  Y0 ; S  Y1 /; HomH.A/ .S  Y0 ; Y1 /Œ Š HomH.A/ .Y0 ; Y1 /:

(3.3) (3.4) (3.5)

The first one is by definition, and the second comes from the functorial nature of S S  D KŒ ˝ . The third one is obtained by applying (3.3) to the pair .S  Y0 ; Y1 /, and then (3.4) in reverse (from right to left) to .Y0 ; Y1 /. Remark 3.3. Another way of seeing (3.5) would be to start with .S  Y0 ; Y1 /, apply (3.4) to pass to .Y0 ; S  Y1 /, and then use (3.3) in reverse to go to .Y0 ; Y1 /. This unfortunately differs from the previous definition by a sign .1/ , which means that a little care is required when applying these isomorphisms. Suppose that the A1 -category A has the property that SY exists for all Y . One can then construct a shift A1 -functor S W A ! A, unique up to isomorphism in H 0 . fun.A; A//, which has the property that the diagram A S



A

I

I

/Q 

SS

/Q

commutes up to isomorphism in H 0 . fun.A; Q//. Namely, consider the full A1 subcategory AQ  A consisting of all objects of the form SY , and correspondingly z  Q of those A1 -modules which are isomorphic, in H 0 .Q/, the subcategory Q to S SY D S SI.Y / for some Y . By construction, the restriction of the Yoneda z hence can be inverted by Theorem 2.9. embedding is a quasi-equivalence AQ ! Q, Given such an inverse K, one sets S D K ı SS ı I. We have explained this for the shift, but the same argument applies to other functorial constructions on A1 -modules. Remark 3.4. The existence of arbitrary shifts S  Y ,  2 Z, and of direct sums Y0 ˚ Y1 , implies that of the tensor products Z ˝ Y for complexes .Z; @Z / with z D H.Z/, finite-dimensional cohomology. Namely, if one takes a basis fŒz i g of Z with  i D jz i j, then M i S Y L

i2I

z gets mapped to i2I SS Y Š Z˝Y under theYoneda embedding. By Remark 3.2, this is also a quasi-representative for Z ˝ Y. i

3 Exact triangles

35

(3e) Cones. Let Y0 ; Y1 be objects of A, and c 2 hom0A .Y0 ; Y1 / a degree zero cocycle (1A .c/ D 0). The abstract mapping cone of c is the A1 -module C D Cone.c/ defined by C .X/ D homA .X; Y0 /Œ1 ˚ homA .X; Y1 /; dC ..b0 ; b1 /; ad 1 ; : : : ; a1 / (3.6)  d  d C1 d D A .b0 ; ad 1 ; : : : ; a1 /; A .b1 ; ad 1 ; : : : ; a1 / C A .c; b0 ; ad 1 ; : : : ; a1 / : A straightforward computation shows that (1.19) is satisfied. For the c-unitality of C , one needs to check that if eX 2 hom0A .X; X/ represents the identity morphism in H.A/, then the endomorphism of H.C .X/; 1C / induced by .b0 ; b1 / 7! 2C ..b0 ; b1 /; eX / is the identity. We already know that this is idempotent, and by looking at the subcomplex of C.X/ where b0 D 0 and its quotient, one sees that it is also invertible, hence the identity. We denote an object quasi-representing the abstract mapping cone by Cone.c/, and call it simply mapping cone of c. As usual, such an object (if it exists) is unique up to canonical isomorphism in H 0 .A/. It is instructive to see how C one.c/ depends on the choice of cocycle c within a fixed cohomology class. If we take another representative cQ D c C 1A .h/, then t 1 .b0 ; b1 / D ..1/jb0 j1 b0 ; .1/jb1 j b1 C 2A .h; b0 //;   t d ..b0 ; b1 /; ad 1 ; : : : ; a1 / D 0; dAC1 .h; b0 ; ad 1 ; : : : ; a1 / for d  2

(3.7)

defines a module homomorphism Cone.c/ ! Cone.c/, Q which by Lemma 1.16 is an isomorphism in H 0 .Q/ (here and in similar cases below, jb0 j is the degree of b0 in homA .X; Y0 /, which means before the shift Œ1 has been applied). Hence, the cohomology class Œc determines Cone.c/ up to non-canonical isomorphism in that category (non-canonical because (3.7) requires a choice of h). By definition, the same thing holds for Cone.c/ as an object of H 0 .A/. Remark 3.5. There is an elementary relation between direct sum, shift, and mapping cone, namely, the cone of the trivial morphism c D 0 2 homA .Y0 ; Y1 / quasirepresents S S Y0 ˚ Y1 . For instance, assume that A is closed under mapping cones, which means that Cone.c/ exists for all c. Taking first an auxiliary object Z and some chain eZ representing the identity, one finds that C one.eZ / is isomorphic to the zero module. Then, for any Y , the cone of the zero map Y ! Cone.eZ / is actually SY , which implies that A is closed under shifts S  ,  > 0 (obviously, it also follows that S Y0 ˚ S Y1 exists for all Y0 ; Y1 ). (3f) Exact triangles. Any abstract mapping cone C D Cone.c/ comes with canonical module homomorphisms  2 hom0Q .Y1 ; C/,  2 hom1Q .C; Y0 /, given in terms of

36

I A1 -categories

(3.6) by 1 .b1 / D .0; .1/jb1 j b1 /,  1 .b0 ; b1 / D .1/jb0 j1 b0 , with vanishing higher order terms. These form a triangle diagram in H.Q/: Y0 `@

ŒI 1 .c/

@@ Œ1 @@ Œ @@

C

/ Y1 ~ ~~ ~~Π~ ~ ~

(3.8)

where the label Œ1 on the arrow reminds us of the fact that the associated morphism has degree one. For future reference, we want to mention alternative representatives of these maps: Lemma 3.6. Let eY0 ; eY1 be chains representing the identity morphisms in H.A/. Then the module homomorphisms Q 2 hom0Q .Y1 ; C /, Q 2 hom1Q .C; Y0 / given by Qd .b; ad 1 ; : : : ; a1 / D .0; dAC1 .eY1 ; b; ad 1 ; : : : ; a1 //; Q d ..b0 ; b1 /; ad 1 ; : : : ; a1 / D dAC1 .eY0 ; b0 ; ad 1 ; : : : ; a1 / are cohomologous to  and , respectively. Proof. We know that the Yoneda embedding is cohomologically unital. Hence, there 1 1 exists an h 2 hom1 Q .Y1 ; Y1 / such that Q .h/ D eY1  I .eY1 / is the difference between the strict identity in Q, and the chain coming from a representative of the identity in A. By taking this as the second summand, and setting the first one to be 1 zero, we get a k D .0; h/ 2 hom1 Q .Y1 ; C/ which clearly satisfies Q .k/ D   Q. The argument for   Q is parallel.  We define an exact triangle to be any diagram in H.A/ of the form Y0 `A

Œc1 

AA Œ1 AA Œc3  AA

Y2

/ Y1 } } } }} ~}} Œc2 

(3.9)

which becomes isomorphic to (3.8) under the Yoneda embedding. This means that if we set c D c1 , then there is a isomorphism Œt W Y2 ! C D Cone.c/ in H 0 .Q/, such that Œ ı Œt D ŒI 1 .c3 / and Œ D Œt ı ŒI 1 .c2 /. As a consequence of our previous discussion of the dependence of Cone.c/ on the choice of chain c, that choice turns out to be irrelevant here (meaning that if one can produce an isomorphism Œt for one choice of c, then also for any other choice). Here is a more concrete criterion for exactness, following the general idea of Lemma 3.1:

37

3 Exact triangles

Lemma 3.7. A triangle (3.9) is exact if and only if the following holds. There are h2 2 hom0A .Y2 ; Y1 /;

h1 2 hom0A .Y1 ; Y0 /;

k 2 hom1 A .Y1 ; Y1 /

satisfying 1A .h1 / D 2A .c3 ; c2 /; 1A .h2 / D 2A .c1 ; c3 /;

(3.10)

1A .k/ D 2A .c1 ; h1 / C 2A .h2 ; c2 / C 3A .c1 ; c3 ; c2 /  eY1 ; where eY1 is a chain representative for the identity; and moreover, for each object X, the following chain complex must be acyclic: homA .X; Y2 /Œ1 ˚ homA .X; Y0 /Œ1 ˚ homA .X; Y1 /; 0 1 1A 0 0 1A 0 A: 2A .c3 ;  / @D@ 2 3 2 A .h2 ;  / C A .c1 ; c3 ;  / A .c1 ;  / 1A

(3.11)

Proof. Suppose that all the data listed above exist. If we set C D Cone.c1 /, then b D .c3 ; h2 / is a cocycle in C.Y2 /, hence yields a module homomorphism t D

C .b/ 2 hom0Q .Y2 ; C/. The complex (3.11) is the mapping cone of the associated chain map (3.1), hence its acyclicity ensures that Œt is an isomorphism in H 0 .Q/. From (1.26) and the definition of C , one sees that Œ2Q .t; I 1 .c2 // D Œ C .2C .b; c2 // is the image under C of .2A .c3 ; c2 /; 2A .h2 ; c2 / C 3A .c1 ; c3 ; c2 // 2 C.Y1 /. By assumption (3.10) this is cohomologous to .0; eY1 /, whose image under C is the module homomorphism Q from Lemma 3.6. As a consequence, Œ D Œ2Q .t; I 1 .c2 //: 1 .Y2 ; Y0 /, Next, using (1.27) one sees that in HomH.Q/

Œ2Q .; t/ D Œ Y0 . 1 .b// D Œ Y0 .c3 / D ŒI 1 .c3 /: This completes the list of requirements for an exact triangle. For the other direction, one essentially reads the argument backwards. Any morphism Œt W Y2 ! C comes, via C , from an element b 2 C .Y2 /. Because ŒI 1 . 1 .b// D Œ2Q .; t/ D ŒI 1 .c3 /, the first component of b is cohomologous to c3 . After adding a suitable coboundary, one can assume that b D .c3 ; h2 / for some h2 . Next, Œ C .0; eY1 / D ŒQ D Œ2Q .t; I 1 .c2 // D Œ C .2C .b; c2 // D Œ.2A .c3 ; c2 /; 2A .h2 ; c2 / C 3A .c1 ; c3 ; c2 //. Choosing a bounding cochain between these two cycles yields the remaining data .k; h1 /. 

38

I A1 -categories

(3g) Another characterization. While the notion of quasi-representing object yields a unified approach to various constructions, it can be cumbersome when it comes to working out specific properties. For this reason, we will temporarily continue our discussion of exact triangles without proofs. This will be remedied later on, when we switch to a more concrete point of view. Consider the strictly unital A1 -category D with objects Z0 ; Z1 ; Z2 ; homD .Zk ; Zk / D K  eZk ; homD .Z0 ; Z1 / D K  x1 ; jx1 j D 0; homD .Z2 ; Z0 / D K  x3 ; jx3 j D 1; homD .Z1 ; Z2 / D K  x2 ; jx2 j D 0; homD .Z2 ; Z1 / D homD .Z1 ; Z0 / D homD .Z0 ; Z2 / D 0I and where the only nontrivial composition maps are 3D .x3 ; x2 ; x1 / D eZ0 ;

3D .x1 ; x3 ; x2 / D eZ1 ;

3D .x2 ; x1 ; x3 / D eZ2 :

Proposition 3.8. A triangle (3.9) in H.A/ is exact if and only if there is an A1 -functor F W D ! A such that F .Zk / D Yk and ŒF 1 .xk / D Œck  on cohomology. It is instructive to compare this with Lemma 3.7. Suppose that we have an A1 functor F as stated above. Choose dk with 1A .dk / D F 1 .xk /  ck . Then h1 D F 2 .x3 ; x2 / C 2A .F 1 .x3 /; d2 /  2A .d3 ; c2 /; h2 D F 2 .x1 ; x3 /  2A .F 1 .x1 /; d3 /  2A .d1 ; c3 /; k D 3A .d1 ; c3 ; c2 /  3A .F 1 .x1 /; d3 ; c2 / C 3A .F 1 .x1 /; F 1 .x3 /; d2 / (3.12)  2A .F 2 .x1 ; x3 /; d2 / C 2A .d1 ; 2A .F 1 .x3 /; d2 /  2A .d3 ; c2 /  F 2 .x3 ; x2 // satisfy all the equations in (3.10), with eY1 D F 1 .eZ1 /. Other components of F can be used to construct a contracting homotopy for the complex (3.11) (which we are not going to write down), and this is one possible way of proving half of Proposition 3.8. The proof that we will give later on is similar in principle, but with better theoretical underpinnings, which reduce the reliance on explicit formulae such as (3.12). Corollary 3.9. If F W A ! B is any A1 -functor, then H.F / maps exact triangles in H.A/ to exact triangles in H.B/. Corollary 3.10. Let F W A ! B be a cohomologically full and faithful A1 -functor. Suppose that we have an exact triangle in H.B/, such that the objects forming it lie in the image of F . Then its preimage is an exact triangle in H.A/. Remark 3.11. Proposition 3.8 is due to Kontsevich. Our D D D3 is in fact part of a family of A1 -categories Dn , n  1, having n objects Z0 ; : : : ; Zn1 . An A1 -functor

3 Exact triangles

39

F W D1 ! A with F .Z0 / D Y0 exists iff Y0 is the zero object in H.A/. Similarly, F W D2 ! A with F .Zk / D Yk exists iff Y0 and Y1 are isomorphic in H 0 .A/ (see [95, p. 193] for a discussion of this). Hence, exact triangles can be viewed as a generalization of the notion of isomorphism from a pair to a triple of objects. Remark 3.12. Here is another noteworthy consequence. Take the basic exact triangle in D consisting of the objects Zk . The Massey product of any three consecutive maps, as defined in Remark 1.2, is the identity up to a sign: hŒx3 ; Œx2 ; Œx1 i D ŒeZ0 ;

hŒx1 ; Œx3 ; Œx2 i D ŒeZ1 ;

hŒx2 ; Œx1 ; Œx3 i D ŒeZ2 : (3.13) By functoriality of Massey products under A1 -functors, the same holds for any exact triangle. For instance, suppose that we take an exact triangle and rescale one of the three morphisms involved by some constant in K n f0; 1g. In general, the resulting triangle will violate (3.13), hence is no longer exact (we say “in general” because the quotient groups in which the Massey products take their values (1.4) may be zero, in which case it is still possible for the modified triangle to be exact). This is also true in the classical framework of triangulated categories, see for instance [20, Corollaire 1.1.10(ii)]. (3h) Triangulated A1 -categories. Call an A1 -category triangulated if it is nonempty (ObA ¤ ;) and • every morphism Œc1  in H 0 .A/ can be completed to an exact triangle (3.9); • for any object Y , there is a Yz such that S Yz Š Y in H 0 .A/. A little explanation is appropriate. In view of Remark 3.5, the first condition implies that every object Y has a shift SY , which makes the second condition meaningful. To put it more functorially, the first condition allows one to construct a shift functor S W A ! A (see Section (3d)), and the second condition asserts that this is a quasiequivalence. Example 3.13. The A1 -categories Q D mod.A/ are triangulated. Proposition 3.14. If A is a triangulated A1 -category, then H 0 .A/ is a triangulated category in the classical sense. Moreover, if A; B are triangulated A1 -categories and F W A ! B is an A1 -functor, then H 0 .F / is an exact functor. Recall that a triangulated category (in the classical sense) [141], [61] is an additive category together with a “translation” self-equivalence T and a class of “distinguished triangles”, which are periodic diagrams of the form    ! Y0 ! Y1 ! Y2 ! T Y0 ! T Y1 ! T Y2 !    :

40

I A1 -categories

In our case, translation is induced by the shift functor, T D H 0 .S/, and distinguished triangles are the exact ones, where we use (3.3) to turn Œc3  into a morphism Y 2 ! S Y0 . (3i) Properties of triangles. Proposition 3.14 says that the class of exact triangles in a triangulated A1 -category satisfies the axiomatic properties of a triangulated structure. In fact, suitably formulated versions of these properties carry over to general A1 -categories. For the benefit of readers with little previous exposure to homological algebra, here is a partial list: Corollary 3.15. Let (3.9) be an exact triangle in an A1 -category A. Then any two subsequent morphisms in that triangle compose to zero, and moreover, by taking Hom’s to the left or right with a fixed object X, we get long exact sequences of vector spaces HomH.A/ .X; Y0 / m

/ HomH.A/ .X; Y1 /

/ HomH.A/ .X; Y2 /;

Œ1

HomH.A/ .Y0 ; X/ o

HomH.A/ .Y1 ; X/ o 1

HomH.A/ .Y2 ; X/:

Œ1

The converse is false, meaning that the properties in the corollary are not enough to characterize exact triangles (counterexamples can be easily constructed using Remark 3.12). Corollary 3.16. Take two exact triangles

Œc3 

Œb0 

? Y0

/ Yz ? 0

ŒcQ3 

Œc1 

ŒcQ1  Œb2 

Y2 _>_ _ _ _ _ _ _ _ _ _/ Yz2 _>> >> > > Œc2 

>> >> 

Y1

ŒcQ2  Œb1 

(3.14)

>> >>  / Yz

1

0 0 .Y0 ; Yz0 /, Œb1  2 HomH.A/ .Y1 ; Yz1 /, which satisfy together with maps Œb0  2 HomH.A/ 2 2 ŒA .cQ1 ; b0 / D ŒA .b1 ; c1 /. Then there is a (not unique) morphism Œb2 , which completes the commutative diagram as shown in (3.14). Moreover, if Œb0 , Œb1  are isomorphisms, then so is Œb2 .

41

3 Exact triangles

Corollary 3.17. Suppose that we have an exact triangle (3.9), and that Y0 admits a shift S Y0 . Then the rotated triangle Y1 aC C

Œc2 

CC Œ1 CC ŒS.c1 / CC

/ Y2 { { {{ {{ }{{ Œc3 

SY0 is again exact.

In the statement of the last corollary, we have implicitly used (3.3) to think of Œc3  as a morphism to SY0 . For Œc1  we similarly use (3.5), whose nontrivial part is the application of the shift functor S, hence the notation ŒS.c1 /. Remark 3.18. The last-mentioned corollary has a simple but useful implication. Suppose that in an exact triangle (3.9), the map Œc1  vanishes. Because the isomorphism class of the abstract mapping cone is independent of the choice of cocycle, the same conclusion as in Remark 3.5 holds, namely that Y2 Š SY0 ˚ Y1 in H 0 .A/. By rotating the exact triangle, one gets similar statements if any of the other maps vanishes. The full list is Œc1  D 0 H) Y2 Š SY0 ˚ Y1 ; Œc2  D 0 H) Y0 Š Y1 ˚ S 1 Y2 ; Œc3  D 0 H) Y1 Š Y2 ˚ Y0 : Corollary 3.19. Given exact triangles Y0 ]

Œa1 

/ Y1

Y1 ]

Œ1

Œa2 

/ Y2

Œ1



Z2

there is also an exact triangle

Œa2 Œa1 

/ Y2

Œ1



Z0

Y0 ]



Z1 I

/ Z1

Z0Y Œ1



Z2 : It is conventional to list the axioms of a triangulated category as (TR1)–(TR4) [61, Chapter 5]. Corollary 3.17 corresponds to (TR2); Corollaries 3.15 and 3.16 are consequences of (TR1)–(TR3); and Corollary 3.19 is an incomplete version of the octahedral axiom (TR4).

42

I A1 -categories

(3j) Generators. Let B be a triangulated A1 -category, and A  B a full A1 z  B subcategory, which is not empty. Consider the smallest full subcategory B which contains A, is closed under isomorphism (in the sense that if X0 Š X1 in z then also X1 2 Ob B), z and is itself triangulated. This H 0 .B/ and X0 2 Ob B, is called the triangulated subcategory of B generated by A (or equivalently, by z by starting with ObA, forming all ObA  ObB). Concretely, one gets Ob B possible mapping cones as well as shifts (upwards and downwards), then repeating the process an arbitrary number of times. The subcategory obtained in this way is z triangulated because, as a special case of Corollary 3.9, a triangle is exact in H.B/ iff it is exact in H.B/. We say that A generates B (or that its objects are generators z D B. for B) if B Let A be a nonempty A1 -category. A triangulated envelope is a pair .B; F / consisting of a triangulated A1 -category B and a cohomologically full and faithful functor F W A ! B, such that the objects in the image of F are generators for B. z Fz / Lemma 3.20. Triangulated envelopes always exist. Moreover, if .B; F / and .B; z such that are two such envelopes for A, there is a quasi-equivalence G W B ! B z G ı F is isomorphic to Fz in H 0 . fun.A; B//. In particular, the triangulated category H 0 .B/ is independent of the choice of envelope up to equivalence (actually, exact equivalence, by the last part of Proposition 3.14). We call this the derived category of A, and denote it by D.A/. The advantage of this definition is that it leaves room for various realizations, using different triangulated envelopes. However, for most applications, this generality is more hindrance than help. We will therefore give up our neutrality and declare in favour of a particular choice of envelope, namely the A1 -category of twisted complexes Tw A, whose definition mimics that of the dg category of chain complexes over an additive category. The mapping cone operation is realized in an explicit way in this category, which makes it easy to study; and we will take this as the starting point for proving all the abstract properties stated above. Remark 3.21. There is at least one alternative canonical construction of triangulated envelope, based on Example 3.13; namely, one takes the full subcategory of Q D mod.A/ generated by the image of the Yoneda embedding. Remark 3.22. The “chain level” or “intrinsic” viewpoint to triangulated structures adopted here arose first in [28]. That paper also introduced twisted complexes, at least for dg categories; the extension to A1 -categories was made in [90]. (3k) Additive enlargement. The first step in the construction of Tw A is the additive enlargement †A. Let A be a non-unital A1 -category. Objects of †A are triples

3 Exact triangles

X D .I; fX i g; fV i g/, written more suggestively as formal direct sums M XD V i ˝ Xi;

43

(3.15)

i2I

where I is a finite set,; fX i gi2I a family of objects of A, and fV i gi2I a corresponding family of finite-dimensional graded vector spaces, called multiplicity spaces. A morphism between two such sums is made up of morphisms between their constituent summands, suitably tensored with linear maps of the associated vector spaces: M  M M j V0i ˝ X0i ; V1 ˝ X1j D homK .V0i ; V1j / ˝ homA .X0i ; X1j /; hom†A i 2I0

j 2I1

ij

(3.16) with the natural grading. We will write morphisms as matrices a D .aj i / indexed by j 2 I1 and i 2 I0 , respectively, and where each entry has the form aj i D P j ik ˝ x j i k 2 homK .V0i ; V1j / ˝ homA .X0i ; X1j /. When writing down formulae, k we will often restrict to the case when the sum has only one term, aj i D  j i ˝ x j i , linearity being tacitly understood. With this in mind, we can define the composition maps on †A. They combine those of A with ordinary composition of linear maps, just like in the tensor product from Remark 1.11: X i ;i .1/G dd d 1    1i1 ;i0 ˝ d†A .ad ; : : : ; a1 /id ;i0 D (3.17) i1 ;:::;id 1 i ;i ˝ dA .xdd d 1 ; : : : ; x1i1 ;i0 /; P i ;i i ;i where G D p 2, so D D 1 is a dg category. We now introduce endomorphisms T of the morphism spaces zi ; Z zj /. With respect to the bases above, these are given by homDz .Z T 1 . 00 ; 10 ; 11 / D .0; 0;  10 /; T 1 .10 ; 11 / D .0; 10 /; T 1 .00 ; 10 / D .10 ; 0/; and vanish in all other cases. Then, idC1z T 1 CT 1 1z is projection onto a subspace D D representing the 1z -cohomology. We apply the Perturbation Lemma, as explained D in Remark 1.13, and get a quasi-isomorphic A1 -category. An explicit computation shows that this is precisely the category D occurring in Proposition 3.8 (note that compositions of order > 3 vanish for degree reasons; hence, the computation stops z  Tw A such that at that point). We therefore get an A1 -functor G W D ! D 1 1 1 z G .Zk / D Zk , ŒG .x1 / D Œc, ŒG .x2 / D Œi, ŒG .x3 / D Œp. Now suppose that we have an exact triangle (3.9) in H.A/. Let C  A be the full subcategory consisting of the three objects in that triangle, and Cz  Tw A the full subcategory which contains the images of those objects under A ,! Tw A and also Cone.c/, for c D c1 . By Lemmas 3.31 and 3.27, the obvious embedding C ,! Cz is a quasi-equivalence, so using Theorem 2.9 one can find an inverse quasi-equivalence K W Cz ! C . This inverse acts trivially on the objects Y0 ; Y1 , takes Cone.c/ to Y2 , and maps Œi, Œp to Œc2 , Œc3  respectively. Hence, F D K ı G W D ! C  A is an A1 -functor with the desired properties. It remains to remove the strict unitality assumption. Suppose that A is only c-unital, and that we are looking at some exact triangle in it. Using Lemma 2.1, we find a formal diffeomorphism ˆ such that AQ D ˆ A is strictly unital. By Corollary 3.9, the image of our triangle under ˆ is still exact. One now applies the previous discussion to get an A1 -functor Fz W D ! AQ with the desired properties, and sets F D ˆ1 ı Fz .  Lemma 3.32. Tw A is generated by its full subcategory A. Proof. Take a twisted complex Y , with its filtration by subcomplexes F i Y . Since ıY is strictly lower-triangular, it induces maps d i 2 hom1TwA .F i Y =F iC1 Y; F iC1 Y / for any i . If one thinks of these as morphisms of degree zero from S 1 .F i Y =F iC1 Y / to F i C1 Y , then their mapping cones are isomorphic to F i Y , so we have exact triangles S 1 .F i Y =Fd iC1 Y /

/ F iC1 Y

Œ1



F i Y:

(3.31)

3 Exact triangles

53

By definition, F i Y =F iC1 Y has vanishing differential, hence is isomorphic to a direct sum of shifted copies of objects of A. By using this and (3.31), one sees Y can be constructed from objects of A by repeatedly applying cones and shifts, as required by the definition of generating subcategory.  Lemma 3.33. An A1 -category A is triangulated iff the embedding A ,! Tw A is a quasi-equivalence. Proof. Suppose that the embedding is a quasi-equivalence. The abstract mapping cone of any degree zero cocycle in A is quasi-represented by an object of Tw A, hence by assumption also by an object of A. The same holds for shifts in either direction, which by definition means that A is a triangulated A1 -category. Conversely, suppose that A is triangulated. By Lemma 3.32, any object of H 0 .Tw A/ can be constructed, up to isomorphism, by starting with objects of H 0 .A/ and repeatedly applying mapping cones and shifts in either direction. However, because A is triangulated, all these operations will stay within the full subcategory of H 0 .Tw A/ consisting of those objects which are isomorphic to some object of H 0 .A/. This means that the embedding is a quasi-equivalence.  From this, together with Proposition 3.29 and Lemma 3.31, one can derive Proposition 3.14. Namely, H 0 .Tw A/ is triangulated in the classical sense; and since H 0 .A/ is an equivalent full subcategory with the same translation functor and corresponding class of distinguished triangles, it is itself triangulated. Lemma 3.34. Let F W A ! B be a cohomologically full and faithful A1 -functor. Assume that B is a triangulated A1 -category, and that the objects in the image of F generate it. Then there is a quasi-equivalence Fz W Tw A ! B whose restriction to A  Tw A is isomorphic to F in H 0 . fun.A; B//. Proof. By assumption, B ,! Tw B is a quasi-equivalence. Take a quasi-equivalence K W Tw B ! B in inverse direction, as given by Theorem 2.9, and define Fz D K ı Tw F . This clearly has the right property under restriction to A, and it is at least cohomologically full and faithful. By assumption, any object of H 0 .B/ can be constructed, up to isomorphism, by starting with objects of the form F .X/ and repeatedly applying mapping cones and shifts in either direction. But because Fz itself respects exact triangles and shifts, see Corollary 3.9, all these operations remain in the subclass of objects isomorphic to those in the image of Fz , which therefore is a quasi-equivalence.  This, together with Lemma 3.32, implies Lemma 3.20.

54

I A1 -categories

(3s) Mapping cones of A1 -modules. We now consider Example 3.13. Suppose that t W M0 ! M1 is a degree zero homomorphism of A1 -modules. Define C D C one.t/ to be the A1 -module C.X/ D M0 .X/Œ1 ˚ M1 .X/; dC ..b0 ; b1 /; ad 1 ; : : : ; a1 / (3.32)  d  D M0 .b0 ; ad 1 ; : : : ; a1 /; dM1 .b1 ; ad 1 ; : : : ; a1 / C t d .b0 ; ad 1 ; : : : ; a1 / : This is obviously a generalization of (3.6), which is the case where Mk D I.Yk /, t D I 1 .c/ come from the Yoneda embedding. There are canonical module homomorphisms  W M1 ! C ,  W C ! M0 of degree zero and one, respectively, defined exactly as in the special case we have just mentioned. Lemma 3.35. The following triangle in H 0 .Q/ is exact: M0 `B

BB Œ1 BB B Œ BB

Œt

/ M1 | | || || Π| | ~

C.

Proof. Let us look at the full embedding Q ,! Tw Q. In the latter category, we have the canonical cone Cone.t/ from (3.28). There is a tautological morphism C ! Cone.t/;

(3.33)

which induces isomorphisms homTwQ .M; C / ! homTwQ .M; Cone.t// for all M 2 Ob Tw Q (in fact, it is enough to check that for M 2 ObQ, since the general case follows from this by a straightforward filtration argument). This in particular implies that (3.33) is an isomorphism. In addition, one checks easily that it relates the natural morphisms i; p between Cone.t/ and its constituents with the ;  defined above. Lemmas 3.27 and 3.31 do the rest.  This shows that any morphism in H 0 .Q/ can be completed to an exact triangle. The existence of shifted (in either direction) modules is a tautology, hence Q is triangulated. To round off our discussion, we take up the point made in Remark 3.21. Since Q is triangulated, one can use Lemma 3.34 to construct a cohomologically full and faithful functor Tw A ! Q. There is also a more concrete construction, compare [82, Section 7]. Namely, combine the Yoneda embeddings for A and Tw A to an A1 -functor I

ITwA A IQ W Tw A ! mod.Tw A/ ! mod.A/ D Q;

whose restriction to A  Tw A is clearly equal to IA .

4 Idempotents

55

Lemma 3.36. IQ is cohomologically full and faithful. Proof. Consider the subset of those objects Y of Tw A with the property that Q Q // Q W HomH.TwA/ .X; Y / ! HomH.Q/ .I.X/; I.Y H.I/

(3.34)

is an isomorphism for all X 2 ObA. Because IQ extends the Yoneda embedding for A, this is clearly true when Y is an object of X. Corollary 3.9 says that IQ maps exact triangles to exact triangles (in fact, one could check by hand that cones are mapped to cones in a strict sense, not just up to isomorphism). One can use the associated long exact sequences from Corollary 3.15 to show that the class of such Y is closed under forming mapping cones. By Lemma 3.32, this implies that all Y have this property. Now modify (3.34) by asking that the bijectivity should be true for all X 2 ObTw A, and repeat the previous argument. 

4 Idempotents (4a) Background. We begin by recalling some elementary notions from category theory. Let A be a category linear over K. Suppose that we have an idempotent endomorphism p 2 HomA .Y; Y / of some object Y . An object Z together with inclusion and retraction morphisms k; r, Zo

k r

/

Y;

rk D eZ ; kr D p

(4.1)

will be called the image of p. The definite article is justified because, given two triples .Z; k; r/ and .Z 0 ; k 0 ; r 0 /, there is a canonical isomorphism Z Š Z 0 . We say that A is split-closed if it contains images of all idempotent endomorphisms. In particular, if A is split-closed and additive, then any idempotent endomorphism gives rise to a direct sum decomposition Y D Z ˚ Z ? , where the summands are the images of p and of the complementary projection p ? D eY  p. A split-closure of a category A is a pair .B; F / where B is a split-closed category, and F W A ! B a full and faithful embedding, such that any object of B is the image of some idempotent endomorphism in A. For any A there is a tautological construction of a split-closure, called the Karoubi completion PA. Objects of PA are pairs .Y; p/, and the morphism spaces are the corresponding summands HomPA ..Y0 ; p0 /; .Y1 ; p1 // D p1 HomA .Y0 ; Y1 /p0 : The embedding A ! PA is Y 7! .Y; eY /. If .B; F / is an arbitrary split-closure of A, then F extends canonically to an equivalence PA ! B. This proves that any two splitclosures are equivalent, in a way which is compatible with the relevant embeddings

56

I A1 -categories

of A. Of course, in practice the general notion of split-closure is somewhat redundant, since one tends to work with PA anyway. Turning to triangulated categories, which are particularly relevant for our purpose, one can prove that a triangulated structure on A induces one on PA [18]. While we do not need this result, the following related but simpler observation will be useful later: Lemma 4.1. Let B be a split-closed triangulated category, and A  B a triangulated full subcategory (this means nonempty, and closed under translations and exact triangles). Let AQ  B be the full subcategory consisting of all objects isomorphic to a direct summand of an object in A, so AQ Š PA. Then AQ is again a triangulated full subcategory of B. Q By definition, each Yzj is isomorphic to Proof. Take a morphism cQ W Yz0 ! Yz1 in A. a direct summand of some object Yj in A. Denote the corresponding inclusion and Q 0 to exact triangles, retraction morphisms by kj , rj . Complete both cQ and c D k1 cr z denoting the third objects by Y2 and Y2 , respectively. There are morphisms r2 ; k2 which complete the commutative diagrams / Yz 2 r2   / Y2

Yz1 r1



Y1

/ T Yz 0 T .r0 /



/ T Y0

YzO 1 and

k1

Y1

/ Yz O 2  k2  / Y2

/ T Yz O 0 T .k0 /

/ T Y0 ,

where T is the translation functor. Since rj kj D idYzj for j D 0; 1, r2 k2 is an automorphism of Yz2 (this is a standard consequence of the axioms of the triangulated structure; in the case where A D H 0 .A/ comes from a triangulated A1 -category, we stated it as Corollary 3.16). It follows that Yz2 is the image of the idempotent endomorphism p D k2 .r2 k2 /1 r2 of Y2 , hence contained in AQ by definition.  (4b) Idempotents and A1 -modules. Consider the ground field K as an A1 -category with a single object , in the obvious way. Of course, the cohomological category of this is again K, now considered as an ordinary category. We will not distinguish notationally between the two; this should cause only minimal confusion. Let A be an A1 -category, A D H.A/ its cohomological category, and Q D mod.A/ the A1 -module category. An idempotent up to homotopy for an object Y is defined to be a non-unital A1 -functor } W K ! A with }. / D Y . Concretely, this is given by elements } d 2 hom1d A .Y; Y /, d  1, satisfying the sequence of equations ( X X } d 1 if d is even, r s1 sr A .} ; : : : ; } / D (4.2) 0 if d is odd r s ;:::;sr 1

4 Idempotents

57

where the sum is over partitions s1 C    C sr D d . Lemma 4.2. Let p 2 HomA .Y; Y / be an idempotent endomorphism in the cohomological category. Then there is an idempotent up to homotopy }, such that Œ} 1  D p. Proof. p defines a non-unital functor K ! A. By Lemma 1.9, the obstructions for lifting this to an A1 -functor lie in HH 2 .K; A/2r , for r > 2. The underlying Hochschild complex is ( if r even; rCs s s r;s s pT  Tp CC .K; A/ D HomA .Y; Y /; @ .T / D .1/ ? pT  Tp if r odd. After splitting HomA .Y; Y / into the images and kernels of left and right composition with p, one sees directly that the only nonzero cohomology groups are HH s .K; A/s D p HomAs .Y; Y /p ˚ p ? HomAs .Y; Y /p ? .  Remark 4.3. Instead of a direct computation using the Hochschild complex, one could use some basic homological algebra. First adjoin a unit to K, so that the functors under discussion become unital; the resulting category of bimodules as in Remark 2.7 is semisimple, hence the higher order Ext groups all vanish. Let } be an idempotent up to homotopy. We associate to it an A1 -module Z, called the abstract image of }, which is defined as follows: Z.X/ D homA .X; Y /Œq is the space of formal polynomials in one graded variable q, jqj D 1, with coefficients in homA .X; Y /. Write ıq for the normalized formal differentiation, which sends q 0 to 0 and all other powers q k to q k1 . With this notation, the differential is X X

b.q/  b.q/ ; 2q r s2 ;:::;sr (4.3) where the last term is antisymmetrization of the power series followed by ıq ; this does not really involve multiplying by 12 , hence also makes sense if K has characteristic 2. The higher order structural maps, for d  2, are 1Z .b.q// D

ıqs2 CCsr rA .} sr ; : : : ; } s2 ; b.q// C .1/jbj

dZ .b.q/; ad 1 ; : : : ; a1 / X X s CCsr r D ıqd C1 A .} sr ; : : : ; } sd C1 ; b.q/; ad 1 ; : : : ; a1 /: r

sd C1 ;:::;sr

58

I A1 -categories

Lemma 4.4. The underlying cohomological module H.Z/ can be described as follows. H.Z/.X / Š p HomA .X; Y /, where p D Œ} 1  2 HomA .Y; Y /, and the module structure H.Z/.X1 / ˝ HomA .X0 ; X1 / ! Hom.Z/.X0 / comes from the ordinary composition p HomA .X1 ; Y / ˝ HomA .X0 ; X1 / ! p HomA .X0 ; Y /. Proof. Z.X/ has a natural increasing filtration, namely Fr D homA .X; Y /q 0 ˚  ˚ homA .X; Y /q r1 . We have usually worked with decreasing filtrations, and for the sake of uniformity, we turn this into one by changing r $ r. The resulting spectral sequence has ( HomAs .X; Y /; r  0; r;s sr E1 D H .FrC1 =Fr / D 0; otherwise. The differential is induced by the terms in (4.3) which contain precisely one ıq or 1=q, hence is given by

p

@

r;s

.b/ D .1/

rCs

?

b; r > 0 odd; pb; r > 0 even; 0; otherwise:

Hence, the spectral sequence degenerates at the E2 stage, and H.Z/.X/ is isomorphic to the only nonzero column E20 D p HomA .X; Y /. In fact, the edge homomorphism of the spectral sequence provides a canonical isomorphism between the two groups, and using that, the desired compatibility with the module structure becomes obvious.  Lemma 4.5. For any A1 -module M there is an isomorphism HomH.Q/ .Z; M/ Š H.M/.Y /p; and this is natural with respect to left composition with module homomorphisms. Proof. Consider the abelian category Mod 0 .A/ as in Remark 2.15. Lemma 4.4 says that as an object of that category, H.Z/ is a direct summand of HomA .; Y /; and since the latter is projective, so is the former. In view of (2.14), this means that the spectral sequence induced by the length filtration on homQ .Z; M/ degenerates at the E2 stage, and that the map HomH.Q/ .Z; M/ ! HomMod 0 .A/ .H.Z/; H.M/Œ/ is an isomorphism. By the classical Yoneda Lemma together with Lemma 4.4, the latter graded group is naturally isomorphic to H.M/.Y /p.  Consider the idempotent endomorphism in H 0 .Q/ obtained by applying the Yoneda embedding to p. Lemma 4.5 implies directly that Z is the image of this endomorphism, in the sense of (4.1). In particular, it follows that up to isomorphism in H 0 .Q/, the abstract image depends only on p, and not on the choice of }.

59

4 Idempotents

Remark 4.6. The definition of Z was suggested by an observation from [24], which says that in suitable categories of complexes, direct summands exist by an infinite resolution trick (I am grateful to François Loeser who pointed this out to me). (4c) Split-closed A1 -categories. Call A split-closed if the abstract image of any idempotent up to homotopy is quasi-represented by some object of A. A split-closure of an A1 -category A is a pair .B; F / consisting of a split-closed A1 -category B together with a cohomologically full and faithful functor F W A ! B such that in H 0 .B/, every object is the image of an idempotent endomorphism of an object of H 0 .A/. In formulating these notions, we have followed the general template from Section 3; but in fact, their meaning can be considerably simplified. First of all, A is split-closed iff H 0 .A/ has that property (in the original, cohomology level, sense). Namely, assume that H 0 .A/ is split-closed, and take an idempotent up to homotopy }, with Œ} 1  D p. By assumption, p has an image Z in H 0 .A/. Using Lemma 4.5, one shows easily that the Yoneda module of Z is isomorphic in H 0 .Q/ to the abstract image of }, hence yields a quasi-representative. The reverse implication is similar, but uses Lemma 4.2 as well. As an immediate consequence, .B; F / is a split-closure of A iff .H 0 .B/; H 0 .F // is a split-closure of H 0 .A/. Lemma 4.7. Every A1 -category has a split-closure. Moreover, if .B; F / and z Fz / are two such closures for A, there is a quasi-equivalence G W B ! B z such .B; z that G ı F is isomorphic to Fz in H 0 . fun.A; B//. Proof. Write …A for the full A1 -subcategory of Q D mod.A/ consisting of all objects that are isomorphic, in H 0 .Q/, to the abstract image Z of some idempotent up to homotopy }. We claim that …A contains all Yoneda A1 -modules Y associated to objects of Y . To see that, take some idempotent up to homotopy such that Œ} 1  is the identity endomorphism of Y (this exists by Lemma 4.2); by Lemma 4.5, the abstract image of } will be isomorphic to the Yoneda module of Y . Given that, the pair .…A; IA W A ! …A  Q/ is a split-closure of A. Indeed, as observed above, it is sufficient to look at the cohomological category, and there the same combination of Lemmas 4.2 and 4.5 yields the desired result. For the second part, let .B; F / be any split-closure of A. Using the Yoneda embedding for B and pullback of A1 -modules, one defines an A1 -functor IB

F

G W B ! mod.B/ ! Q: Since F is cohomologically full and faithful, G ı F is isomorphic to IA , see (2.13). This implies that G is cohomologically full and faithful when restricted to objects of the form F .Y /; but by definition, every object of H 0 .B/ is the image of an idempotent

60

I A1 -categories

endomorphism of some F .Y /, hence G is cohomologically full and faithful on the whole of B. A similar argument shows that the image of G is contained in …A, and that G is in fact a quasi-equivalence B ! …A. Of course, now that we have a quasiequivalence between an arbitrary split-closure and …A, one can use Theorem 2.9 to construct quasi-equivalences between any two such closures.  Lemma 4.8. The split-closure of a triangulated A1 -category is again triangulated. Proof. Let A be a triangulated A1 -category. By Example 3.13, Q D mod.A/ is triangulated, and through the Yoneda embedding, H 0 .A/ is identified with a triangulated subcategory (in the classical sense) of H 0 .Q/. Moreover, H 0 .…A/ is the subcategory consisting of objects which are isomorphic to some direct summand of an object of H 0 .A/, hence is again a triangulated subcategory of H 0 .Q/ by Lemma 4.1. In other words, if we have an exact triangle in H.Q/ and two of the objects involved lie in …A, then so does the third one. Because of Corollary 3.10, this implies that …A itself is a triangulated A1 -category.  Let A be an arbitrary A1 -category; take a triangulated envelope of it, and then the split-closure of that, denoting the outcome by B. For instance, taking the standard choices both times yields B D ….Tw A/. We call H 0 .B/ the split-closed derived category, and denote it by D  .A/. In parallel with the discussion in Section (3j), we should also mention a related notion. Let B be a split-closed triangulated A1 -category, and A  B a full A1 -subz  B which category, which is not empty. Consider the smallest full subcategory B 0 contains A, is closed under isomorphism in H .B/, and is itself triangulated and splitclosed. We call this the subcategory of B split-generated by A. Concretely, one gets z by starting with ObA, forming mapping cones and shifts, repeating that process Ob B arbitrarily often, and finally taking images under all idempotent endomorphisms of the previously constructed objects. We say that A split-generates B (or that its objects z D B. For instance, ….Tw A/ is split-generated by the are split-generators of B) if B objects of A. As an immediate consequence of the uniqueness results in Lemmas 3.20 and 4.7, we get: Corollary 4.9. Let B be a split-closed triangulated A1 -category, and let A  B be a full subcategory which split-generates it. Then there is a quasi-equivalence ….Tw A/ ! B, which induces an equivalence of triangulated categories, H 0 .B/ Š D  .A/.  Remark 4.10. The split-closure …A has one aesthetically unappealing property, which is that it does not contain A itself as a full subcategory (it does, of course, contain a subcategory quasi-isomorphic to A). We will now outline an alternative construction, which is a variant of unbounded twisted complexes. This is more

4 Idempotents

61

concrete and stays closer to the original inspiration, but at least in the version presented here, it requires strict unitality. Let A be a strictly unital A1 -category. We will assume that it has a zero object O, such that homA .O; X/ and homA .X; O/ are trivial groups. Define another A1 -category †A as follows. Objects of †A are formal infinite sequences X D .X 0 ; X 1 ; X 2 ; : : : / with X i 2 ObA. A morphism a 2 homk†A .X0 ; X1 / is an infinite matrix a D .aj i /;

kCj i aj i 2 homA .X0i ; X1j /;

which is “essentially lower-triangular”, in the sense that there exists some N (depending on a) such that aj i D 0 if j < i  N . Composition maps are defined using matrix multiplication, in the same way as for the additive enlargement. Note that each X i piece is thought of as being shifted by i, which gives rise to some signs. In fact, to simplify the formulae below, we adopt the modified sign conventions from Remark 3.26. Consider pairs .X; ıX / consisting of X 2 Ob†A and a ıX 2 hom1†A .X; X/, which is strictly lower-triangular in the sense that ıXj i D 0 for j  i, and which satisfies (3.19). We define an A1 -category Tw A whose objects are such pairs, and where the morphisms are deformed by differentials as in (3.20). One can identify A with the full A1 -subcategory of Tw A consisting of sequences .X 0 ; X 1 D O; X 2 D O; : : : /. Given a homotopy idempotent } of Y 2 ObA, we define an object .Z; ıZ / of Tw A as follows:

„ } for all i; if i even, and j D i C 1  0,

Zi D Y

1

ji ıZ D

} 1  eY } j i 0

if i odd, and j D i C 1  0, if i C 1 < j  0, in all other cases.

The generalized Maurer–Cartan equation is equivalent to (4.2). Define r 2 hom0Tw A .Y; Z/;

r 00 D eY ;

k 2 hom0Tw A .Z; Y /; k 0i D } 1i ; i1;i h 2 hom1 D .1/i eY Tw A .Z; Z/; h

where all coefficients other than those mentioned are 0. Then 1Tw A .r/ D 0;

1Tw A .k/ D 0;

2Tw A .k; r/ D } 1 ;

2Tw A .k; r/  1Tw A .h/ D eZ :

(4.4)

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I A1 -categories

This means that in the cohomological category, Z is the image of the idempotent endomorphism Œ} 1  of Y . Hence, if we take the full A1 -subcategory of Tw A consisting of all objects of A together with those of type (4.4), then this is a splitclosure of A. To make the connection with the previous approach, note that the Yoneda A-module associated to Z is precisely the abstract image Z.

5 Twisting (5a) The abstract operation. Fix an A1 -category A and object Y in it. Given an A1 -module M over A, we can define another module C D TY M by setting   C.X/ D M.Y / ˝ homA .X; Y / Œ1 ˚ M.X/; 1C .c0 ˝ b; c1 / D ..1/jbj1 1M .c0 / ˝ b C c0 ˝ 1A .b/; 1M .c1 / C 2M .c0 ; b//; dC ..c0 ˝b; c1 /; ad 1 ; : : : ; a1 / D .c0 ˝dA .b; ad 1 ; : : : ; a1 /; dM .c1 ; ad 1 ; : : : ; a1 / C dMC1 .c0 ; b; ad 1 ; : : : ; a1 //

for d  2.

(5.1) The c-unitality of C can be proved in the same way as for (3.6). The map M 7! TY M is part of an A1 -functor TY W Q ! Q, where as usual Q D mod.A/, called the abstract twist functor along Y . The image of a pre-module homomorphism t 2 homQ .M0 ; M1 / is tQ D TY1 .t/ 2 homQ .TY M0 ; TY M1 /; tQ1 .c0 ˝ b; c1 / D ..1/jbj1 t 1 .c0 / ˝ b; t 1 .c1 / C t 2 .c0 ; b//; tQd ..c0 ˝ b; c1 /; ad 1 ; : : : ; a1 / D .0; t d .c1 ; ad 1 ; : : : ; a1 / C t d C1 .c0 ; b; ad 1 ; : : : ; a1 //

for d  2.

The higher order terms TYd , d  2, are zero (so TY is in fact a dg functor). By looking at (2.10) one sees that it is also strictly unital. One can decompose the abstract twist construction into two steps, as follows. First, take the tensor product M.Y / ˝ Y in the sense of Section (3c). This comes with an abstract evaluation homomorphism  W M.Y / ˝ Y ! M;  d .c0 ˝ b; ad 1 ; : : : ; a1 / D dMC1 .c0 ; b; ad 1 ; : : : ; a1 /

(5.2)

and TY M D Cone./ is its cone in the sense of (3.32). In particular, there are associated module homomorphisms ;  which together with  form an exact triangle

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5 Twisting

in H.Q/, M.Y / ˝eLY

LLL Œ1 LLL LLL Œ

Π

/M z z zz zzΠz z|

(5.3)

TY M .

Remark 5.1. There is a simple generalization of the twist operation to more than one twisting object. For instance, given two objects YC ; Y and an A1 -module M, we define C D TY ;YC M to be  C.X/ D M.Y / ˝ homA .X; Y / ˚ M.YC / ˝ homA .X; YC //Œ1 ˚ M.X/; 1C .c0; ˝ b ; c0;C ˝ bC ; c1 / D ..1/jb j1 1M .c0; / ˝ b C c0; ˝ 1A .b / C .1/jbC j1 1M .c0;C ˝ bC / C c0;C ˝ 1A .bC /; 1M .c1 / C 2M .c0; ; b / C 2M .c0;C ; bC //; with higher order terms defined in a similar fashion. If homA .Y ; YC / D 0, this is the same A1 -module as TY .TYC M/. If homA .Y ; YC / is merely acyclic, there is an obvious embedding of one A1 -module into the other with acyclic quotient, so the two operations still yield isomorphic objects of H 0 .Q/.

(5b) Twisting ofYoneda modules. Given two objects Y0 ; Y1 of A, denote by TY0 .Y1 / any object which quasi-represents TY0 .Y1 /, and call it the result of twisting Y1 along Y0 . If this exists, it comes with a canonical morphism Œi W Y1 ! TY0 .Y1 / in H 0 .A/, which turns into Œ under the Yoneda embedding. Now assume that H.A/ is closed under finite direct sums and shifts S  ,  2 Z. Additionally, assume that HomH.A/ .Y0 ; Y1 / is a finite-dimensional graded vector space. Then (Remark 3.4) one can form the tensor product HomH.A/ .Y0 ; Y1 / ˝ Y0 , and this comes with a canonical evaluation morphism Œf  to Y1 , which is the preimage of Œ under the Yoneda embedding. More concretely, if one chooses a basis Œb i  of HomH.A/ .Y0 ; Y1 /, with  i D jb i j, and thinks of the tensor product as the direct sum of shifted copies of Y0 , then Œf  is the morphism whose restriction to the i-th summand is the image of Œb i  under the isomorphism i

i

 0 .Y0 ; Y1 / Š HomH.A/ .S  Y0 ; Y1 / HomH.A/

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I A1 -categories

from (3.5). If TY0 .Y1 / exists in A, it sits in a triangle Œf 

HomH.A/ .Y0 ; Y1 / ˝ Y0

hQQQ QQQŒ1 QQQ Œp QQQQ

/ Y1 z z zz zzŒi z |zz

(5.4)

T Y0 Y1

where Œp corresponds to Œ under the Yoneda embedding. By applying Corollary 3.10, one sees that (5.4) is exact. Finally, let us look at the case when A is triangulated and c-finite (cohomologically finite). By the latter condition, we mean that for any two objects Y0 ; Y1 , the space HomH.A/ .Y0 ; Y1 / is finite-dimensional. The triangulated nature of A implies closedness under direct sums and shifts (Remark 3.5). More importantly, the evaluation morphism Œf  has a mapping cone in A, sitting in its own standard exact triangle. By Corollary 3.16, the image of this cone under the Yoneda embedding is isomorphic to TY0 .Y1 /. Hence, all objects TY0 .Y1 / exist in A. Using the same argument as in Section (3d), one can now make the twist functorial. This means that for each Y 2 ObA there is a twist functor TY W A ! A, unique up to isomorphism in H 0 . fun.A; A//, which has the property that the diagram A 

TY

/A

I

Q

TY



I

(5.5)

/Q

commutes up to isomorphism in H 0 . fun.A; Q//. An important special case is when B D Tw A is the category of twisted complexes over a c-finite A1 -category. Recall from Lemma 3.36 that the Yoneda embedding A ! Q D mod.A/ extends naturally to a cohomologically full and faithful functor IQ W B ! Q. Fix Y0 2 ObA and Y1 2 ObB. By definition, the Yoneda embedding takes TY0 .Y1 / 2 ObB to the abstract twist TY0 .Y1 / as defined in mod.B/. Moreover, the abstract twist construction is compatible with restriction mod.B/ ! Q. The upshot is that the notions of twist in B and Q are compatible: Q Y .Y1 // Š TY .I.Y Q 1 // I.T 0 0

in H 0 .Q/:

(5.6)

Alternatively, one could derive this from (5.4) and Corollary 3.16, but the argument above provides a canonical isomorphism. This is not only slightly more precise, but it also allows one to make the isomorphism functorial, in the same sense as in (5.5).

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5 Twisting

Remark 5.2. It is straightforward to generalize this discussion to the setting from Remark 5.1. The resulting objects TY0; ;Y0;C .Y1 / sit in exact triangles HomH.A/ .Y0; ; Y1 / ˝ Y0; ˚ HomH.A/ .Y0;C ; Y1 / ˝ Y0;C

/ Y1 ww w w ww ww w w w{ w

iSSS SSS Œ1 SSS SSS SS

TY0; ;Y0;C Y1 :

(5c) Detecting quasi-representatives. Given objects Y0 ; Y1 ; Y2 of an A1 -category A, it is useful to have an explicit criterion for detecting when Y2 can be taken as a quasi-representative for TY0 Y1 . One could base this on Lemma 3.1, which specifies the data needed to define a morphism Y2 ! TY0 Y1 ; however, rather than doing that, we prefer to think in terms of a map going in the opposite direction. This choice, as well as the technical details of our formulation, are dictated by the intended geometric application (see Section 17). Lemma 5.3. (i) Take a pair .c; k/ consisting of c 2 homA .Y1 ; Y2 /; k W homA .Y0 ; Y1 / ! homA .Y0 ; Y2 /Œ1; which satisfy 1A .c/ D 0; 1A .k.c0 //  k.1A .c0 // C 2A .c; c0 / D 0 for all c0 2 homA .Y0 ; Y1 /:

(5.7)

This determines a module homomorphism t W TY0 .Y1 / ! Y2 , with the property that Œ2Q .t; / D ŒI 1 .c/ 2 HomH.Q/ .Y1 ; Y2 /. (ii) Given a pair .c; k/ as in (i), and arbitrary  2 homA .Y1 ; Y2 /Œ1;

W homA .Y0 ; Y1 / ! homA .Y0 ; Y2 /Œ2; one can form a new solution of (5.7) by setting cQ D c C 1A ./;   Q 0 / D k.c0 / C .1/jc0 j1 1 . .c0 //  .1 .c0 // C 2 .; c0 / ; k.c A

A

A

and the homomorphism tQ associated to that is cohomologous to the original t.

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I A1 -categories

(iii) Suppose that .c; k/ are as in (i). In addition, assume that for each X 2 ObA we are given maps dW lW W W

homA .X; Y1 / ! homA .X; Y2 /; homA .Y0 ; Y1 / ˝ homA .X; Y0 / ! homA .X; Y2 /Œ1; homA .X; Y1 / ! homA .X; Y2 /Œ1; homA .Y0 ; Y1 / ˝ homA .X; Y0 / ! homA .X; Y2 /Œ2

satisfying 1A .d.c1 // C d.1A .c1 // D 0; d.2A .c0 ; b// C 1A .l.c0 ; b// C .1/jbj1 l.1A .c0 /; b/ C l.c0 ; 1A .b// D 0; .1A .c1 //  1A ..c1 // C d.c1 /  2A .c; c1 / D 0; .1/jbj1 .1A .c0 /; b/ C

.c0 ; 1A .b//  1A . .c0 ; b//

C .2A .c0 ; b// C l.c0 ; b/  2A .k.c0 /; b/  3A .c; c0 ; b/ D 0 for all c0 ; b; c1 , and such that the following chain complex is acyclic: D D homA .Y0 ; Y1 / ˝ homA .X; Y0 /Œ2 ˚ homA .X; Y1 /Œ1 ˚ homA .X; Y2 /;  @.c0 ˝ b; c1 ; c2 / D .1/jbj1 1A .c0 / ˝ b C c0 ˝ 1A .b/; (5.8)  1 2 1 A .c1 / C A .c0 ; b/; A .c2 / C d.c1 / C l.c0 ; b/ : Then the homomorphism t defined by .c; k/ induces an isomorphism in H 0 .Q/. In fact, it is sufficient to verify this condition for a subset of objects which form a quasiequivalent full subcategory AQ  A; it will then follow automatically for all remaining objects. Proof. (i) A straightforward computation shows that if we define t as follows, it has all the desired properties: t d ..c0 ˝ b; c1 /; ad 1 ; : : : ; a1 / D dAC1 .c; c1 ; ad 1 ; : : : ; a1 / C dAC1 .k.c0 /; b; ad 1 ; : : : ; a1 / C dAC2 .c; c0 ; b; ad 1 ; : : : ; a1 /: 1 Q (ii) One defines a u 2 hom1 Q .TY0 Y1 ; Y2 / which satisfies Q .u/ D t  t , as in (i) but using .; / instead of .c; k/. (iii) In order for Œt to be an isomorphism, we need the mapping cones of t 1 to be acyclic. Concretely, these cones are the chain complexes

z D homA .Y0 ; Y1 / ˝ homA .X; Y0 /Œ2 ˚ homA .X; Y1 /Œ1 ˚ homA .X; Y2 /; D  Q 0 ˝ b; c1 ; c2 / D .1/jbj1 1 .c0 / ˝ b C c0 ˝ 1 .b/; 1 .c1 / C 2 .c0 ; b/; @.c A A A A  1A .c2 / C 2A .c; c1 / C 2A .k.c0 /; b/ C 3A .c; c0 ; b/ :

67

5 Twisting

z namely Using .; / one can construct an isomorphism D ! D, .c0 ˝ b; c1 ; c2 / 7! .c0 ˝ b; c1 ; c2 C .c1 / C

.c0 ˝ b//:

Hence the acyclicity of (5.8) implies the desired statement. The last remark is obviQ is a quasi-equivalence ous, since the restriction functor Q D mod.A/ ! mod.A/ (Lemma 2.8).  Lemma 5.4. Fix .c; k/ as in Lemma 5.3 (i). Suppose that in addition, there are  1 P g D j g2j ˝ g1j 2 homA .Y0 ; Y1 / ˝ homA .Y2 ; Y0 / ;  0 P p D j p2j ˝ p1j 2 homA .Y0 ; Y1 / ˝ homA .Y1 ; Y0 / ;  1 P q D j q2j ˝ q1j 2 homA .Y0 ; Y1 / ˝ HomK .homA .Y0 ; Y1 /; homA .Y0 ; Y0 // ; satisfying P j jg1 j1 1 A .g2j / ˝ g1j C g2j ˝ 1A .g1j / D 0; j .1/ P j P P j j jp1 j1 1 2 A .p2j / ˝ p1j C j p2j ˝ 1A .p1j /; j g2 ˝ A .g1 ; c/ D j .1/  P j  2 j j 3 j g2 ˝ A .g1 ; k.  // C A .g1 ; c;  / P P D j q2j ˝ 1A .q1j .  //  j q2j ˝ q1j .1A .  // P P j C j .1/jq1 jCjj1 1A .q2j / ˝ q1j .  / C j p2j ˝ 2A .p1j ;  / P C j b j ˝ .eY0 ˝ b j;_ /; where: eY0 is a representative of the identity for Y0 ; b j is a basis of homA .Y0 ; Y1 /; b j;_ the dual basis; and .eY0 ˝ b j;_ / 2 HomK .homA .Y0 ; Y1 /; homA .Y0 ; Y0 // the linear map which sends b j to eY0 , killing all other basis elements. Then, g gives rise to a module homomorphism u W Y2 ! homA .Y0 ; Y1 / ˝ Y0 of degree one, which is such that Œ2Q .u; t/ D Œ is the map from (5.3). Proof. Define pre-module homomorphisms u W Y2 ! homA .Y0 ; Y1 / ˝ Y0 ; P ud .b; ad 1 ; : : : ; a1 / D j g2j ˝ dAC1 .g1j ; b; ad 1 ; : : : ; a1 /; and v W TY0 .Y1 / ! homA .Y0 ; Y1 / ˝ Y0 ; P v d ..c0 ˝ b; c1 /; ad 1 ; : : : ; a1 / D j p2j ˝ dAC1 .p1j ; c1 ; ad 1 ; : : : ; a1 / P C j q2j ˝ dAC1 .q1j .c0 /; b; ad 1 ; : : : ; a1 / P C j p2j ˝ dAC2 .p1j ; c0 ; b; ad 1 ; : : : ; a1 /

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I A1 -categories

  

P j

g2j ˝ dAC2 .g1j ; c; c1 ; ad 1 ; : : : ; a1 /

j

g2j ˝ dAC2 .g1j ; k.c0 /; b; ad 1 ; : : : ; a1 /

j

g2j ˝ dAC3 .g1j ; c; c0 ; b; ad 1 ; : : : ; a1 /

P

P

of degree 0 and 1, respectively. Using the assumptions, one computes that 1Q .u/ D 0, and that w D 2Q .u; t/  1Q .v/ is given by w d ..c0 ˝ b; c1 /; ad 1 ; : : : ; a1 / D .1/jc0 j c0 ˝ dAC1 .eY0 ; b; ad 1 ; : : : ; a1 /: By the same argument as in Lemma 3.6, this is cohomologous to .



Remark 5.5. Suppose we are in the situation described by parts (i) and (iii) of Lemma 5.3; assume in addition that A is closed under finite direct sums and shifts, and that HomH.A/ .Y0 ; Y1 / is finite-dimensional, so that HomH.A/ .Y0 ; Y1 / ˝ Y0 makes sense. By construction, we then have the following exact triangle in H.A/: HomH.A/ .Y0 ; Y1 / ˝ Y0

hPPP PPP Œ1 PPP PPP PP

Œf 

/ Y1 ~ ~~ ~~ ~ ~ ~ Œc

(5.9)

Y2 .

From this point of view, the twisting procedure recedes into the background: one simply has a list of conditions, given by Lemma 5.3, which allows one to decide that a triangle of the specific form (5.9) is exact. Note, however, that the third morphism in the triangle is characterized only indirectly, by the fact that its image under theYoneda embedding is Œ ı Œt1 . It is here that Lemma 5.4 becomes relevant. Suppose that we have additional data as specified there, and consider the morphism X Œg D Œgj2  ˝ Œgj1  W Y2 ! HomH.A/ .Y0 ; Y1 / ˝ Y0 j

in H.A/. Under the Yoneda embedding, this maps to the module homomorphism Œu, which by construction satisfies Œu ı Œt D Œ. Hence, the third map in (5.9) is precisely Œg. (5d) Functoriality. Let F W A ! B be a cohomologically full and faithful functor between A1 -categories. Take Y0 ; Y1 2 ObA. Lemma 5.6. Suppose that both twists TY0 Y1 and TF .Y0 / F .Y1 / exist. Then there is a canonical isomorphism in H 0 .B/, F .TY0 Y1 / Š TF .Y0 / F .Y1 /:

(5.10)

5 Twisting

69

Proof. We start with a simpler preliminary version of the argument. By Corollary 3.16, the exact triangle (5.4) characterizes TY0 .Y1 / up to isomorphism in H 0 .A/. From this and Corollary 3.9 one can easily derive that the two objects in (5.10) are isomorphic. However, this does not give a canonical choice of isomorphism, and to get the sharper statement one has to argue a little more carefully. Along the same lines as in (1.23), one has for any Y 2 ObA and M 2 Ob.mod.B// a canonical module homomorphism s W TY .F  M/ ! F  .TF .Y / M/: Namely, s 1 maps .c0 ˝ b; c1 / to ..1/jc0 jCjbj1 c0 ˝ F 1 .b/; .1/jc1 j c1 /, and for d  2, s d maps ..c0 ˝b; c1 /; ad 1 ; : : : ; a1 / to ..1/4 c0 ˝F d .b; ad 1 ; : : : ; a1 /; 0/, where 4 D ja1 jC  Cjad 1 jCjbjCjc0 jd . By looking at d D 1, one sees that Œs is an isomorphism in H 0 .mod.A//. Applying this to Y D Y0 , M D .IB ı F /.Y1 / yields TY0 .F  ı IB ı F /.Y1 / Š F  .TF .Y0 / .IB ı F /.Y1 //: (5.11) By (2.13), the left-hand side of (5.11) is canonically isomorphic to TY0 .IA .Y1 //, which by definition of TY0 is canonically isomorphic to IA .TY0 .Y1 //, hence to .F  ı IB ı F /.TY0 Y1 /. By applying the definition of TF .Y0 / to the right-hand side, one finds that it is canonically isomorphic to .F  ıIB /.TF .Y0 / F .Y1 //. The conclusion is that the two objects in (5.10) become canonically isomorphic after applying F  ı IB . Again using (2.13), one sees that F  ı IB is cohomologically full and faithful on the image of F , hence also on the slightly larger class of objects of H 0 .B/ which are isomorphic to some F .X/. Since the two sides of (5.10) lie in that class, we get a canonical isomorphism between them.  The isomorphism constructed in this way is natural with respect to Y1 (in fact, with some more effort one could extend it to a canonical isomorphism between F ıTY0 and TF .Y0 / ı F in the functor category H 0 . fun.A; B//). It is also compatible with the canonical maps Y1 ! TY0 Y1 and F .Y1 / ! TF .Y0 / F .Y1 /. If one further assumes that our A1 -categories are closed under finite direct sums and shifts, and that the relevant Hom space is finite-dimensional, then the image of the exact triangle (5.4) under H.F / is canonically isomorphic to the corresponding triangle for the objects F .Y0 /, F .Y1 /. (5e) Composition of twists. Take a c-finite triangulated A1 -category A. Fix objects Y1 ; : : : ; Ym . Start with an arbitrary object X D Xm and twist it successively, forming Xk1 D TYk Xk . By definition, there are exact triangles Œij 

/ Xk1 Xk fNN o NNN o o NNN o wooo Œ1 Zk ˝ Y k

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with Zk D HomH.A/ .Yk ; Xk /. By repeatedly applying Corollary 3.19, one gets a combined exact triangle Œi1 ııŒim 

X bFF FF FF F

Y

/ TY : : : TY .X/ m 1 mm m m mmm m v mm Œ1

(5.12)

where the object Y is constructed from Zk ˝ Yk in a sequence of exact triangles, hence lies in the triangulated A1 -subcategory of A generated by the Yk . Now assume that Œi1  ı    ı Œim  D 0. Then (5.12) splits as discussed in Remark 3.18, so X ˚ TY1 : : : TYm .X/Œ1 Š Y:

(5.13)

Obviously, the usefulness of this observation in any given situation depends on whether one can prove that the composition of the Œik  vanishes. Here are two particularly simple cases: Corollary 5.7. Let A be a c-finite triangulated A1 -category. Suppose that there are Y1 ; : : : ; Ym such that for all objects X, TY1 : : : TYm X Š 0

(5.14)

is the zero object of H 0 .A/ (equivalently, the endomorphism group of TY1 : : : TYm X in that category is zero). Then the Yk generate A, and moreover, that category is split-closed. Proof. The first claim follows directly from (5.13). For the second one consider the split-closure …A, which is obviously again c-finite, and is triangulated by Lemma 4.8. From the definition of twisting, one sees that TYk preserves direct sums, which implies that (5.14) holds for all objects X of …A. Hence the Yk also generate …A, which means that the embedding A ,! …A is a quasi-equivalence.  Corollary 5.8. Let A be a c-finite triangulated A1 -category, which moreover is split-closed. Suppose that there are Y1 ; : : : ; Ym and a  ¤ 0 such that for all X, TY1 : : : TYm .X/ Š S  X in H 0 .A/. Then the Yk split-generate A. Proof. The finite-dimensionality assumption implies that for any X, there is an s > 0 s such that HomH.A/ .X; X/ D 0. Take the given collection of objects Yk and concatenate it with itself s many times. In the resulting exact triangle (5.12), the morphism X ! .TY1 : : : TYm /s .X/ Š S s X necessarily vanishes, which by (5.13) implies that X is split-generated by the Yk . 

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(5f) Twisted complexes. Let B be a strictly unital A1 -category, such that the chain level hom spaces are finite-dimensional, and take A D Tw B. In this context, twist functors have explicit representatives. Namely, given Y0 ; Y1 2 ObA, take the tensor product homA .Y0 ; Y1 / ˝ Y0 , where the first factor is thought of as a chain complex with boundary operator @ D 1A , as in Section (3o). By definition, homA .homA .Y0 ; Y1 / ˝ Y0 ; Y1 / D homA .Y0 ; Y1 /_ ˝ homA .Y0 ; Y1 /:

(5.15)

The 1A -operator on this morphism group is .c _ ˝ b/ 7! .1/jbj @_ .c _ / ˝ b C c _ ˝ @.b/, where @_ is the dual differential in the usual sense. In particular, if we identify (5.15) with EndK .homA .Y0 ; Y1 // in such a way that c _ ˝ b goes over to the endomorphism a 7! hc _ ; aib, then the identity endomorphism is a cocycle. Denote the corresponding element of (5.15) by f , and form the mapping cone Cone.f / as defined in Section (3p). Under the Yoneda embedding, homA .Y0 ; Y1 / ˝ Y0 becomes homA .Y0 ; Y1 / ˝ Y0 ; I 1 .f / is the abstract evaluation map  as defined in (5.2); and therefore Cone.f / D TY0 .Y1 /. This means that in this kind of A1 -category A, we can set (5.16) TY0 Y1 D Cone.f /: Remark 5.9. It is maybe instructive to make this even more explicit, at least in the simple case when Y0 ; Y1 are objects of B.PChoose a basis fb i g of homB .Y0 ; Y1 /, indexed by i 2 I , and write 1B .b i / D j ˇ j i b j . Then, TY0 .Y1 / is the twisted complex M i Y D S jb jC1 Y1 ˚ Y0 ; (5.17) i2I

with the differential ıY D . j i ˝aj i / consisting of ’s which are the identity between the various shifted copies of K, and

 .1/

a

ji

D

 i C1 j i

ˇ eY0

i i

.1/ b 0;

if i; j 2 I; if i 2 I; j D ; otherwise.

(5.18)

Here, is the index for the added row and column corresponding to the Y0 summand in (5.17) (alternatively, if one follows Remark 3.26 in changing the identifications i (3.29), then the sign .1/ disappears from both lines in (5.18), leaving ˇ j i eY0 and b i , respectively). The existence of canonical representatives (5.16) has many practical advantages. For instance, the A1 -functor TY0 can be defined by writing down a direct formula, as opposed to the indirect construction used in Section (5b).

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Remark 5.10. Let AQ be a c-finite triangulated A1 -category. Apply the Perturbation Lemma to replace it by a quasi-isomorphic category which has 1 D 0, hence has finite-dimensional chain level hom spaces. Next, use Lemma 2.1 to make the category strictly unital. Denote the outcome by B, and set A D Tw B. By construction, B is Q hence also a triangulated A1 -category, which in turn implies quasi-isomorphic to A, Q Using this, one can often reduce the fairly general that A is quasi-equivalent to A. context of c-finite triangulated A1 -categories to the technically simpler case we have just discussed. (5g) The adjoint twist. The twist operation has a dual, or more properly speaking adjoint. Even though this is equally important in principle, it will be used only rarely in our exposition, hence we only give the bare definition. Take objects Y0 ; Y1 of an A1 -category A, and define an A1 -module C _ by setting C _ .X/ D homA .X; Y1 / ˚ HomK .homA .Y1 ; Y0 /; homA .X; Y0 //Œ1; 1C _ .c; ˇ/ D .1A .c/; 1A ı ˇ C .1/jˇ j1 ˇ ı 1A C 2A .; c//; dC _ ..c; ˇ/; ad 1 ; : : : ; a1 /

(5.19)

D .dA .c; ad 1 ; : : : ; a1 /; dA .ˇ.  /; ad 1 ; : : : ; a1 / C dAC1 .  ; c; ad 1 ; : : : ; a1 /: Any quasi-representative of C _ will be called the adjoint twist of Y1 along Y0 , denoted by TY_0 Y1 . Now assume that A is c-finite and triangulated. Then, such a quasirepresentative exist for all Y0 ; Y1 . It sits in an exact triangle TY_0 Y1

hQQQ QQQ Œ1 QQQ QQQ QQ

/ Y1 nnn n n nn nnnŒf _  n n nw n

(5.20)

HomH.A/ .Y1 ; Y0 /_ ˝ Y0

involving the “dual evaluation map” Œf _ , which is again a combination of all possible morphisms Y1 ! Y0 . (5h) Spherical objects. Let A be a c-finite triangulated A1 -category. Following an idea originally due to Kontsevich, we say that Y0 2 ObA is a spherical object of dimension n 2 Z if it satisfies the following conditions: R n • (Poincaré duality) There is a linear map W HomH.A/ .Y0 ; Y0 / ! K such that for all Y1 , the pairing composition

R

n  n HomH.A/ .Y1 ; Y0 /˝HomH.A/ .Y0 ; Y1 / ! HomH.A/ .Y0 ; Y0 / ! K

is nondegenerate;

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• (Spherical endomorphism ring) The space of endomorphisms of Y0 fits into a short exact sequence ŒeY0 

R

0 ! K ! HomH.A/ .Y0 ; Y0 / !  K ! 0: If n ¤ 0, the last condition is equivalent to saying that HomH.A/ .Y0 ; Y0 / Š KŒt=t 2 , where t has degree n. For n > 0 this is just the ordinary cohomology ring of S n , which explains the terminology. We shall see later on that Lagrangian spheres (as well as homology spheres) become spherical objects in the Fukaya category. Lemma 5.11. If Y0 is spherical, TY0 is a quasi-equivalence from A to itself. Proof. Given two objects Yk of A (k D 1; 2), let C be the mapping cone (in the ordinary chain complex sense) of the map TY10 W homQ .Y1 ; Y2 / ! homQ .TY0 Y1 ; TY0 Y2 /. This carries a three-step decreasing filtration F  C , whose associated graded spaces are C =F 1 C D homQ .Y1 ; homA .Y0 ; Y2 / ˝ Y0 /Œ1; F 1 C =F 2 C D homQ .Y1 ; Y2 /Œ1 ˚ homQ .Y1 ; Y2 / ˚ homQ .homA .Y0 ; Y1 / ˝ Y0 ; homA .Y0 ; Y2 / ˝ Y0 /; F 2 C D homQ .homA .Y0 ; Y1 / ˝ Y0 ; Y2 /Œ1: Due to cancellations on the F 1 C =F 2 C level, the resulting spectral sequence has the following nonzero terms: rCsC1  if r D 0; HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y1 ; Y0 /  .Y ; Y / ˝ Hom 2 H.A/ 0 rCs E1rs D if r D 1; ˝ HomH.A/ .Y0 ; Y0 / ˝ HomH.A/ .Y0 ; Y1 /_   _ rCs1 HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y0 ; Y1 / if r D 2:

†

The E1 level differential @rs is induced by Œk , which means that it consists of composition maps in H.A/ or their duals. For instance, @1 is surjective, since HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y0 ; Y1 /_ ? ?Id˝Œe ˝Id Y0 y HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y0 ; Y0 / ˝ HomH.A/ .Y0 ; Y1 /_ ? ? 1 y@ HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y0 ; Y1 /_

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is the identity. Similarly, @0 is injective, since we get an isomorphism after composing with the Poincaré map: HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y1 ; Y0 / ? ? 0 y@ HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y0 ; Y0 / ˝ HomH.A/ .Y0 ; Y1 /_ ? ?Id˝R ˝Id y HomH.A/ .Y0 ; Y2 / ˝ HomH.A/ .Y0 ; Y1 /_ : Dimension-counting then shows that the E2 page of the spectral sequence is zero. By construction, this implies that TY0 is cohomologically full and faithful. It remains to prove surjectivity on isomorphism classes of objects. Write B  A for the full A1 -subcategory consisting of objects isomorphic to TY0 .Y1 / for some Y1 . From (5.4) and the sphericality of Y0 , it follows that TY0 .S n1 Y0 / Š Y0 ; hence Y0 lies in B. Using this and again (5.4), one sees that B generates A. On the other hand, Corollary 3.9 shows that B is itself a triangulated A1 -category, hence the inclusion B ,! A must be a quasi-equivalence.  Remark 5.12. In fact, the adjoint twist TY_0 is an inverse of TY0 , up to isomorphism in H 0 . fun.A; A//. We will not prove this (see [134, Proposition 2.10] for the parallel theorem in the context of classical derived categories), but we do want to explain a partial result, namely that TY0 TY_0 .Y1 / Š Y1 (5.21) in H 0 .A/. To see this, pass to the module category Q D mod.A/ and take C _ as in (5.19). The cohomology of C _ .Y0 / sits in a long exact sequence :: : HomH.A/ .Y1 ; Y0 /Œ1 ? ? y HomH.A/ .Y0 ; Y1 /_ ˝ HomH.A/ .Y0 ; Y0 /Œ1 ? ? y

(5.22)

H.C _ .Y0 // :: : where the first homomorphism is again a dual composition map. By Poincaré duality, R this becomes an isomorphism after projection W HomH.A/ .Y0 ; Y0 / ! K. Using

5 Twisting

75

the sphericality of the endomorphism ring of Y0 , we conclude that the restriction of the second map in (5.22) to the subspace HomH.A/ .Y0 ; Y1 /_ ˝ K  ŒeY0  is an isomorphism. In forming TY0 C _ as the cone of the abstract evaluation map, one may therefore replace C _ .Y0 / by the quasi-isomorphic subcomplex homA .Y0 ; Y1 /_ D HomK .homA .Y0 ; Y1 /; K  eY0 /. The outcome of that replacement is the A1 -module M.X/ D .homA .Y0 ; Y1 /_ ˝ homA .X; Y0 // ˚ homA .X; Y1 / ˚ HomK .homA .Y0 ; Y1 /; homA .X; Y0 //Œ1; 1M . ˝ b; c; ˇ/ D ..1/jbjCj j . ı 1A / ˝ b C  ı 1A .b/; 1A .c/; 1A ı ˇ C .1/jˇ j1 ˇ ı 1A C 2A .c;  / C 2A .eY0 ; b/ ˝ .  //; dM .. ˝ b; c; ˇ/; ad 1 ; : : : ; a1 / D . ˝ dA .b; ad 1 ; : : : ; a1 /; dA .c; ad1 ; : : : ; a1 /; dA .ˇ.  /; ad 1 ; : : : ; a1 / C dAC1 .  ; c; ad 1 ; : : : ; a1 / C dAC1 .eY0 ; b; ad 1 ; : : : ; a1 / ˝ .  //: There is a module homomorphism M ! Y1 , which consists of . ˝ b; c; ˇ/ 7! .1/jcj c with vanishing higher order terms. The kernel of this is the subcomplex of M.X/ where c D 0; using the fact that eY0 is the identity on the cohomology level, one sees easily that this is acyclic. Hence, in H 0 .Q/ we have isomorphisms TY0 C _ Š M Š Y1 . After using the definitions to return to H 0 .A/, this establishes (5.21). (5i) Exceptional collections. Throughout the following discussion, A will always be a c-finite triangulated A1 -category. A finite ordered family Y D .Y1 ; : : : ; Ym / of objects of A is called an exceptional collection if the following two conditions are satisfied: • Each Yk is an exceptional object, which by definition means that its endomorphism space is spanned by the identity: HomH.A/ .Yk ; Yk / D K  ŒeYk : • On the cohomology level, the morphisms between different Yk are directed, in the sense that HomH.A/ .Yk ; Yl / D 0 for k > l. If in addition, the Yk generate A in the sense of Section (3j), one says that Y is a full exceptional collection. Lemma 5.13. Suppose that Y is a full exceptional collection. Then for any X one has an isomorphism in H 0 .A/, TY1 : : : TYm .X/ Š 0:

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Proof. Set Xm D X, Xk1 D TYk .Xk / as in Section (5e). Our aim is to show that for all X, X0 Š 0. Consider first the case where X D Yl for some l. As long as k > l, we have HomH.A/ .Yk ; X/ D 0, hence Xk1 Š Xk Š    Š X by (5.4). In the next step, the triangle reduces to ŒeYl 

Yl aD D

DD Œ1 DD DD

/ Xl Š Yl tt tt t tt t z t

Xl1 ,

hence Xl1 Š Cone.eYl / Š 0. From there on, all the lower Xk , k < l, are immediately seen to be zero. F D TY1 : : : TYm is an A1 -functor from A to itself, hence preserves exact triangles by Corollary 3.9. This means that if we have two objects X0 ; X1 such that F .X0 /, F .X1 / are both zero, then the same holds for the mapping cone X2 D Cone.X0 ! X1 / of any morphism between them. Somewhat more trivially, F is also compatible with shifts up to isomorphism. Since the Yk generate A, this implies the desired result.  This argument in fact gives an explicit Postnikov decomposition of an arbitrary object X into shifted copies of the Yk (this is well-known in the framework of exceptional collections in general triangulated categories, see for instance [66, §2]). Namely, with Zk D HomH.A/ .Yk ; Xk / as before, there is a ladder consisting of exact triangles on the left and commutative triangles on the right, 0 ŠO X0 UU Œ1 UUUU UUU* XO 1

iii iiii t iii i

Z1 ˝ Y 1 Œ1



:: : :: : O

Xm1 O

(5.23) Œ1



Zm1 ˝ Ym1

i iiii tiiii UUUUŒ1 UUUU U* iiii tiiii

Xm D X .

Œ1



Zm ˝ Y m

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Remark 5.14. By looking at Corollary 5.7, one sees that if A admits a full exceptional collection, it is split-closed. In fact, slightly more is true: if Qfin  Q D mod.A/ is the full subcategory consisting of those A1 -modules M such that H.M.X// is finite-dimensional for each X 2 ObA, then the Yoneda embedding gives a quasiequivalence A ,! Qfin . The corresponding theorem in the classical framework is [27, Corollary, p. 530], which says that if T is a triangulated category with a full exceptional collection, then any cohomological functor from T to the category of finite-dimensional K-vector spaces is representable (due to the limitations of triangulated category theory, the proof is much harder than in the case we are dealing with here). (5j) Mutation. One can modify a given exceptional collection Y D .Y1 ; : : : ; Ym / by applying the following transformations: • Replace each Yk by an object Yzk isomorphic to it in H 0 .A/, and form the resulting collection Yz ; • Given .1 ; : : : ; m / 2 Zm , take S  Y D .S 1 Y1 ; : : : ; S m Ym / where S is the shift functor on A; • Finally, the k-th elementary mutation (1  k < m) is the operation Y 7! U D Lk Y , where Uj D Yj for j ¤ k; k C 1, and the remaining two objects are Uk D TYk YkC1 ;

UkC1 D Yk :

(5.24)

For the moment, let us work on the level of isomorphism classes of exceptional collection, which means that we consider transformations of the first kind to be trivial. Then, the second and third kinds are invertible. The inverse of the latter is Y 7! V D L1 Y , where as before Vj D Yj for j ¤ k; k C 1, and k Vk D YkC1 ;

VkC1 D TY_kC1 Yk :

Next, there are braid relations Lj Lk Y Š Lk Lj Y for jj  kj  2, Lj Lj C1 Lj Y Š Lj C1 Lj Lj C1 Y : The outcome is that one gets an action of the group Zm Ì Br m on the set of isomorphism classes of exceptional collections of length m. The subgroup Zm acts by shifts; the standard generators ˇ1 ; : : : ; ˇm1 of the braid group Br m act by elementary mutations; and the semidirect product is formed with respect to the permutation representation of the symmetric group Symm , considered as a quotient of Br m . Moreover, if an exceptional collection is full, the same holds for all collections in its Zm Ì Br m orbit.

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The proofs are easy arguments with exact triangles, which can be carried out in the framework of triangulated categories without reference to A1 -structures; see for instance [66, Proposition 2.4.2]. We emphasize that the Lk are not functors: they act on collections of objects within A, but not on the morphisms between these objects. (5k) The Koszul dual collection. Let Y D .Y1 ; : : : ; Ym / be a full exceptional collection in A. For k  l, write Yk;l D TYk TYkC1 : : : TYl1 .Yl /; which in the simplest case means Yk;k D Yk . Define the (right) Koszul dual collection Š Y Š to be the one formed by the objects YmC1k D Y1;k , 1  k  m. It is easy to see that this is again a full exceptional collection, obtained from the original Y by mutating along the “global half-twist” 1=2 D ˇm1 .ˇm2 ˇm1 / : : : .ˇ2 : : : ˇm1 /.ˇ1 : : : ˇm1 / 2 Br m

(5.25)

(our composition in the braid group is from right to left, which means that the mutation corresponding to (5.25) begins with Lm1 , followed by Lm2 down to L1 , and so on). Composition of the maps Œi in the relevant exact triangles (5.4) yields canonical morphisms in H 0 .A/, Œij;k;l  W Yk;l ! Yk1;l !    ! Yj;l for all j < k  l; we extend that to j D k by setting Œik;k;l  to be the identity. As a special case, we have for each k a canonical morphism Œi1;k;k  W Yk ! Y1;k D Š YmC1k . We will now discuss some properties of the Koszul dual collection. All of them are basically consequences of the long exact sequences induced by the triangles (5.4). For the sake of brevity, we write Hom rather than HomH.A/ ; also, the customary brackets around cohomology level morphisms will be omitted, so that ij;k;l stands for Œij;k;l . Š Lemma 5.15. (i) Hom.Yl ; YmC1k / is one-dimensional (generated by i1;k;k ) for k D l, and zero for k ¤ l. Š (ii) Hom.X; YmC1k / Š Hom.Yk ; Xk /_ , where Xk D TYkC1 : : : TYm .X/ as in (5.23). Š Š (iii) Hom.Ymk ; YmC1k / Š Hom.Yk ; YkC1 /_ Œ1.

Proof. We omit (i), which is straightforward. To derive the more general statement (ii) from that, we go through the decomposition (5.23) starting from both ends, and Š apply Hom.; YmC1k /, the outcome being that Š Š Š / Š Hom.Xm1 ; YmC1k / Š    Š Hom.Xk ; YmC1k /; Hom.X; YmC1k Š Š 0 Š Hom.X1 ; YmC1k / Š    Š Hom.Xk1 ; YmC1k /:

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The remaining exact triangle, relating Xk to Xk1 , then yields Š Š / Š Hom.Yk ; Xk /_ : Hom.Xk ; YmC1k / Š Hom.Hom.Yk ; Xk / ˝ Yk ; YmC1k Š For (iii), set X D Ymk in (ii). Then Xk is the cone of the map i1;kC1;kC1 W YkC1 ! Š Ymk , so that Hom.Yk ; Xk / Š Hom.Yk ; YkC1 /Œ1. 

Lemma 5.16. Take the degree one morphism Zk ˝ Yk ! ZkC1 ˝ YkC1 from (5.23), and use the isomorphism (ii) above to think of it as a map Š Š Hom.X; YmC1k /_ ˝ Yk ! Hom.X; Ymk /_ ˝ YkC1 :

(5.26)

Using (iii) above, this can be expressed as follows: take dual bases cl and cl_ of Š Š Hom.Yk ; YkC1 / and Hom.Ymk ; YmC1k /Œ1. For each l, take multiplication with _ Š Š cl , which is a linear map Hom.X; Ymk / ! Hom.X; YmC1k /, dualize it, and then combine it with cl to get a map between the groups in (5.26). Finally, add up over all l to get the correct result. Sketch of proof. Suppose first that X is simply the cone of some morphism Zk ˝ Yk ! ZkC1 ˝ YkC1 . In that case, all the isomorphisms from the previous Lemma can be written down easily, and one can verify the result by an explicit computation (which can be carried out either abstractly, using the standard tools of triangulated categories, or in terms of twisted complexes). Now consider a general object X. Take the canonical map X ! Xk1 , and complete it to an exact triangle X 0 ! X ! Xk1 . Again by applying the standard exact triangles, one sees that X 0 ! X induces an isomorphism between the groups (5.26) associated to X and its analogue for X 0 , 0 which commutes with the map between them. Similarly, if we set X 00 D XkC1 , then 0 00 the map X ! X induces isomorphisms on (5.26). Hence, it is sufficient to prove 0 commutativity for X 00 . But that satisfies TYk TYkC1 X 00 D Xk1 D 0, hence is a cone of the kind considered above.  (5l) A Beilinson type spectral sequence. Suppose that we have an object X, with Postnikov decomposition (5.23). Let K be a cohomological functor. This is a contravariant functor from H.A/ to graded K-vector spaces, which is compatible with finite direct sums and shifts, and takes exact triangles to long exact sequences; the precise definition can be found e.g. in [61], [143]. There is a spectral sequence converging to K.X/, whose starting term is  _ rCs E1rs D K rCs .Zmr ˝ Ymr / Š Zmr ˝ K.Ymr / : (5.27) The differential @r 1 is the image under K of the morphism Zmr1 ˝ Ymr1 ! Zmr ˝ Ymr from (5.23). To set up the spectral sequence, one can take the standard

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definition of the Atiyah–Hirzebruch spectral sequence [138, Chapter 15], and replace cell decomposition and generalized homology theory by Postnikov decomposition and cohomological functor, respectively; or if one prefers, there is a formulation in terms of exact couples [61, p. 263]. We will apply this general construction to obtain the following: Proposition 5.17. For any pair of objects U0 , U1 of H.A/, there is a spectral sequence converging to Hom.U0 ; U1 /, whose starting term is  rCs Š / ˝ Hom.Ymr ; U1 / : (5.28) E1rs D Hom.U0 ; YrC1 Choose dual bases as in Lemma 5.16, for k D m  r  1. Then the differential @r W E1r ! E1rC1; is X @.u0 ˝ u1 / D .1/jcl j.ju0 jCju1 j/ .cl_  u0 / ˝ .u1  cl /: (5.29) l

Proof. We set X D U0 and use K D Hom.; U1 /. After taking into account Š _ the isomorphism Zmr D Hom.Ymr ; Xmr /_ Š Hom.X; YrC1 /, the initial term (5.27) turns into (5.28), and Lemma 5.16 yields the formula (5.29) (strictly speaking, the sign issue is somewhat fuzzy, since we have not fully explained how the spectral sequence gets set up; but have no need for full precision at this point).  Remark 5.18. The original example of the spectral sequence (5.28) arises from Beilinson’s resolution of the diagonal on projective space, see [19] or [104, p. 245]. It was generalized to exceptional collections of coherent sheaves in [67], and from there, the extension to general triangulated categories is fairly straightforward. A complete exposition can be found in [66], in particular Sections 2.7.3 and 3.3.2 (note that due to the use of left Koszul duals rather than right ones, the E1 term of their spectral sequence is slightly different from ours). The connection with Koszul duality is explained in [26], [25]; see also Section (5o) below. (5m) Directed A1 -categories. Up to now we have taken an extrinsic viewpoint, where one studies exceptional collections and their mutations inside a fixed triangulated A1 -category. The alternative is to restrict attention to a special class of A1 -categories, whose objects themselves form an exceptional collection in the derived category. In this intrinsic context, a mutation will be a move that changes such a category into another one with an equivalent derived category. A directed A1 -category is a strictly unital A1 -category A! with m objects, together with an ordering of those objects as .Y1 ; : : : ; Ym /, such that

0

homA! .Yi ; Yj / D

for i > j; K  e Yi for i D j; finite-dimensional over K for i < j:

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Because of the strict unitality condition, the only non-tautological composition maps are dA! W homA! .Yid 1 ; Yid / ˝    ˝ homA! .Yi0 ; Yi1 / ! homA! .Yi0 ; Yid /Œ2  d ; i0 <    < id :

(5.30)

Hence, A! can be completely described by the finitely many elements of K which are matrix coefficients of (5.30). Let A be the triangulated envelope of A! . By definition, the images of the Yk under the cohomologically full and faithful functor A! ! A form an exceptional collection. Since by construction, A is generated by objects of A! , this collection is full. Remark 5.19. Let us take more explicitly A D Tw A! . Directedness allows one to make several minor technical simplifications in the definition of twisted complexes. First of all, any such complex can be written as 

XD

d M



V i ˝ Yi ; ıX D .ıXj i /

(5.31)

iD1

for some finite-dimensional graded vector spaces V 1 ; : : : ; V d and degree 1 elements ıXj i 2 HomK .V i ; V j / ˝ homA .Yi ; Yj /. The diagonal terms are of the form ıXi i D @V i ˝ eYi , where .V i ; @V i / is a chain complex of vector spaces. The next observation is that the lower-triangularity condition on ıX is automatically satisfied, hence can be omitted from the definition of a twisted complex in this L particular case. Namely, one can filter (5.31) by ik V i ˝ Y i , and then refine that by using the degree filtration of the leading chain complex .V k ; @V k /. This yields a filtration of X, with respect to which ıX satisfies the desired lower triangularity condition. Finally, any twisted complex X is isomorphic in H 0 .A/ to some Xz whose differential has vanishing diagonal terms, ı izi D 0. Namely, the Postnikov decomposition X (5.23) says that, up to isomorphism, one can reconstruct X from X0 D 0 through a sequence of mapping cones   Xk Š Cone S 1 Xk1 ! Zk ˝ Yk :

(5.32)

Suppose that we have an object Xzk1 isomorphic to Xk1 , which is a twisted complex consisting only of summands Y1 ; : : : ; Yk1 , and for which the differential has vanishing diagonal terms. Let Xzk be the cone of the morphism S 1 Xzk1 ! Zk ˝Yk corresponding to (5.32). This is isomorphic to Xk and has the same properties as Xzk1 , so by iterating the construction we end up with the desired Xz D Xzm .

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(5n) Directed subcategories. Temporarily, we will restrict ourselves to the class of A1 -categories which have the property that all hom spaces are finite-dimensional, since that simplifies some technical details. Let Y D .Y1 ; : : : ; Ym / a (not necessarily exceptional) collection of objects in such a category A. One then defines the associated directed A1 -subcategory A! D A! .Y / as the unique directed A1 -category with objects Yk , such that homA! .Yi ; Yj / D homA .Yi ; Yj / for i < j , and whose nontrivial composition maps (5.30) agree with their counterparts in A. This construction is functorial: suppose that F W A ! B is an A1 -functor, and let B ! be the directed A1 -subcategory associated to the image collection F .Y /. Then there is a unique strictly unital A1 -functor F ! W A! ! B ! , whose restriction to strictly ascending chains of objects equals F . If F is cohomologically full and faithful, F ! is a quasi-isomorphism. A word of warning: if A is strictly unital, one can identify A! with the subcategory of A consisting of homA .Yi ; Yj / for i < j , as well as the subspaces K  eYk for each k. However, in the general c-unital case, A! is not strictly speaking a subcategory. To remedy that, one has the following result: Lemma 5.20. Let Y be a collection of objects in A, and A! the associated directed A1 -subcategory. Then there is an A1 -functor H W A! ! A acting trivially on objects, which induces isomorphisms HomH.A! / .Yi ; Yj / Š HomH.A/ .Yi ; Yj / for all i < j . Proof. Use Lemma 2.1 to find a formal diffeomorphism ˆ such that AQ D ˆ A ! is strictly unital. Let AQ be the directed subcategory of AQ associated to the same ! Q collection Y. Because of strict unitality, we have an inclusion functor HQ W AQ ! A. ! ! ! Q On the other hand, there is an induced formal diffeomorphism ˆ W A ! A . Set H D ˆ1 ı HQ ı ˆ! .  Lemma 5.21. Let Y ; Y 0 be two collections of objects in A, with the property that each Yj is isomorphic to Yj0 in H 0 .A/. Then the associated directed subcategories A! .Y /, A! .Y 0 / are quasi-isomorphic (by a strictly unital A1 -functor, which preserves the ordering of the objects). Proof. Let B  A be the full A1 -subcategory that contains only the objects Yj , B 0 the corresponding subcategory with objects Yj0 , and C the one that contains both types of objects. By assumption, the inclusions of B and B 0 into C are quasi-equivalences. Using Theorem 2.9, one can therefore construct a quasi-equivalence B ! B 0 . From the construction, it is easy to see that this will map each Yj to Yj0 , hence is really a quasi-isomorphism. Now pass to directed subcategories.  Lemma 5.22. Let Y be a full exceptional collection in a triangulated A1 -category A. Take A! to be the associated directed A1 -subcategory, and H W A! ! A an

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A1 -functor as in Lemma 5.20. Then there is a quasi-equivalence G W Tw A! ! A, whose restriction to A!  Tw A! is isomorphic to H inside H 0 . fun.A! ; A//. In particular, D.A! / is equivalent to H 0 .A! / as a triangulated category. Proof. The fact that Y is an exceptional collection means that H is cohomologically full and faithful. Since A is triangulated and generated by the objects of Y , it is in fact a triangulated envelope of A! . The same holds for Tw A! ; now use Lemma 3.20.  One application of this goes as follows. Take a directed A1 -category A! , and the full exceptional collection in A D Tw A! formed by the objects of A! itself. Transform this collection by applying mutations, as in Section (5j), and let B ! be the directed A1 -subcategory of A associated to the mutated collection. Lemma 5.22 yields a quasi-equivalence Tw B ! ! Tw A! , hence an equivalence of triangulated categories D.B ! / Š D.A! /. As originally observed by Kontsevich, this defines an action of the group Zm Ì Br m on the set of quasi-isomorphism classes of directed A1 -categories with m objects, with the property that all categories lying on the same orbit have equivalent derived categories (actually, the central element .1; : : : ; 1/ 2 Zm  Zm Ì Br m acts trivially, so the action descends to the quotient Zm1 Ì Br m ). We will also need a version of Lemma 5.22 for collections which are not exceptional: Lemma 5.23. Let Y be a collection of objects in A, each of which is spherical. In Tw A, consider the collection U D Lk Y , defined by the same formula as in (5.24). Denote by B ! , C ! the directed A1 -subcategories associated to these collections. Then Tw B ! is quasi-equivalent to Tw C ! . Proof. In view of Lemma 5.6, we may replace the A1 -category A by any quasiequivalent one, and Y , U by the corresponding collections in the new category. Because of that, we may assume that A is strictly unital and has 1A D 0 (this is purely for convenience). We want to differentiate the notation a little, writing Yzk for the object of B ! corresponding to Yk in A. These objects forms a full exceptional collection Yz in z be the k-th basic mutation of this collection. The z D Lk X B D Tw B ! . Let U ! inclusion H W B ! A induces an A1 -functor Tw H W B ! Tw A, which satisfies Tw H .TYzi Yzj / D TYi Yj

for i < j .

(5.33)

On the left-hand side we are carrying out the twist inside B, and the right-hand side is the corresponding construction in Tw A. Both times we use the explicit formulae from Section (5f), and then the equality reduces to the fact that homB ! .Yzi ; Yzj / D z / D U . The homA .Yi ; Yj /. By definition, (5.33) means in particular that Tw H .U next step is to show that the maps .Tw H /1 W homB .Uzi ; Uzj / ! homTwA .Ui ; Uj /

for i < j

(5.34)

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are quasi-isomorphisms. This is nontrivial only when either i or j is equal to k. Most of these cases are dealt with by observing that one side is built out of homB ! .Yzp ; Yzq / for p < q, while the other consists of the corresponding homA .Yp ; Yq / spaces; they therefore coincide by the definition of B ! . The only remaining case is (i D k; j D k C 1), where the groups in (5.34) are homB .TYzk YzkC1 ; Yzk / D homA .Yk ; YkC1 /_ Œ1; homTwA .TYk YkC1 ; Yk / D homA .YkC1 ; Yk / ˚ homA .Yk ; YkC1 /_ Œ1 ˝ homA .Yk ; Yk /: Because 1A D 0, the first group has zero differential; on the second one, the differential is the map .2A /0 W homA .YkC1 ; Yk / ! homA .Yk ; YkC1 /_ ˝ homA .Yk ; Yk / obtained by partially dualizing the composition. Sphericality means that R .id ˝ / ı .2A /0 W homA .YkC1 ; Yk / ! homA .Yk ; YkC1 /_ Œnk  is an isomorphism (nk is the dimension of Yk as a spherical object). Hence .2A /0 is injective, R and by dimension-counting, its image is a subspace complementary to ker.id˝ / D homA .Yk ; YkC1 /˝KeYk . But the map in (5.34) is precisely .Tw F /1 D .0; id ˝ eYk /, hence is a quasi-isomorphism. z in B. This Let Cz ! be the directed A1 -subcategory associated to the collection U comes with a natural embedding into B, and by combining that with Tw H one gets an A1 -functor G W Cz ! ! Tw A. This is in fact strictly unital; it maps to the directed A1 -subcategory C !  Tw A; and by the previous computation, it yields a quasiisomorphism between Cz ! and C ! . Hence Tw C ! is quasi-equivalent to Tw Cz ! ; but by our general discussion of mutation, the latter category is quasi-equivalent to Tw B ! .  To extend the construction of directed subcategories to the case where A is only c-finite, one can proceed as follows: using the Perturbation Lemma, pass to a quasiisomorphic A1 -category AQ where the actual homAQ spaces are finite-dimensional, and then apply the previous construction to that category. The result, still denoted by A! , again comes with an A1 -functor H W A! ! A as in Lemma 5.20. The Q but drawback is that A! is no longer canonical, since it depends on the choice of A, it is at least unique up to quasi-isomorphism, which is sufficient for most purposes. In particular, Lemmas 5.22 and 5.23 will hold as before. (5o) Koszul dual collections revisited. Let A! be a directed A1 -category, and A D Tw A! . Recall from Lemma 3.36 that there is a canonical cohomologically full and faithful A1 -functor IQ W A ! Q D mod.A! /. By definition, the image of Š Š YmC1k 2 ObA under this functor is the A1 -module YmC1k D TY1 : : : TYk1 .Yk /.

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It is straightforward to write down what such a module actually looks like. The underlying graded vector spaces are M M Š .Yl / D homA .Yir ; Yk / ˝ homA .Yir1 ; Yir / ˝    YmC1k r i1 ;:::;ir (5.35) ˝ homA .Yi1 ; Yi2 / ˝ homA .Yl ; Yi1 /Œr where the sum is over r  0 and l  i1 < i2 <    < ir < k; in particular, the r D 0 summand is just homA .Yl ; Yk /. Next, X .brC1 ˝    ˝ b1 / D .1/ brC1 ˝    ˝ bnCpC1 1Y Š mC1k p;n (5.36) ˝ pA .bnCp ; : : : ; bnC1 / ˝ bn ˝    ˝ b1 ; where  D jb1 j C    C jbn j  n; and for d  2, dY Š

.brC1 ˝    ˝ b1 ; ad 1 ; : : : ; a1 / X d 1Cn brC1 ˝    ˝ bnC1 ˝ A .bn ; : : : ; b1 ; ad 1 ; : : : ; a1 /: D

mC1k

n1

Now consider the simple A1 -module SSk , defined by setting SSk .Yk / D K, S Sk .Yl / D 0 for all other l, with 2S Sk .; eYk / D IdK , and all other structural maps set to zero. There is a canonical module homomorphism Š ! SSk : t W YmC1k Š .Yk / D homA .Yk ; Yk / D The first order piece t 1 is the identity map from YmC1k d K  eYk to S Sk .Yk /; and the higher order maps t , d  2, all vanish identically.

Lemma 5.24. Œt is an isomorphism in H 0 .Q/. Proof. The complex (5.35) admits an obvious increasing length filtration F r . Change r $ r to make it decreasing, and consider the associated spectral sequence, which Š converges to H.YmC1k .Yl //. The starting term is E1r;s D

 M

s

HomH.A/ .Yir ; Yk / ˝    ˝ HomH.A/ .Yl ; Yi1 /

i1 ;:::;ir

where the ik are as in (5.35), and differential X .1/CjbnC1 j brC1 ˝    ˝ bnC2 bnC1 ˝    ˝ b1 @r;s .brC1 ˝    ˝ b1 / D n

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with  as in (5.36). One recognizes this as a reduced bar resolution; more precisely, if one takes the sum over all l, then the outcome is the standard projective resolution of the simple object H.SSk / in the abelian category Mod 0 .H.A// of graded H.A/-modules. Hence, the E2 term of the spectral sequence is one-dimensional (spanned by ŒeYk ) for k D l, and vanishes otherwise. It is straightforward to set up a comparison argument with the trivial filtration of SSk .Yl /, the outcome being Š that t 1 W YmC1k .Yl / ! SSk .Yl / induces an isomorphism on cohomology for any l. The desired result follows from this by Lemma 1.16.  Remark 5.25. We outline how this point of view leads to an alternative construction of the spectral sequence (5.28). Any U 2 ObQ comes with a natural decreasing filtration by submodules ( U.Yk / k  m  r; r .F U/.Yk / D 0 otherwise. The graded pieces are F r U=F rC1 U Š U.Ymr / ˝ SSmr , where U.Ymr / is considered as a chain complex of vector spaces. Now set U D U1 , and consider the induced filtration of homQ .U0 ; U1 /. The starting term of the resulting spectral sequence is  rCs E1rs D H.homQ .U0 ; S Smr // ˝ H.U1 .Ymr // :

(5.37)

Now suppose that U0 ; U1 are images of twisted complexes under the cohomologically full and faithful embedding IQ W A ! Q. Then, using Lemmas 2.12 and 5.24, one can bring (5.37) into the form (5.28). (5p) Unital A1 -modules. Let P D mod.A! /fin be the A1 -category of finite and strictly unital A1 -modules. Finite-dimensionality means that M.Yk / is a graded vector space of finite (total) dimension; strict unitality means that 2M .b; eYk / D b, and dM .b; ad 1 ; : : : ; a1 / D 0 for d  3 whenever one of the aj is a unit; moreover, one only allows module pre-homomorphisms with an analogous vanishing property. Because of directedness, the latter condition implies that the morphism spaces homP .M0 ; M1 / are finite-dimensional. The Yoneda embedding I, and its extension IQ to twisted complexes, naturally land in P . On the other hand, the embedding P ! Q is cohomologically full and faithful; this follows by looking at the E1 terms of the spectral sequences coming from the length filtration, which are the full (for Q) and reduced (for P ) variants of the standard bar complex computing module Ext groups as in (2.14). Corollary 5.26. P is a triangulated envelope of A! .

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Proof. From the observations made above, it follows that I W A! ! P is a cohomologically full and faithful A1 -functor. Moreover, P is closed under taking mapping cones in the sense of (3.32), hence is triangulated by the same argument as in Lemma 3.35. It remains to show that the image of I generates P , or equivalently, that IQ W Tw.A! / ! P is a quasi-equivalence. But we know from Lemma 5.24 that Q and on the other hand, any module can the simple modules lie in the image of I, easily be represented as an iterated cone formed from simple ones.  Corollary 5.27. Any directed A1 -category is quasi-isomorphic to another one which is also directed, and which is actually a dg category (d D 0 for d > 2). Proof. The I.Yk / form an exceptional collection in P . Take the associated directed A1 -subcategory, which is straightforward since P is strictly unital and has finitedimensional hom spaces. Vanishing of higher order composition maps is a general property of A1 -modules.  Of course, we already knew that any A1 -category can be transformed into a quasi-isomorphic dg category (Corollary 2.14). The point of the result above is that in the directed case, one can do that while retaining finite-dimensionality of hom spaces; this would be impossible in the more general context.

6 Z=2-actions (6a) Basic notions. This section is about A1 -categories with involutions, in the simplest possible sense. A naive Z=2-action on an A1 -category A is an A1 -functor A W A ! A whose terms of order  2 vanish, and which satisfies A ı A D IdA . In fact, this just consists of an involution on objects and corresponding maps on morphisms A W homA .X0 ; X1 / ! homA .A X0 ; A X1 /, such that dA .A .ad /; : : : ; A .a1 // D A .dA .ad ; : : : ; a1 //: An A -fixed object is one such that A X D X. The fixed A1 -subcategory Afix is the full A1 -subcategory formed by these objects. The invariant A1 -subcategory Ainv has the same objects as Afix , but the morphisms are now the invariant parts homAinv .X0 ; X1 / D homA .X0 ; X1 /Z=2 , with the correspondingly restricted composition maps. We need passage to the invariant part to be homologically well-behaved, hence assume from now on that char.K/ ¤ 2. Several times later on, we will encounter instances of the following elementary question. Set A D H 0 .A/, and let A be the Z=2-action induced by A . Take two A -fixed objects X0 ; X1 , which are isomorphic in A. When can the isomorphism be

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made Z=2-invariant? The obstruction is the class Œy D A .x 1 /  x 2 fy 2 AutA .X0 / W y 1 D A .y/g= AutA .X0 /

(6.1)

where x is any isomorphism from X0 to X1 , and AutA .X0 / D HomA0 .X0 ; X0 / acts on itself by twisted conjugation, meaning that z acts by y 7! A .z/1 yz. The simplest situation is when X0 is free, by which we mean that HomA .X0 ; X0 / Š R ˚ R is the direct sum of two copies of the same graded algebra R, with A acting by exchanging the two summands. In that case, the set in (6.1) consists of pairs y D .r; r 1 /, and each of them can be trivialized by twisted conjugation with z D .r 1 ; e/. Hence: Lemma 6.1. Suppose that we have two A -fixed objects, one of which is known to be free. Then, if they are isomorphic in H 0 .A/, they are also isomorphic in  H 0 .Ainv /. (6b) Equivariant twisted complexes. As a preliminary step, we need to discuss the equivariant version †eq A of the additive enlargement. An object of †eq A is an object of †A, which means a triple .I; fX i g; fV i g/ written as a formal sum as in (3.15), together with the following additional data: an involution i ! i of the indexing set I , such that X i D A .X i /; and graded maps of vector spaces V i W V i ! V i , such that V i ı V i D Id for all i. Morphism spaces and composition maps are defined as in the non-equivariant version. Each morphism space carries an induced involution, which in the notation from (3.16) is obtained by taking the tensor products of i

j

V j ı  ı V i W homK .V0i ; V1j / ! homK .V0 ; V1 / 1

0

and i

j

A W homA .X0i ; X1j / ! homA .X0 ; X1 /: An equivariant twisted complex is a pair .X; ıX / consisting of X 2 Ob†eq A and a differential ıX lying in the invariant part hom1†eq A .X; X/Z=2 , with the usual conditions (existence of a filtration, which now should be by equivariant subcomplexes F i X ; and the Maurer–Cartan equation). The morphism spaces are inherited from †eq A, and composition maps are as in (3.20), the outcome being an A1 -category Tweq A carrying a naive Z=2-action which is trivial on objects. The simplest equivariant twisted complex would be a single A -fixed object X of A (more precisely, one takes a fixed one-element set I D f g, a one-dimensional vector space V  D K with trivial map V  D Id, and X  D X). A less trivial example is a direct sum X ˚ A X of an arbitrary object of A and its image; the differential is still trivial, but the involution i 7! i exchanges the two summands. Note that objects of the first kind give rise a full embedding of Afix into Tweq A, compatible with the Z=2-actions. Generalizing this, any twisted complex in Ainv can

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89

be viewed as an equivariant twisted complex (with trivial involutions on the sets I and vector spaces V i ), and in this way Tw.Ainv / becomes a full A1 -subcategory of .Tweq A/inv . Finally, one can just forget about the involutions, and this defines a full and faithful A1 -functor Tweq A ! Tw A. Let us mention the equivariant versions of some familiar operations. Direct sum can be defined in the obvious way. Next, given a finite-dimensional graded vector space Z with a Z=2-action Z and an equivariant twisted complex X, one can form the equivariant tensor product Y D Z ˝ X by taking the tensor products Z ˝ V i with the combined Z=2-action Z ˝ V i , and the differential ıY D idZ ˝ ıX . This includes the shift operation S , defined as usual by taking Z D KŒ1 with trivial Z=2action. Another interesting case is that of the nontrivial character of Z=2, meaning that Z D K but Z D idZ . Then, the tensor product just reverses the sign of all the V i , x A general tensor product Z ˝X is essentially a process which we denote by X 7! X. x one for each invariant (anti-invariant) basis a finite sum of shifted copies of X and X, element of Z. Finally, if c 2 homTwA .Y0 ; Y1 / is an invariant degree zero cocycle, then the mapping cone, defined by equipping SY0 ˚ Y1 with the differential (3.28), is again an equivariant twisted complex. This shows that .Tweq A/inv is triangulated (Tweq A itself is not). Lemma 6.2. Let Y; Y0 ; Y1 2 ObA be A -fixed, with the additional assumption that Y is free, and that the Z=2-action on HomH.A/ .Y0 ; Y1 / is trivial. Suppose that in D.A/ D H 0 .Tw A/, Y is isomorphic to the cone of some morphism Y0 ! Y1 . Then, the same statement holds in D.Ainv /. Proof. By assumption, HomH.A/ .Y0 ; Y1 / is Z=2-invariant, so without changing the isomorphism class of the cone, we can assume that the relevant cocycle c 2 hom0A .Y0 ; Y1 / is itself Z=2-invariant. In this case, we can form Cone.c/ as an object of Tweq A. In fact, it lies in the image of the embedding Tw.Ainv /  Tweq A. Since the forgetful functor H 0 .Tweq A/ ! H 0 .Tw A/ is full and faithful, we get an isomorphism Y Š Cone.c/ in H 0 .Tweq A/. Next, because Y is free, we can use Lemma 6.1 to make the isomorphism invariant, which means that it comes from an isomorphism in H 0 .Tw.Ainv //.  Lemma 6.3. Suppose that A is c-finite. Let X be an A -fixed object. Suppose that X quasi-represents the direct sum Y ˚ A Y of some object Y and its A -image, such that moreover HomH.A/ .Y; A Y / D 0. Suppose further that in D.A/, we have an isomorphism TY1 : : : TYm .Y / Š S.A Y /: (6.2) Here Y1 ; : : : ; Ym are A -fixed objects of A, with the property that the Z=2-action on HomH.A/ .Yj ; Yk / is trivial for all j < k, and S is the shift. Then X lies in the triangulated subcategory of Tw.Ainv / generated by the Yk .

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Proof. The most transparent situation is where A is strictly unital and 1A D 0 (which in particular implies that the homA spaces themselves must be finite-dimensional). By iterating (5.16), one finds that the object X on the left-hand side of (6.2) has the form M C DY ˚ homA .Yjr ; Y /Œ1 ˝ homA .Yjr1 ; Yjr /Œ1 ˝     (6.3)    ˝ homA .Yj1 ; Yj2 /Œ1 ˝ Yj1 ; L where the is over all r  1 and j1 < j2    < jr . Rather than writing down the full differential ıC explicitly, we will only need to know its general shape, which can be easily deduced from the construction. Firstly, • ıC strictly decreases the length of the tensor product expressions. In particular, it vanishes on the first summand Y . This means that C can be written as the cone of a morphism U ! Y , where .U; ıU / 2 ObTw A contains all the terms in (6.3) except the first one, together with the corresponding pieces of the differential. Secondly, • ıU is compatible with the filtration of C given by considering only summands with j1  j for some j . Moreover, the diagonal terms (the induced differentials on the associated graded spaces) with respect L to this filtration are homA .Yjr1 ; Yjr / ˝ @Zj ˝ eYj , where @Zj is some differential on Zj D    ˝ homA .Yj ; Yj2 /. Here, the sum is over all r and j2 <    < jr with j2  j (a closer look would show that @Zj is a form of the bar differential, but we will not use that fact). This, together with the assumption that homA .Yj ; Yk / is entirely Z=2-invariant for j < k, implies that each coefficient of ıU is the tensor product of some vector space homomorphism and an A -invariant morphism in A. Hence, U lies in the subcategory Tw.Ainv /  Tw A; and another look at the statement above shows that it actually lies in the triangulated subcategory of Tw.Ainv / generated by the Yk . In view of (6.2), we have an exact triangle in H.Tw A/ of the form U cGG

GG GG G Œ1 GG

/ wY w w ww ww w w{

S.A Y / .

By assumption, there are no nontrivial morphisms from Y to A Y of any degree in H.A/, hence the exact triangle splits, meaning that X Š Y ˚ A Y Š U

(6.4)

in H .Tw A/. All objects in (6.4) are equivariant twisted complexes, and the middle one is free. The argument concluding the proof of Lemma 6.2 then yields the desired isomorphism in H 0 .Tw.Ainv //. 0

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To extend the result to general A, removing the technical restrictions introduced above, one proceeds as follows. Choose data as in Remark 1.13, but taking care that the splitting and homotopy on the complexes homA .X0 ; X1 / are compatible with the Z=2-actions. The only instances where this causes any potential difficulty is when both X0 and X1 are A -invariant, but in those cases one simply splits the complex into its invariant and anti-invariant summands, and proceeds separately for each part, then finally putting the result together. Then, the formulae (1.18) after the Perturbation Lemma 1.12 preserve the existing symmetry, which means that we get an equivariant quasi-isomorphism F W A ! B, where B is an A1 -category with naive Z=2-action and vanishing 1B . Next, we want to make B strictly unital, as in Lemma 2.1. The first step of the proof of that lemma is trivial: since 1B D 0, there is a unique (automatically Z=2-invariant) representative of the cohomological units, and one can take ˆ2 D 0. The rest of the construction does not involve any choices, see (2.4), hence is naturally compatible with our involution, yielding an equivariant z D ˆ B is strictly unital. Take G D ˆ ı F formal diffeomorphism ˆ such that B and consider the images of X, Y and the Yk under it. Clearly z G X Š G Y ˚ G .A Y / Š G Y ˚ Bz .G Y / in H 0 .B/,  HomH.B/ z .G Y // D 0; z .G Y; B  HomH. z .G Yi ; G Yj / is Z=2-invariant, for i < j . B/

In view of Lemma 5.6, the analogue of (6.2) also holds, hence we are in the special case z inv / considered above, meaning that G X lies in the triangulated subcategory of Tw.B z inv is also a quasigenerated by the G Yk . But since the invariant part G inv W Ainv ! B isomorphism, it induces an equivalence of derived categories (Lemma 3.25), and the desired result follows.  (6c) A1 -modules. In parallel with the discussion of twisted complexes above, one can also introduce an equivariant version of A1 -modules. An equivariant A1 -module over A is an A1 -module M together with a module homomorphism  M W M ! A M, which has vanishing terms of order  2, and such that the compo sition of M and A .M /, in either order, is the identity module endomorphism. Concretely, the additional data consists of a family of maps M W M.X/ ! M.A .X//, such that the maps for X and A .X/ are mutual inverses, and which are compatible with A in the sense that dM .M .b/; A .ad 1 /; : : : ; A .a1 // D M .dM .b; ad 1 ; : : : ; a1 //: Define morphisms and their composition as in the non-equivariant case, and denote the resulting A1 -category by Q D mod eq .A/. This carries an induced naive Z=2-

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action, which acts trivially on objects, namely for t 2 homQ .M0 ; M1 / one sets .Q t/d .b; ad 1 ; : : : ; a1 / D M1 .t d .M0 .b/; A .ad 1 /; : : : ; A .a1 ///: If Y is an A -fixed object of A, the associated Yoneda module Y carries a natural equivariant structure, Y D A on Y.X/ D homA .X; Y /. This defines a cohomologieq cally full and faithful A1 -functor IA W Afix ! Q, the equivariantYoneda embedding, which strictly commutes with the Z=2-actions (one can extend this to a cohomologically full and faithful functor Tweq A ! Q, by an equivariant version of Lemma 3.36). On the other hand, if M is an equivariant A1 -module, then by considering only the Z=2-invariant part of the spaces M.X/ for A -fixed X, one gets an A1 -module M inv over Ainv . This, together with the corresponding restriction of Z=2-invariant natural transformations, defines the fixed part A1 -functor Qinv ! mod.Ainv /. Finally, one can just forget about the maps M , which yields a full and faithful A1 -functor Q ! mod.A/. Given a chain complex of vector spaces Z with a Z=2-action and an equivariant module M, the tensor product Z ˝ M is again equivariant. Secondly, given a Z=2invariant homomorphism between two equivariant modules, the cone in the sense of (3.32) is again an equivariant module. Finally, if Y is an A -fixed object and M is an equivariant module, then the twisted module TY M is naturally equivariant, and we have an exact triangle (5.3) in H.Qinv /. Lemma 6.4. Let Y0 , Y1 , Y2 be A -fixed objects, with the property that the Z=2-action on HomH.A/ .Y0 ; Y1 / is trivial. Suppose that TY0 .Y1 / Š Y2

(6.5)

in H 0 .Q/, with the isomorphism belonging to the Z=2-invariant part of the morphism group in that category. Then a parallel isomorphism (6.5) holds in H 0 .mod.Ainv //, where both theYoneda modules and the twisting are taken with respect to Ainv . Finally, if A is c-finite, then TY0 .Y1 / Š Y2 in D.Ainv /. Proof. To properly distinguish between the various categories, let us denote equivarieq ant Yoneda A1 -modules as IA .Y /, and their counterparts in mod.Ainv / as IAinv .Y /. eq We apply the fixed part functor Qinv ! mod.Ainv /. This takes IA .Y2 / to IAinv .Y2 /. eq On the other hand, the image of TY0 .IA .Y1 // is the A1 -module C over Ainv given by  Z=2 C.X/ D homA .Y0 ; Y1 / ˝ homA .X; Y0 /Œ1 ˚ homA .X; Y1 /Z=2 ; with the usual structure maps (5.1). Since the anti-invariant part of homA .Y0 ; Y1 / is acyclic, replacing the first summand with homA .Y0 ; Y1 /Z=2 ˝ homA .X; Y0 /Z=2 yields a quasi-isomorphic submodule, which is an isomorphic object in mod.Ainv /

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by Lemma 1.16. But by definition, that submodule is just TY0 .IAinv .Y1 //. The last statement follows immediately, in view of (5.6).  Remark 6.5. We close with some slightly vague general remarks. The naive approach to group actions on A1 -categories is at odds with the philosophy of homotopy algebraic structures, but still, it works quite well for finite groups and ground fields K of characteristic 0 (or suitably large characteristic p  0). We will reconsider the general situation in Section (10b); however, the definition proposed there, while more sophisticated than the one used here, is at most a first approximation. Our use of fixed objects is also somewhat naive. In a proper categorical formulation, one should instead consider “equivariant objects”, which are fixed under the group action only up to preferred isomorphisms. The cohomology level version of this will appear briefly later on (Section (14b)), but defining an appropriate cochain level notion is not straightforward. Formally, the advantage of such an approach would be that one could define both Tweq A and mod eq .A/ as the subcategories of fixed objects for the induced actions of Tw A and mod.A/, respectively. However, the ad hoc definitions made here are sufficient for our purpose.

II Fukaya categories The main aim of this chapter is to explain the definition of Fukaya’s A1 -category, in the comparatively simple special case of an exact symplectic manifold, using the classical analytical methods pioneered by Gromov and Floer. One can argue that this approach is now obsolete, having been supplanted by the fundamental work of Fukaya–Oh–Ohta–Ono [60], [56], which allows one to define Fukaya categories in complete generality. However, the general definition is quite involved, both analytically and algebraically, and much of that sophistication would be unnecessary in the case we are discussing here. For the benefit of non-specialist readers, we begin with the original definition of Floer cohomology groups for exact Lagrangian submanifolds [44]. It has been known since the early 1990s that the Floer groups admit operations indexed by a suitable class of Riemann surfaces with marked boundary points, making them into an open string TQFT (the idea seems to have originated with Donaldson; some papers dealing with it, or its closed string analogue, are [119], [106], [135], [131]). The Fukaya A1 -category, first mentioned in [52], appears when one allows families of Riemann surfaces with varying complex structure (actually it uses only discs with marked boundary points, hence should be regarded as the first piece of a more complicated theory, which is still largely unexplored). What we mean by the classical approach is to perturb the pseudo-holomorphic map equation by a zero order inhomogeneous term, so that the moduli spaces of solutions become smooth (thus avoiding the use of virtual fundamental chains). As in [116], the inhomogeneous term varies with the complex structure of the Riemann surface, and we will spend some time explaining what the precise requirements are. Once this has been clarified, there is nothing especially hard or new about the basic analytic results, which we therefore state with minimal sketches of proofs. Rather more emphasis will be placed on the issues of grading and signs, since both are important for our applications. For the grading we use Kontsevich’s formalism of graded Lagrangian submanifolds [90], [125], which retains slightly more information than the older relative gradings due to Floer and Viterbo. The general framework for studying sign issues in Floer theory was set up in [48], and the case of Lagrangian Floer cohomology is treated in [60]; but we still find it worth while to rework part of the theory. Other issues addressed here are equivariant Fukaya categories for symplectic manifolds with involutions, and the (partially) combinatorial nature of the Fukaya category for a surface. The last-mentioned topic has appeared already in several contexts: in homological mirror symmetry for elliptic curves [90], [108], [107], in Floer homology for curves on surfaces [135], [35], and in Legendrian knot theory [30], [43]. We concentrate on the regularity argument which relates the analytic and combinatorial approaches, since that has been less well served in the literature.

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7 A little symplectic geometry (7a) Exact symplectic manifolds with corners. Our basic setup involves manifolds with corners; the advantage (over manifolds with boundary) being that this class is closed under taking products, which is convenient when talking about fibrations (see Chapter III). We recall some relevant terminology. In a manifold with corners, consider the set of corner points of some fixed codimension k  1. Connected components of this set are called connected boundary strata, and the closures of such strata inside the whole manifold are called connected boundary faces. Throughout, we impose the condition that these faces should have no selfintersections; this means that each codimension l corner point lies in exactly 2l  1 different connected boundary faces of all codimensions 1  k  l. Dropping connectedness, we define a boundary face to be a union of pairwise disjoint connected boundary faces of the same codimension; and a boundary stratum to be a disjoint union of connected boundary strata of the same codimension, having pairwise disjoint closures. An exact symplectic manifold with corners .M; øM ; M ; IM / is a compact smooth manifold with corners M , equipped with a symplectic form øM , a one-form M satisfying dM D øM , and an øM -compatible almost complex structure IM . These should satisfy the following two convexity conditions: the negative Liouville vector field defined by øM .; XM / D M must point strictly inwards along all boundary faces of M ; and the boundary must be weakly IM -convex, which means that IM -holomorphic curves cannot touch @M unless they are completely contained in it. This definition is a compromise solution between our various needs. Its weakness is that the two types of convexity are not well tied to each other, which makes things clumsy on occasion. The subscripts in øM , M , IM will be omitted whenever the discussion concentrates on a single manifold. An isomorphism of exact symplectic manifolds with corners is a diffeomorphism  W M ! N which satisfies   N D M C d.some function/, hence also   øN D øM . We equip the space Iso.M; N / of all such isomorphisms with the C 1 -topology. For M D N , this becomes a topological group Aut.M /. Note that any isotopy in Aut.M / is automatically Hamiltonian, because of the exactness assumption. We will also use a slightly more restrictive, but technically simpler, notion of automorphism. Namely, let Aut.M; @M / be the group of diffeomorphisms  which are the identity near @M , and such that    D  C d.some function vanishing near @M /. A Lagrangian submanifold L  M (unless otherwise specified, these are always assumed to be closed and disjoint from @M ) is called exact if jL is an exact oneform. This class of submanifolds is preserved by isotopies in Aut.M /, as well as by the flow of the negative Liouville vector field. A deformation of exact symplectic manifolds with corners is simply a one-parameter family .øs ;  s ; I s /, s 2 Œ0I 1, of data on a fixed manifold M , which for each s

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satisfies the conditions stated above. There is a weak analogue of Moser’s Lemma in this context, which can be stated as follows. Let M; N be exact symplectic manifolds with corners. An embedding  W M ! N is called exact conformally symplectic if  N D e c M C d (some function) for some c 2 R. Then, Lemma 7.1. If two exact symplectic manifolds with corners can be deformed into each other, there is an exact conformally symplectic embedding of one into the other. Proof. Take a deformation, and look at the s-dependent vector field obtained from @ s =@s as in the standard Moser argument. This does not necessarily point inwards along the boundary faces, but that can be remedied by adding a large multiple of X s . Integrating the resulting vector field yields the desired embedding.  One can be a little more precise. Consider the isotopy class of the embedding  which we have just constructed, inside the space of all exact conformally symplectic embeddings. This is independent of all choices; it is unchanged under homotopies of the deformation rel endpoints; and it is compatible with compositions of deformations. This implies in particular that if Q is the embedding obtained by running the deformation backwards, then Q ı  is isotopic to the identity. (7b) The contact type boundary case. To supplement the discussion above, we will now look at a more familiar class of symplectic manifolds with boundary. Some foundational references are [42], [100]. An exact symplectic manifold with contact type boundary is a compact manifold with boundary (but no corners) M , with a symplectic form øM and one-form M satisfying dM D øM , such that XM points inwards along @M . A basic construction enlarges such an M to a noncompact exact symplectic manifold, by attaching an infinite cone in a canonical way: y ; ø y ;  y / D .M; øM ; M /[@M .RC @M; d.e r .M j@M //; e r .M j@M //; (7.1) .M M M where r is the radial variable of the cone (the convention throughout this book is that R˙  R are closed, meaning that they include 0). The differentiable structure along the “seam” is defined using the collar neighbourhoods of @M provided by the Liouville flows on both sides. Hence, the negative Liouville fields XM on M and y . Similarly, @r on RC  @M glue together to give the negative Liouville field on M note that there is a unique (in the sense of germs) function hM W U ! .0I 1, defined in a neighbourhood U  M of @M , such that 1 hM .1/ D @M

and XM :hM D hM :

(7.2)

y by setting This can be smoothly extended to a function hMy on U [ .RC  @M /  M r hMy .r; y/ D e on the cone.

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An appropriate notion of isomorphism between two manifolds of type (7.1) is y ! Ny such that O   y   y D d (some compactly given by diffeomorphisms O W M N M supported function). In particular, such a map preserves the Liouville vector fields y for some C  0, it will @r outside a compact subset. Hence, on ŒC I 1/  @M  M be of the form O y/ D .r  w.y/; .y// .r; where W @M ! @N is a contact diffeomorphism, and w 2 C 1 .@M; .1I C / the function such that  .N j@N / D e w .M j@M /. We say that O is of contact type near y ; Ny / of all such isomorphisms carries a natural topology; infinity. The space Iso.M this allows isotopies that change the contact diffeomorphism at infinity, hence are not y / D Iso.M y;M y /. It is a standard compactly supported. As usual, we write Aut.M fact that a version of Moser’s Lemma holds in this context: s s ; M /, s 2 Œ0I 1, be a smooth family of exact symplectic strucLemma 7.2. Let .øM tures with contact type boundary on M . Then there is an exact symplectic isomory0 phism, of contact type near infinity, between the resulting noncompact manifolds M 1 y and M . 

Let M be an exact symplectic manifold with contact type boundary. Following a well-known strategy, see for instance [76], we consider øM -compatible almost complex structures J which, in some neighbourhood of @M , are invariant under the flow of XM and satisfy dhM ı J D M ; (7.3) where hM is as in (7.2). Let us call such almost complex structures of contact type near the boundary. Given a Riemann surface S with complex structure IS , and a J holomorphic map u W S ! M , consider the function  D hM ı u W u1 .U / ! .0I 1. Due to (7.3) this satisfies  D d.d ı IS / D d.dhM ı J ı du/ D d.M ı du/ D u øM  0:

(7.4)

By the maximum principle, if  attains its maximal possible value 1 at some (interior) point of S, it is necessarily locally constant there. Therefore, choosing any IM D J which is of contact type near the boundary makes @M weakly convex, hence turns M into an exact symplectic manifold with corners. We will need a couple of variations on this basic convexity argument. First of all, it applies to families of almost complex structures, in the following sense: Lemma 7.3. Let .J z /z2S be a family of øM -compatible almost complex structures on M , parametrized by some Riemann surface S. Suppose that each J z is of contact type near @M . Then the almost complex structure on S  M given by J.z; x/ D IS .z/  J z .x/ makes S  @M weakly convex. 

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Secondly, we can look at pseudo-holomorphic curves in completions (7.1). Every almost complex structure on M which is of contact type near the boundary has a y , which is invariant under r-translations on the cone. canonical extension JO to M O This satisfies dhMy ı J D My , so by using the same computation as before. One gets: Lemma 7.4. Let S be a compact connected Riemann surface with boundary, and y a JO -holomorphic map such that u.@S/  M . Then u.S/  M as uW S ! M well.  There is also an integrated version of the maximum principle argument, which can be used to give an alternative proof of the previous lemma, but which also yields somewhat more general results. Namely, let N be an arbitrary exact symplectic manifold, M an exact symplectic manifold with contact type boundary (of the same dimension), and  W M ,! int.N / an embedding such that !M D  !N , M D  N . For simplicity, we will write this as an inclusion. Take an !N -compatible almost complex structure J , whose restriction to M is of contact type near the boundary. Let S be a compact connected Riemann surface with boundary, and u W S ! N n int.M / a J -holomorphic map such that u.@S/  @M . Clearly, R R R R 0  S u !N D @S u M D @S u .dhM ı J / D @S dhM ı du ı IS : (7.5) If  is a tangent vector along @S which is positive with respect to the boundary orientation, then IS  points inwards, and du.IS / does not point towards the interior of M ; hence dhM .du.IS //  0. This contradicts (7.5) unless u is constant. As an easy application, we get the following: Lemma 7.5. Let S be a compact connected Riemann surface with boundary, and u W S ! N a J -holomorphic map such that u.@S/  int.M /. Then u.S/  int.M / as well. 1 Proof. Replace M by the slightly smaller subset M D M n hM ..e  I 1/, for some  > 0. By suitably choosing , we may achieve that J jM is still of contact type near the boundary, that u.@S/ lies in int.M /, and that u intersects @M transversally. Hence, S D u1 .N n int.M // is again a compact Riemann surface with boundary, whose boundary gets mapped to @M . Applying the previous argument (to each connected component), we find that u.S /  M , hence that u.S/  int.M /. 

Finally, given any exact symplectic manifold with corners N , one can round off the corners in the following way. Let F1 ; : : : ; Fd be the codimension one boundary faces of N . For each Fj choose a smooth function fj W N ! RC such that Fj D fj1 .0/, and with dfj < 0 on the outwards pointing normals to Fj , so that in particular

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XN :fj > 0 in a neighbourhood of Fj . Set f D f1 f2    fd . We claim that XN :f > 0 at all points of int.N / which lie sufficiently close to the boundary. One can rewrite the inequality as X X :fj N > 0: (7.6) fj j

Suppose that we have a sequence of points in int.N / converging to a limit which lies in Fj iff j 2 E, for some nonempty subset E  f1; : : : ; d g. Along this sequence, the j 2 E summands in (7.6) each go to C1, while the remaining summands are bounded, so the inequality will hold eventually. Having established our claim, one sees immediately that Lemma 7.6. M D f 1 .ŒıI 1//  N , for any sufficiently small ı (and carrying the restrictions of øN , N ) is an exact symplectic manifold with contact type boundary. Moreover, the negative Liouville flow ( for large times) will compress the whole of N into int.M /. 

8 Classical Floer theory (8a) Riemann surfaces. At this point, we fix an exact symplectic manifold with corners M , which will be the “target space” throughout the following discussion. Write ø D øM ,  D M , I D IM for the given structures on M . Let SO be a compact Riemann surface with boundary, and † a finite set of boundary points, divided into two parts † D † t †C . We call S D SO n † a pointed-boundary Riemann surface, and the points of † , †C (which are retained as parts of the datum) its incoming respectively outgoing points at infinity. Note that SO can be recovered canonically from S . We will use the following notation for the simplest such surfaces: • D is the closed unit disc in C; • H is the closed upper half plane, with one incoming point at infinity; x is the same surface as H , but with the point at infinity considered to be an • H outgoing one; • Z D R  Œ0I 1 with coordinates .s; t/, an incoming point s D 1 and an outgoing point s D C1. Isomorphisms of pointed-boundary Riemann surfaces are biholomorphic maps which preserve the distinction between incoming and outgoing points at infinity. A set of Lagrangian labels for S is a family fLC g of exact Lagrangian submanifolds LC  M , indexed by the connected components C  @S . Every  2 †  SO lies in the closure of two boundary components C ;0 and C ;1 (which may

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coincide); hence, in the presence of a set of Lagrangian labels, there is an associated pair of Lagrangian submanifolds .L ;0 ; L ;1 /. Our convention is that for  2 † , C ;1 comes before  in the natural boundary orientation, and C ;0 after ; and the opposite for  2 †C . (8b) Floer cohomology. Fix some field K of characteristic 2. For each pair .L0 ; L1 / of exact Lagrangian submanifolds in M there is a Floer cohomology group HF pr .L0 ; L1 /, which is a finite-dimensional K-vector space (pr stands for “preliminary” and serves to distinguish this from the more elaborate theory to be introduced later on, where char.K/ can be arbitrary and the Floer groups are graded). To any pointed-boundary Riemann surface S with Lagrangian labels one can associate a homomorphism O O ˆS W HF pr .L C ;0 ; L C ;1 / ! HF pr .L  ;0 ; L  ;1 / (8.1)

C 2†C

 2†

(the contravariant convention may appear strange, but recall that we are dealing with Floer cohomology). These maps obey a version of the open string TQFT axioms (see e.g. [122]), the most important of which is the following composition law. Let S1 ; S2 be two pointed-boundary Riemann surfaces with Lagrangian labels. Suppose that   there are points at infinity 1C 2 †C 1 , 2 2 †2 , which are equipped with the same (ordered) pair of Lagrangian submanifolds. Form another pointed-boundary Riemann surface S D S1 #S2 by gluing them together as in Figure 8.1 (due to another TQFT property, topological invariance, the details of how this is done do not matter). S carries induced Lagrangian labels, and ˆS D .Id ˝    ˝ ˆS1 ˝    ˝ Id/ ı .Id ˝    ˝ ˆS2 ˝    ˝ Id/;

(8.2)

where the rule is that one inserts the 2 -factor of the output of ˆS2 into the 1C -input component of ˆS1 . (8c) The Donaldson–Fukaya category. It is a general fact that one can associate to any open string TQFT a Frobenius category. In the Floer theory context this yields a K-linear category H F .M /pr , called (preliminary version of) the Donaldson– Fukaya category. Objects are exact Lagrangian submanifolds L  M . The space of morphisms L0 ! L1 is the Floer group HF pr .L0 ; L1 /. Composition HF pr .L1 ; L2 / ˝ HF pr .L0 ; L1 / ! HF pr .L0 ; L2 /

(8.3)

is the triangle product, which in TQFT terms is the invariant ˆS for S a disc minus three boundary points (one incoming, two outgoing), labelled with L0 ; L1 ; L2 . The identity morphisms are ˆH 2 HF pr .L; L/, where we put the label L on @H . The

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S1

S2 1C 2

S

Figure 8.1

associativity of composition, and the fact that ˆH is a unit, follow from the gluing axiom of the TQFT. The Frobenius property of H F .M /pr is as follows. For each L, the invariant ˆHx gives a linear map HF pr .L; L/ ! K, and by combining this with composition one gets a nondegenerate pairing HF pr .L1 ; L0 / ˝K HF pr .L0 ; L1 / ! K for any L0 ; L1 . More directly, the surface defining the pairing is an infinite strip with two outgoing points at infinity. By attaching such strips, one can trade incoming points at infinity for outgoing ones. This means that the distinction between the two kinds is somewhat artificial. One could indeed drop it (thus bringing the axiomatics into line with the original formulation of TQFTs [7]), but we prefer not to do that, for technical reasons which will become clear later on. We should also mention two properties which are more specific to Donaldson– Fukaya categories. First, for any L there is a canonical isomorphism of Frobenius algebras, the Piunikhin–Salamon–Schwarz (PSS) map [106] H  .LI K/ Š HF pr .L; L/:

(8.4)

The best way to understand (8.4) is as part of a natural generalization of the TQFT framework, where the surfaces S carry additional marked boundary points, which are then treated in a way similar to Gromov–Witten theory. Secondly, each isotopy fLs g0s1 of exact Lagrangian submanifolds gives rise to a distinguished element in HF pr .L0 ; L1 /, which has the following properties: it depends only on the homotopy class of fLs g rel endpoints; it is the identity for

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103

the constant isotopy; and composition of isotopies (as paths) corresponds to taking the triangle product of the associated elements. As a direct consequence, all of these elements are isomorphisms in H F .M /pr . It follows that isotopic Lagrangians give rise to isomorphic objects of the Donaldson–Fukaya category, hence that Floer cohomology groups are isotopy invariant. Formally, the construction is based on another extension of the basic TQFT structure, in which one allows the Lagrangian boundary conditions to vary smoothly along the points of @S . (8d) Strip-like ends. We will now start reviewing the basic underpinnings of Floer theory. This is all familiar material, as attested by the references listed at the start of the chapter, but it is a good warmup exercise for the discussion of Fukaya categories. First of all, while pointed-boundary Riemann surfaces are sufficient for formulating the TQFT, a slightly more rigid structure is better for its actual construction. Denote by Z ˙ D R˙  Œ0I 1 the semi-infinite strips. Let S be a pointed-boundary Riemann surface, with points at infinity †˙ . A set of strip-like ends for S consists of proper holomorphic embeddings  W Z ˙ ! S, one for each  2 †˙ , satisfying  1 .@S/ D R˙  f0I 1g

and

lim  .s;  / D ;

s!˙1

and with the additional requirement that the images of the  are pairwise disjoint. The Riemann surface structure on Z ˙ extends to the one-point compactification y ˙ D Z ˙ [ f˙1g, and each strip-like end extends to a holomorphic embedding Z y ˙ ! SO mapping ˙1 to . O W Z There are several reasons why these ends are needed. One reason is that the W 1;p (p > 2) norms used as the analytic framework for the theory are not conformally invariant. Of equal importance is the fact that the inhomogeneous terms which appear in the definition of Floer cohomology involve the standard coordinates on Z D R  Œ0I 1, so to define the operations (8.1) one needs similar structures on the ends of more general Riemann surfaces. Yet another point is that one can make the gluing process from Figure 8.1 more precise: given 1C , 2 and a gluing length l > 0, set S10 D S1 n  C ..lI 1/  Œ0I 1/; 1

S20 D S2 n  2 ..1I l/  Œ0I 1/; and

(8.5)

S D S1 #l S2 D .S10 t S20 /= ; where the identification is  2 .s  l; t/  C .s; t/ for .s; t/ 2 Œ0I l  Œ0I 1. 1

(8e) Floer data, perturbation data. Denote by J the space of all ø-compatible almost complex structures J on M which agree with the given I near the boundary. If T is any manifold, write C 1 .T; J/ for the space of smooth T -parametrized families

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J D fJ.t/g t 2T of almost complex structures which agree with I on T  U , where U is some neighbourhood of @M . Similarly, we write H D Cc1 .int.M /; R/ for the space of smooth functions on M vanishing near the boundary, and C 1 .T; H / for the functions on T  M which vanish on some set T  U as before. Let L0 ; L1  M be a pair of exact Lagrangian submanifolds. A Floer datum consists of H 2 C 1 .Œ0I 1; H / and J 2 C 1 .Œ0I 1; J/, with the following property: if X is the timedependent Hamiltonian vector field of H and  its “flow”, then  1 .L0 / intersects L1 transversally. Let S be a pointed-boundary Riemann surface with Lagrangian labels. Suppose that we have chosen strip-like ends for it, and also a Floer datum .H ; J / for each of the pairs of submanifolds .L ;0 ; L ;1 / associated to the points at infinity  2 †. A perturbation datum for S is a pair .K; J /, where K 2 1 .S; H / satisfies K./jLC D 0 for all  2 T C  T .@S/.

(8.6)

In words: K is a one-form with values in the space of functions on M , such that if one takes a boundary point belonging to some connected component C  @S , and a vector  tangent to the boundary at this point, then K./ 2 H is a function which vanishes along LC  M . The second piece of the perturbation datum is simply a family of almost complex structures J 2 C 1 .S; J/. In addition, K and J must be compatible with the chosen strip-like ends and Floer data, in the sense that   K D H .t/dt;

J. .s; t// D J .t/

(8.7)

for each  2 †˙ and .s; t/ 2 Z ˙ (note that conversely, one can consider a Floer datum as a translation-invariant perturbation datum on S D Z, setting K.s; t/ D H.t/dt ). The notions of Floer datum and perturbation datum determine the class of inhomogeneous terms that we will add to the pseudo-holomorphic map equation. (8f) Inhomogeneous pseudo-holomorphic maps. Suppose that we have a pair L0 ; L1 of exact Lagrangian submanifolds, and a Floer datum .H; J /. As before let X be the time-dependent vector field of H . Define C.L0 ; L1 / to be the set of maps y W Œ0I 1 ! M with y.0/ 2 L0 , y.1/ 2 L1 , and dy=dt D X.t; y.t//. These correspond bijectively to points in  1 .L0 / \ L1 , hence there are only finitely many of them. Floer’s equation for u 2 C 1 .Z; M / is ( @s u C J.t; u/.@ t u  X.t; u// D 0; (8.8) u.s; 0/ 2 L0 ; u.s; 1/ 2 L1 : We write MZ .y0 ; y1 / for the set of solutions of (8.8) with the asymptotic conditions lims!C1 u.s;  / D y1 , lims!1 u.s;  / D y0 , where y0 ; y1 2 C.L0 ; L1 /. Let us say that the convergence is in C 1 sense with exponential speed (due to the transverse

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intersection assumption, all reasonable convergence conditions are equivalent). There is an R-action on MZ .y0 ; y1 / by translation in the s-variable. The action is free  whenever y0 ¤ y1 , and we then denote the quotient space by MZ .y0 ; y1 /; in the exceptional case y0 D y1 , when MZ .y0 ; y1 / is a point (the only element being the  stationary solution u.s; t/ D y0 .t/), we set MZ .y0 ; y1 / D ;. More generally, take a pointed-boundary Riemann surface S with Lagrangian labels. Equip it with strip-like ends, Floer data .H ; J / for each point at infinity, and a compatible perturbation datum .K; J /. K determines a vector-field-valued oneform Y 2 1 .S; C 1 .TM //: for each  2 T S, Y./ is the Hamiltonian vector field of K./. The inhomogeneous pseudo-holomorphic map equation for u 2 C 1 .S; M / is ( Du.z/ C J.z; u/ ı Du.z/ ı IS D Y .z; u/ C J.z; u/ ı Y.z; u/ ı IS ; (8.9) u.C /  LC for all C  @S, where IS is the complex structure on S. This can be written more succinctly as .Du  Y /0;1 D 0, or as @N J .u/ D .z; u.z// with  D Y 0;1 (as a comparison with [115] shows, we are not using the widest possible class of inhomogeneous terms ). Floer’s equation is clearly a special case of (8.9), with S D Z and a translationinvariant datum .K; J /. Conversely, (8.7) implies that (8.9) reduces to (8.8) on the strip-like ends of S. Therefore, it makes sense to consider the set MS .fy g/ of solutions of (8.9) with convergence conditions lim u. .s;  // D y

s!˙1

for  2 †˙ , y 2 C.L ;0 ; L ;1 /.

(8.10)

(8g) Energy. Start with L0 ; L1 and a Floer datum .H; J /. Because of exactness, there are functions hLk 2 C 1 .Lk ; R/ with dhLk D jLk . The H -perturbed action functional on the path space fy 2 C 1 .Œ0I 1; M / W y.0/ 2 L0 ; y.1/ 2 L1 g is Z  AH .y/ D y   C H.t; y.t// dt C hL1 .y.1//  hL0 .y.0//: (8.11) The critical point set of AH is precisely C.L0 ; L1 /, and (8.8) is the negative gradient flow equation in an LR2 metric. Therefore, for any u 2 MZ .y0 ; y1 / one has the a priori bound E.u/ D Z j@u=@sj2 D AH .y0 /  AH .y1 /. To generalize this to (8.9) one chooses, for each connected component C  @S, a function hLC 2 C 1 .LC ; R/ with the same property as before. Then for u 2 MS .fy g/, Z Z X def 2 1 E.u/ D jDu  Y j D

A .y /  R.z; u.z//: (8.12) H

2 S

2†˙

S

The ambiguity (of a constant, or local constant in the disconnected case) in the choice of each hLC cancels out when summing over , so that the first term makes sense.

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As for the second one, the curvature integrand R 2 2 .S; H / is given in local coordinates .s; t/ on S by   R D @s K.@ t /  @ t K.@s / C fK.@s /; K.@ t /g ds ^ dt: Here f; g is the Poisson bracket. Since K vanishes over the ends of S, the integral is bounded independently of the map u, hence so is E.u/ for u 2 MS .fy g/. (8h) Cauchy–Riemann operators. Let S be a Riemann surface with boundary, E ! S a vector bundle with a symplectic structure øE and compatible complex structure JE , and F  Ej@S a Lagrangian subbundle. We write C 1 .S; E; F / for the space of sections whose restriction to the boundary lies in F , and C 1 .S; 0;1 S ˝E/ for the E-valued .0; 1/-forms (with no boundary conditions). With less smoothness, 1;p .S; E; F / of locally W 1;p sections for all 2  p < one has well-defined spaces Wloc 1 (the boundary condition obviously makes sense for p > 2, since such sections 1;2 are continuous; for p D 2 one defines it via the restriction map Wloc .S; E/ ! p L2loc .@S; E/, see e.g. [29]), and Lploc .S; 0;1 ˝ E/ of locally L .0; 1/-forms. Given S any (not necessarily complex or symplectic) connection r on E, one defines the associated Cauchy–Riemann operator to be the .0; 1/ part of covariant differentiation: @N r D r 0;1 W C 1 .S; E; F / ! C 1 .S; 0;1 S ˝ E/:

(8.13)

1;p This extends to Wloc .S; E; F / ! Lploc .S; 0;1 S ˝ E/. If S is compact, the extension is a Fredholm operator. To have a reasonable analytic theory of (8.13) for non-compact surfaces, one needs to assume that S has strip-like ends, and that the asymptotic behaviour of .E; F; r/ over those ends is sufficiently well controlled. The precise formulation requires some preliminary definitions. To each end  of our surface S, associate a vector bundle E ! Œ0I 1, equipped with a symplectic structure øE , a compatible complex structure JE , a symplectic connection r , and Lagrangian subspaces ƒ ;k  E ;k for k D 0; 1. We call .E ; øE ; JE ; r ; ƒ ;0 ; ƒ ;1 / the limiting datum at . It will be often convenient to assume that this datum is nondegenerate, by which we mean that r -parallel transport along Œ0I 1 carries ƒ ;0 to a subspace which is transverse to ƒ ;1 . Let  ˙ W Z ˙ ! Œ0I 1 be the projections in t-direction. We want our bundle E to come with preferred isomorphisms

  E Š . ˙ / E

(8.14)

which are asymptotically compatible with the limiting data. This means that if we identify the two sides of (8.14), the differences øE  øE , JE  JE , r  r decay exponentially fast (in the C 1 -topology, say) as s ! ˙1; and F  .s;k/ ! ƒ ;k in the same sense (within the Grassmannian of Lagrangian subspaces of .E /k ). A

8 Classical Floer theory

107

@N r -operator which satisfies (8.14) at the ends is called admissible. If in addition all limiting data are nondegenerate, we say that @N r itself is nondegenerate. The trivialization on the ends determines a metric which is constant in s-direction, which one uses to define spaces W 1;p .S; E; F / and Lp .S; 0;1 S ˝ E/ of globally W 1;p and Lp sections. Then (8.13) extends to a continuous operator @N r W W 1;p .S; E; F / ! Lp .S; 0;1 S ˝ E/:

(8.15)

If the nondegeneracy condition introduced above holds, these extensions are Fredholm, with kernel and cokernel which are independent of p. The more general approach is to introduce spaces W 1;pI and LpI with some weight   0 over the ends (elements of these spaces are exponentially decaying sections; for  D 0, they reduce N to the ordinary Sobolev spaces considered before). For any admissible @-operator, there is an 0 > 0 such that if  is chosen in .0I 0 /, the extension @N r W W 1;pI .S; E; F / ! LpI .S; 0;1 S ˝ E/

(8.16)

is Fredholm, and its kernel and cokernel are independent of the weight (compare [45, Proposition 4.1]). If the Cauchy–Riemann operator is nondegenerate, the same statement holds with  2 Œ0I 0 /, which means that in this case, the kernel and cokernel of (8.16) coincide with those of (8.15). On occasion, we will use Sobolev spaces W k;p with higher derivatives k > 1, as well as their weighted versions W k;pI , but the same discussion holds for them as well. (8i) Linearization. The solution sets MS .fy g/ of (8.9) fit into a standard function space framework. Fix 2 < p < 1. Let BS be the Banach manifold of maps u W S ! M which are locally W 1;p , and which converge to some collection fy g in W 1;p -sense on the strip-like ends. The tangent space of this manifold at a point u is just W 1;p .S; E; F /, where E D u TM and F jC D u T LC for each component C  @S . Equip E with the almost complex structure JE .z/ D J.z; u.z//, where J is part of our perturbation datum. There is a natural Banach vector bundle ES ! BS  N whose fibres are the spaces Lp .S; 0;1 S ˝u TM /. One can think of @ as a smooth section of this bundle, and its zero set is then obviously the union of the moduli spaces MS .fy g/ (thus equipping them with an appropriate topology). The linearization at a point in the moduli space, denoted by DS;u W .T BS /u ! .ES /u ;

(8.17)

is a nondegenerate Cauchy–Riemann operator, hence Fredholm. One says that u is regular if DS;u is onto. If this holds for all u 2 MS .fy g/, then the moduli space itself is called regular. This implies that it is smooth of the appropriate dimension, given by the Fredholm index of DS;u at each point. Finally, if this holds for all fy g,

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we say that the perturbation datum .K; J / itself is regular (the same terminology is used for Floer data). To identify DS;u with an explicit operator @N r of the kind considered above, equip E with the symplectic structure pulled back from ø. The connection r is not unique, since only its .0; 1/-component enters the definition of the operator, but one possible way of making the choice is as follows. Let uQ W S ! S  M be the graph of u. Fix a Z 2 C 1 .S  M; T S  ˝ TM / such that Z.z; u.z// D du.z/, and for which .Z  Y /0;1 D 0 everywhere. To any such Z one can associate a connection r on E. For that, take Xz 2 C 1 .S  M; TM /, denote by X 2 C 1 .S; E/ its pullback by u, Q and set z rX D uQ  ŒZ; X: (8.18) z and really defines a connection, To see that this is independent of the choice of X, 0 e choose an arbitrary torsion-free connection r on TM ! S  M , and denote by r 0 its pullback by u. Q With the help of this one can rewrite (8.18) as  e0 hZ; i r X D r0 X  uQ  .r X for all  2 C 1 .S; T S/ (and where hZ; i 2 C 1 .S  M; TM / is the natural pairing T S ˝ .T  S ˝ TM / ! TM ). This shows that r is of the form r 0 plus a zero order term, which depends on the derivative of Z along the graph of u. Next, using local holomorphic coordinates z D s C it on S to make things more explicit, one computes the .0; 1/-part of r to be   e0 hZ; @s i  J r e0 hZ; @ t i .rX /0;1 D 12 r@0s X C J r@0t X  r X X   e0 J /du.@ t / D 12 r@0s X C J r@0t X C uQ  .r X    12 uQ  rX0 Y.@s / C J Y.@ t / : This is indeed the standard form of the linearized operator in the theory of pseudoholomorphic maps, with the last summand added to take into account the inhomogeneous term in (8.9), so DS;u D @N r . To satisfy the admissibility condition, we choose Z in such a way that, if s C it are the standard coordinates on some strip-like end  W Z ˙ ! S , then Z.@s / ! 0 exponentially (in the C 2 -topology) as s ! ˙1. In that case, the asymptotic structure of .E; øE ; JE ; F / and r is described by the following limiting data: the vector bundle E D y  TM with the symplectic structure inherited from ø, and the almost complex structure JE .t/ D J .t; y.t//; the Lagrangian subspaces ƒ ;k D T .L ;k /y.k/ ; and the symplectic connection r obtained by linearizing the “flow” of H along its integral curve y . Remark 8.1. There are some special cases in which the linearized operator has a more straightforward description. Suppose that J is independent of z 2 S and integrable, and that Y D 0. Then TM and its pullback E are naturally holomorphic bundles, and

8 Classical Floer theory

109

DS;u is the canonical associated Dolbeault operator. Another such situation is when u  x is constant, and Y together with its first derivatives vanishes at the point x. The second derivatives D 2 Y can then be viewed as an element of 1 .S; sym2 .TMx /_ /, hence as giving a symplectic connection on the trivial bundle E D S  TMx . Then DS;u is the .0; 1/-part of this connection, taken with respect to the complex structure JE .z/ D J.z; x/. (8j) Definition of the TQFT. For each pair .L0 ; L1 / of exact Lagrangian submanifolds, choose a regular Floer datum .HL0 ;L1 ; JL0 ;L1 /. Define the Floer cochain group CF pr .L0 ; L1 / to be the K-vector space with one basis vector for each y 2 C .L0 ; L1 /. The boundary operator on that group is X  #MZ .y0 ; y1 / y0 ; (8.19) @.y1 / D y0

where # means counting the number of isolated points mod 2. HF pr .L0 ; L1 / is the cohomology of .CF pr .L0 ; L1 /; @/. Next, for each pointed-boundary Riemann surface S with Lagrangian labels fLC g and strip-like ends, choose a regular perturbation datum .KS ; JS / which is compatible with the previous Floer data. Define a map O O CF pr .L C ;0 ; L C ;1 / ! CF pr .L  ;0 ; L  ;1 /; C ˆS W

C 2†C

C ˆS .˝ C y C / D

X

 2†

#MS .fy  ; y C g/ .˝  y  /:

(8.20)

fy  g

and take ˆS to be the induced map on cohomology. Non-expert readers should keep in mind that, in spite of the rather summary treatment here, each piece of this construction is a major undertaking. To even define @ and C ˆS , one needs (i) transversality results, which say that regularity is a generic condition for Floer data and perturbation data; and (ii) compactness results, which ensure that in the regular case, the relevant moduli spaces have only finitely many isolated points. After that, suitable gluing theorems show that (iii) @2 D 0, and that (iv) the C ˆS are chain maps. Further steps are (v) that the resulting theory is independent of all choices, up to canonical isomorphism; and (vi) that it satisfies the composition axiom (8.2). We will not say anything about the first four issues, since these are well covered in the literature, and will anyway come up again in the construction of the Fukaya category (Sections (9k), (9l)). Concerning the last two steps (which are actually mutually intertwined), a few comments may be appropriate. To start off (v), one first shows that the ˆS are independent of the conformal structure of S , the choice of strip-like ends, and the perturbation datum, as long as the Floer data at the ends are kept fixed. To do that, one takes a family of conformal structures and perturbation data over Œ0I 1 interpolating between the two given ones,

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and uses the union of all the resulting moduli spaces to define a chain homotopy. (This can be considered as a first, and rather trivial, occurrence of parametrized moduli spaces, which we will discuss extensively later on.) Next, turning to (vi), take surfaces S1 , S2 equipped with strip-like ends, Lagrangian boundary conditions, and perturbation data. We want to glue them together along points at infinity 1C ; 2 to form S D S1 #l S2 , as in (8.5). Assume that the Floer data associated to the two ends 1C , 2 coincide, so that one naturally gets a perturbation datum on S. Assuming moreover that suitable regularity conditions hold, and that l  0, one can show that the composition law is true on the chain level. This equality, of course, only holds for this very specific choice of conformal structure and perturbation datum on S . However, we have already proved that the cohomology level map ˆS is independent of those choices. (For Fukaya categories, which are chain level structures, this kind of argument would no longer apply, which means that we will be forced to exert much more control over the perturbation data used in the construction.) Given that, the remaining part of (v), which concerns independence of the choice of Floer data, is entirely formal. Consider the infinite strip Z with Lagrangian labels Lk attached to its boundary components Ck D R  fkg, k D 0; 1. At the two ends 0 D 1, 1 D C1, equip the pair .L ;0 ; L ;1 / D .L0 ; L1 / with regular Floer data .H j ; J j / D .HLj 0 ;L1 ; JLj0 ;L1 /, and choose a regular compatible perturbation datum. This gives rise to a map ˆZ W HF pr .L0 ; L1 /1 ! HF pr .L0 ; L1 /0 between the resulting Floer cohomology groups, traditionally called a continuation map [118]. The homotopy argument mentioned above shows that it is canonical; and an application of the composition law shows that it is an isomorphism. Remark 8.2. In many cases, it is possible to use Floer data and perturbation data with vanishing inhomogeneous terms. Namely, whenever L0 t L1 , one can find a regular Floer datum with H D 0. Analogously, consider a Riemann surface S with strip-like ends and Lagrangian boundary conditions fLC g. Suppose that any pair of Lagrangian submanifolds associated to a point at infinity intersect transversally, and that there is no point of M which lies on all of the LC ; the second condition excludes constant solutions of (8.9). Then one can find a regular perturbation datum with K D 0. (8k) Moving boundary conditions. Let S be a Riemann surface with strip-like ends. A set of moving boundary conditions for S consists of a family of exact Lagrangian submanifolds Lz  M depending smoothly on z 2 @S , but locally constant for z 2 @S \  .Z ˙ /, so that each point at infinity still has a pair of Lagrangian submanifolds .L ;0 ; L ;1 / associated to it. In the definition of a perturbation datum

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8 Classical Floer theory

.K; J /, (8.6) should be replaced by the following slightly more complicated condition: d.K./jLz / D ø.; @ Lz / 2 1 .Lz / for any  2 T .@S/z .

(8.21)

Here, @ Lz is any section of C 1 .TM jLz / tangent to the infinitesimal deformation of the Lagrangian submanifold, so the right-hand side of (8.21) is the standard (exact) one-form corresponding to such a deformation. One can then consider the same equation (8.9) as before. The condition on K allows one to integrate out boundary terms in (8.11), and the rest of the theory goes through without any significant adjustments, leading to generalized maps (8.1) which satisfy obvious variants of the TQFT axioms. In fact, the increase in generality is much smaller than it might appear at first sight. To see that, suppose that all connected components of @S are open intervals. Then there is a family of automorphism z 2 Aut.M; @M /, parametrized by z 2 S, such that z .Lz / is locally constant, and .@=@s/  .s;t/ D 0 over the strip-like ends. The coordinate change u.z/ 7! z .u.z// (8.22) preserves the class of equations (8.9) but reduces moving boundary conditions to constant ones. Nevertheless, thinking in terms of moving boundary conditions is often convenient, as the following application shows. Let fLs g0s1 be an exact Lagrangian isotopy. Take S D H with the moving boundary condition L .s/ , where 2 C 1 .R; Œ0I 1/ satisfies .s/ D 0 for s  0, .s/ D 1 for s  0. This gives rise to a characteristic element ˆH 2 HF pr .L0 ; L1 /;

(8.23)

which is in fact the one mentioned in Section (8c). All the properties stated there follow from standard homotopy and gluing arguments. For instance, by composing (8.23) with the triangle product, one gets isomorphisms HF pr .L; L0 / ! HF pr .L; L1 / and HF pr .L1 ; L/ ! HF pr .L0 ; L/ for an arbitrary L. In fact, by inspecting the gluing argument, and applying a transformation (8.22), one can reduce these to the previously considered continuation maps. (8l) Evaluation at boundary points. Suppose that we are given a pointed-boundary Riemann surface with Lagrangian labels, and an additional finite set of marked points …  @S , divided into positive and negative ones … ; …C . One can then introduce a generalization of the TQFT maps (8.1), O O ˆS;… W HF pr .L C ;0 ; L C ;1 / ˝ H  .LCzC I K/

C 2†C

!

O

 2†

z C 2…C

HF .L  ;0 ; L  ;1 / ˝ pr

O z  2…

H  .LCz I K/:

(8.24)

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II Fukaya categories

Here, Cz is the boundary component to which z belongs. In addition to the usual gluing together of points at infinity, one can now form boundary connected sums,   which fuse together two surfaces S1 ; S2 at points z1C 2 …C 1 , z2 2 …2 where the respective Lagrangian boundary conditions coincide. For the associated operators (8.24), this corresponds to composition applied to the relevant factor H  .LCz I K/. The simplest case is that of a compact S . Assuming regularity, the moduli space MS of solutions of (8.9) will be a smooth closed manifold. The productQof evaluation maps evz .u/ D u.z/, z 2 …, then gives rise to a homology class in z LCz . One uses the Künneth formula and Poincaré duality (for the … factors) to read this as a multilinear map (8.24). However, if one sets the inhomogeneous term equal to zero, then a straightforward energy argument shows that MS contains only constant maps, which means that the resulting invariants are not really interesting. For instance, taking S D D with Lagrangian label L, one gets

„ the unit 1 2 H



.LI K/; j… j D 1; j…C j D 0; the integration map H  .LI K/ ! K; j… j D 0; j…C j D 1; the identity H  .LI K/ ! H  .LI K/; j… j D j…C j D 1; the cup product; j… j D 1; j…C j D 2:

(8.25)

To give a brief indication of the more general situation, assume that † D fg consists of a single outgoing end, and … D fzg P of a single negative marked point. The idea then is that if one takes a Floer cocycle k mk yk 2 CF pr .L ;0 ; L ;1 /, the corresponding formal linear combination of evaluation maps evz W M.yk / ! LCz is class (or rather its Poincaré a cycle on Lz in the ordinary topological sense, whoseP dual) will be the image of the Floer cohomology class Œ k mk yk  under (8.24). An equivalent but maybe technically cleaner approach, which goes back to [120], is to use a Morse homology model for H  .LCz I K/. This means that the coefficients of the cochain map underlying (8.24) are determined by counting points in the intersections u u M.yk / \ ev1 z .W .x//, where W .x/ is the unstable manifold of rhz for some x with Lagrangian choice of Morse function hz on LCz . For instance, by taking S D H label L, one defines the PSS map (8.4). All properties stated in Section (8c) now follow from (8.25) and the composition laws. For instance, the inverse map is just given by S D H with one positive marked point. In some sense, this is actually the only application, since one can use gluing and the PSS isomorphism to replace marked points z by points at infinity  where L ;0 D L ;1 , which reduces (8.24) to the more basic version (8.1).

9 The Fukaya category (preliminary version)

113

9 The Fukaya category (preliminary version) (9a) Families. Let SO be some smooth oriented surface with boundary, † D †C t † a finite set of boundary points, and S D SO n †. Take a fibre bundle  W S ! R whose fibre is S , and whose structure group is the group of oriented diffeomorphisms of SO which are the identity on †. Each fibre Sr then has a natural compactification SO r , and these fit together canonically into a smooth proper fibre bundle O W SO ! O such that R. Moreover, each point of †˙ gives rise to a section  W R ! @S, S ˙ O S D S n .R/. We will often identify the points of † with the corresponding sections. Now suppose that S comes equipped with an almost complex structure O hence makes each fibre Sr IS on the fibrewise tangent bundle, which extends to S, into a pointed-boundary Riemann surface. In that situation, we call S a family of pointed-boundary surfaces. A set of strip-like ends for S ! R consists of proper embeddings  W R Z ˙ ! S fibered over R, one for each  2 †˙ , which restrict to strip-like ends on each fibre. These will automatically extend to smooth, fibrewise holomorphic embeddings O whose restriction to ˙1 2 Z y ˙ ! S, y ˙ is the corresponding section . Any RZ family admits strip-like ends, essentially because the set of choices of ends for a single pointed-boundary Riemann surface is contractible (this is in contrast to the choice of local coordinate around an interior point on a Riemann surface, where there is an essential S 1 freedom, leading to gravitational descendants). In a slightly different direction, suppose that we have fixed a set of ends f g. It is then often convenient to use local trivializations of S ! R which make the ends constant, in the following sense. Choose a point r0 2 R and let S D Sr0 be the fibre over it, with its strip-like ends S; D  .r0 ;  / W Z ˙ ! S . For a sufficiently small neighbourhood r0 2 U  R, there is a fibrewise diffeomorphism ‰ W U  S ! SjU which is the identity on fr0 g  S, and such that  .r; s; t / D ‰.r; S; .s; t//

for r 2 U , .s; t/ 2 Z ˙ .

(9.1)



on S, parametrized by r 2 U , which Then ‰ IS is a family of complex structures S vary only on the compact subset S n  .Z ˙ /. (9b) Kodaira–Spencer maps. Let S be a Riemann surface with strip-like ends. N Take the standard @-operator on the tangent bundle, with boundary conditions in the totally real subbundle T .@S/. In the terminology of Section (8h), this is admissible (but fails to be nondegenerate), and therefore becomes Fredholm on weighted Sobolev spaces with small weights  > 0, see (8.16). Actually, we want to allow a slightly larger domain W k;pI .S; T S; T .@S//C W k;pI .S; T S; T .@S//

(9.2)

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consisting of those vector fields Z that, over each strip-like end, are of the form z with c 2 R and Z z 2 W k;pI . This adds finitely many dimensions   Z D c @s C Z (one for each end, corresponding to the constants c ); therefore, @N still extends to a Fredholm operator @N C W W k;pI .S; T S; T .@S//C ! W k1;pI .S; 0;1 S ˝ T S /:

(9.3)

Denote the cokernel of this by TS ; it is independent of both .k; p/ and (suitably small)  > 0. If S is stable (has no continuous families of automorphisms), (9.3) is injective, O and TS has dimension j†j  3 .S/. Let S ! R be a family of Riemann surfaces with strip-like ends. Choose a point r0 2 R, write S D Sr0 for the fibre, and take a local trivialization ‰ which makes the strip-like ends constant, as in (9.1). Differentiating the family .‰  IS / of pullback complex structures on S yields, for each tangent vector in .T R/r0 , an element of NC Cc1 .S; End0;1 .T S// D Cc1 .S; 0;1 S ˝T S /. By projecting to coker.@ / one defines the Kodaira–Spencer map

r0 W .T R/r0 ! TS :

(9.4)

This is independent of the trivialization, and (although that is not immediately visible from our definition, due to our preference for function spaces which are closely tied to the strip-like end structure) also of the choice of strip-like ends. Call S ! R infinitesimally miniversal if (9.4) is bijective for each r0 2 R. To justify the name, one would have to prove that such families have a local versality property. This is not difficult, but we have no actual use for it. Interested readers are referred to the algebro-geometric deformation theory of pointed Riemann surfaces [72, p. 94], which is entirely analogous. (9c) Pointed discs. We now restrict discussion to the class of surfaces relevant to Fukaya categories. A .d C 1/-pointed disc, d  0, is a pointed-boundary Riemann surface S whose compactification SO Š D is a disc, and which has one incoming point at infinity and d outgoing ones. Our convention is to number the points at infinity, respecting their cyclic order around the boundary of the disc, so that the incoming point is † D f0 g and the outgoing ones are †C D f1 ; : : : ; d g; if S has strip-like ends, we number these correspondingly by 0 ; : : : ; d . Connected components of @S will be denoted by C0 ; : : : ; Cd , respecting the cyclic order and where 0 sits between Cd and C0 (see Figure 9.1 for the case d D 2). If S carries Lagrangian labels, these are correspondingly denoted by L0 ; : : : ; Ld . The pairs of Lagrangian submanifolds associated to the points at infinity are then ( k D 0; .L0 ; Ld / .L k ;0 ; L k ;1 / D (9.5) .Lk1 ; Lk / k > 0:

9 The Fukaya category (preliminary version)

C2

115

2

C1

0

1

C0 Figure 9.1

In the stable range d  2, .d C1/-pointed discs have no nontrivial automorphisms, and there is a universal family of them, denoted by S d C1 ! Rd C1 :

(9.6)

Its construction is straightforward: let Conf d C1 .@D/  .@D/d C1 be the configuration space of .d C 1/-tuples of points on the circle whose numbering is compatible with their cyclic order. Aut.D/ Š PSL2 .R/ acts freely and properly on this configuration space, and one sets Rd C1 D Conf d C1 .@D/= Aut.D/;

d C1 SO D Conf d C1 .@D/ Aut.D/ D:

(9.7)

d C1 ! Rd C1 admits sections k .Œz0 ; : : : ; zd / D Œz0 ; : : : ; zd ; zk ; The projection SO S d C1 and following the general procedure, we set S d C1 D SO n k k .Rd C1 /. Riemann uniformization (in a parametrized form) shows that (9.6) is indeed a universal family, in the sense that every family of .d C 1/-pointed discs is the pullback of S d C1 by a unique smooth map from its base into Rd C1 . The analogous infinitesimal result, which is also not difficult to prove, says that (9.6) is infinitesimally miniversal. Define a universal choice of strip-like ends to be a choice, for every d  2, of a set of strip-like ends fkd C1 g0kd for S d C1 ! Rd C1 . It is, of course, universal in the sense that any family of .d C 1/-pointed discs inherits strip-like ends through its classifying map.

(9d) Tree-speak. Gluing operations involving several pointed discs are naturally described in terms of trees. We assemble here all the relevant terminology, including some for later use. By a d -leafed tree, d > 0, we mean a properly embedded planar tree T  R2 which has d C 1 semi-infinite edges: one of them is singled out and called the root, and the others are the leaves. The semi-infinite edges are also called exterior, and the others interior ones. We denote by Ve.T / the set of vertices, and by

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Edint .T / the set of interior edges. Recall that a flag in a graph is a pair consisting of a vertex and an adjacent edge, f D .v; e/. We write Flint .T / for the set of interior flags (those where the edge is interior). We will orient T by flowing “upwards” from the root to the leaves, Figure 9.2. A flag f D .v; e/ is called positive if the orientation of e points away from v, and negative otherwise. To every interior edge e correspond two flags f ˙ .e/, one positive and the other negative. Similarly, an exterior edge has a single flag belonging to it, which is positive or negative depending on whether the edge is a leaf or the root. On occasion, we will find it useful to number the flags adjacent to a given vertex v by f0 .v/; : : : ; fjvj1 .v/, where jvj is the valency. This numbering always starts with the unique negative flag f0 .v/ and continues anticlockwise with respect to the given planar embedding.

root leaves

Figure 9.2

A d -leafed tree is called stable if jvj  3 for all vertices v, and semistable if jvj  2. (9e) Trees and gluing. Suppose that we have a d -leafed tree T ; for each vertex v, a jvj-pointed disc Sv with strip-like ends; and for each interior edge e, a gluing length le > 0. From this data, the gluing construction produces a new .d C 1/-pointed disc Sl with strip-like ends. Note first that there is a canonical bijection between flags f D .v; e/ and points at infinity in Sv , such that negative/positive flags correspond to incoming/outgoing points: namely, we associate to the k-th point at infinity of Sv the flag fk .v/. We adapt the notation accordingly, writing f for the strip-like end of Sv around the point at infinity which corresponds to f . The first part of the gluing process is to cut off a piece from each end belonging to an interior flag:  a  Sv0 D Sv n f .Zl˙e / ; (9.8) f D.v;e/ 2Flint .T /

9 The Fukaya category (preliminary version)

117

where Zle D .1I le /  Œ0I 1 or ZlCe D .le I 1/  Œ0I 1 is used, depending on whether the flag is negative or positive.`After that, one identifies the stubs of these cut-off ends with each other: set Sl D v Sv0 = , where f  .e/ .s  le ; t / f C .e/ .s; t/

(9.9)

for each interior edge e and .s; t/ 2 Œ0I le   Œ0I 1. The strip-like ends of Sl are inherited from those associated to the exterior flags of T . More precisely, if we number those flags by f0 ; : : : ; fd starting with the root and proceeding in cyclic order, then the k-th strip-like end of Sl is a Sv0 ! Sl : (9.10) l;k D fk W Z ˙ ! v

In the trivial case where T has just one vertex, the gluing process does nothing (gives back the original surface); if it has two vertices, it is equal to the basic connected sum operation from (8.5); and generally, it can be viewed as a finite sequence of such operations, performed in an arbitrary order. Remark 9.1. The surface Sl inherits a thick–thin decomposition (this comes from the gluing process and is not intrinsic to the surface, hence differs from the decomposition of the same name provided by hyperbolic geometry). The thin part is the union of the strip-like ends and of finite strips of length le : Slthin D

d a

l;k .Z ˙ / t

D

a

Sv n

v

f C .e/ .Œ0I le   Œ0I 1/;

e2Edint .T /

kD0

Slthick

a

jvj1 a

 fk .v/ .Z ˙ / :

kD0

As we will see later, it is better to work with gluing parameters which are inverse exponentials of the lengths; more precisely e 2 .1I 0/ is related to the corresponding length by le D  log.e /=, so that e ! 0 corresponds to le ! 1. The glued surfaces S D Sl fit together into a family of .d C 1/-pointed discs, S ! R D .1I 0/Ed

int .T /

;

and this comes with strip-like ends R Z ˙ ! S which are fibrewise given by (9.10). In a sense, one can also allow the limiting case in which certain gluing parameters are zero. The rule then is that the degenerate edges (those with vanishing gluing parameter) are omitted in both steps of the gluing process, (9.8) and (9.9). The result is no longer a single pointed disc, but rather a disjoint union of such discs, described

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by the tree Tx obtained from T by collapsing all nondegenerate edges; this in particular means that to each vertex vN of Tx corresponds a connected component which is a jvjN pointed disc (in the trivial case where all gluing parameters are zero, one gets back the disjoint union of the Sv ). Adding these fibres to S gives a partial compactification x which is a smooth manifold with corners and comes with a submersion (clearly, S, x D .1I 0Edint .T / . not a fibration) Sx ! R A straightforward generalization of the gluing construction is to start with a family Sv ! Rv of jvj-pointed discs with strip-like ends, for each vertex v. Gluing then produces a new family Y int Rv : S ! R D .1I 0/Ed .T /  v

x where the gluing parameters As before this has a partial compactification Sx ! R are allowed to become zero. (9f) Deligne–Mumford–Stasheff compactifications (a.k.a. associahedra). As a set, the Deligne–Mumford–Stasheff compactification of Rd C1 is a x d C1 D RT : (9.11) R T

Q Here, the union is over all stable d -leafed trees T , and one sets RT D v2Ve.T / Rjvj . To put a topology and smooth structure on (9.11), we make some universal choice of strip-like ends, and use the gluing construction. Namely, take a stable tree T and attach to each vertex v the universal family S jvj ! Rjvj . Gluing these together yields a family of .d C 1/-pointed discs S ! R D .1I 0/Ed

int .T /

 RT ;

(9.12)

which by definition is classified by a smooth gluing map  T W R ! Rd C1 :

(9.13)

As explained before, one can allow (some or all of) the gluing parameters to become zero. The result is a degenerate glued surface, which is a disjoint union of pointed discs governed by a collapsed tree Tx . Clearly, this determines a point in the Tx -stratum of (9.11), hence defines a canonical extension of the gluing maps x D .1I 0Edint .T /  RT ! R x d C1 : N T W R int

(9.14)

On the subset f0gEd .T /  RT this represents the trivial (most degenerate) gluing operation, hence just gives the identity on the T -stratum in (9.11). The topology of

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the Deligne–Mumford–Stasheff compactification is the one induced from the maps x d C1 to an arbitrary space is continuous (9.14) for all T : that is to say, a map from R iff its compositions with all the N T are continuous. The differentiable structure is characterized by the analogous property, replacing “continuous” by “smooth”. This x d C1 into a compact manifold with corners, in a way which is independent makes R of the universal choice of strip-like ends; and moreover, (9.11) is the decomposition into the interior and connected boundary strata. The properties we have just stated are not immediately obvious from the conx d C1 is Hausdorff). struction (indeed, it may not even be clear that the topology on R x d C1 as The best way of verifying them is to start with the alternative definition of R x 0;d C1 , x part of the real locus M 0;d C1 .R/ of the complex Deligne–Mumford space M d C1 x which is explained in [58], [60]. When introduced in this way, R comes with a canonical structure of smooth compact manifold with corners. x d C1 with the smooth structure coming from its embedding Lemma 9.2. Equip R x into M 0;d C1 .R/. Then the extended gluing maps N T are smooth, and are local diffeomorphisms near the subset where all the gluing parameters are zero. x 0;d C1 .R/  M x 0;d C1 is Sketch of proof. Recall first that the embedding Rd C1  M achieved by “doubling” a pointed disc to a punctured CP1 with an antiholomorphic involution. Strip-like ends on a disc can be doubled to give local holomorphic coordinates near the punctures of the CP1 . By complexifying the gluing parameters, one can then extend (9.14) to maps .N T /C W int.D/Ed

int .T /

x 0;d C1 ;  RT ! M

(9.15)

where int.D/ D D n@D is the open unit disc. More precisely, if we fix a point of RT , then the complexified gluing construction produces a holomorphic family of stable int pointed discs over int.D/Ed .T / . (9.15) will be the classifying map of this family; hence it is holomorphic with respect to the gluing parameters, and smooth on the open subset where all the gluing parameters are nonzero. To conclude that it is smooth everywhere, use the following elementary consequence of Cauchy’s formula: let Ux D int.D/q  Rp , U D .int.D/ n f0g/q  Rp , and for r 2 Rp , Uxr D int.D/q  frg. Suppose that we have a function g W Ux ! C which is smooth on U , and whose restriction to each Uxr is holomorphic. Then g is smooth. Finally, to prove that N T is a local diffeomorphism, we need to compute its derivative with respect to the gluing parameters, at points where all those parameters are zero. Again, complex geometry (more precisely, the deformation theory of Riemann surfaces with nodes) makes this an easy exercise.  With Lemma 9.2 at hand, it is straightforward to see that the topology and smooth x d C1 , as originally defined (in terms of gluing maps (9.14) for an arbitrary structure of R

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x d C1  universal choice of strip-like ends), agree with those given by the embedding R x 0;d C1 .R/. This implies all the properties stated above. M Define S T ! RT to be the disjoint union of the pullbacks of the universal fibrations S jvj ! Rjvj , v 2 Ve.T /; the fibres of S T will be disjoint unions of jvjpointed discs. The universal family S d C1 ! Rd C1 admits a partial compactification, which set-theoretically is a x d C1 : S T ! R (9.16) Sxd C1 D T

This can be equipped with a topology and smooth structure, by using gluing maps on the total spaces of families. By definition, the gluing map  T associated to (9.12) is covered by a fibrewise isomorphism  T W S ! S d C1 : As we observed when explaining the general gluing construction, S has a natural x By comparing this with (9.16), one sees that  T partial compactification Sx ! R. x T W Sx ! Sxd C1 . As before, the topology and smooth structure induced extends to  by these maps are the same as those obtained by embedding Sxd C1 into the universal x 0;d C1 . family of stable curves over M (9g) Choosing strip-like ends consistently. As before, we make a universal choice of strip-like ends, and consider the gluing construction associated to a stable d -leafed tree T . The resulting family (9.12) carries two sets of strip-like ends, which a priori may be different. The first set is inherited from the universal choice of strip-like ends on each S jvj through the gluing construction; and the second set is pulled back from the universal choice on S d C1 via the classifying map (9.13). We call our universal x x choice of strip-like Qendsjvjconsistent if, for every T , there is an open subset U  R containing f0g  v R , such that the two abovementioned sets of strip-like ends agree over U D Ux \ R. In future, we will refer to such U or Ux as subsets where the gluing parameters are sufficiently small. Lemma 9.3. Consistent universal choices of strip-like ends exist. The proof of this is simple and somewhat tedious, so we will only explain the basic ideas. The first noteworthy point is that the consistency condition is of inductive nature. Let T be a d0 -leafed stable tree with at least two vertices (the one-leafed tree gives rise to an empty consistency condition). Consistency for T partially determines the strip-like ends fkd0 C1 g on S d0 C1 in terms of the fkd C1 g for d D jvj1 < d0 . The next observation is that the consistency condition for a single tree T can be satisfied. By inspecting the proof of Lemma 9.2, one sees that the restriction of N T to a subset

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Ux where the gluing parameters are small is an open embedding, whose image is a x eC1 . In particular,  T jU is also an embedding. neighbourhood of the T -stratum in R d C1 Hence, if we assume that the fk g for d < d0 have already been fixed, one can certainly choose fkd0 C1 g so that its pullback under  T jU , for a single specific T , agrees with the set of strip-like ends on SjU obtained from the gluing process. The final piece of the argument consists in checking that the conditions imposed by various T do not contradict each other. This is again a recursive argument, based on the combinatorics of the boundary strata in the Deligne–Mumford–Stasheff space. For concreteness, we will explain it in the simplest nontrivial case of d0 D 4. Suppose that we have already chosen fk3 g and fk4 g, in such a way that the consistency condition is satisfied for stable 3-leafed trees. There are five stable 4-leafed trees T with two vertices (of valency 3 and 4, necessarily), giving rise to five extended gluing maps x5: N T W .1I 0  R3  R4 D .1I 0  fpointg  .1I 1/ ,! R x 5 is a pentagon. After restricting to a subset UxT where the gluing parameters are R small, each gluing map is an embedding covering a neighbourhood of one boundary face (see Figure 9.3). Consistency requires that fk5 g should be determined by

x5 R

images of gluing maps overlaps

Figure 9.3

fk3 g; fk4 g on the intersection of these neighbourhoods with the interior R5 . What needs to be shown is that these five conditions can be met at the same time. Any two neighbourhoods N T1 .UxT1 /, N T2 .UxT2 / are either disjoint or overlap near a vertex of the pentagon, which corresponds to a stable 4-leafed tree T3 with three vertices. In the first case, the consistency conditions clearly do not interfere with each other; in

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the second case, using the gluing map N T3 and the consistency between fk3 g and fk4 g, one can verify that the strip-like ends induced from the T1 and T2 gluing construction agree for small values of the gluing parameters, so that the associated consistency conditions for fk5 g can be simultaneously satisfied. (9h) Moduli spaces for families. Fix an exact symplectic manifold with corners M . A set of Lagrangian labels for a family S ! R is a locally constant family of Lagrangian labels for the fibres. As in the case of a single surface, once one has a choice of labels, it is natural to associate to each  2 †˙ a pair of exact Lagrangian submanifolds .L ;0 ; L ;1 /, which are the labels of the adjacent boundary components (in any fibre). Suppose that we have equipped S ! R with such labels, and with a set of strip-like ends. Suppose also that for each pair .L ;0 ; L ;1 / we have chosen a Floer datum .H ; J /. Then, a perturbation datum for our family is a pair .K; J / consisting of K 2 1S=R .S; H / and J 2 C 1 .S; J/; which satisfy the conditions from Section (8e), including (8.7), when restricted to any fibre. The notation may require some explanation: K is a smooth family of one-forms on the fibres, with values in the space H of Hamiltonian functions on M . Equivalently, if T S v ! S denotes the tangent bundle along the fibres, one can think of K as a section of its dual bundle pulled back to S  M (with additional conditions near S  @M and on the strip-like ends). Fix a perturbation datum for S ! R. Then, for each fy g 2† , y 2 C.L ;0 ; L ;1 /, one can consider the moduli space  .r; u/ W r 2 R; and u 2 C 1 .Sr ; M / satisfies (8.9), (8.10) MS .fy g/ D : for the restriction of the perturbation datum to the fibre Sr (9.17) Using a local trivialization over some small open subset U  R which makes the strip-like ends constant, one can define a (trivial) Banach fibre bundle BSjU ! U whose fibres are the W 1;p map spaces BSr from Section (8i). There is also a Banach vector bundle ESjU ! BSjU , which over each r 2 U restricts to the Lp space ESr . N The inhomogeneous @-equation (8.9) yields a section BSjU ! ESjU , whose zero-set is the union of MSjU .fy g/ for all fy g. The derivative of this section at a point .r; u/ of the moduli space is called the extended linearized operator DS;r;u W .T BSjU /.r;u/ D T Rrd C1  .T BSr /u ! .ESr /u :

(9.18)

The second component is just the ordinary linearized operator DS;u from (8.17) with S D Sr , while the first one is the derivative of (8.9) with respect to changes of r, keeping u fixed. Note that the extended operator is again Fredholm. The appropriate notion of regularity for .r; u/ is surjectivity of DS;r;u (as opposed to the

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stronger requirement of surjectivity of DSr ;u , which would be regularity of u taken as an inhomogeneous pseudo-holomorphic map with fixed domain Sr ). As usual, we say that a moduli space MS .fy g/ or a perturbation datum .K; J / is regular if the regularity condition holds for all .r; u/ concerned. Remark 9.4. Our formulation bypasses a technical difficulty, which appears whenever one looks at pseudo-holomorphic maps with varying domains. There is no canonically defined Banach manifold BS for the whole family S ! R, since changes of trivialization induce only Banach homeomorphisms (not differentiable maps) between the locally defined spaces BSjU . However, elliptic regularity ensures that the operator (9.18), and in the regular case the manifold structure of MS .fy g/, are independent of the local trivialization. (9i) Choosing perturbation data consistently. At this point, we need to briefly revisit the gluing construction. Suppose that we have a d -leafed tree T , and for each vertex v 2 Ve.T /, a jvj-pointed disc Sv with strip-like ends and Lagrangian labels. Recall that there is a bijection between flags f D .v; e/ and points at infinity of Sv . Hence, in our situation there is a pair of exact Lagrangian submanifolds associated to each flag. We say that the Sv are compatibly labeled if for each e 2 Edint .T /, the two flags f ˙ .e/ D .v ˙ .e/; e/ have the same pair of Lagrangian submanifolds associated to them. One can picture this situation by labeling the connected components of R2 n T with exact Lagrangian submanifolds (see Figure 9.4). The family of .d C 1/pointed discs S obtained by gluing the Sv together then inherits Lagrangian labels in an obvious way. Assume in addition that the Sv are equipped with perturbation data .Kv ; Jv /, such that the associated Floer data on each pair of ends f ˙ .e/ are the same. They then determine a perturbation datum .K; J / on S through the gluing x construction. This datum extends smoothly to the natural partial compactification S, in the sense that there are x Kx 2 1S= x R x .S; H /

and

x J/ JN 2 C 1 .S;

(9.19)

whose restriction to S equals .K; J / (the space of fibrewise one-forms makes sense x is a submersion). Over the most in the partial compactification, because Sx ! R ` x x JN / is just the disjoint degenerate point 0 2 R, where the fibre is S0 D v Sv , .K; union of the given data .Kv ; Jv /. Finally, all these observations remain true if we glue together families Sv instead of single surfaces. Fix a universal choice of strip-like ends. Also, for each pair .L0 ; L1 / of exact Lagrangian submanifolds in the symplectic manifold M under consideration, fix a Floer datum .HL0 ;L1 ; JL0 ;L1 /. Then, a universal choice of perturbation data is the choice, for every d  2 and every .d C 1/-tuple .L0 ; : : : ; Ld / of exact Lagrangian submanifolds, of a perturbation datum .KL0 ;:::;Ld ; JL0 ;:::;Ld /

(9.20)

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II Fukaya categories

Lv2 ;2 v2

Lv1 ;3

Lv1 ;2 Lv2 ;1

v1

Lv1 ;1

Lv1 ;0

Lv2 ;0 the tree T

the pointed discs associated to Ve.T /, with their Lagrangian labels

Lv2 ;1

Lv1 ;3 D Lv2 ;2

Lv1 ;3

Lv2 ;1

Lv1 ;2 D Lv2 ;0 Lv1 ;0

Lv2 ;0

Lv1 ;1

Lv1 ;1 Lv1 ;0

the compatibility conditions

the glued surface, with labels Figure 9.4

on the family S d C1 ! Rd C1 with Lagrangian labels .L0 ; : : : ; Ld /, which over the strip-like ends of that family reduces to the Floer data for the pairs (9.5). From now on assume that our universal choice of strip-like ends is consistent. Suppose that we have made a universal choice of perturbation data. Let T be a stable d -leafed tree; associate to its vertices the universal families S jvj ; and equip them with compatible Lagrangian labels. Then the glued family S ! R carries two perturbation data, which a priori may be different (more precisely, they will always be equal over the strip-like ends, but might differ on the remaining part of the surfaces). The first one is induced from the perturbation data on the S jvj through the gluing construction, and the other one is pulled back from the universal choice of perturbation datum on S d C1 (with the correct labels) by the classifying map  T . We say that the universal choice of perturbation data is consistent if, in all such situations, the following conditions are satisfied: • There is a subset U  R where the gluing parameters are sufficiently small, such that the two abovementioned perturbation data agree on the thin parts of the surfaces Sr , r 2 U . This refers to the thick–thin decomposition which

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arises from the gluing process, see Remark 9.1. • Let .K; J / be the first perturbation datum on S (the one obtained by gluing), and x JN / its extension to the partial compactification Sx as in (9.19). We ask that .K; x the other datum (obtained by pullback from S d C1 ) also extends smoothly to S, int .T / Ed T x x JN / over the subset f0g and that the extension agrees with .K; R  R where all the gluing parameters are zero. To get a better idea of what consistency means, suppose that we have a sequence x d C1 n Rd C1 . For r1 ; r2 ;    2 Rd C1 converging to a point in the T -stratum of R large k one can then write rk D  T .fe;k g; frv;k g/

(9.21)

where rv;k 2 Rjvj , v 2 Ve.T /, and e;k 2 .1I 0/, e 2 Edint .T /. As k ! 1, each sequence rv;k converges, and the e;k go to 0. More geometrically, consider the pointed discs : Sk D SrdkC1 ; Sv;k D Srjvj v;k Then (9.21) says that if we glue together the Sv;k as indicated by the edges of T , leaving (as k ! 1, increasingly long) finite strips Œ0I  log.k;e /=  Œ0I 1 between any two, the outcome is isomorphic to Sk . Equip the Sk with some Lagrangian labels .L0 ; : : : ; Ld /, and the Sv;k with the corresponding consistent labels. By definition, a universal choice of perturbation data gives a preferred perturbation datum on each stable pointed disc with Lagrangian labels, hence in our case on Sk and Sv;k . Consistency says that for large k, the perturbation data on Sk “converge to” those ` on v Sv;k `; this statement makes sense because by definition, Sk is a quotient of a subset of v Sv;k . Moreover, on the finite strips (and also on the strip-like ends) of Sk , this convergence is trivial, in the sense ` that the perturbation data there will be equal to those on the corresponding parts of v Sv;k for k  0. Lemma 9.5. Consistent universal choices of perturbation data exist.



This is proved by the same type of inductive argument as Lemma 9.3. We will limit ourselves to a single observation about the proof. In general, a surface S D Srd0 C1 can lie in the image of more than one gluing map  T , and these give rise to different thick–thin decompositions. However, one can choose for each T an open subset int U T  .1I 0/Ed .T /  RT where the gluing parameters are small, in such a way that for each r 2 Rd C1 there is a maximal tree T for which r 2  T .U T / (maximal in the sense that all other trees Tx with that property would be obtained by contracting edges of T ). As a consequence of consistency in the choice of strip-like ends, the thick–thin decomposition of S induced by  T is maximal (the decompositions x x x coming from other  T with r 2  T .U T / being obtained from it by forgetting some of the finite thin strips). With respect to our consistency requirement, the maximal

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decomposition imposes the strictest condition on the perturbation datum. This has the following implication for the strategy used in proving Lemma 9.5. Suppose that we have already chosen perturbation data (9.20) for d < d0 , so that the consistency condition is satisfied. Then, the construction for S d0 C1 starts with the subsets  T .U T / for the most complicated (maximal number of edges) trees T , which correspond to x d0 C1 , and proceeds by increasing dimension of the boundary strata of vertices of R the compactification. Remark 9.6. At first sight (especially comparing the discussion above with its counterpart in Section (9g)), it might seem more natural to impose a stronger consistency condition, which is that the two perturbation data on the glued families S ! R should agree for small values of the gluing parameter. To see where the drawback lies, take a tree T with 2 vertices of valency 3 each. To each vertex we associate the family S 3 ! R3 , which in fact consists of the unique 3-pointed disc S. In the notation of (8.5), the resulting glued 4-pointed discs are S #l S. Choose all labels equal to the same exact Lagrangian submanifold. If a universal perturbation datum satisfies the stronger consistency condition, it equips the discs S#l S with perturbation data which are the same on both “halves” (more precisely, on both connected components of the thick part). When trying to achieve regularity for the moduli spaces of inhomogeneous pseudo-holomorphic maps within this class of perturbation data, multiple-cover type problems arise. Our definition of consistency bypasses this difficulty. (9j) The definition. To define the Fukaya category (in its preliminary version) F .M /pr of an exact symplectic manifold with corners M , we need to pick additional data as follows: • a consistent universal choice of strip-like ends fkd C1 g; • A Floer datum .HL0 ;L1 ; JL0 ;L1 / for each pair .L0 ; L1 / of exact Lagrangian submanifolds in M ; and • a consistent universal choice of perturbation data .KL0 ;:::;Ld ; JL0 ;:::;Ld / (compatible with the previous choices of strip-like ends and Floer data). These are subject to an additional regularity condition, which we will explain now. For each .d C 1/-tuple (d  2) of exact Lagrangian submanifolds .L0 ; : : : ; Ld /, and points y0 2 C.L0 ; Ld /, yk 2 C .Lk1 ; Lk / (1  k  d ), consider M d C1 .y0 ; : : : ; yd / D MS d C1 .y0 ; : : : ; yd /:

(9.22)

This is the moduli space (9.17) of solutions of (8.9) for the family S d C1 ! Rd C1 equipped with the Lagrangian labels .L0 ; : : : ; Ld /, our chosen perturbation datum, and where the asymptotic condition is convergence to the yk on the strip-like ends. Points .r 2 Rd C1 ; u W Srd C1 ! M / of (9.22) will be called inhomogeneous pseudoholomorphic polygons. We extend the notation to d D 1, setting M2 .y0 ; y1 / D

9 The Fukaya category (preliminary version)

127

 .y0 ; y1 / to be equal to the moduli space of solutions of Floer’s equation (mod MZ translation). Then, the requirement is that all these moduli spaces should be regular. For d D 1, we have already mentioned that this is satisfied for a generic choice of the Floer data. The same statement holds for d  2, within the class of consistent universal choices of perturbation data. Objects of F .M /pr are exact Lagrangian submanifolds L  M . Morphisms

homF .M /pr .L0 ; L1 / D CF pr .L0 ; L1 / are given by Floer chain spaces. The first order map 1 D @ is the boundary operator, defined as before (8.19) by counting isolated points of M 2 .y0 ; y1 /. The higher order compositions are similarly given by X #M d C1 .y0 ; : : : ; yd / y0 : (9.23) d .yd ; : : : ; y1 / D y0

Compactness and gluing arguments show that this is well-defined (the zero-dimensional part of each M d C1 .y0 ; : : : ; yd / is a finite set) and satisfies the A1 -associativity equation (1.2), so that F .M /pr is an ungraded A1 -category. By definition, the map on cohomology induced by 2 is the triangle product (8.3). Hence, the underlying cohomological category is indeed the previously defined H F .M /pr , which also shows that F .M /pr is cohomologically unital. Remark 9.7. We have not explicitly defined ungraded A1 -categories, but the meaning should be obvious. Recall that by assumption, our ground field has char.K/ D 2, so the grading is not needed to define any signs. With that in mind, the elementary theory from Chapter I goes through with cosmetic changes (the single exception is Corollary 5.8). The reader should keep in mind that the current setup is only a temporary simplification: all applications will ultimately take place in a context where gradings are well-defined. Remark 9.8. The results summarized in the previous paragraph are well-known to experts, and their proofs, while somewhat lengthy, do not go beyond the standard techniques of pseudo-holomorphic curve theory. In the rest of this section, we will give a rapid sketch of the main steps, which hopefully conveys an idea of what those techniques are. For readers interested in digging into the details, the following are some possible indications. Inhomogeneous terms for families of Riemann surfaces seem to have been first used in [116]. The basic reference for transversality arguments is [49]. The original papers of Floer [46], [45], [44] contain some of the basic analysis required to deal with Lagrangian boundary conditions, as do [103], [50]. The spaces Rd C1 and their compactifications are considered in [58], which also contains much material related to Fukaya categories. The most complicated piece

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of the puzzle is probably the gluing procedure which is used to prove that the A1 associativity equations hold; this is explained in [60, §18]. The argument generalizes the more familiar proof of @2 D 0 in Floer theory, for which see [44] and [119] (in the Lagrangian and Hamiltonian versions, respectively). (9k) Transversality. Fix a consistent universal choice of strip-like ends, and a Floer datum for each pair of exact Lagrangian submanifolds. Suppose that we have already chosen perturbation data (9.20) for all d  d0 . Take some Lagrangian labels .L0 ; : : : ; Ld0 /. We want to modify the given perturbation datum .K; J / D .KL0 ;:::;Ld0 ; JL0 ;:::;Ld0 / slightly, in such a way that the associated moduli spaces (9.22) become regular. Formally, an infinitesimal deformation of .K; J / is given by a pair .ıK; ıJ / 2 1S d0 C1 =Rd0 C1 .S d0 C1 ; H /  C 1 .S d0 C1 ; T JJ /

(9.24)

where .ıK; ıJ / vanishes on the strip-like ends of each surface Srd0 C1 , and also satisfies ıK./jLk D 0 for each tangent vector  to the connected component Ck  @Srd0 C1 . Such an infinitesimal deformation can be easily “exponentiated” to an actual one, (9.25) .K 0 ; J 0 / D .K C ıK; J exp.J  ıJ //: To retain consistency with the choices of perturbation data for d < d0 , we have to restrict the space of allowed deformations somewhat. Fix, for each d0 -leafed stable int tree T , an open subset U T  .1I 0/Ed .T /  RT where the gluing parameters are T small. Provided that the U are taken sufficiently small, one can choose an open subset   S d0 C1 such that for each r 2 Rd0 C1 , the intersection r D  \ Srd0 C1 is nonempty and has the following properties: r is disjoint from the strip-like ends of Srd0 C1 ; and if r 2  T .U T / for some T , then r lies in the thick part of the resulting decomposition of Srd0 C1 . Consider the subspace T of pairs .ıK; ıJ / as in (9.24) which vanish outside , and which are decay asymptotically with respect to the partial compactification Sxd0 C1 . The last-mentioned condition means that .ıK; ıJ / extends smoothly to a pair .ıK; ıJ / defined on Sxd0 C1 , which vanishes (together with its derivatives of arbitrary order) on Sxd0 C1 n S d0 C1 . For any such .ıK; ıJ /, the deformed perturbation datum (9.25) still satisfies consistency with respect to the gluing maps  T . Namely, since .K 0 ; J 0 / D .K; J / outside , the condition on the thin parts of the surfaces Srd0 C1 are still satisfied; and the other piece of the consistency requirement concerns the asymptotic behaviour with respect to Sxd0 C1 , which again is the same for .K 0 ; J 0 / as for the original .K; J /. It remains to prove that this class of deformations is sufficient to achieve regularity of the moduli spaces of inhomogeneous pseudo-holomorphic .d0 C 1/-gons with labels .L0 ; : : : ; Ld0 /. This is a standard application of the Sard–Smale theorem.

9 The Fukaya category (preliminary version)

129

Associated to any point .r; u/ 2 M d0 C1 .y0 ; : : : ; yd0 / in the moduli space for the original perturbation data .K; J / is a universal linearized operator DT ;S d0 C1 ;r;u . This contains (9.18) but has additional variables which express how .du  Y /0;1 changes if we modify .K; J /. To make this rigorous, one should really choose a local trivialization of the family S d0 C1 , see Remark 9.4, but we suppress that for simplicity. The explicit formula is then x d0 C1 W T  T Rrd C1  .T B d0 C1 /u ! .E d0 C1;r /u ; D S ;r;u S S r

.ıK; ıJ; ; X/ 7! .ıY /0;1 C ıJ ı 12 .du  Y / ı IS C DS d0 C1 ;r;u .; X/;

(9.26)

where ıY is the fibrewise one-form with values in Hamiltonian vector fields obtained from ıK; and IS is the complex structure on S D Srd0 C1 . The main step is to prove the surjectivity of (9.26). Note that since the last summand is Fredholm, the image is automatically closed. An element of the cokernel of (9.26) is a 2 Lq .S; .0;1 S ˝    u TM / /, 1=p C 1=q D 1, which in particular must satisfy DS;u D 0. By elliptic regularity is smooth at least in int.S/ (by looking more closely at its properties, one can in fact prove that is smooth up to the boundary, but we do not need that here). Moreover, it satisfies Z h.ıY /0;1 ; i D 0 (9.27) S

for all ıY . By construction, there is a nonempty open set r  S of points where ıK can be chosen in an essentially arbitrary way. In particular, for any z 2 r \ int.S/ and any element ˛ 2 0;1 .S/z ˝ TMu.z/ , there is a sequence of .ıK; ıJ / such that the associated ıY converge weakly to a ı-function ız ˝ ˛. By inserting that sequence into (9.27) one sees that z D 0. One therefore knows that vanishes on r , and the fact that it is zero everywhere then follows from unique continuation for solutions of  D 0: DS;u

Remark 9.9. The sketch proof which we have just given omits at least one technical point, which is routine but not trivial. The space T of admissible .ıJ; ıK/ is a Fréchet space. To apply Sard–Smale, one needs to restrict further to a dense Banach subspace, for instance by using a version of Floer’s C 1 -topology which takes into account the required asymptotic behaviour with respect to Sxd0 C1 . Doing that reduces the domain of (9.26) to a dense subspace in the first two variables. However, since we still have the third variable, which is a Fredholm operator, surjectivity is not affected. (9l) Compactness. The moduli spaces of inhomogeneous pseudo-holomorphic polygons have natural Gromov compactifications, whose structure follows to a cerx d C1 (which one can in fact think of as the trivial case tain extent that of Rd C1  R

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M D L0 D    D Ld D point). We will define these compactifications as sets, state their main topological properties without proof, and then explain how this implies that the maps d form an A1 -structure. Fix Lk and yk as in the definition of (9.22), as well as a semistable d -leafed tree T . Suppose that for each vertex v 2 Ve.T / we have a jvj-pointed disc Sv . If jvj  3, the disc determines a point rv 2 Rjvj , and there is a unique isomorphism Sv Š Srjvj v . In the remaining case jvj D 2, one has Sv Š Z but the isomorphism is fixed only up to translations in s-direction. Additionally, the discs are supposed to carry Lagrangian labels, so that for each flag f D .v; e/ there is a distinguished pair of Lagrangian submanifolds .Lf;0 ; Lf;1 /. As in our previous discussion of gluing, the labels should be mutually compatible, so that the resulting labels on the glued .d C 1/-pointed disc are the given Lk . Finally, for each f we want to have a point yQf 2 C.Lf;0 ; Lf;1 /, subject to the condition that yQf C .e/ D yQf  .e/ , and so that the points associated to the flags nearest the root and leaves are y0 ; : : : ; yd , in the natural order. An inhomogeneous pseudo-holomorphic stable polygon is a collection of maps uv W Sv ! M , v 2 Ve.T /, satisfying the following conditions: • If jvj  3, we can think of uv as a map Srjvj v ! M , and the requirement is that it should be a solution of the relevant inhomogeneous pseudo-holomorphic map equation: .rv ; uv / 2 Mjvj .yQf0 .v/ ; : : : ; yQfjvj1 .v/ /; where f0 .v/; : : : ; fjvj1 .v/ are the flags around v in their natural order. This is with respect to the perturbation datum taken from our universal choice. • If jvj D 2, then uv W Z ! M should satisfy uv 2 M 2 .yQf0 .v/ ; yQf1 .v/ /; which means that it is a non-stationary solution of Floer’s equation, for the Floer data which is part of our setup. Since this equation is invariant under translations, the statement is independent of the choice of isomorphism Sv Š Z. Define MT .y0 ; : : : ; yd / to be the set of all equivalence classes of such inhomogeneous pseudo-holomorphic stable polygons. The equivalence relation is isomorphism of each pointed disc Sv , with corresponding equality between the maps uv . Concretely, this means that if we have a v with jvj D 2, then moving the associated uv by a translation does not affect the point fuv g 2 M T .y0 ; : : : ; yd /. As a set, the Gromov compactification is defined to be the union over all T , a x d C1 .y0 ; : : : ; yd / D M M T .y0 ; : : : ; yd /: (9.28) T

In contrast with (9.11), there are infinitely many semistable T , since one can always insert more two-valent vertices. However, for any choice of .y0 ; : : : ; yd /, only finitely

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9 The Fukaya category (preliminary version)

many strata of (9.28) will be actually nonempty. To see this, one sums up the energies (8.12) of each component to get X v

E.uv / D AHL0 ;Ld .y0 / 

d X

AHLk1 ;Lk .yk / C .curvature term/:

kD1

The curvature term carries contributions only from the vertices with jvj  3, and after looking at the consistency condition for our universal choice of perturbation data, one concludes that there is a uniform bound for it over the whole space (9.28). On the other hand, there is an  > 0 such that each component uv with jvj D 2 contributes at least  to the left-hand side of (8.12), namely: one looks at all pairs of elements y0 ; y1 2 C.Li ; Lj / for 0  i < j  d , and defines  to be the minimum over all the action differences AHLi ;Lj .y0 /  AHLi ;Lj .y1 / which are positive. These two facts together give a bound on the number of two-valent vertices of T for those trees that contribute nonempty strata to (9.28). From now on, we assume that all the moduli spaces involved are regular. Write M d C1 .y0 ; : : : ; yd /p for the p-dimensional part of M d C1 .y0 ; : : : ; yd /, which is an open and closed subset of that space, and analogously M T .y0 ; : : : ; yd /p . There is a natural stable map topology on (9.28), with respect to which the whole space is compact. This topology is such that each x d C1 .y0 ; : : : ; yd /p D M

a

M T .y0 ; : : : ; yd /pj Ed

int .T /j

(9.29)

T

is an open and closed subset, and a p-dimensional topological manifold with corners. More precisely, the part corresponding to the one-vertex tree, which can be identified with M d C1 .y0 ; : : : ; yd /p , is the interior, and all the others are codimension j Edint .T /j boundary strata. Questions about the differentiable structure of (9.29) are rather subtle: each stratum is certainly a smooth manifold, and one can define gluing maps which yield collar neighbourhoods of the boundary strata. However, the choice of gluing parameters is less clear, which means that these maps may not match up as x d C1 . Fortunately, the construction of Fukaya categories only well as in the case of R uses moduli spaces of dimension p  1, for which this issue is irrelevant. Considering first the case p D 0, one sees that only the one-vertex tree contributes to (9.29). The compactness of that space means that Md C1 .y0 ; : : : ; yd /0 is a finite set, which justifies the point-counting in the definition (9.23). Next, for p D 1, the boundary points come from semistable trees with exactly two vertices, which are the T D T .m; n/ from Figure 9.5. Let .r1 ; u1 ; r2 ; u2 / 2 M d C2m .y0 ; y1 ; : : : ; yn ; y; Q ynCmC1 ; : : : ; yd /0  M mC1 .y; Q ynC1 ; : : : ; ynCm /0

(9.30)

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be a point in the T .m; n/-stratum, which means that yQ 2 C.Ln ; LnCm /. The number mod 2 of such .r1 ; u1 ; r2 ; u2 /, taking the union over all possible y, Q is precisely the coefficient of y0 in the expression d C1m .yd ; : : : ; ynCmC1 ; m .ynCm ; : : : ; ynC1 /; yn ; : : : ; y1 /: Summing over m and n takes into account all boundary points, hence must yield an even number, which is exactly the statement of the A1 -associativity equation (1.2) with K-coefficients.

v2 v1

.n C 1/-st outgoing edge

jv1 j D d C 2  m jv2 j D m C 1 Figure 9.5

It is instructive to look at the last-mentioned case of the Gromov compactification from a more down-to-earth perspective. Suppose that we have a sequence .r i ; ui / 2 M d C1 .y0 ; : : : ; yd /1 , converging to some boundary point (9.30). This covers two related but distinct phenomena: • The unstable situation occurs when m D 1 or d . To keep notation simple, we discuss only the first of these. The ui are maps SrdiC1 ! M , and u1 a map Srd1C1 ! M , for r i ; r1 2 Rd C1 . In fact, the r i converge to r1 , and the ui to Q ynC1 / is a solution u1 on compact subsets. The other component u2 2 M 1 .y; of Floer’s equation, and appears as a “bubble at infinity” in the following way. Let d C1  i D nC1 j .fr i g  Z C / W Z C ! SrdiC1 be the .n C 1/-st outgoing strip-like end of SrdiC1 . Then there are  i ! C1, such that the rescaled maps uQ i .s; t/ D ui . i .s C  i ; t// converge to u2 on compact subsets. The statement makes sense because the uQ i are partial solutions of Floer’s equation (this follows from condition (8.7) imposed on any perturbation datum), which are defined on subsets that exhaust Z as i ! 1. • In the remaining stable situation 1 < m < d , r i converges to the point .r1 ; r2 / x d C1 . By definition of the topology, this means that there are convergent in @R

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10 Some basic properties

sequences rki ! rk and a sequence of gluing parameters i ! 0, such that r i D  T .i ; r1i ; r2i / for large i. Denoting the associated pointed discs by S1i D S diC2m ; r1

S2i D S mC1 i r2

it follows that S i is obtained by gluing together S1i , S2i as in (8.5), with the gluing length l i D  log.i /=. Restricting ui to the pieces of S i inherited from the gluing will yield sequences of partially defined maps on S1 ; S2 , which converge on compact subsets to u1 ; u2 . This makes sense because of the consistency in the choice of perturbation data, which requires the perturbation data used on S i and Ski to be related.

10 Some basic properties (10a) Well-definedness. Abstractly, we need to consider systems of mutually equivalent categories, indexed by some set I . Here are three versions of this notion (only the middle one is not obvious): • A strict system is a family of categories Ai , i 2 I , together with functors F i1 ;i0 W Ai0 ! Ai1 , such that F i i D IdAi and F i2 ;i1 ı F i1 ;i0 D F i2 ;i0 . • A coherent system consists of categories Ai and functors F i1 ;i0 , as well as functor isomorphisms T i2 ;i1 ;i0 W F i2 ;i1 ı F i1 ;i0 ! F i2 ;i0 , subject to the following two conditions: first, F i i D IdAi , and analogously T i2 ;i1 ;i0 is the identity whenever i2 D i1 or i1 D i0 ; secondly, for all .i3 ; i2 ; i1 ; i0 /, the following diagram commutes: R

F L

F i3 ;i2

i3 ;i2

ıF

i2 ;i1

ıF

i1 ;i0

F i1 ;i0

.T i3 ;i2 ;i1 /

.T i2 ;i1 ;i0 /

/ F i3 ;i1 ı F i1 ;i0



T i3 ;i1 ;i0

F i3 ;i2 ı F i2 ;i0

T i3 ;i2 ;i0

(10.1)

 / F i3 ;i0

(L and R are left and right composition, acting on natural transformations; these are the classical counterparts of the A1 -constructions discussed in Section (1e)). • A weak system is a family of categories Ai , any two of which are equivalent, without specifying a choice of equivalence. Suppose that we have a category Atot and a family of full subcategories Ai , such that each embedding H i W Ai ! Atot is an equivalence. Temporarily fix some i and

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choose, for each X 2 ObAtot , an isomorphic object K i .X/ 2 ObAi together with a preferred isomorphism S i 2 HomAtot .K i .X/; X/, making the obvious choices if X already belongs to Ai . It is straightforward to turn K i into a functor Atot ! Ai , which satisfies K i ı H i D IdAi . Then S i becomes a natural transformation H i ı K i ! IdAtot , with the properties that RH i .S i / and LK i .S i / are both trivial (identity) natural transformations. Having done this for each i, we can then turn the Ai into a coherent system of categories by setting F i1 ;i0 D K i1 ı H i0 ;

T i2 ;i1 ;i0 D LK i2 .RH i0 .S i1 //:

The notion of strict system carries over to A1 -categories without any changes, and so do weak systems, replacing equivalence by quasi-equivalence. For a coherent system, one requires A1 -functors F i1 ;i0 W Ai0 ! Ai1 such that F i i D IdAi , together with cohomology classes of natural transformations ŒT i2 ;i1 ;i0  2 HomH 0 . fun.Ai0 ;Ai2 // .F i2 ;i1 ı F i1 ;i0 ; F i2 ;i0 / which should again be the identity ŒEF i2 ;i0  whenever i2 D i1 or i1 D i0 , and such that the analogue of (10.1), with left and right composition functors as defined in Section (1e), is commutative in H 0 . fun.Ai0 ; Ai3 //. As before, a coherent system can be constructed from a fixed A1 -category Atot and a family of quasi-equivalent full A1 -subcategories Ai . Namely, denoting by H i the embedding, one first constructs a quasi-equivalence K i W Atot ! Ai with K i ı H i D IdAi , as in the discussion preceding Theorem 2.9. We know from Lemma 1.7 that the map H.LK i / W HomH. fun.Atot ;Atot // .H i ı K i ; IdAtot / ! HomH. fun.Atot ;Ai // .K i ; K i / (10.2) is an isomorphism. Choose an ŒS i  which, under (10.2), is mapped to the identity natural transformation ŒEK i . By first looking at LK i ı RH i D RH i ı LK i and then applying Lemma 1.7 once more, it follows that H.RH i /.ŒS i / is ŒEH i . The rest of the construction is the same as for ordinary categories. Turning to Floer theory, recall that if we are given an exact symplectic manifold with corners M , the definition of F .M /pr involves additional choices, as summarized in Section (9j). Let I be the set of all possible such choices, and denote by fF .M /pr;i g the resulting family of A1 -categories. If we pass to the cohomological level, then the H F .M /pr;i form a strict system in the sense defined above. This is a consequence of the well-definedness of Floer cohomology and its TQFT structure, which we briefly discussed in Section (8j). To deal with F .M /pr itself, we find it convenient to apply the formal trick described above. Namely, define a total Fukaya category F .M /pr;tot whose objects are pairs .L; i/ consisting of an exact Lagrangian submanifold L and an index i 2 I . The morphism spaces are Floer complexes as usual, where one chooses a regular Floer datum separately for each pair ..L0 ; i0 /; .L1 ; i1 //, subject to the condition that whenever i0 D i1 D i, then the datum is the same as for .L0 ; L1 /

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10 Some basic properties

as objects of F .M /pr;i . Similarly, the compositions of order d  2 are defined by suitable choices of strip-like ends and perturbation data, such that if all the objects involved have the same index i, then the choice is the same as for F .M /pr;i . This involves a slight modification of the standard setup, since we allow the strip-like ends on a .d C 1/-punctured disc to depend on the labels .Lk ; ik /. However, it is straightforward to adapt the consistency condition from Section (9g) appropriately, using gluing of compatibly labeled families as in Section (9i). The outcome is that our total category comes with canonical full and faithful embeddings F .M /pr;i ! F .M /pr;tot :

(10.3)

For any fixed exact Lagrangian submanifold L and choice of indices .i0 ; i1 /, the objects .L; i0 / and .L; i1 / are isomorphic in H F .M /pr;tot . This is a cohomology level statement, whose proof comes down to the standard continuation map argument. As a consequence, the embeddings (10.3) are quasi-equivalences. One can then use this to turn fF .M /pr;i g into a coherent system of A1 -categories (strictly speaking, ungraded A1 -categories; however, Remark 9.7 applies here and to similar situations later on this section). Moreover, this statement carries over to any algebraic construction one can do with the Fukaya category, as long as it is functorial and preserves quasiequivalences (taking the derived category comes to mind). The remaining question is to what extent the functors and natural transformations which make up the coherent system are themselves canonical; we leave that to the interested reader. Remark 10.1. In the framework of ordinary categories, there is a converse construction which starts with a coherent system of categories Ai and constructs a common category Atot with equivalences Ai ! Atot . This apparently fails in the A1 -context, the reason being that our definition of coherent system is inherently too weak, since it remains at the level of cohomological functor categories H. fun/. (10b) Autoequivalences. This topic is closely related to the previous one (in fact, one could give a unified treatment in terms of groupoids). We need to discuss what it means for a group G to act on a category A: • A strict G-action is a family of functors F g W A ! A, g 2 G, satisfying F e D IdA , F g2 g1 D F g2 ı F g1 . • A coherent G-action consists of functors F g together with isomorphisms T g2 ;g1 W F g2 ı F g1 ! F g2 g1 , such that F e D IdA , T g2 ;g1 is the identity whenever g2 D e or g1 D e, and with commutative diagrams F g3 ı F g2 ı F g1

RF g1 .T g3 ;g2 /

/ F g3 g 2 ı F g 1

LF g3 .T g2 ;g1 /



T g3 g2 ;g1

F g 3 ı F g 2 g1

T g3 ;g2 g1

 / F g3 g2 g1 :

(10.4)

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II Fukaya categories

• A weak G-action is a family of functors F g , such that F e is isomorphic to IdA and F g2 g1 to F g2 ı F g1 , without specifying any choice of isomorphisms. Suppose that we have a category Afree with a strict G-action fˆg g, and a full subcategory A such that the embedding H W A ! Afree is an equivalence. Choose K W Afree ! A and a functor isomorphism S W H ı K ! IdAfree as in Section (10a). One then gets an induced coherent G-action on A: F g D K ı ˆg ı H;

T g2 ;g1 D LKıˆg2 .Rˆg1 ıH .S//:

(10.5)

It should now be obvious how to adapt these definitions to the A1 -framework (note that in this case, there is another possibility which is even more basic than strict actions, namely naive group actions as in Section 6; however, those will not play any role in the present discussion). To show that the analogue of (10.5) in the A1 context satisfies the coherence relations, one argues as follows. Both sides of the 1 diagram corresponding to (10.4) have the form L1Kıˆg3 .Rˆ g2 ıˆg1 ıH .something//, the expression in brackets being 2Q .S; L1H ıK .L1ˆg2 .R1 g1 .S//// and ˆ

2

2Q .L1ˆg2 .R1 g1 .S//; R1 ˆ

g 1 ˆg2 ıH ıKıˆ 2

2

(10.6)

.S//

respectively, where Q D fun.Afree ; Afree /. Applying Lemma 1.4 with G1 D H ı K;

1

G2 D ˆg2 ı H ı K ı ˆg2 ;

T1 D S;

T2 D L1ˆg2 .R1 g1 .S// ˆ

2

shows that on the level of H.Q/, the two terms in (10.6) coincide. Remark 10.2. Correspondingly to what we said in Remark 10.1, there is a converse construction in the framework of ordinary categories, which turns a coherent group action on A into a strict one on an equivalent category Astrict (see Section (14b) for the special case G D Z=2); and again, this apparently fails in the A1 -context, at least with the definition of coherence given above. Fix an exact symplectic manifold with corners M , and let G D Aut.M; @M / be its automorphism group rel boundary. Any  2 G induces a canonical automorphism of the entire Floer TQFT, hence in particular of H F .M /pr , and these form a strict G-action. On the chain level, we proceed as follows. Define a Fukaya-type (ungraded) A1 -category F .M /pr;free whose objects are pairs .L; / consisting of an exact Lagrangian submanifold L  M together with a  2 G. Morphism spaces CF pr ..L0 ; 0 /; .L1 ; 1 // and the composition maps between them are defined by choosing Floer and perturbation data for each finite collection of objects, and we ask

10 Some basic properties

137

that these choices be equivariant with respect to the G-action .L; / D ..L/; ı/. This poses no problems because the action is free on objects, and the outcome is that F .M /pr;free carries a strict G-action, which consists of A1 -functors  having vanishing higher order (d > 1) terms. One can clearly embed F .M /pr into F .M /pr;free as a full subcategory, and the embedding is a quasi-equivalence since any two .L; 0 /, .L; 1 / are isomorphic on the cohomological level. Hence we get an induced coherent G-action on F .M /pr . (10c) Hamiltonian isotopies. Our next topic is a higher-order generalization of continuation maps, in the sense of Section (8k). We will give only a brief sketch, omitting technical aspects. We will work in the A1 -category F .M /pr;free introduced above. Let . s /, 0  s  1, be an isotopy in G D Aut.M; @M /, going from  0 D Id to  1 D . One can associate to each .L; / a cochain T 0 2 CF pr ..L; /; .L; // by following the construction of (8.23), which means taking the upper half-plane H with a moving Lagrangian boundary condition given by the isotopy  s .L/, and equipping it with a perturbation datum which is compatible with the given Floer datum. Note that the choice of perturbation datum necessarily breaks the holomorphic automorphism group of H . It may therefore be more natural to think of H as a surface with an additional marked point in its interior, say z D i, which stabilizes it. Following that indication, we consider more generally a .d C 1/-pointed disc S with an additional interior marked point z and a set of strip-like ends, together with a collection of objects .Lk ; k /, k D 0; : : : ; d . Choose 0 D s0  s1      sd C1 D 1, and equip S with moving boundary conditions which on each component Ck , 0  k  d , go from  sk .Lk / to  skC1 .Lk /. Having done that, one chooses a perturbation datum on S whose behaviour over the incoming and outgoing strip-like ends is determined by the Floer data for the pairs ..L0 ; 0 /; .Ld ; d // respectively . sk .Lk1 ; k1 /;  sk .Lk ; k //. Note that by definition of F .M /pr;free , the latter are induced from the Floer data for ..Lk1 ; k1 /; .Lk ; k // via  sk . We want all this to depend smoothly on the complex structure and choice of marked point, which means that rather than looking at a single .S; z/, one considers the moduli space Rd C1;1 of all .d C 1/-pointed discs with one additional interior marked point, and the universal family over it. This space has a Deligne–Mumford type compactification  a  Y x d C1;1 D R Rjvj Rjv j;1  (10.7) .T;v /

v¤v

whose strata are indexed by d -leafed trees with one preferred vertex v (this may be of any positive valency, while the other vertices should be at least three-valent).

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II Fukaya categories

The choices of strip-like ends, of the fsk g, and of the perturbation data must satisfy a suitable consistency condition with respect to (10.7), which we will not formulate explicitly. One can then look at the associated moduli space of pairs .r 2 Rd C1;1 ; u W Srd C1;1 ! M /, where u is an inhomogeneous pseudo-holomorphic map. Counting points with suitable asymptotic behaviour in that space defines maps T d W CF pr ..Ld 1 ; d 1 /; .Ld ; d // ˝       ˝ CF pr ..L0 ; 0 /; .L1 ; 1 // ! CF pr ..L0 ; 0 /; .Ld ; d //: Here, we have again used the fact that CF pr . sk .Lk1 ; k1 /;  sk .Lk ; k // can be canonically identified with CF pr ..Lk1 ; k1 /; .Lk ; k //. The fT d g satisfy equations X  d mC1 1 .ad /; : : : ; 1 .anCmC1 /; T m .anCm ; : : : ; anC1 /; an ; : : : ; a1 / X m;n C T d mC1 .ad ; : : : ; anCmC1 ; m .anCm ; : : : ; anC1 /; an : : : ; a1 / D 0 m;n

(10.8) (Figure 10.1 shows the degenerations corresponding to the two types of summands, emphasizing the boundary conditions, both moving and non-moving). Since the A1 functor  has zero higher order terms, and we do not currently have to worry about signs, (10.8) is just the condition for T to be a natural transformation from the identity to  . As already observed at the end of Section (8k), each T 0 is an isomorphism in H F .M /pr;free , hence by Lemma 1.6, T is an isomorphism in the cohomological category of A1 -functors. Transferring that result to the quasi-equivalent A1 -category F .M /pr , we have: Proposition 10.3. Let  2 Aut.M; @M / be an automorphism which is isotopic to the identity. Then the A1 -functor induced by  is isomorphic to the identity functor in H. fun.F .M /pr ; F .M /pr //.  In other words, the coherent action of Aut.M; @M / on F .M /pr descends to a weak action of 0 .Aut.M; @M //. Remark 10.4. Take the general setting of a weak action of a group  on a category A. Let HH 0 .A; A/ be the set of functor isomorphisms from IdA to itself, turned into an abelian group by composition. The obstruction to making the action coherent is a group cohomology class, more precisely an element of H 2 .I HH 0 .A; A/ /. Now suppose that  D 0 .G/, G D Aut.M; @M /, and for simplicity take A D H F .M /pr to be the Donaldson–Fukaya category. In that case, the obstruction class is inherited from a canonical class (which exists for any topological group) in H 2 .0 .G/I 1 .G//, through a homomorphism 1 .G/ ! HH 0 .A; A/ which is defined using the methods of [123], [1]. There is presumably a parallel statement on the level of A1 -categories,

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10 Some basic properties

 s .L3 / (s2  s  1)

 1 .L4 /

 s .L2 / (s1  s  s2 )

 s .L1 / (0  s  s1 )

 1 .L3 /

L1 L0  s .L4 / (s3  s  1)

 s .L3 / (s2  s  s3 )

 s2 .L3 /  s2 .L2 /

 s .L0 / (0  s  s1 )

 s .L1 / (s1  s  s2 )

 s2 .L1 /

Figure 10.1

but we have not explored the details. In any case, even that is still only part of the story. Just as in Remark 10.1, our definition of coherent group action on an A1 -category should be considered as a first approximation; one would expect the definitive concept to involve the whole homotopy type of G, rather than just the homotopy groups of degree  1. (10d) The contact type boundary case. Continuing from Section (7b), we discuss the situation for exact symplectic manifolds with contact type boundary. Given such a manifold .M; ø; /, one can change the definition of J to be the space of all compatible almost complex structures which are of contact type near the boundary. This means that in the resulting notions of Floer datum and perturbation datum, the family of almost complex structures J.t / or J.z/ is no longer constant near @M . Due to Lemma 7.3 and the fact that the inhomogeneous terms vanish near the boundary, solutions of the equations (8.8), (8.9) will still remain in the interior of M . Defining F .M /pr in this way, one finds that it is essentially independent of any choice of almost complex structure (in the sense that different choices fit into a coherent system, of course).

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Proposition 10.5. Let M be an exact symplectic manifold with contact type boundary, N an exact symplectic manifold with corners, and  W M ,! int.N / an embedding, such that  !N D !M ,  N D M Cd (function). Then F .M /pr is quasi-isomorphic to the full A1 -subcategory of F .N /pr consisting of those exact Lagrangian submanifolds which lie inside .int.M //. Proof. Without affecting F .N /pr at all, one may change N by the derivative of a function vanishing near @N . In this way, one reduces the situation to the case when  N D M . Having done that, we think of  simply as an inclusion. Fix an almost complex structure J on N which agrees with IN near @N , and such that J jM is of contact type near @M . When defining the subcategory of F .N /pr under discussion, one can choose all the almost complex structures to be equal to J in a neighbourhood of N nint.M /; and the Hamiltonian functions and perturbation data, so that they vanish in a neighbourhood of N n int.M /. As a result, Lemma 7.5 applies, showing that all inhomogeneous pseudo-holomorphic polygons remain inside int.M /. In this case, the two categories are actually equal. The quasi-isomorphism for general choices of data then follows from the previous considerations about well-definedness.  Corollary 10.6. Let M , N be exact symplectic manifolds with contact type boundary, y ; Ny the noncompact manifolds obtained by attaching cones as in (7.1). Suppose and M y ! Ny . Then the categories that there is an exact symplectic isomorphism O W M pr pr F .M / , F .N / are quasi-equivalent. Proof. Consider finite parts of the completions, y  D M [@M .Œ0I    @M /; M

Ny  D N [@N .Œ0I   @N /;

y  /  Ny  . First of all, the O M with 0     chosen in such a way that N  .  y (and similarly, that of N is Fukaya category of M is quasi-equivalent to that of M quasi-equivalent to Ny  ). This is easy to see, since the two manifolds can be identified with each other, up to a conformal rescaling of the symplectic form and its one-form primitive, by using the negative Liouville flow. If one then rescales the Floer data and perturbation data accordingly, the two Fukaya categories will actually be the same. y  / is quasi-isomorphic to Now apply Proposition 10.5, which says that F .M  a certain full subcategory of F .Ny /. This subcategory certainly contains all exact Lagrangian submanifolds which lie inside N itself, but up to isotopy this is everything, hence we get a quasi-equivalence as desired.  In view of Lemma 7.2, it follows that: Corollary 10.7. F .M /pr is invariant under deformations of M in the class of exact symplectic manifolds with contact type boundary. 

10 Some basic properties

141

y /pr , Remark 10.8. As a minor variation, one could directly define categories F .M using the natural class of almost complex structures JO on the completion. Lemma 7.4 y /pr . One advantage of this would then show that F .M /pr is quasi-equivalent to F .M y / of exact symplectic approach is that it yields a coherent action of the group Aut.M automorphisms which are of contact type at infinity, on F .M /pr . Corollary 10.9. Let N be an exact symplectic manifold with corners. Round off the corners as in Lemma 7.6, with sufficiently small ı, to get an exact symplectic manifold with contact type boundary M  int.N / . Then F .M /pr is quasi-equivalent to F .N /pr . This follows immediately from Proposition 10.5 and the fact that every exact Lagrangian submanifold can be moved into the inside of M . Corollary 10.10. Up to quasi-equivalence, the Fukaya category of an exact symplectic manifold with corners is invariant under deformations. s s Proof. Take N and a deformation .!N ; N ; INs /. By inspecting the proof of Lemma 7.6, one sees that there is a M  int.N / such that for all s, the restriction of s s ; N / to M yields the structure of an exact symplectic manifold with contact type .!N boundary. Corollary 10.7 shows that F .M /pr is independent of s, and Corollary 10.9 allows one to carry over the result to F .N /pr . 

Corollary 10.11. Let  W N ! Nz be an isomorphism of exact symplectic manifolds with corners. Then there is a quasi-equivalence from F .N /pr to F .Nz /pr , which maps each exact Lagrangian submanifold L to .L/. Before giving the proof, we should clear up a possible source of confusion. If the isomorphism is compatible with the almost complex structures, in the sense that  .IN / D INz near the boundary, the formal arguments in Section (10a) are sufficient to prove the result (in fact, they give a somewhat sharper statement, since the resulting quasi-equivalence is automatically canonical up to isomorphism in the relevant functor category). The difficulty in the general case arises precisely from the fact that our notion of isomorphism, as defined in Section (7a), ignores almost complex structures. Proof. Round off the corners to get an exact symplectic manifold with contact type boundary, M  int.N /. The restriction jM is clearly an exact symplectic embedding. By inverting the quasi-equivalence from Corollary 10.9, and then applying Proposition 10.5, one gets an A1 -functor F .N /pr ! F .M /pr ! F .Nz /pr

(10.9)

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II Fukaya categories

which is cohomologically full and faithful. By choosing ı small, one can arrange that any exact Lagrangian submanifold of Nz can be moved inside .int.M // by the negative Liouville flow, and then (10.9) is a quasi-equivalence. As defined, this maps Q where L Q is isotopic to L. However, since .L/ and .L/ Q are isomorphic L to .L/, pr z objects of in H F .N / , a purely algebraic argument allows one to modify the functor so that L really goes to .L/.  (10e) The homotopy method. There is another way of proving well-definedness theorems in Floer theory, which is quite direct (it historically precedes continuation maps) but can be technically tricky to implement. We give an exposition of this method because of its intrinsic importance, even though it is not strictly necessary for our immediate purposes. Suppose that for some exact symplectic manifold with corners M , we are given two versions of the Fukaya category F .M /pr;s , s D 0; 1. Both should be based on the same universal choice of strip-like ends, so that the difference arises from the Floer data and perturbation data involved. We further assume that for each pair .L0 ; L1 / there exists a smooth homotopy of Floer data .HLs 0 ;L1 ; JLs 0 ;L1 /;

0s1

(10.10)

interpolating between those used for F .M /pr;0 and F .M /pr;1 . Note that while the first assumption is merely a technicality, the second one is quite restrictive since it prohibits birth-death phenomena. To explain this, let us fix .L0 ; L1 / and omit the subscripts from now on. Let X s be the Hamiltonian vector field associated to H s . This still depends on another time variable t 2 Œ0I 1. We denote by  s;t the family of Hamiltonian diffeomorphisms obtained by integrating in the latter direction, meaning that  s;0 D IdM and @ s;t =@t D X s;t . s;t /. Since all the (10.10) are supposed to be Floer data, the intersections  s;1 .L0 / \ L1 are transverse, and therefore the intersection points vary smoothly with s. To put it differently, let CF pr .L0 ; L1 /s be the Floer cochain group defined using (10.10) for some s 2 Œ0I 1, and C.L0 ; L1 /s its standard set of generators, consisting of maps y s W Œ0I 1 ! M such that y s .0/ 2 L0 , y s .1/ 2 L1 , and dy s =dt D X s;t .y s .t//. Similarly, let C.L0 ; L1 /Œ0I1 be the set of smooth maps y W Œ0I 12 ! M such that each restriction y s D y.s;  / lies in C .L0 ; L1 /s , and CF pr .L0 ; L1 /Œ0I1 the K-vector space generated by that set. Then the restriction C.L0 ; L1 /Œ0I1 ! C .L0 ; L1 /s to any s is a bijection, so that we have induced isomorphisms CF pr .L0 ; L1 /Œ0I1 Š CF pr .L0 ; L1 /s :

(10.11)

It would be naive to expect these by themselves to give rise to an A1 -functor. Instead, one needs to consider suitable moduli spaces of inhomogeneous pseudo-holomorphic maps with an added s-parameter.

143

10 Some basic properties

Let us begin with the first order (cochain map) construction, which is quite transparent and serves as a model for the general case. Given y0 ; y1 2 C.L0 ; L1 /Œ0I1 ,   write MZ .y0 ; y1 /s D MZ .y0s ; y1s / for the moduli space of non-stationary solutions of Floer’s equation with respect to the datum .H s ; J s /, having limits y0s ; y1s , divided  by translation as usual. Similarly, write MZ .y0 ; y1 /Œ0I1 for the space of pairs .s; u/,  s where s 2 Œ0I 1 and u 2 MZ .y0 ; y1 / . For each   1, we now introduce a related moduli space H  .y0 ; y1 /Œ0I1 whose points are -tuples ..s1 ; u1 /; : : : ; .s ; u // where  .yQii ; yQi /s for some y0 D yQ0 ; yQ1 ; : : : ; yQ D y1 . s1      s , and each u 2 MZ  One can picture elements of H .y0 ; y1 / abstractly as chains of 2-pointed discs, where each disc has a variable sk associated to it, see Figure 10.2. For ease of writing, we formally define H 0 .y0 ; y1 /Œ0I1 to be a point if y0 D y1 , and empty otherwise.

1

2 s1  s2

 s2      s

Figure 10.2

One might think of our construction as a parametrized moduli problem for the trivial family Œ0I 1Z ! Œ0I 1, except that unlike the setup in Section (9h), the Floer data associated to the ends vary with the parameter s. Nevertheless, the basic structure  remains the same, in particular one can associate to each .s; u/ 2 MZ .y0 ; y1 /Œ0I1 an extended linearized operator, which as in (9.18) takes the additional variable into account. Regularity is then defined as surjectivity of this operator, and implies local smoothness of the moduli space. This will hold everywhere if the homotopy (10.10) is chosen generically, which we assume to be the case from now on. To see that the spaces H  .y0 ; y1 /Œ0I1 are well-behaved, one additionally needs to prove that their boundary faces, namely the submanifolds of products  Y

 MZ .yQi1 ; yQi /Œ0I1

iD1

cut out by one or more equations of the form si D siC1 , are themselves regular. Using a slightly more sophisticated version of the standard transversality argument, one shows that this can be achieved provided that the .si ; ui / are pairwise different. Fortunately, that condition is automatically satisfied in the present context, because each ui strictly decreases the H si -perturbed action functional. An energy argument shows that H  .y0 ; y1 /Œ0I1 is empty for sufficiently large . Moreover, for a fixed , a version of Gromov compactness with added s-parameter shows that there are only finitely many isolated points in H  .y0 ; y1 /Œ0I1 . We can

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II Fukaya categories

therefore define a map  W CF pr .L0 ; L1 /1 ! CF pr .L0 ; L1 /0 by X #H  .y0 ; y1 /Œ0I1 y00 : .y11 / D

(10.12)

0

Lemma 10.12.  is an isomorphism of (ungraded) chain complexes. The idea of the proof is straightforward. Write H  .y0 ; y1 /Œ0I1;1 for the union of one-dimensional connected components of H  .y0 ; y1 /Œ0I1 . This is a one-manifold with boundary, the boundary points being those where (i) s1 D 0, (ii) s D 1, or (iii) si 1 D si for some i. Passing to the Gromov compactification Hx  .y0 ; y1 /Œ0I1;1 adds more boundary points, corresponding to the breakup of one of the ui into a pair   of Floer trajectories, which is an isolated point of MZ .yQi1 ; y/ Q si  M Z .y; Q yQi /si for some y. Q In an algebraic count of boundary points, these precisely cancel out the contributions from case (iii) above, leaving an equality XX XX   #MZ .y0 ; y/ Q 0 #H  .y; Q y1 /Œ0I1 C #H  .y0 ; y/ Q Œ0I1 #MZ .y; Q y1 /1 D 0 0 yQ

0 yQ

which just says that  is a chain map. To see that it is an isomorphism, one argues  roughly as follows. If the moduli spaces MZ .y0 ; y1 /Œ0I1 have no isolated points,  D Id by definition. In the next simplest case, where isolated points appear exactly for one value of the parameter s, we have  D Id C (lower-triangular), where the second summand is strictly decreasing for the filtration by values of the AH s -action. Finally, one uses the fact that (10.12) is well-behaved with respect to composition of homotopies to reduce the general case to the ones we have discussed. For a complete argument along these lines, see the references listed in Remark 10.14 below. The analogous construction for pseudo-holomorphic .d C 1/-gons, d  2, is technically a little more complicated. For each .d C 1/-tuple of exact Lagrangian submanifolds .L0 ; : : : ; Ld / choose L0 ;:::;Ld ;k ; 1  k  d;

and

.KLs 0 ;:::;Ld ; JLs 0 ;:::;Ld /; 0  s  1;

(10.13)

with the following properties (while discussing these, we will omit the subscripts L0 ; : : : ; Ld ). Each k is a smooth function Rd C1  Œ0I 1 ! Œ0I 1, with the property that k .r; s/ D s for s D 0; 1. We call these delay functions. The .K s ; J s / are a homotopy of perturbation data on the universal family S d C1 , which at the endpoints s D 0; 1 should reduce to the data used for F .M /pr;s , and which are compatible with the Floer data (10.10) up to a change of s-variable determined by our delay functions. Explicitly, on the image of each strip-like end kd C1 W Rd C1  Z ˙ ! S d C1 one wants to have .kd C1 / .K s ; J s / ( (10.14) for k D 0; .HLs 0 ;Ld .t/dt; JLs 0 ;Ld .t// D .HLsQ k1 ;Lk .t/dt; JLsQ k1 ;Lk .t// for k > 0; where sQ D k .r; s/:

10 Some basic properties

145

There is a little clash of notations here: t is the second coordinate on the half-infinite strips Z ˙ ; however, s is not the first coordinate (which plays no role throughout the following argument), but rather the variable governing the homotopy. The fact that the Floer data (10.14) at incoming and outgoing ends belong to generally different values sQ ¤ s affects the way in which one glues the surfaces together. We will now indicate how to adapt the notion of consistent choice of perturbation data, originally introduced in Section (9i), to this situation. Let T be a stable d -leafed tree. For each vertex v 2 Ve.T / fix some jvj-pointed disc Sv , and let rv 2 Rjvj be the corresponding modulus. Equip these discs with mutually compatible Lagrangian labels. Fix also s 2 Œ0I 1. One can then associate to each vertex a parameter value sv 2 Œ0I 1, by the following recursive rule: first, if v is the vertex nearest the root, set sv D s. Second, let e be any interior edge, connecting the vertex v  .e/ to v C .e/ (we recall from Section (9d) that edges are oriented away from the root), and suppose that the associated flag f  .e/ D .v  .e/; e/ is the ke -th positive flag adjacent to v  .e/ with respect to the canonical ordering. Then svC .e/ D ke .rv .e/ ; sv .e/ /;

(10.15)

where ke is one of the delay functions associated to the jv  .e/j-tuple of Lagrangian submanifolds which are labels of Sv .e/ . Now glue together the Sv with some gluing parameters fe g, and let r 2 Rd C1 be the moduli point which classifies the resulting .d C1/-pointed disc S. Recall that the outgoing points at infinity of S can be identified with the leaves of T . If the flags associated to the leaves are fk1 .v1 /; : : : ; fkd .vd /, then we equip those points at infinity with the numbers s 1 D k1 .rv1 ; sv1 /; : : : ; s d D kd .rvd ; svd / 2 Œ0I 1. The first part of the consistency condition, which concerns only the delay functions, says that for small values of the gluing parameters, these numbers should match those naturally associated to S , namely 1 .r; s/; : : : ; d .r; s/. We have suppressed the Lagrangian labels on both sides of the equality, but the choices are the natural ones: each Svk carries Lagrangian labels because we have chosen compatible ones for all the vertices, and S inherits labels from these through the gluing process. Next, equip each Sv with the perturbation datum (10.13) for the time sv . Conditions (10.14) and (10.15) together ensure that these data have the same behaviour over corresponding strip-like ends, so that they can be glued together, yielding a perturbation datum on S. On the other hand, we could equip the same disc with the perturbation datum pulled back from (10.13) by the classifying isomorphism S Š Srd C1 . The consistency condition requires the same relation between these two data as in Section (9i): equality on the thin parts of S for small values of the gluing parameter, and asymptotic agreement (defined in terms of the partial compactification of the universal family) on the thick parts. After having made a choice of (10.13) which is consistent in that sense, one can introduce the associated moduli spaces. For given .L0 ; : : : ; Ld / and y0 2 C.L0 ; Ld /Œ0I1 , yk 2 C .Lk1 ; Lk /Œ0I1 , let M d C1 .y0 ; : : : ; yd /Œ0I1 be the space

146

II Fukaya categories

of triples .r; s; u/, where r 2 Rd C1 , s 2 Œ0I 1, and u W Srd C1 ! M is a solution of the inhomogeneous pseudo-holomorphic map equation with perturbation datum (10.13) on S, whose limits are: y0s over the incoming strip-like end, and yksQ , sQ D L0 ;:::;Ld ;k .r; s/, over the outgoing ends. In parallel with the previously considered H  .y0 ; : : : ; yd /Œ0I1 , there is also another family of closely related moduli spaces H T .y0 ; : : : ; yd /Œ0I1 . Their definition partially resembles the compactifications discussed in Section (9l), and we will borrow some notation from there. Fix a semistable d -leafed tree T . A point of H T .y0 ; : : : ; yd /Œ0I1 consists of the following data: • For each vertex with jvj  3, an rv 2 Rjvj and hence a jvj-pointed disc Sv . For the remaining vertices, we set Sv D Z. There is a unique way of equipping the Sv with compatible Lagrangian labels, such that the induced labels on the glued .d C 1/-pointed disc are the given Lk . Having done that, the next piece of data is • For each v a number sv 2 Œ0I 1, satisfying an inequality version of (10.15). Namely, for each interior edge e, ( k .rv .e/ ; sv .e/ / if jv  .e/j  3; svC .e/  (10.16) if jv  .e/j D 2: sv .e/ Next, recall that in this situation, there is a pair .Lf;0 ; Lf;1 / of Lagrangian submanifolds associated to a flag f . • For each f , we want to specify a yQf 2 C .Lf;0 ; Lf;1 /Œ0I1 , subject to the following conditions. If f is the unique flag associated to the root of T , then yQf D y0 . Similarly, for the d flags associated to the leaves, the yQf will be y1 ; : : : ; yd . And if f ˙ .e/ are the two flags belonging to an interior edge, then yQf  .e/ D yQf C .e/ . • Finally, for each v a map uv W Sv ! M , which is a non-stationary solution of (8.8) if jvj D 2, and a solution of (8.9) if jvj  3. The relevant Floer data and perturbation data are given by (10.10) and (10.13), respectively, with s D sv ; and the limiting values over the strip-like ends should be the yQf associated to the flags f around v. Equivalently, (  .sv ; uv / 2 MZ .yQf0 .v/ ; yQf1 .v/ /Œ0I1 for jvj D 2; .rv ; sv ; uv / 2 Md C1 .yQf0 .v/ ; : : : ; yQfd .v/ /Œ0I1 for jvj  3: Figure 10.3 shows a schematic picture of the surfaces Sv with their Lagrangian labels, and resulting inequalities between the sv . As usual, collections fuv W Sv ! M g which differ only by a translation of each unstable component Sv Š Z will be considered

147

10 Some basic properties

L6 v4

sv4  L3 ;L5 ;L6 ;2 .rv2 ; sv2 /

L6 sv2  L0 ;L3 ;L6 ;2 .rv1 ; sv1 /

v2

L5

L5 sv5  L3 ;L5 ;L6 ;1 .rv2 ; sv2 / L5

L3

L6

L3

v5

L4

L3 v1

sv3  L0 ;L3 ;L6 ;1 .rv1 ; sv1 / L3

L0 v3

sv6  sv3

L0

L3 L0

v6

L2

L1 Figure 10.3

the same. Supposing that the choices made in (10.13) are suitably generic, one defines in analogy with (10.12) multilinear maps G d W CF pr .Ld 1 ; Ld /1 ˝    ˝ CF pr .L0 ; L1 /1 ! CF pr .L0 ; Ld /0 ; X G d .yd1 ; : : : ; y11 / D #H T .y0 ; : : : ; yd /Œ0I1 y00 :

(10.17)

T

Proposition 10.13. The map G 1 D  from (10.12) and the higher order terms G d from (10.17) constitute an (ungraded) A1 -functor G W F .M /pr;1 ! F .M /pr;0 : Note that by Lemma 10.12, the linear parts G 1 are isomorphisms (if one identifies the Floer cochain spaces CF pr .L0 ; L1 /s for s D 0; 1 via (10.11), then G is a formal

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II Fukaya categories

diffeomorphism in the sense of Section (1c)). In particular, F .M /pr;0 and F .M /pr;1 are quasi-isomorphic A1 -categories. The proof strategy for Proposition 10.13 is basically the same as for Lemma 10.12. The boundary points of the one-dimensional moduli spaces H T .y0 ; : : : ; yd /Œ0I1;1 occur where either (i) for the vertex v nearest to the root, sv D 0; (ii) for one of the vertices v nearest to the leaves, sv D 1; or (iii) equality holds in one of the conditions (10.16). In the first two cases, one divides the tree into pieces as shown in Figure 10.4, and concludes that the relevant algebraic contributions are precisely the two terms in the A1 -functor equation (1.6). The contribution from (iii) again cancels out against the additional points introduced by the Gromov compactification. Of course, this

sv1 D 0

sv5 D 1

(i)

(ii) Figure 10.4

formal outline does not constitute a complete proof. However, most of the analysis is not really more sophisticated than the one used to define the Fukaya category. We will therefore limit ourselves to discussing one particular aspect of the transversality issue, namely the one which motivates the introduction of delay functions. To bring out that point more clearly, we temporarily restrict ourselves to trivial delay functions, k .r; s/ D s. To start with, one can choose the perturbation data in (10.13) generically, and then all the spaces M d C1 .y0 ; : : : ; yd /Œ0I1 are regular. The remaining problem is to ensure that the boundary faces of H T .y0 ; : : : ; yd /Œ0I1 , which are subsets of the products Y M jvj .yQf 0 .v/ ; : : : ; yQf jvj1 .v/ /Œ0I1 (10.18) v2Ve.T /

cut out by equalities svC .e/ D sv .e/ ;

(10.19)

are still regular. This is made somewhat easier by the fact that only moduli spaces of dimension  1 need to be considered. For instance, in the zero-dimensional situation,

11 Indices and determinant lines

149

the desired statement is that if fuv W Sv ! M g 2 H T .y0 ; : : : ; yd /Œ0I1 is such that each .rv ; sv ; uv / is an isolated point of the relevant factor in (10.18), then (10.19) never holds for any edge e. Standard transversality arguments show that this can be achieved by a generic choice of perturbation data, except if the two maps associated to the relevant vertices coincide: Sv .e/ Š SvC .e/

and

uv .e/ D uvC .e/ :

(10.20)

Unfortunately, unlike at the comparable point in our previous discussion of the map , this remaining case cannot possibly be excluded, because there are regular pseudoholomorphic .d C 1/-gons which have the same limit on the incoming end as on one (or several) of the outgoing ones. Now allow nontrivial delay functions, where instead of (10.19) one considers having some of the inequalities (10.16) become equalities. Generically the following will be true: if .r; s; u/ is an isolated point of some moduli space Md C1 .y0 ; : : : ; yd /Œ0I1 , d  2, then the delay functions satisfy k .r; s/ ¤ s for all k D 1; : : : ; d . This condition obviously excludes pathological behaviour like (10.20), hence allows one to complete the transversality argument. We omit the parallel considerations for one-dimensional moduli spaces, which are only slightly more involved. Remark 10.14. The homotopy method for Floer homology already appears in the original paper of Floer [44]. The extension to Fukaya categories follows Fukaya– Oh–Ohta–Ono [60, Theorem 15.19], except that we have replaced virtual perturbation theory by a more explicit transversality argument. As an aside, we point out that when one is dealing with Floer homology in the non-exact situation, the transversality issue becomes more serious, and only indirect solutions have been found (see [94], or [78] for the model case of Morse–Novikov theory).

11 Indices and determinant lines (11a) Determinant lines. The discussion in this section will be largely centered around determinant lines, which are one-dimensional real vector spaces associated to Fredholm operators. Actually, rather than such a line itself, only the set of its two possible orientations will be of interest to us. In particular, whenever we say that an isomorphism between two determinant lines (or analogous objects to be introduced later on) is canonical, this is always meant up to multiplication with positive real numbers. With that verbal precaution taken, let D be a Fredholm operator between two real Banach spaces. The determinant line of D is defined by tensoring together the top exterior powers of the kernel and cokernel, more precisely det.D/ D top .coker.D/_ / ˝ top .ker.D//I

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II Fukaya categories

since we are only interested in orientations, dualizing the cokernel is redundant, but we still prefer to keep it that way, so that the formulae remain comparable to those for complex determinant lines. There is a canonical isomorphism det.D1 / ˝ det.D2 / Š det.D1 ˚ D2 /;

(11.1)

defined by thinking of all vectors as odd and using the Koszul sign rule. Explicitly, if _ fvk;i g are bases for ker.Dk /, and fwk;i g bases for coker.Dk /_ , the isomorphism is  V _  V _ V V i w1;i ˝ i v1;i ˝ i w2;i ˝ i v2;i (11.2) V _ V V V _ 7! .1/dim.coker D2 /index.D1 / i w2;i ^ i w1;i ˝ i v1;i ^ i v2;i : This is strictly associative, and also commutative in a graded sense, meaning that the composition of the two isomorphisms, det.D1 / ˝ det.D2 / Š det.D1 ˚ D2 / D det.D2 ˚ D1 / Š det.D2 / ˝ det.D1 /, differs from the naive exchange of factors by a sign .1/index.D1 /index.D2 / . For a moment, let us consider linear maps D W H ! K between finite-dimensional vector spaces. There is a unique way of choosing identifications tD W det.D/ ! det.0/ D top .K _ / ˝ top .H /, which is compatible with composition of D with isomorphisms on the left and right, and obeys the following three additional rules: if D D 0, tD is the identity map; if D D id W H ! H for a one-dimensional vector space H , then tD W R D det.id/ ! H _ ˝ H maps 1 7! v _ ˝ v, for hv _ ; vi > 0; and finally, the isomorphisms are compatible with direct sums, in the sense that det.D1 ˚ D2 / 

tD1 ˚D2

Š

det.D1 / ˝ det.D2 /

tD1 ˝tD2

/ det.01 ˚ 02 / 

Š

/ det.01 / ˝ det.02 /

commutes (0k is the zero map between the same vector spaces as Dk ). Alternatively, one can define tD directly for any D W H ! K, as follows. Take a basis e1 ; : : : ; em of H and a basis f1 ; : : : ; fn of K, such that D.e1 / D f1 ; : : : ; D.ek / D fk , D.ekC1 / D 0; : : : ; D.em / D 0, as well as their dual bases ei_ ; fi_ . Then tD W

_ .fn_ ^    ^ fkC1 / ˝ .ekC1 ^    ^ em / 2 det.D/ _ 7! .fn ^    ^ f1_ / ˝ .e1 ^    ^ em / 2 det.0/:

(11.3)

With this in mind, we return to the general situation, so let F .H; K/ be the space of all Fredholm operators between real Banach spaces H , K. The determinant line bundle det ! F .H; K/ has fibres det.D/, and its topological structure can be characterized by the following properties. First, if H; K are finite-dimensional, then the isomorphisms tD should give rise to a trivialization of det. Secondly, the maps

11 Indices and determinant lines

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det.D1 /˝det.D2 / Š det.D1 ˚D2 / defined above are isomorphisms of determinant line bundles, over F .H1 ; K1 /  F .H2 ; K2 /. Finally, if we restrict to the subspace of surjective operators, then the equality det.D/ D top .ker.D// gives rise to an isomorphism between the determinant line bundle and the top exterior power of the kernel vector bundle (this is a slight variation on the familiar definition of the determinant line bundle by stabilization). Instead of fixed Banach spaces, one can consider two Banach vector bundles with a Fredholm mapping between them, and then get a determinant line bundle on the base space. Remark 11.1. The sign convention introduced here is that of [87] (see also [77, Appendix D]), and is called “point of view II” in [36, Volume I, pp. 62 and 362]. It is maybe instructive to mention the other possible convention is: replace the rightV V V _whatV _ hand side of (11.2) by i w1;i ^ i w2;i ˝ i v1;i ^ i v2;i (removing the sign), and invert the ordering of the fi_ in (11.3). This gives rise to another line bundle det 0 with the same fibres as det; the two bundles are isomorphic, but not trivially so (the identity det ! det0 is not a continuous map). (11b) Line bundles on path spaces and spectral flow. Let N be a closed manifold and †  N a hypersurface, carrying a real line bundle  ! † with a preferred isomorphism (11.4) . ˝2 /_ Š † : Here † is the normal bundle, hence † inherits a co-orientation from (11.4). From these data, one obtains a locally constant function I W P ! Z and a real line bundle ı ! P on the space P of smooth paths Œ0I 1 ! N whose endpoints lie outside †. In cohomological terms, this is done by combining the Thom isomorphism and the evaluation map P  Œ0I 1 ! N : namely, if S W H  .†/ Š H C1 .N; N n †/ ! H C1 .P  Œ0I 1; P  f0I 1g/ Š H  .P /; then I D S.1/ and similarly w1 .ı/ D S.w1 .//. In fact, I is the intersection number with Œ†, hence can be interpreted geometrically as a count of intersection points. We will now explain how to construct an explicit representative for ı along the same lines. Take a path ˛ 2 P which intersects † transversally. At each of the finitely many crossing points s 2 ˛ 1 .†/  .0I 1/ we have a nonzero normal vector ˛ 0 .s/, hence by (11.4) a nondegenerate quadratic form q˛ .s/ P on ˛.s/ . The sign of this is the local intersection number with †, so that I.˛/ D s sign.q˛ .s//. We define O sign.q .s// ı˛ D ˛.s/ ˛ (11.5) s

where  1 is the inverse (or dual) bundle, and the convention is that the tensor product is taken in increasing order over the crossing times s. To make the ı˛ into the fibres

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of a line bundle, one needs to define parallel a transport map ı˛0 ! ı˛1 for a generic homotopy f˛r g, 0  r  1. The nontrivial aspect of this is to prescribe what happens under a generic bifurcation (birth or death) in the intersections ˛r \ †. Suppose that a birth occurs at .r; Q sQ /. At that point, .d=ds/2 ˛r is a well-defined nonzero normal vector to †, so that we have an associated nonzero quadratic form qQ on Q D ˛rQ .Qs / . Take a sufficiently small  > 0, set r˙ D rQ ˙ , and consider the two newborn intersection points at times .rC ; s˙ /, s < sC . These will lie close to the bifurcation point, and if one then identifies the fibres of  at those points with Q by using a Q The signs at the local trivialization, then the sign of q˛rC .s˙ / will be ˙sign.q/. other intersection points are carried over from ˛r , and therefore one has a preferred isomorphism O sign.q˛ .s// O sign.q˛ .s// Q Q ı˛rC Š ˛r .s/ r ˝ Q sign.q/ ˝ Q sign.q/ ˝ ˛r .s/ r : (11.6)   sQs

Taking v, Q vQ 1 to be a nonzero vector in Q resp. the dual vector in Q 1 , one defines the parallel transport map ı˛r ! ı˛rC as the insertion of vQ 1 ˝ vQ or vQ ˝ vQ 1 , depending on the sign of q, Q into (11.6). For death processes the converse rule is used. To show that this actually defines a line bundle on path space one should verify a consistency condition for “homotopies of homotopies”. This is straightforward, and we omit it. The prototypical example is the following one. Let N D sym2 .Rn /_ be the space of quadratic forms on Rn , and N D †0 [ †1 [    its stratification by the corank, with †0 the (open dense) subset of nondegenerate forms. The null spaces form vector bundles  d ! †d , which come with tautological isomorphisms .sym2  d /_ Š †d . By applying the construction above to † D †1 D †1 [ †2 [    and  D  1 , one obtains a function I and line bundle ı on the space P of paths in N with endpoints in †0 . There is a slight complication because † is singular (in codimension two) and  is defined only on its smooth part. To address that, one first defines ı for paths ˛ which remain inside †0 [ †1 , and then proves that it extends to all of P by a computation involving the link of the next stratum †2 . We omit the details. The relation to determinant line bundles comes through the spectral flow construction. Fix an inner product on Rn , as well as a function 2 C 1 .R; Œ0I 1/ 0 with .s/ D 0 for s  0, .s/ D 1 for s  0, and .s/ > 0 for all s where .s/ 2 .0I 1/. For every ˛ 2 P we get a Fredholm operator d˛ D d=ds C˛. .s// 2 F .W 1;2 .R; Rn /; L2 .R; Rn //. Lemma 11.2. I.˛/ D index.d˛ /; moreover, ı is canonically isomorphic to the determinant line bundle of the family fd˛ g. Proof. Take some ˛ 2 P which is transverse to the strata †d , which in particular means ˛ 1 .†d / D ; for all d  2. Denote by ˛.s/ the negative eigenspace of the symmetric matrix ˛.s/. One first extends the domain of d˛ by adding the linear

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span of the functions e s v, where is any negative eigenvalue of ˛.1/ and v the corresponding eigenvector. The resulting operator W 1;2 .R; Rn / ˚ ˛.1/ ! L2 .R; Rn / is surjective (since one can integrate from 1 to s to find preimages), and its kernel can be identified with ˛.0/ by evaluating at some point near 1. This yields dim˛.0/ D index.d˛ / C dim˛.1/ ;

.˛.0/ / Š det.d˛ / ˝ top .˛.1/ /; top

(11.7)

where the isomorphism is canonical. At each of the finitely many times sj (1  j  m) where ˛ crosses †1 , we have a one-dimensional nullspace ˛.sj / equipped with a nondegenerate quadratic form q˛ .sj /, which is simply the restriction of ˛ 0 .sj /. It is easy to see that there are isomorphisms ( ˛.sj / ˚ ˛.sj C / Š ˛.sj  / if q˛ .sj / is positive, or ˛.sj C / Š ˛.sj / ˚ ˛.sj  / if it is negative, where  > 0 is small. At all other times ˛.s/ varies smoothly with s, and by following up the changes one gets X sign.q˛ .sj // C dim˛.1/ ; dim˛.0/ D j

top .˛.0/ / Š

O

sign.q .sj //

˛.sj / ˛

˝ top .˛.1/ /;

j

which together with (11.7) yield the desired equality and isomorphism. Everything being linear algebra, it is easy to check that the isomorphism is compatible with the parallel transport maps from (11.6).  We will now reformulate the previous argument in a way which is better suited to infinite-dimensional generalizations. First, note that the definition of ı˛ and d˛ can be trivially extended to paths ˛ parametrized by arbitrary closed intervals C . Decompose C into C1 [ C2 by cutting at a point s 2 int.C / such that ˛.s/ … †. By definition, I.˛/ D I.˛jC1 / C I.˛jC2 / and there is a canonical isomorphism ı˛ Š ı˛jC1 ˝ ı˛jC2 . On the other hand, one can see from (11.7) that index.d˛ / D index.d˛jC1 / C index.d˛jC2 /; det.d˛ / Š det.d˛jC1 / ˝ det.d˛jC2 /:

(11.8)

After a small deformation which does not change the intersection points ˛.sj / or the sign of the intersection there, we may assume that for each j there is a neighbourhood

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Œsj I sj C in which the ˛.s/ are simultaneously diagonalizable. Decompose Œ0I 1 into intervals as follows: C0 D Œ0I s1  ; C1 D Œs1  I s1 C ; C2 D Œs1 C I s2  ; :: : C2m D Œsm C I 1: Thanks to simultaneous diagonalizability, the operators d˛jC2j 1 can be reduced to scalar ones and then analyzed explicitly. The outcome is that each of them is either injective or surjective, with one-dimensional kernel or cokernel (depending on the sign of the intersection) which can be identified directly with ˛.sj / . For the other intervals there is a preferred deformation of d˛jC2j to an invertible operator, simply by contracting ˛jC2j inside N n † to a constant path with value ˛.s/, for some s 2 C2j . To conclude the argument, one applies (11.8) to put the pieces back together. N (11c) Determinant lines of @-operators. We need to recall some familiar index theory results, which can be seen as two-dimensional analogues of (11.8). Let Sk (k D 1; 2) be two Riemann surfaces with strip-like ends, carrying Cauchy–Riemann operators @N rk which are nondegenerate in the sense of Section (8h). Suppose that S1 has an outgoing end 1C , and S2 an incoming end 2 , such that the limiting data over the two ends agree. One can then glue together the vector bundles and other data to obtain a nondegenerate Cauchy–Riemann operator @N r D @N r1 #l @N r2 on the surface S D S1 #l S2 for l  0. This is unique up to a small perturbation, in particular the determinant line det.@N r / is independent of the choices made in the construction. The gluing theorem says that index.@N r / D index.@N r1 / C index.@N r2 /; det.@N r / Š det.@N r / ˝ det.@N r /; 1

(11.9)

2

where the second part is a canonical isomorphism, which moreover is associative with respect to double gluing operations. The standard approach to proving (11.9) is to first assume that both constituent operators are surjective. Then, for l  0, one can patch together elements of ker.@N rk / to approximate solutions of @N r D 0, which after orthogonal projection yields an isomorphism ker.@N r / Š ker.@N r1 / ˚ ker.@N r2 /I

(11.10)

a compactness argument applied to the limit l ! 1 shows that coker.@N r / D 0, and the desired statement follows. The surjectivity assumption can be removed by using a finite-dimensional stabilization argument.

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Occasionally we will also apply another gluing procedure, which is more commonly used in Gromov–Witten theory. There are two slightly different versions, depending on whether the gluing takes place at interior points or at boundary points, but we will discuss both simultaneously. Start again with two surfaces Sk with strip-like ends, carrying operators @N rk acting on vector bundles Ek , with Lagrangian subbundles Fk . Instead of asymptotic isomorphisms at the ends, one now supposes that there are zk 2 Sk (either both interior or both boundary points) and an identification of the fibres E1;z1 Š E2;z2 which respects the symplectic and complex structure (and in the boundary case, also maps the Lagrangian subspaces to each other). Forming the connected sum (in the ordinary sense if the zk are interior; or otherwise as a boundary connected sum) yields a Riemann surface S with an operator @N r . The analogue of (11.10) is now only defined for pairs .X1 ; X2 / of kernel elements which take on the same value at the points zk , and the resulting version of (11.9) is: for boundary points, index.@N r / D index.@N r1 / C index.@N r2 /  n; det.@N r / ˝ top .F1;z1 / Š det.@N r1 / ˝ det.@N r2 /

(11.11)

and for interior points, index.@N r / D index.@N r1 / C index.@N r2 /  2n; det.@N r / ˝ top .E1;z1 / Š det.@N r1 / ˝ det.@N r2 /:

(11.12)

One can use the orientation of E1;z1 given by its symplectic form to remove the

top .E1;z1 / term from (11.12). Note also that in the interior point case, the construction of S and @N r depends on an additional angular parameter, which measures the identification of the tangent spaces T .Sk /zk . However, the isomorphism in (11.12) is continuous with respect to that parameter; therefore, varying the angle yields an S 1 family of operators @N r along which the determinant line bundle has trivial monodromy. Remark 11.3. Gluing formulae such as (11.9) appear in basically every construction of Floer homology type theories, see for instance [119], [48], [37, Section 3.3]. Beyond that, there is a lot of literature about cut-and-paste properties of determinant lines bundles in general, much of it based on Atiyah–Patodi–Singer type boundary conditions rather than the (simpler but more restrictive) tubular ends framework. See e.g. [122, Lecture 2] for an introduction; [105], [121] for general theorems; and [77, Appendix D] for the Riemann surface case. N (11d) @-operators on line bundles. On a closed Riemann surface, line bundles of negative degrees cannot have nontrivial holomorphic sections. A standard argument extends this to Riemann surfaces with boundary, and to operators of the form

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@N C forder zero termg. Namely, let S be a compact connected Riemann surface with nonempty boundary; E ! S a hermitian line bundle; F  Ej@S a real subbundle; N and @N r an associated @-operator. After choosing a trivialization of E, we can view F as a finite collection of maps 1 ; : : : ; l W S 1 ! RP1 , one for each boundary component. Write .j / for their degrees. Suppose that we have a nonzero solution of @N r X D 0. Its zeros are isolated, each of them has finite order .z/, and an easy winding number argument shows that X X 2 .z/ C .z/ D .1 / C    C .l /: (11.13) z2int.S/

z2@S

In particular, Lemma 11.4. If .1 / C    C .l / < 0, then @N r is injective.



There is also a version of this for surfaces S with strip-like ends, and nondegenerate @N r -operators on them. For simplicity, assume that over each end  W Z ˙ ! S , the pullback of .E; r/ is trivial, and the boundary subbundle is locally constant. After rotating the trivialization   E Š Z ˙  C by a constant if necessary, we write F  .s;0/ D R;

F  .s;1/ D e i R

(11.14)

for some  2 .I 0/. One can now get maps j as before, one for each boundary component of the compactification SO , by following F and, near each point at infinity, taking the homotopy between the two lines in (11.14) which goes through e it  R, t 2 Œ0I 1. Lemma 11.5. If .1 / C    C .l /  j† j < 0, then @N r is injective. To prove this, consider a nonzero solution of @N r X D 0. Over each strip-like end, this operator is simply the standard @s C i@ t with boundary conditions (11.14), hence X .s; t/ D X. .s; t// can be Fourier-expanded as (P c ;m e . C m/z ;  2 † ; X .s; t/ D Pm>0 .  m/z ;  2 †C ; m0 c ;m e where the coefficients c ;m 2 R decay rapidly as m ! 1. Write ./ for the least m such that c ;m ¤ 0. This determines the topological behaviour of the nonzero function X j .fsg  Œ0I 1/ for sufficiently large jsj, and the resulting correction terms to (11.13) yield X X X 2 .z/ C .z/ C ./ D .1 / C    C .l /: z2int.S/

z2@S

2†

Since ./ > 0 for  2 † , the desired result follows.

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Remark 11.6. The fact that nontrivial solutions of @N r X D 0 have only isolated zeros, each of which has finite multiplicity, is a consequence of Aronszaijn’s unique continuation theorem: see [49] for interior points, and [91] for boundary points. The same theorem (or a more direct Taylor series computation) also shows that the local multiplicities are all positive. Finally, we should mention that Lemma 11.5 holds even without the assumption of triviality over each strip-like ends. The principle of the proof remains the same, but X can no longer be written down explicitly, so the definition of ./ relies instead on a more sophisticated analysis of the asymptotic behaviour of X ; see [114]. (11e) Index theory and the Lagrangian Grassmannian. Let V D .V; øV / be a 2n-dimensional vector space with a symplectic structure, and Gr.V / Š Un =On its Grassmannian of (unoriented) linear Lagrangian subspaces. A map B ! Gr.V / is the same as a Lagrangian subbundle of the trivial symplectic vector bundle BV ! B, so cohomology classes of Gr.V / can be thought of as relative characteristic classes. The ones that concern us are the Maslov class  2 H 1 .Gr.V //, and the second Stiefel– Whitney class w2 2 H 2 .Gr.V /I Z=2/ induced from the ordinary Grassmannian of all unoriented subspaces (the first Stiefel–Whitney class is the mod 2 reduction of , hence redundant). The pairing with  is an isomorphism 1 .Gr.V // ! Z. For n  3, w2 similarly gives an isomorphism 2 .Gr.V // ! Z=2. The exceptional low-dimensional cases are: for n D 1, U1 =O1 D S 1 so 2 .Gr.V // D 0; and for n D 2, U2 =O2 D S 1 Z=2 S 2 , so 2 .Gr.V // Š Z. The last-mentioned group is detected by the Euler class on the double cover U2 =S O2 , and since that reduces to the second Stiefel–Whitney class modulo 2, it is still true that w2 W 2 .U2 =O2 / ! Z=2 is onto. We will also need to look at the based and free loop spaces  Gr.V /, L Gr.V /. These have Z many connected components k Gr.V / or Lk Gr.V /, distinguished by the Maslov index ./ D h; Œi D k. For n  3, one easily sees that 1 .k Gr.V // Š Z=2 and 1 .Lk Gr.V // Š 2 .Gr.V //  1 .Gr.V // Š Z=2  Z. In particular, H 1 .Lk Gr.V /I Z=2/ Š Z=2 ˚ Z=2 (11.15) with explicit generators T .w2 / and (the mod 2 reduction of) U./. The notation here is that .T; U / are the components of the pullback map H  .Gr.V // ! H  .S 1  L Gr.V // Š H 1 .L Gr.V // ˚ H  .L Gr.V //. To make the connection with index theory, one proceeds as follows. Fix an øV -compatible complex structure IV on V , a compact Riemann surface S with one boundary component, and an oriented identification @S Š S 1 . For any  2 L Gr.V /, N let DS; be the @-operator acting on sections of the trivial complex vector bundle E D S  V , with boundary conditions given by the Lagrangian subbundle F  Ej@S , .F /exp.2 i s/ D .s/. By considering the family of operators DS; for varying , we

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get a determinant line bundle det S ! L Gr.V /. Lemma 11.7. index.DS; / D n .S/ C ./. Moreover, the isomorphism class of the determinant line bundle is given by w1 .det S / D T .w2 / C .T ./  1/U./:

(11.16)

Proof. The first statement is the well-known Riemann–Roch theorem for surfaces with boundary. The second one could be similarly viewed as an instance of the general index theorem for families of “Real” operators [9]. Alternatively, there is a less direct but more explicit argument, based on the following ingredients: (i) Pinching off the boundary. Suppose that S has genus > 0. One can then think of it as the connected sum of a disc D and a closed Riemann surface of the same N genus. The @-operator on the closed surface is obviously C-linear, hence has a canonical trivialization of its determinant line. From (11.12) one therefore obtains an isomorphism det S Š detD . (ii) The direct sum formula. Let  W Gr.V /  Gr.W / ! Gr.V ˚ W / be the direct sum operation .ƒ1 ; ƒ2 / 7! ƒ1 ˚ ƒ2 , and Lk;l  W Lk Gr.V /  Ll Gr.W / ! LkCl Gr.V ˚ W / the induced map on free loop spaces. By (11.1), .Lk;l / detS;V ˚W Š detS;V  det S;W (here, we have added the vector spaces concerned to the notation, so as to clarify the relationship). On the other hand, using the Whitney sum formula, .Lk;l / .T .w2 // D T .w2  1 C 1  w2 C   / D T .w2 /  1 C 1  T .w2 / C U./  T ./ C T ./  U./ D .T .w2 / C l U.//  1 C 1  .T .w2 / C k U.//;  .Lk;l / .U.// D U. / D U./  1 C 1  U./; hence .Lk;l / .T .w2 / C .k C l  1/U.// D .T .w2 / C .k  1/U.//  1 C 1  .T .w2 / C .l  1/U.//: In view of the expression for the generators in (11.15), the computation above also shows that Lk;l  induces an injective map on H 1 .I Z=2/, as long as at least one of the two spaces V; W has dimension  6 (in fact,  4 is enough, but we will not insist on that). (iii) n D 1 computations. Suppose that n D 1, in which case the Maslov number ./ reduces to the winding number considered in Section (11d). For instance, if S D D and ./ D 1, then DS; is injective by Lemma 11.4, hence invertible by the index

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formula. Next, consider the case where S D D and ./ D 0. (11.13) shows that any X 2 ker.DS; / is a nowhere zero function. This implies that any two kernel elements are proportional, since their quotient would be a holomorphic function on S with real boundary values, hence necessarily constant. This means that evaluation at a boundary point induces a canonical isomorphism ker.DS; / Š .0/. On the other hand, the index formula shows that the operator is onto, hence det.DS; / Š .0/ as well. (iv) A higher-dimensional case. Consider L0 Gr.V /, with n  3, and suppose that S D D. The second summand in (11.15) comes from the subspace of constant loops Gr.V /  L0 Gr.V /. If  is constant, .E; F / obviously decomposes into onedimensional summands, and the argument from (iii) then shows that det.DS; / Š

top ..0//, which implies that both sides of (11.16) agree when restricted to Gr.V /. On the other hand, one can construct a map S 2 ! Gr.V / which pairs nontrivially with w2 and which, if thought of as a loop S 1 !  Gr.V /, yields a nontrivial determinant line bundle over S 1 . This is carried out in detail in [60, Chapter 6] or [34], and we will not repeat it here. The two observations together show that w1 .det S / D T .w2 / C U./ on L0 Gr.V /. With this at hand, one proves (11.16) as follows. (i) shows that it is enough to consider S D D. From (iii), we know the equality to be true on L0 Gr.C/, L1 Gr.C/. In view of (ii), if the equality holds on Lk Gr.V ˚ C/, it also holds on Lk Gr.V / and on LkC1 Gr.V /; moreover, provided that dim.V /  6, one can also argue in the converse direction. By starting from (iv) and using these observations, the general case follows readily.  Remark 11.8. After trivializing the family of Hilbert spaces W 1;2 .D; E; F /, which is always possible by Kuiper’s theorem, the DD; (keeping .0/ fixed) define a map .Un =On / ! F .H; H /. As explained in [34], this should be viewed (up to a constant shift in the index) as an approximation to the Bott periodicity homotopy equivalence F .H; H / ' Z  BO1 ' .U1 =O1 /. (11f) The Arnol’d stratification. The product Gr.V /2 D Gr.V /  Gr.V / has a Schubert-type stratification by relative position. Each stratum †d D f.ƒ0 ; ƒ1 / W dim.ƒ0 \ ƒ1 / D d g carries a rank d vector bundle  d formed by the intersections, and this comes with a canonical isomorphism .sym2  d /_ Š †d :

(11.17)

One can apply the path space construction from Section (11b) to Gr.V /2 , with † D †1 and  D  1 . The outcome is a locally constant integer function IZ , the Maslov index for paths, and a real line bundle ıZ , both living on the space P 2 Gr.V / of paths . 0 ; 1 / W Œ0I 1 ! Gr.V /2 satisfying 0 .s/ t 1 .s/ for s D 0; 1. As in the

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previously considered case of quadratic forms, which can in fact serve as local model, there is an easily overcome complication involving †2 . It is worth while to make this a little more explicit. Take a path . 0 ; 1 / and some s 2 Œ0I 1. Choose families of linear maps k;r;s W k .s/ ! k .r/ for k D 0; 1 and jr  sj small, such that k;s;s D Id. The crossing form is the quadratic form on

0 .s/ \ 1 .s/ defined by d  !V .0;r;s .v/; 1;r;s .v//: q0 ;1 .s/ W v 7!  dr rDs It is independent of the k;r;s . In terms of (11.17), if . 0 .s/; 1 .s// 2 †d for some d > 0, the crossing form is the quadratic form corresponding to the normal component of . 00 .s/; 01 .s//. For a generic choice of paths, the spaces 0 .s/ \ 1 .s/ will be at most one-dimensional, and the crossing forms on the one-dimensional spaces will be nonzero. One then obtains IZ . 0 ; 1 / and .ıZ /0 ;1 by applying the general mechanism (11.5). In parallel with Section (11b), we will now explain the index-theoretic meaning of these invariants. Choose a function as in the discussion there. To each . 0 ; 1 / 2 N P 2 Gr.V / associate the operator DZ;0 ;1 , which is the @-operator on the trivial vector bundle E D Z  V over the infinite strip, with boundary conditions given by the Lagrangian subbundle F0 ;1  Ej@Z, .F0 ;1 /.s;k/ D k . .s//. Because of the transversality condition at the endpoints, this is a nondegenerate Cauchy–Riemann operator, hence Fredholm. Then Lemma 11.9. IZ . 0 ; 1 / D index.DZ;0 ;1 /; and .ıZ /0 ;1 is canonically isomorphic to det.DZ;0 ;1 /. The strategy for this is the same as in the second version of our proof of Lemma 11.2, so we will not repeat the whole argument. Instead of the elementary ODE computation yielding (11.8), one uses directly the gluing formula (11.9). The remaining task is to find enough examples where the index and determinant line can be written down explicitly. Here is one approach (several others are possible, and some of them are technically a little simpler). Suppose that V D R2n D Cn with the standard symplectic structure and standard complex structure IV D i; that 1 .s/ D Rn f0g for all s; and that 0 .s/ is a graph fx2 D ˛.s/x1 g, where ˛.s/ is a family of symmetric   nxn 0 matrices. Equivalently, this means that 0 .s/ D exp.iA.s//Rn with A.s/ D ˛.s/ . 0 0 If we then change the trivialization of E by applying the family of symplectic linear maps ˆ.s; t/ D exp..1  t/iA. .s/// for .s; t/ 2 Z, the operator becomes DZ;0 ;1 X D .@s  J.s; t/iA. .s///X C J.s; t/@ t X C ˆ.s; t/1 .@s ˆ.s; t//X; (11.18) 1 where J.s; t/ D ˆ.s; t/ iˆ.s; t/ is a family of compatible complex structures on R2n . One can deform J.s; t / continuously back to the constant family i ; this gives

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rise to a homotopy of (11.18) through Fredholm operators, hence the index and (up to canonical isomorphism) the determinant line remain the same. Similarly, the last term in (11.18), which is compactly supported and of order zero, hence compact, can be deformed away. The outcome is the operator z Z X D .@s C A. .s///X C i@ t X D with constant boundary conditions X.s; 0/; X.s; 1/ 2 Rn  f0g. An elliptic regularity argument shows that each element of the kernel extends to a smooth function on x t/; and R2 with the periodicity property X.s; t C 2/ D X.s; t/, X.s; t/ D X.s; similarly for the cokernel. After restricting to these subspaces in the domain and z Z into a direct range, one can apply Fourier transform in t-direction, which splits D sum of one-dimensional differential operators of the form   ˛. .s// l  Id with l D 1; 2; : : : : (11.19) d=ds C ˛. .s// and d=ds C l  Id 0 We have previously studied the index theory of such operators in Section (11b). For (11.19), the outcome is that the only nontrivial contribution comes from the firstmentioned summand, and more specifically from those s where ˛.s/ has a nonempty null space. An easy computation shows that these contributions are precisely the crossing forms for . 0 ; 1 /. Remark 11.10. The stratification of the Lagrangian Grassmannian was introduced in [2]. The first part of Lemma 11.9 is Floer’s index theorem [45]. In general, our exposition has followed closely that in [112], [113]. The last argument above can be viewed as a linear version of the reduction from Floer theory to Morse theory [47]. (11g) A variation. We will now modify the previous construction slightly, replacing the strip Z by the upper half-plane H . Denote by P  Gr.V / the space of paths

W Œ0I 1 ! Gr.V / with the following two properties: • .0/, .1/ intersect transversally. • The pair . ; .1//, where the second component denotes the constant path with value .1/, has negative definite crossing form at s D 1. This space carries a natural locally constant function IH and line bundle ıH . To reduce this to the previous setup, one notes that by the second condition above, there is an s 0 < 1 such that .s/ t .1/ for all s 2 Œs 0 I 1/; in particular, the restrictions of . ; .1// to Œ0I s 0  give rise to an element of P 2 Gr.V /. One can choose s 0 so that it depends smoothly on , and then restriction defines a map P  Gr.V / ! P 2 Gr.V /. Take IH , ıH to be the pullbacks of IZ , ıZ under this map. More explicitly, writing

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q instead of q;.1/ , we have IH . / D

X

sign.q .s//;

snCm i.yk //  index.DS2 ;u2 /. Note that here, the indices are those of the ordinary linearized operators on the surfaces Sk D .Sk /rk , not their extended versions involving parameter spaces. (12e) Floer cohomology for branes. We need one elementary piece of notation. Fix a coefficient field K. If ı is any one-dimensional real vector space, we denote by jıjK the one-dimensional K-vector space generated by the two orientations of ı, with the relation that the sum of the two generators is zero. Obviously, an R-linear isomorphism c W ı1 ! ı2 induces a K-linear isomorphism jcjK W jı1 jK ! jı2 jK . This is called the K-normalization of ı and c, respectively. Normalization also works in families, turning real line bundles into local coefficient systems with fibre K, and similarly for isomorphisms between them. One can think of it as passing to associated bundles with respect to the homomorphism R ! f˙1g  K , t 7! t=jtj. The upgraded version of Floer cohomology associates to any pair .L#0 ; L#1 / of exact Lagrangian branes in M a finite-dimensional graded K-vector space HF  .L#0 ; L#1 / (one recovers HF pr .L0 ; L1 / from this by taking char.K/ D 2 and summing up all the

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graded pieces). We will now go through some of the fundamental properties discussed in Section 8, and explain how to adapt them to the presence of brane structures. • Triangle product. There is a graded, associative, and unital product HF  .L#1 ; L#2 / ˝ HF  .L#0 ; L#1 / ! HF  .L#0 ; L#2 /:

(12.11)

• Change of brane structure. If we apply the same operation to all the branes involved, the Floer cohomology group remains unchanged, and the same applies to (12.11). Here, the “operation” may be to change the grading by a constant in Z, or to tensor the Pin structures by the restriction of a common line bundle ˇ ! M . A little less obviously, there are canonical isomorphisms (justifying the name “shift”) HF  .L#0 ; S  L#1 / Š HF C .L#0 ; L#1 /:

(12.12)

These are again compatible with the product, in the sense that the following diagram commutes: HF  .L#1 ; S  L#2 / ˝ HF  .L#0 ; L#1 / 

Š

HF C .L#1 ; L#2 / ˝ HF  .L#0 ; L#1 /

/ HF  .L# ; S  L# / 0 2 

Š

(12.13)

/ HF C .L# ; L# /: 2 0

• PSS isomorphism. There is a canonical isomorphism HF  .L# ; L# / Š H  .LI K/, which turns (12.11) (for L#0 D L1# D L#2 D L# ) into the ordinary cup product. More generally, if one twists L# by any two line bundles ˇ0 ; ˇ1 , the outcome is the cohomology of L with values in the corresponding local coefficient system: HF  .L# ˝ ˇ0 ; L# ˝ ˇ1 / Š H  .LI jˇ0_ ˝ ˇ1 jK /:

(12.14)

• Duality. For any L# , Poincaré duality and the PSS isomorphism give rise to a map R n # # top n top L# W HF .L ; L ˝ .T L// Š H .LI j T LjK / Š H0 .LI K/ ! K: (12.15) The degree n pairings constructed from this, R HF  .L#1 ; L#0 ˝ top .T L0 // ˝ HF  .L#0 ; L#1 / ! K; ha2 ; a1 i D L# a2 a1 0

are nondegenerate and graded symmetric. The latter property requires a little explanation. Using the preferred isomorphism top .T Lk / Š M jLk , one can think of a2 as an element of HF  .L#1 ; L#0 ˝ M jL0 /, and of a1 as an element of

12 The Fukaya category (complete version)

181

HF  .L#0 ˝M jL0 ; L#1 ˝ top .T L1 //. With this in mind, a1 a2 2 HF  .L#1 ; L#1 ˝

top .T L1 //, and so the following equation makes sense: ha1 ; a2 i D .1/ja1 jja2 j ha2 ; a1 i: • Isotopy invariance. An isotopy of exact Lagrangian branes from L#0 to L#1 gives rise to a preferred element of HF 0 .L#0 ; L#1 /. The properties of these elements remain as in Section (8k), in particular they can be used to prove that HF  is isotopy invariant. We define the Donaldson–Fukaya category H F .M / by taking exact Lagrangian branes as objects and Floer cohomology groups as morphisms, with composition given by (12.11). This is a K-linear Z-graded category (which reduces to H F .M /pr if one takes char.K/ D 2 and forgets the grading). Due to the twisted form of the Poincaré duality pairing, it will not in general be a Frobenius category, except if M is trivial. (12f) Determining the grading and signs. With all the preparations we have made, the definition of HF  .L#0 ; L#1 / is fairly straightforward. Choose a regular Floer datum and define the graded Floer cochain group by setting, for each k 2 Z, M jo.y/jK : (12.16) CF k .L#0 ; L#1 / D i.y/Dk

Let u 2 MZ .y0 ; y1 / be a non-stationary solution of Floer’s equation. There is an  obvious short exact sequence R ! T MZ .y0 ; y1 /u ! T MZ .y0 ; y1 /u , where the R term corresponds to the translational direction (more precisely, our convention is that the positive generator is @u=@s). This allows one to identify the top exterior powers of  Tu MZ .y0 ; y1 /u and Tu MZ .y0 ; y1 /u , hence by (12.2) yields a preferred orientation  of MZ .y0 ; y1 / relative to o.y0 / ˝ o.y1 /_ . In particular, if u is an isolated point of  MZ .y0 ; y1 /, we get an isomorphism cu W o.y1 / ! o.y0 /:

(12.17)

The .y0 ; y1 /-coefficient of the boundary operator @ is defined as the sum over K-normalizations of all such cu , X @y0 ;y1 D jcu jK W jo.y1 /jK ! jo.y0 /jK : (12.18) u

This is a fancy (basis-free) way of saying that each u contributes with ˙1, depending on the sign of cu . Another look at (12.2) shows that @ has degree 1 as desired. To see why (12.18) is a sensible thing to do, take a pair of solutions u1 2 MZ .y0 ; y/, Q u2 2 MZ .y; Q y1 /. After translating u2 sufficiently to the right, one can

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glue them together and obtains some u 2 MZ .y0 ; y1 /. In terms of the Gromov com pactification (9.28), u is a point of the one-dimensional moduli space MZ .y0 ; y1 /, lying close to the boundary point .u1 ; u2 / of the compactification. From the structure of the gluing map g, one sees that its differential Dg.u1 ;u2 / has the following property: .@u1 =@s; @u2 =@s/ goes to the translational tangent vector @u=@s, while .@u1 =@s; @u2 =@s/ goes to a vector pointing away from the boundary point .u1 ; u2 / (moving .u1 ; u2 / in the opposite direction .@u1 =@s; @u2 =@s/ corresponds to pulling apart the two pieces, which is the same as letting the gluing length to go infinity).  .y0 ; y1 /u obtained from From this and (12.4) it follows that the orientation of T MZ cu1 ı cu2 W o.y1 / ! o.y0 / through the isomorphism (12.2) always points away from the Gromov boundary. Hence, the two boundary points belonging to the same connected component contribute with opposite maps jcu1 ı cu2 jK , which implies @2 D 0. As usual, HF  .L#0 ; L#1 / is defined to be H.CF  .L#0 ; L#1 /; @/. Now take a .d C 1/-pointed disc S, with brane labels .L#0 ; : : : ; L#d /. Choose regular Floer data and perturbation data. At every isolated point u 2 MS .y0 ; : : : ; yd /, the relative orientation (12.2) gives rise to an isomorphism cu W o.yd /˝  ˝o.y1 / ! o.y0 /. Proceeding as in (12.18), we use these to define a map C ˆS W CF  .L#d 1 ; L#d / ˝    ˝ CF  .L#0 ; L#1 / ! CF  .L#0 ; L#d /: To prove that C ˆS is a chain map, one looks at the two possible kinds of boundary points of the Gromov compactification: Q ynC2 ; : : : ; yd /  • In the first case, which is .u1 ; u2 / 2 MS .y0 ; : : : ; yn ; y;  MZ .y; Q ynC1 /, the image under Dg of the translational vector .0; @u2 =@s/ is a tangent vector to MS .y0 ; : : : ; yd / which points away from the boundary. Hence, the local orientation of that moduli space arising from cu1 ı cu2 under (12.2) differs from the standard outwards-pointing one by .1/ times the Koszul sign in (12.4) (where index.DS2 ;u2 / D 1).  .y; Q y0 /  MS .y; Q y1 ; : : : ; yd /, the image of • In the second case .u1 ; u2 / 2 MZ .@u1 =@s; 0/ points towards the boundary, and the Koszul sign is trivial. Together, this yields X .1/ C ˆS .yd ; : : : ; @.ynC1 /; yn ; : : : ; y1 / D @.C ˆS .yd ; : : : ; y1 //; (12.19) n

where D i.ynC2 /C  Ci.yd /. A similar but simpler argument along the same lines shows that the composition law (8.2) holds (without any signs). As in Section (8b), one uses the cohomology level maps ˆS induced by C ˆS to define the identity morphism in HF 0 .L# ; L# / (d D 0), the triangle product (d D 2), and continuation maps (d D 1). It is maybe also worth while mentioning the construction of chain homotopies based on families .K s ; J s /, s 2 Œ0I 1, of perturbation data on S . One considers the

12 The Fukaya category (complete version)

183

resulting parametrized moduli spaces, and uses isolated points in them to define a map h W CF  .L#d 1 ; L#d /˝  ˝CF  .L#0 ; L#1 / ! CF  .L#0 ; L#d /Œ1. The local orientation of moduli spaces comes from (12.8), and gluing considerations rely on (12.10), with R D Œ0I 1. With respect to the previous argument, the only difference is that in the second case, the Koszul sign convention introduces another sign .1/dim.R/ D 1. The outcome is that X C ˆ1S  C ˆ0S D .1/ h.yd ; : : : ; @.ynC1 /; yn ; : : : ; y1 / C @.h.yd ; : : : ; y1 //; n

(12.20) where C ˆkS are the chain maps derived from .K k ; J k /, and is as in (12.19). This is all one needs to establish the basic well-definedness properties. Following the discussion from Section (12c), it is no problem to generalize the theory to other surfaces S, as long as each boundary circle of SO contains an odd number of points at infinity; and this leads to a TQFT in a suitable sense. If one wants to allow surfaces which do not have this property, one needs to either impose suitable orientation assumptions on the branes, or else use the trick from (12.6). As an example, take x (the end being outgoing) with brane label L# . (12.7) says that each isolated elS DH # # ; ƒy;1 ˝ top .ƒy;1 // ! K, ement u 2 MS .y/ gives rise to an isomorphism cu W o.ƒy;0  # and these can be combined into a degree n chain map CF .L ; L# ˝ top .T L// ! K. The induced map on cohomology is precisely (12.15). (12g) The Fukaya category. It is maybe useful to recall that the appearance of the A1 -associativity relations (1.2) in the Fukaya category is in great part due to x d C1 . the recursive structure of the Deligne–Mumford–Stasheff compactifications R More precisely, each stable d -leafed tree with two vertices, T .m; n/ in the notation from Figure 9.5, labels a codimension one boundary face x T .m;n/ Š R x d C2m  R x mC1  @R x d C1 I R

(12.21)

those faces correspond to the summands with 1 < m < d in (1.2). As one can see from (12.8), the definition of the A1 -structure requires a choice of orientations of the spaces Rd C1 . Once one has made that choice, a comparison of orientations in (12.21) gives rise to a sign for each .m; n/. The aim is to arrange things in such a way that these signs, together with the Koszul signs arising in (12.10) and some other auxiliary ones entering the definition of d , will finally produce the correct version of (1.2). We start with the definition (9.7) of Rd C1 as a quotient of Conf d C1 .@D/ by Aut.D/. Take the subspace of Conf d C1 .@D/ consisting of configurations .z0 ; : : : ; zd / where z0 ; z1 ; z2 are fixed points. The natural boundary orientations of .z3 ; : : : ; zd /  .@D/d 2 define an orientation of that subspace. On the other hand, it is a global slice for the Aut.D/-action, hence can be identified with Rd C1 by projection. We equip Rd C1 with the induced orientation. An easy but

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tedious computation, which we omit, shows that in (12.21), the product orientation x d C2m  R x d C1 by .1/ , x mC1 differs from the boundary orientation of @R of R D m.d  n/ C m C n:

(12.22)

Suppose that we have made all the choices necessary to set up the preliminary version F .M /pr , as explained in Section (9j). The first few steps in the definition of the actual Fukaya category F .M / are obvious: objects are exact Lagrangian branes in M , the morphism spaces are hom.L#0 ; L#1 / D CF  .L#0 ; L#1 /; and the first order operation is the Floer boundary map with an added sign, 1 .a/ D .1/jaj @.a/: Note that this uses the Floer datum associated to the underlying pair of Lagrangian submanifolds .L0 ; L1 /. In principle, one could make the Floer data (and perturbation data) depend on the brane structures, too. At the moment, this would just be an unnecessary complication, but we will adopt it later on for some variants of the Fukaya category; see Section (14e). Next, take branes .L#0 ; : : : ; L#d /, for some d  2, and fix y0 2 C.L0 ; Ld /, yk 2 C.Lk1 ; Lk /. Using (12.8) and the given orientation of Rd C1 , one sees that the moduli spaces M d C1 .y0 ; : : : ; yd / are of dimension i.y0 /  i.y1 /  i.yd /C2d , and are oriented relative to o.y0 /˝o.y1 /_ ˝  ˝o.yd /_ . Now suppose that .r; u/ is a point in a zero-dimensional moduli space. Then we have i.y0 / D i.y1 / C    C i.yd / C 2  d , which is the correct grading for the operation d , and moreover we get a preferred isomorphism cr;u W o.yd / ˝    ˝ o.y1 / ! o.y0 /:

(12.23)

Define the .y0 ; : : : ; yd / coefficient of d to be the sum of the corresponding normalizations jcr;u jK , as in (12.18), but with an additional sign .1/ ,  D i.y1 / C 2 i.y2 / C    C d i.yd /:

(12.24)

We point out right away that, due to the convention in (1.3), the underlying cohomological category indeed coincides with the previously defined H F .M /. Proposition 12.3. The d form an A1 -structure. Proof. As in the simpler examples carried out in the previous section, this is based on a gluing argument, starting with a pair of isolated points .rk ; uk /, k D 1; 2, as in (9.30). We will consider only the stable case 1 < m < d , and consciously ignore the unstable one (the extra argument required to take that case into account follows closely the proofs of (12.19) and (12.20), and is left to the reader as an exercise).

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12 The Fukaya category (complete version)

Take the isomorphisms crk ;uk , and compose them by inserting the output of the second into the .n C 1/-st input of the first one. In terms of orientation spaces, this corresponds to pairing together o.y/ Q and its dual. For brevity, we denote the outcome by c. Q By the general discussion above, cQ induces an orientation of the one-dimensional moduli space M d C1 .y0 ; : : : ; yd /. Using some small parameter , glue together the .rk ; uk / to a point .r; u/ in that space. In terms of the Gromov compactification (9.28), .r; u/ lies close to the boundary point .r1 ; u1 ; r2 ; u2 /, hence there is a canonical local orientation pointing towards that boundary. The two orientations differ by a sign .1/ . Once one has determined that sign, it follows from the construction that the sum of normalizations .1/ jcj Q K over all pairs .rk ; uk / is zero. To get from that to the contribution of .r1 ; u1 ; r2 ; u2 / to the .y0 ; : : : ; yd / coefficient of d C1m .ad ; : : : ; m .anCm ; : : : ; anC1 /; an ; : : : ; a1 /, one still has to add the appropriate signs from (12.24), 1 D i.y1 / C 2 i.y2 / C    C .n C 1/ i.y/ Q C .n C 2/ i.ynCmC1 / C    C .d C 1  m/ i.yd /   D i.y1 / C 2 i.y2 / C    C .n C 1/ i.ynC1 / C    C i.ynCm / C 2  m C .n C 2/ i.ynCmC1 / C    C .d C 1  m/ i.yd /; 2 D i.ynC1 / C 2 i.ynC2 / C    C m i.ynCm /: The remaining claim is that    C 1 C 2 D zn C 2 i.y1 / C    C .d C 1/ i.yd / ;

(12.25)

where the first term on the right gives rise to the standard sign from (1.2), and the second one is independent of .m; n/. This, together with the argument above, establishes the associativity equations. To carry out the necessary computation, consider the diagram (12.10) with R1 D Rd C2m and R2 D RmC1 , but now replacing R D .1I 0/  R1  R2 by Rd C1 , through the map (9.13) associated to T .m; n/. Correspondingly, we set MS1 D M d C2m , MS2 D M mC1 and replace MS by M d C1 . At each corner of the diagram sits a one-dimensional real vector space, and we orient them as follows. • In the upper right-hand corner, take cQ together with the orientation of Rd C1 , and then use the horizontal arrow to transfer this to an element of top T M d C1 . In terms of our previous discussion, this is precisely the orientation of the moduli space induced by c. Q • In the lower left-hand corner, the vector space is canonically identified with R, and we use the horizontal arrow to transfer this to a trivialization of the lower right-hand corner. By definition of the crk ;uk , this is the same as taking the orientations of Rd C2m and RmC1 and combining them with cr1 ;u1 ˝ cr2 ;u2 .

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Next, consider whether these orientations are compatible with the vertical arrows in the diagram. • If we take the trivial orientation of the moduli spaces in the lower left corner, and map it via " which is the linearization of the gluing map, we get the orientation of M d C1 pointing towards the Gromov boundary, which by definition differs from the one introduced above by .1/ . • In the right-hand column, the only difference between the two orientations comes from the gluing map (9.13) between the bases of the relevant families, hence is equal to .1/ . ı Finally, the whole diagram P commutes up to a known Koszul sign .1/ , in this case ı D m.d  m  1/ C m k>nCm i.yk /. We hence know that  D ı C (mod 2). A straightforward computation yields (12.25).  Remark 12.4. There are two artificial ingredients in our construction, namely the orientations of Rd C1 (or rather the requirements that the boundary strata give rise to orientation differences (12.22), which determine those orientations up to an irrelevant overall sign change), and (12.24). By adjusting those terms, one can get any of the other current sign conventions for A1 -structures; the geometry by itself does not single out any preferred convention. For another proof of Proposition 12.3, see [60, §22]. (12h) The shift operation and the A1 -structure. We will now show that the shift operation on exact Lagrangian branes, defined pointwise by (11.36), does indeed correspond to the correct algebraic notion; in more explicit terms, it represents the appropriate A1 -module, defined in Section (3d) as a special case of (3.2). The cohomology level consequence of this fact was already stated (without proof) in our discussion of Floer cohomology, see (12.12) and (12.13). Consider a pair L#0 ; L#1 of exact Lagrangian branes, and the shifted version SL #1 of the latter one. Take some y 2 C .L0 ; L1 /. A useful notational distinction is to write yQ for the same path y thought of in connection with the pair of branes .L#0 ; SL #1 /. This means that we have indices and orientation spaces i.y/, Q o.y/ Q as well as i.y/, o.y/. Lemma 11.21 says that i.y/ Q D i.y/  1, and that there is a canonical isomorphism y W o.y/ ! o.y/: Q

(12.26)

From the proof of that Lemma, we recall that in index-theoretic terms, (12.26) is a consequence of the fact that the orientation operator DH;yQ is the boundary connected sum (in that order) of DH;y and an operator DD; of index n  1 on the disc. In fact, it is maybe better to restrict the domain of DD; to the space of functions which vanish at some fixed point of @D, since that reduces the index to 1 and simultaneously removes the additional top .F / term in the gluing formula (11.11).

12 The Fukaya category (complete version)

187

 .y0 ; y1 / is an isolated Floer trajectory, and uQ the Our basic claim is that if u 2 MZ  same map considered as an element of MZ .yQ0 ; yQ1 /, then the associated isomorphisms (12.17) commute with (12.26), meaning that

y0 ı cu D cuQ ı y1 :

(12.27)

Both sides are the result of thinking of DH;yQ0 as being glued together from DZ;u (the linearized operator for u), DH;y1 and DD; . Since these occur in the same order both times, (12.27) follows from the associativity property of (11.9) and (11.11). We use the K-normalization of (12.26), multiplied with .1/i.y/ , to identify CF  .L#0 ; SL #1 / Š CF C1 .L#0 ; L#1 /:

(12.28)

By definition, this anticommutes with @, hence commutes with 1 . Now consider a .d C 1/-tuple of branes .L#0 ; : : : ; L#d /, d  2, and an isolated point u 2 M d C1 .y0 ; : : : ; yd /. We want to shift L#d to SL #d , and introduce additional notation yQ0 , yQd , uQ as before. The parallel statement to (12.27) says that the multilinear maps cu and cuQ between orientation spaces satisfy y0 ı cu D .1/? cuQ ı .yd ˝ id ˝    ˝ id/; where ? D i.y1 / C    C i.yd 1 /. This is the same argument as before, except that in relating the two sides, one has to move DD; past DH;y1 ; : : : ; DH;yd 1 in the order of composition, which gives rise to the Koszul-type sign. As a consequence, d commutes with (12.28), which is precisely the definition of shifted object in a general A1 -category. To see that, one notes that besides ?, the other relevant sign quantities are: i.yd / and i.y0 / D i.y1 / C    C i.yd / C 2  d , from the definition of (12.28) for yd and y0 ; and moreover, i.y1 / C    C d i.yd / and i.y1 / C    C d i.yQd /, from (12.24). Ultimately all signs cancel out, yielding the desired result. (12i) Automorphisms. All the observations made in Section 10 concerning F .M /pr carry over to F .M /. Of course, some statements have to be changed a little to accommodate the brane structures, but that is usually quite straightforward. As a practical demonstration we will consider one of the more appealing examples, namely the action of the symplectic automorphism group on the Fukaya category, originally discussed in Section (10b). Let us start with some generalities. Take an arbitrary symplectic manifold .M; ø/ 2 . A with an almost complex structure IM and quadratic complex volume form M graded automorphism of M is a pair .;  # /, consisting of a symplectic automorphism  and a map  # W Gr.TM / ! R which satisfies exp.2 i  # .ƒ// D

˛M .D.ƒ// ˛M .ƒ/

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for all ƒ 2 Gr.TM /. Composition of such automorphisms is defined by 2# ı 1# D .2 ı 1 ; 2# ı D1 C 1# /. Graded automorphisms act naturally on the set of Lagrangian branes: if .L; ˛ # ; P # / is a brane, the image .L/ inherits the grading x 7!  # .T L 1 .x/ /C˛ # . 1 .x//, as well as the pushforward Pin structure . 1 / P # . This is not quite yet the most general story. Define a brane automorphism to be a triple  # D .;  # ; ˇ/, where .;  # / is as before and ˇ ! M is a real line bundle. The action of such automorphisms on branes now includes a twist of the Pin-structure, P # 7! . 1 / .P # ˝ ˇjL/:

(12.29)

One relevant example is the shift operation S, which in view of Remark 11.18 can be written as the action of .IdM ;  # D 1; ˇ D M /. From now on, assume that M is an exact symplectic manifold with corners, and consider the group Aut gr .M; @M / of those .;  # / such that the underlying  lies in Aut.M; @M /. We equip this with the topology induced from Aut.M; @M /  C 1 .Gr.TM /; R/. There is a canonical strict action of Aut gr .M; @M / on H F .M /, and this descends to a weak action of 0 .Aut gr .M; @M //. For the Fukaya category itself, the same construction as in Sections (10b) and (10c) yields a coherent action of Aut gr .M; @M / on F .M /, which again descends to a weak action of 0 .Aut gr .M; @M //. Now include general brane automorphisms, but without distinguishing between isomorphic line bundles on M . This defines an extension of Aut gr .M; @M / by H 1 .M I Z=2/, which we denote by Aut # .M; @M /. The natural actions of this group on Fukaya categories, or even Donaldson–Fukaya categories, will in general be only weak actions, simply because (12.29) requires a choice of representative ˇ within the isomorphism class Œˇ. This phenomenon is not terribly important in practice, but traces of it will be visible to the attentive eye later on, when we discuss involutions (Section 14). Remark 12.5. To see that the choice of representative ˇ is indeed relevant, consider the following situation: we have three exact Lagrangian branes L#k D .Lk ; ˛k# ; Pk# / (k D 0; 1; 2), of which the last two are isomorphic (share the same Lagrangian submanifold and grading, and have isomorphic but not necessarily strictly equal Pin structures). A choice of Pin isomorphism g W P1# ! P2# determines an isomorphism of Floer cohomology groups, HF  .L#0 ; L#1 / Š HF  .L#0 ; L#2 /:

(12.30)

For simplicity, assume that L0 intersects L1 D L2 transversally, so that the Hamiltonian terms in the relevant Floer data can be chosen to be zero. Fix an intersection point y, and let ƒ#k D ƒ#k;y (k D 0; 1; 2) be the linear branes obtained from the L#k at that point. Choose a path 2  .Gr.TMx /; ƒ0 ; ƒ1 / in the right homotopy class, and a Pin structure P# on the associated vector bundle F ! Œ0I 1. One can identify

12 The Fukaya category (complete version)

189

.ıH / with both orientation spaces o.ƒ#0 ; ƒ#1 / and o.ƒ#0 ; ƒ#2 /, see Lemma 11.11. However, this identification depends on a choice of auxiliary data, namely isomorphisms f0 W .P# /0 ! P0# and fk W .P# /1 ! Pk# for k D 1; 2. To see the chain map underlying (12.30), one should take f2 D gy ı f1 and consider the induced map of orientation spaces. Flipping the sign of g affects that of f2 , hence (following the discussion in Remark 11.19) changes the identification .ıH / Š o.ƒ#0 ; ƒ#2 /, thus ultimately multiplying (12.30) by 1. One way to improve the situation would be to consider Aut # .M; @M / not as a group but as a slightly more refined object, a “weak 2-group”, which takes the internal symmetry groups H 0 .M I Z=2/ of the line bundles into account. This meshes nicely with general category theory, which would argue that the “automorphism group of a category” is anyway an object of this kind. In these terms, the computation from Remark 12.5 shows that the automorphism Idˇ of the trivial line bundle ˇ D M R gives rise to the natural transformation E from the identity functor to itself. (12j) Additional remarks. Fukaya categories have many other features which we cannot discuss here. However, two of them deserve some rudimentary mention, if only because their cohomology level counterparts have already figured in our discussion of Floer theory. The first one is the PSS isomorphism HF  .L# ; L# / Š H  .LI K/. The corresponding chain level statement says that if L# is a single exact Lagrangian brane, the A1 -algebra B D homF .M / .L# ; L# / obtained by looking only at this particular object is quasi-isomorphic to the one obtained by doing classical topology on L (for instance, as the dg algebra of singular cochains, or equivalently using Morse homology and Fukaya’s gradient tree operations [54]). An early form of this statement appears in [58], which basically aims at equality of the relevant algebraic structures, through the construction of an isomorphism of moduli spaces. An alternative approach, which is less analytically tricky but only proves quasi-isomorphisms, would be to use a mixture of pseudo-holomorphic polygons and gradient trees, in the manner of [33] and its precursors. There are also minor variations allowing twisted coefficients, which include chain level versions of (12.14). The second topic is the analogue of the Frobenius nature of the Donaldson– Fukaya category, so we will assume that M is trivial. For any A1 -category A there is a canonical A1 -functor D W A ! Q D mod.A/, called the abstract Serre functor. The image of an object Y under D is the dual A1 -module Y _ , which satisfies Y _ .X/ D homA .Y; X/_ , with similar formulae for the structure maps. We say that A is an n-dimensional weak Calabi–Yau category if, inside H. fun.A; Q//, there is

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II Fukaya categories

an isomorphism of degree n from the Yoneda embedding I to D. Natural transformations I ! D arise naturally from classes in the dual Hochschild cohomology HH.A; A_ /. In our case A D F .M /, the relevant class is defined by considering pointed discs with only outgoing points at infinity, together with a single added interior marked point, in the style of Section (10c). The fact that the associated natural transformation is an isomorphism then follows from Lemma 1.6 and Poincaré duality in Floer cohomology.

13 Polygons on surfaces (13a) Automatic regularity. Let S be a connected Riemann surface with strip-like z of all complex structures j on S which agree with the ends. Consider the space R z S ! R z given j0 D IS outside some compact subset. The product SQ D R has tautological complex structures on the fibres, which makes it into a family of Riemann surfaces with strip-like ends. Moreover, this family carries a natural action of the group GQ of those diffeomorphisms of S which, near infinity on each strip-like end, restrict to a translation in s-direction. Equip S and hence SQ with Lagrangian labels, in such a way that L ;0 t L ;1 for each point at infinity . Choose the trivial perturbation datum K D 0, J D IM (constant family of almost complex structures). Take some pseudo-holomorphic map u 2 MS .fy g/, and consider .j0 ; u/ as an element of the corresponding parametrized moduli space MSQ .fy g/. The infinitesimal action of GQ defines a linear map d 0 from the Lie algebra LGQ to the Zariski tangent space T Zar MSQ .fy g/.j0 ;u/ (we say “Zariski tangent space” since we have made no regularity assumption). On the other hand, this Zariski tangent space is, by definition, the kernel of the extended linearized operator d 1 D DS;j Q 0 ;u . One can therefore think of the two operators as boundary maps in a complex d0 d1 zj  .T BS /u ! 0 ! LGQ ! T R .ES /u ! 0: (13.1) 0

z and GQ as infinite-dimensional spaces has been rather informal, but Our treatment of R at this point, we want to make things more precise at least on the tangent space level, by choosing suitable Hilbert space completions. As defined before, the Lie algebra of GQ would consist of smooth vector fields Z on S, tangent to the boundary, which near infinity on each strip-like end satisfy   Z D c @s for some constant c 2 R. Instead, we take its weighted Sobolev completion with .k; p/ D .2; 2/ and some small  > 0, which is W 2;2I .S; T S; T .@S//C in the notation from (9.2). Similarly, the tangent z would naively be the space of compactly supported endomorphisms of space of R T S which anti-commute with the complex structure. We interpret these as smooth sections of the (complex) tensor product line bundle 0;1 S ˝ T S, and complete to

13 Polygons on surfaces

191

W 1;2I .0;1 S ˝ T S /. Finally, we use the same Sobolev topology for the linearized operator DS;u , which is the second component of d 1 . The completion of (13.1) then has the form W 2;2I .S; T S; T .@S//C d0



1;2I W 1;2I .S; 0;1 .S; E; F / S ˝ TS/ ˚ W

(13.2)

d1



L

2I

.S; 0;1 S ˝ E/

where E D u TM , F  Ej@S is given by the tangent spaces of the relevant Lagrangian submanifolds, and N Du ı Z/; d 0 .Z/ D .2IS ı @Z; d 1 .Y; X/ D 12 IM ı Du ı Y C DS;u .X/: Since we have made an ad hoc choice of Sobolev spaces, one should first check that these are indeed bounded operators, but that is straightforward. In fact, rather more is true: the standard Cauchy–Riemann operator @N on T S with boundary conditions T .@S/ is admissible, hence becomes Fredholm in weighted Sobolev spaces, see the discussion in Section (9b). DS;u itself is a nondegenerate @N r -operator, hence also Fredholm. The other components are of order zero, therefore compact. The conclusion is that (13.2) is a Fredholm complex. Choose some family S ! R (this time, with finite-dimensional base), which is an infinitesimally miniversal deformation of S D Sr0 in the sense of Section (9b). Let G be the group of holomorphic automorphisms of S . Lemma 13.1. The cohomology groups of (13.2) have the following interpretation: H 0 and H 1 are, respectively, the kernel and cokernel of a map LG ! ker.DS;r0 ;u /; and H 2 D coker.DS;r0 ;u /. z To turn Proof. Informally, the idea is that R is a local slice for the GQ -action on R. this into a rigorous argument, choose a local trivialization of S which is constant on z hence a map of the strip-like ends. This provides a local embedding of R into R,

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II Fukaya categories

their tangent spaces. We then have a commutative diagram with exact columns, / T Rr  .T BS /u 0

0 

LGQ D



LGQ

d0



/ TR zj  .T BS /u 0  / TR zj =T Rr 0 0

DS;r0 ;u

/ .ES /u D

d1



/ .ES /u

(13.3)

 / 0.

By definition, T Rr0 projects isomorphically to the cokernel of (9.3), which implies that the bottom row in (13.3) is a complex with H 0 D LG , H 1 D 0. The long exact sequence in cohomology yields the desired statement.  Lemma 13.2. If M is two-dimensional and u is not constant, DS;r0 ;u is onto for any infinitesimally miniversal family S ! R. Proof. u is a holomorphic map between Riemann surfaces and not constant, hence Du W T Sz ! TMu.z/ is an isomorphism except at finitely many points. It follows that the first component of d 1 has dense image. By the Fredholm nature of the second component, this means that d 1 is onto. One then appeals to the last part of Lemma 13.1.  We refer to Lemma 13.2 as an “automatic regularity” statement, since it holds for any IM and does not require perturbations of the pseudo-holomorphic map equation. Remark 13.3. Algebro-geometrically minded readers may find he following sheaftheoretic interpretation helpful. For simplicity, we pass to the analogous situation of a holomorphic map u W S ! M , where S is a closed Riemann surface and M a complex manifold. Then (13.2) becomes 1;2  .S; u TM / ! L2 .S; 0;1 W 2;2 .S; T S/ ! W 1;2 .S; 0;1 S ˝T S /˚W S ˝u TM /;

which in fact computes the hypercohomology of the two-step complex of coherent sheaves fDu W T S ! u TM g. If u is not constant, Du is injective (as a sheaf map), so we are actually computing the cohomology of the quotient u TM=T S (shifted by one). Finally, if M is a surface, this quotient is a torsion sheaf, hence has vanishing cohomology in degree one. (13b) Combinatorial Floer cohomology. Let M be a two-dimensional exact symplectic manifold with corners. A connected exact Lagrangian submanifold in M is a

13 Polygons on surfaces

193

R simple closed curve S 1 Š L  int.M / satisfying L  D 0, where  2 1 .M / is the distinguished primitive for the symplectic form ! (in the case of a curve with several connected components, the integral over each component should vanish). Note that by Stokes’ theorem, this condition rules out null-homologous curves. Fix a collection L D .L1 ; : : : ; Lm / of exact Lagrangian submanifolds, which are in general position to each other; this means that any two intersect transversally, and that there are no triple intersections. When defining the Fukaya category F .M /pr in the preliminary (ungraded and unsigned) version of Section 9, one can choose the Floer data .HLi0 ;Li1 ; JLi0 ;Li1 /;

i0 < i1

(13.4)

to be trivial (H D 0, J D IM ), and the same for the perturbation data .KLi0 ;:::;Lid ; JLi0 ;:::;Lid /;

i0 <    < id :

(13.5)

There are two aspects involved in checking that this is allowed. The first one is taken care of by Lemma 13.2, which ensures that the moduli spaces of Floer trajectories resp. holomorphic polygons associated to (13.4), (13.5) are regular; there are no constant maps in these moduli spaces because the Lagrangian labels Lik on the boundary are all different and in general position. The second point is that (13.5) is compatible with the consistency condition for universal choices of perturbation data, see Section (9i). This is a consequence of the following elementary observation: take a stable d -leafed tree T , and for each vertex, a jvj-pointed disc Sv with Lagrangian labels and striplike ends. We assume that the labels are compatible, so that all the Sv can be glued to a .d C 1/-pointed disc S with Lagrangian labels. Then, if the labels on S are .Li0 ; : : : ; Lid / with i0 <    < id , those on each constituent Sv are an ordered subset thereof, hence are of the same form. Next, assuming that (13.4) and (13.5) have been chosen trivial, we claim that the directed A1 -subcategory F .L/pr;!  F .M /pr , in the sense of Section (5n), is independent of the complex structure IM . In other words, for fixed i0 <    < id and intersection points y0 2 Li0 \ Lid , yk 2 Lik1 \ Lik (1  k  d ), the number of isolated points in Md C1 .y0 ; : : : ; yd / does not change under a homotopy of IM . This is proved by a straightforward cobordism argument of moduli spaces, which can be viewed as the trivial case of the discussion in Section (10e): due to Lemma 13.2, the parametrized moduli spaces M d C1 .y0 ; : : : ; yd /Œ0I1 are smoothly fibered over Œ0I 1, hence contain no isolated points, so that the A1 -functor from Proposition 10.13 is the identity. Having established that F .L/pr;! is topological in principle, the next step is to obtain an explicit formula for it. This is well-trodden ground (see the references at the beginning of this chapter), so we will be brief. Let d C1  R2 be an oriented convex .d C 1/-gon (for d D 1, this is a 2-gon or half-moon), with corners and edges numbered in the same way as for .d C 1/-pointed discs. Given points y0 ; : : : ; yd as

194

II Fukaya categories

before, consider the set Immd C1 .y0 ; : : : ; yd / of positively oriented smooth immersions d C1 ! M such that the k-th corner gets mapped to yk , and the k-th edge to Lik (Figure 13.1). We insist that the maps must be immersions up to the boundary,

y4 Li3 Li4

y3

u

y0

Li2 y2

Li0 Li1 y1 Figure 13.1

which means that the image near each corner is a wedge-shaped region with angle <  at the vertex. The group of diffeomorphisms of d C1 which fix each corner acts on Immd C1 .y0 ; : : : ; yd /, and the quotient can be identified with the zero-dimensional part of Md C1 .y0 ; : : : ; yd /. The proof of this goes as follows: for convenience, assume that IM has been chosen in such a way that near each intersection point of two of our curves, there are local holomorphic coordinates in which the curves become the real and imaginary axis in C. This is not a problem, since we already know that F .L/pr;! is independent of IM (alternatively, one can work with arbitrary IM at the cost of using a little more analysis, as demonstrated in [114]). An index argument [43] proves that any isolated u 2 M d C1 .x0 ; : : : ; xd / is an immersion, and that its image near the points at infinity is wedge-shaped. This, with a little more consideration of the convergence ratio of u over the strip-like ends, implies that one can compactify it to an immersion uQ W d C1 ! M . The converse direction, where one starts with an immersion and constructs a holomorphic map, depends on the fact that if we have any complex structure on d C1 and remove the corners, what remains is a .d C 1/-punctured disc; this is easily reduced to the uniformization theorem for the closed disc. Remark 13.4. It is natural to ask whether the full A1 -subcategory of F .M /pr consisting of a generic collection L also has a combinatorial description. This seems likely, but the answer ought to be rather more complicated than the one we have

13 Polygons on surfaces

195

given, because of the contributions coming from constant holomorphic polygons at the intersection points, where Lemma 13.2 fails (their moduli spaces have excess dimension: they are smooth, but not regular). (13c) Brane structures. Now suppose that M carries a quadratic complex volume form, which in this dimension is just a nowhere vanishing C 1 quadratic differential 2 M W TM ˝C TM ! C. The associated squared phase map can be thought of as a trivialization of the real projectivized tangent bundle, ˛M W RP .TM / D Gr.TM / ! 1 .1/, which S 1 . A good way of visualizing this is through the real subbundle M D ˛M gives rise to an unoriented foliation on the surface. In those terms, a Lagrangian brane in M is an embedded closed curve L equipped with • a real subbundle H of the pullback   .TM jL/, where  W Œ0I 1  L ! L is the projection, satisfying H j.f0g  L/ D M jL and H j.f1g  L/ D T L. • a real line bundle ˇ ! L. H is a homotopy of subbundles of TM jL, or more geometrically, a way of simultaneously rotating all the tangent spaces to L so that they become parallel to M . The obstruction to existence of such a homotopy is the global rotation number of L relative to M (the requirement that this has to vanish again excludes null-homologous simple closed curves). To relate this discussion to the general notion of grading, note that if we have such an H , there is a unique function a W Œ0I 1  L ! R with a.0; x/ D 0 and exp.2 i a.r; x// D ˛M .Hr;x /, and then ˛ # .x/ D a.1; x/ is a grading for L. As for ˇ, we know from Example 11.20 that it is equivalent to a choice of Pin-structure. Let L# be a family of exact Lagrangian branes L#k D .Lk ; Hk ; ˇk /, 1  k  m, in general position. We will now add gradings and signs to our previous construction, leading to a combinatorially defined directed A1 -category. For that, we first of all associate to each intersection point y 2 Li0 \ Li1 (i0 < i1 ) an absolute index i.y/ and orientation space o.y/. Combine the homotopies Hi0 .1  ; y/, Hi1 .; y/ to a path in RP .TMy / going from .T Li0 /y to .T Li1 /y . Let ^.y/ 2 R n Z be the total angle by which this path rotates, and set i.y/ D Œ^.y/= C 1; o.y/ D Hom.ˇi0 ;y ; ˇi1 ;y / ˝ .T Li1 /y˝i.y/ :

(13.6)

Having that, we define the morphism spaces CF  .L#i0 ; L#i1 / as usual (12.16). Next, take an immersed polygon uQ W d C1 ! M with corners at y0 ; : : : ; yd . An easy computation using (13.6) shows that the existence of such a polygon implies the equality i.y0 / D i.y1 / C    C i.yd / C 2  d . Use the boundary orientation of d C1 at the corners to orient .T Lid /y0 and .T Lik /yk , k D 1; : : : ; d , as indicated by the arrows in Figure 13.1 (note that the conventions at y0 and y1 ; : : : ; yd are different). Parallel transport along the sides of d C1 defines isomorphisms ˇik ;yk ! ˇik ;ykC1 and ˇid ;yd ! ˇid ;y0 . By looking at (13.6), one sees that these two elementary facts

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II Fukaya categories

together yield an isomorphism o.yd / ˝    ˝ o.y1 / ! o.y0 /, which we denote by cQuQ . Define Q d W CF  .L#id 1 ; L#id / ˝    ˝ CF  .L#i0 ; L#i1 / ! CF C2d .L#i0 ; L#id / by adding up all the normalizations jcQuQ jK . An elementary argument, based on decompositions of polygons with one concave corner, shows that these maps satisfy the A1 -associativity equation (1.2). Denote by Fz .L# /! the directed A1 -category defined in this way. Example 13.5. Consider M D S 1  Œ1I 1, drawn in Figure 13.2 as a square with the two vertical sides identified, with the curves L0 ; L1 on it. We use the constant complex volume form corresponding to the foliation M by horizontal lines. Choose the trivial homotopy from M jL0 to T L0 . In the other case, the angle between M jL1 and T L1 is everywhere < =2, hence there is a unique “short” homotopy between the two, which we adopt. Finally, take the trivial line bundles ˇ0 ; ˇ1 . Then i.y0 / D 0,

L1 y1

v

u

y0

L0

Figure 13.2

i.y1 / D 1, o.y0 / D R (canonically, since the index is even) and o.y1 / D .T L1 /y1 . The convention from Figure 13.1 says that when computing the contribution of the 2-gon u to the boundary operator, one should identify o.y1 / Š R using the orientation shown in Figure 13.2. The other 2-gon v requires the opposite identification; therefore their contributions cancel out, yielding HF 0 .L#0 ; L#1 / D K;

HF 1 .L#0 ; L#1 / D j.T L1 /y1 jK :

Because L0 ; L1 are isotopic (as exact Lagrangian branes), the general theory says that there is a canonical isomorphism HF  .L#0 ; L#1 / Š H  .L1 I K/. And indeed, H 0 .L1 I K/ Š K canonically, while the natural generators for H 1 .L1 I K/ are just the orientations of L1 , so that H 1 .L1 I K/ Š j.T L1 /y1 jK . The reader is encouraged to experiment with the effect of changing the brane structures, and compare the results with (12.12), (12.14). We will now relate the combinatorial construction above to the directed subcategory of the actual Fukaya category, F .L# /!  F .M /. The objects are clearly the

197

13 Polygons on surfaces

same; so are the morphism spaces, since (13.6) is merely a reformulation of (11.35); and as previously observed, there is a canonical bijection between points .r; u/ 2 M d C1 .y0 ; : : : ; yd / of zero-dimensional moduli spaces and immersed .d C 1/-gons uQ up to reparametrization. The only difference is that when constructing the A1 structure maps d according to the general prescription from Section (12g), one uses isomorphisms cr;u defined analytically by a gluing argument for determinant lines (12.23). More precisely, d is obtained by adding up contributions j.1/ cr;u jK , Q 2 f˙1g the sign where  is as in (12.24). With that in mind, denote by  d .u/ difference between cQuQ and .1/ cr;u . Q D  d .kd ; : : : ; k1 / is actually determined by the absolute We claim that  d .u/ indices kj D i.yj / of the intersection points involved. Note first that the sign difference depends only on the local behaviour of our geometric data near the immersed polygon, and secondly that it remains the same if we pass from M to some covering (taking the preimage of the Lik , and choosing an arbitrary lift of the polygon). Using this, one can reduce the determination of the sign to a situation where the polygon is embedded rather than immersed. In that case, one can trivialize the line bundles ˇik along its edges, which shows that  d .u/ Q is the same for all choices of line bundles. The only remaining local topological data are the absolute indices, which establishes our claim. We have one more a priori piece of information, which is that both d and d Q satisfy the A1 -associativity equations. This imposes some fairly strict constraints on the signs  d . For instance, by looking at decompositions of the kind indicated in Figure 13.3 (i) one sees that  d .kd ; : : : ; knC1 ; kn C 1; : : : ; k1 / 1 .kn / D

(i)

(ii) Figure 13.3

 d .kd ; : : : ; knC1 C 1; kn ; : : : ; k1 / 1 .knC1 / and  d .kd C 1; kd 1 ; : : : ; k1 / 1 .kd / D  d .kd ; : : : ; k1 / 1 .k1 C    C kd C 2  d /. By an elementary manipulation, this implies that  d .kd ; : : : ; k1 / .k1 /.k2 / : : : .kd /.k1 C    C kd C 2  d / ; D d .2  d /  .0; : : : ; 0/

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II Fukaya categories

where .k/ D  1 .0/ 1 .1/     1 .k  1/ for k  0 (resp. .k/ D  1 .k/ 1 .k C 1/ : : :  1 .1/ for k < 0). A similar argument, applied to Figure 13.3 (ii), shows that  d .kd ; : : : ; kn C1; : : : ; k1 / 1 .kn / D  d 1 .kd ; : : : ; kn CknC1 ; : : : ; k1 / 2 .knC1 ; kn /, from which it follows that  d .0; : : : ; 0/ D  2 .0; 0/d 1 .2  d /. As a consequence, F .L# /! and Fz .L# /! are related by a simple linear isomorphism, which multiplies each Floer chain group CF k by .k/ 2 .0; 0/. Since that is enough for our purpose, we will not try to determine the remaining sign explicitly.

14 Symplectic involutions (14a) Equivariant branes. Let .M; øM / be a symplectic manifold, equipped with 2 an almost complex structure IM and quadratic complex volume form M , and carrying 2 an involution M which preserves .øM ; IM ; M /. We want to combine the natural action of M on Lagrangian submanifolds with a change of Pin structure. This causes some complications on a purely formal level, which we will now address. A pre-equivariant brane consists of: its principal component, which is a La# grangian brane L#C D .LC ; ˛C ; PC# /; its shadow, which is a Pin structure P# on # # the graded Lagrangian submanifold .L D M .LC /; ˛ D ˛C ı M /; and addi#  # top tionally, a pair of isomorphisms fC W PC ! M .P ˝ .T L //, f W P# !  M .PC# ˝ top .T LC //, such that the composition fC

 .f /˝Id M 

  top .P# ˝ top .T L // ! PC# ˝ top .T LC / ˝ M

.T L / Š PC# ; PC# ! M (14.1)  where the last step uses the obvious isomorphism M .T L / Š T LC , is equal to the nontrivial automorphism e 2 ker.Pinn ! On / of the Pin structure. To keep the terminology concise, we will sometimes speak of “the pre-equivariant brane L#C ”, but it is always intended that some specific choice of the auxiliary data P# and f (or fC ; any one of the two isomorphisms determines the other uniquely, of course) has been made.

Remark 14.1. Every Lagrangian brane L#C can be made pre-equivariant in an obvious  way: set P# D M .PC# ˝ top .T LC //, with f the identity, and adjust fC accordingly. There is a natural involution  on pre-equivariant branes, which exchanges the C and  parts; in abbreviated notation, .L#C / D L# . This relies on the fact that the  composition in the other order, .M .fC /˝Id/ıf , is also e. An equivariant brane is defined to be a fixed point of this involution. Concretely, this is given by an invariant Lagrangian submanifold L, M .L/ D L, together with a M -invariant grading ˛ # ,  and a Pin-structure P # together with an isomorphism f W P # ! M .P # ˝ top .T L//,

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14 Symplectic involutions

whose “square”  .f /˝Id M

f

  top ! M .P # ˝ top .T L// ! P # ˝ top .T L/ ˝ M

.T L/ Š P # (14.2) P# 

is e. Given some M -invariant Lagrangian submanifold L, we will now look at the conditions under which it admits an equivariant brane structure. Suppose that the Maslov class in H 1 .L/ vanishes, so that L admits a grading ˛ # . If L is connected, the difference ˛ # ı M  ˛ # is a M -anti-invariant constant function, hence automatically zero. On the other hand, if L has two connected components exchanged by M , one can adjust ˛ # on one of them to make the difference vanish. The general case follows from these two, the upshot being that there are no additional obstructions to getting an invariant grading. The Pin part of the story is a little more interesting. Most obviously, this can be viewed as a step-by-step process: first equip L with a Pin structure, then try to find an  isomorphism f W P # ! M .P # ˝ top .T L//, and finally determine its square. The obstructions lie, respectively, in H 2 .LI Z=2/, H 1 .LI Z=2/, and H 0 .LI Z=2/, but the latter two are secondary obstructions, which depend on the previously made choices, and do not assume all values. Alternatively, one can combine all these aspects into a single and more canonical obstruction, which is the equivariant cohomology class eq

eq

2 w2 .L/ C w1 .L/u C u2 2 HZ=2 .LI Z=2/:

(14.3)

Here, u is the generator of H  .BZ=2I Z=2/ D Z=2Œu; and wk .L/ are the equivariant Stiefel–Whitney classes of T L, which depend on its structure as an equivariant vector bundle. In the case where M jL is fixed point free, there is a stabilization trick which reduces our question to a standard one about non-equivariant Pin strucx be the covering of the quotient tures, hence explains (14.3). Namely, let  W L ! L x x L D L=.Z=2/, and ˇ ! L the associated real line bundle. Suppose that we have a x ˚ ˇ ˚ ˇ ˚ ˇ, which means that Pin structure Px # on T L eq

2 2 x x x x w2 .T L˚ˇ˚ˇ˚ˇ/ D w2 .T L/Cw 1 .T L/w1 .ˇ/Cw1 .ˇ/ 2 H .LI Z=2/ (14.4)

must be zero. Take the pullback   Px # , which is a Pin structure on T L ˚ R3 , and let P # be its restriction to T L. The pullback has an obvious automorphism covering (the action on the frame bundle of) D.M jL/ ˚ .IdR3 /. We want to compose this with an auxiliary automorphism g covering IdT L ˚ .IdR3 /, and then restrict the composition to P # . By (11.30) and Lemma 11.14, the choice of g corresponds to an orientation of the bundle R3 ˝ top .T L ˚ R3 /, which is the same as an orientation of T L itself. After taking this factor into account (for instance, by choosing local orientations and patching together the resulting local automorphisms of P # ), the  outcome is that P # comes with a natural automorphism f W P # ! M .P # ˝ top .T L//.  x# The original automorphism of  P was an involution by definition; g 2 D e by

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the rank.F anti / D 3 case of Lemma 11.14; and the two commute, which means that (14.2) has the desired property. The argument runs equally well in reverse, showing that the existence of an equivariant brane structure implies the vanishing of (14.4),  x which indeed corresponds to (14.3) under HZ=2 .L/ Š H  .L/. As for the general (non-free) case, one can turn an arbitrary Z=2-action into a free one through the Borel construction; and this immediately shows that the vanishing of (14.3) is a necessary condition for the existence of equivariant brane structures on L. Sufficiency can also be proved in this way, but requires a little more discussion of Pin structures on the Borel construction, which we omit. To complement the previous discussion, we now take a more direct look at the situation along the fixed point set Lfix of an invariant Lagrangian submanifold L. Split T LjLfix D F inv ˚ F anti into its DM -invariant and anti-invariant parts. Suppose that we have a Pin structure P # on L together with an isomorphism f W P # !  .P # ˝ top .T L//. By (11.30), the restriction of f to Lfix is given by a section of Pin.T L ˝ top .T L// Z=2 S. top .T L// which covers D.M jL/ ˝ Id 2 O.T L ˝

top .T L//. Appealing again to Lemma 11.14, one finds that such sections correspond to orientations of .F anti ˝ top .T L// ˚ top .T L/, or equivalently trivializations of

top .F anti /˝ rank.F

anti /

˝ top .F inv /˝ rank.F

anti /C1

! Lfix :

(14.5)

Moreover, the square of f will be e if 12 rank.F anti /.rank.F anti / C 1/ is even, and e if it is odd. Here, square is meant in the sense of (14.2), which is why compared to anti (11.29) we have an additional sign .1/rank.F / ; this comes from det.D.M jL// via  the appearance of the pairing M . top .T L// ˝ top .T L/ ! R. As an aside, note that large parts of this argument could equally be carried out by starting from the abstract obstruction theory (14.3), and noticing that  eq ˇ eq w2 .L/ C w1 .L/u C u2 ˇLfix   D w2 .F / C rank.F anti /w1 .F anti / C .rank.F anti / C 1/w1 .F inv / u  rank.F anti /.rank.F anti / C 1/  C C 1 u2 2 H  .Lfix I Z=2/Œu: 2 Ultimately, only one special case will be relevant for our applications, and we record this for future reference: Lemma 14.2. Let L be a M -invariant Lagrangian submanifold of M . Suppose that L is connected, and that its fixed point set Lfix is again connected, nonempty, and has codimension 1. Moreover, suppose that we have a Pin structure P # on L  which is isomorphic to M .P # ˝ top .T L//. Then, choices of isomorphism f W P # !  M .P # ˝ top .T L// correspond bijectively to co-orientations of Lfix  L. Moreover, the square of any such isomorphism, taken in the sense of (14.2), is e. 

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14 Symplectic involutions

Clearly, Z=2-equivariant graded symplectic automorphisms of M act on the set of pre-equivariant branes, in a way which strictly commutes with the involution , in particular preserves equivariant branes. One can also act by equivariant line bundles ˇ ! M : given such a line bundle and a pre-equivariant brane, one replaces P˙# with P˙# ˝ ˇjL˙ , and uses the isomorphisms ˇjL˙ Š  .ˇjL / given by the equivariant structure of ˇ to turn f˙ into isomorphisms between the new pair of Pin structures. For instance, take ˇ D R  M with the nontrivial Z=2-action (corresponding to the nontrivial real character of Z=2): tensoring with that is the same as multiplying both f˙ by e. Returning to general ˇ, apply this action to an equivariant brane, and look at the fixed point set Lfix  L. Then, the effect is to change the trivialization of (14.5) by the sign of the Z=2-action on ˇjLfix . (14b) The equivariant Donaldson–Fukaya category. It is worth while to see how the definitions made above fit into the more abstract discussion in Sections (10b) and (12i). Suppose that we are given a category A with a coherent Z=2-action. This consists of an auto-equivalence E W A ! A and a functor isomorphism T W E 2 ! IdA , satisfying the single coherence condition RE .T / D LE .T /, or more concretely TE.X / D E.TX / 2 HomA .E 3 .X/; E.X//

for all X:

(14.6)

One can then define a new category Astrict as follows. Objects are of the form .XC ; FC ; F /, where X˙ are objects of A and F˙ W X˙ ! E.X / isomorphisms, such that E.F / ı FC W XC ! E 2 .XC / equals TX1 (because of (14.6), a correC sponding property, starting with X , will then hold automatically). Morphisms from .X0;˙ ; F0;˙ / to .X1;˙ ; F1;˙ / in Astrict are simply morphisms from X0;C to X1;C in A, which immediately shows that Astrict is equivalent to A. However, we now have a strict Z=2-action, given by the functor E strict which exchanges the C and  parts of the objects, and maps 1 ı E.ˆ/ ı F0; 2 HomA .X0; ; X1; /: ˆ 2 HomA .X0;C ; X1;C / 7! F1;

In particular, an object of Astrict which is invariant under E strict is given by a single object X of A, together with an isomorphism F W X ! E.X/ such that E.F / ı F D TX1 . If .X0 ; F0 / and .X1 ; F1 / are two such objects, then the morphism group HomAstrict .X0 ; X1 / D HomA .X0 ; X1 / carries a natural involution, namely E strict .ˆ/ D F11 ı E.ˆ/ ı F0 . Let us pass to the geometric situation, where M is an exact symplectic manifold 2 with corners, carrying a quadratic complex volume form M . Suppose that we have an involution M of M which preserves all the given structure (øM , IM , and the oneform M ; this will be a standing assumption whenever we talk about involutions of # exact symplectic manifolds). Make it into a brane automorphism M by adding the

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trivial grading (equal to zero) and the line bundle ˇ D M . On general grounds, this defines a unique auto-equivalence of the Donaldson–Fukaya category, which will be our E (we have previously considered this construction for automorphisms which are equal to the identity near the boundary; however, all one needs, besides exactness, is the weaker condition that the automorphism preserves IM near the boundary, which is clearly satisfied here). One can use the isomorphism M jL Š top .T L/ from Remark 11.18 to write the action of E on objects as    .L; ˛ # ; P # / 7! M .L/; ˛ # ı M ; M .P # ˝ top .T L// : (14.7) # #  Since M ı M D .IdM ; 0; M ˝ M M / is trivial, E 2 D E ı E is isomorphic to  the identity functor. More precisely, a trivialization of M ˝ M M gives rise to an 2 isomorphism T W E ! Id. There is an obvious such trivialization, defined via the  isomorphism top .DM / W M ! M M , but we wish to multiply it with 1 and take the associated T . The single nontrivial coherence condition (14.6) comes down to the simple fact that our trivialization is itself Z=2-invariant (under the action which consists of pulling back by M and exchanging the two factors). It is now obvious that our definition of pre-equivariant branes is simply an instance of the abstract algebraic construction carried out above. In particular, there is a Klinear graded category H F .M /strict whose objects are pre-equivariant branes, with morphism spaces HF  .L#0;C ; L#1;C /. This is equivalent to H F .M /, and its only advantage is that it carries a strict, rather than coherent, action of Z=2. On objects, this action is given by the previously described involution ; on morphisms, one sets

Š

 W HF  .L#0;C ; L#1;C /  ! HF  .L#0;C ˝ top .T L0;C /; L#1;C ˝ top .T L1;C // Š

 ! HF  .M .L#0;C ˝ top .T L0;C //; M .L#1;C ˝ top .T L1;C /// Š

 ! HF  .L#0; ; L#1; / (14.8) where the first isomorphism uses the fact that the top .T Lk;C / are canonically isomorphic to the restrictions of a common line bundle M , compare (14.7); the second one is functoriality under graded symplectic automorphisms; and the third one comes from composition with f0; ; f1; . There are a few other things one can say about the Z=2-action on H F .M /strict , basically equivariant versions of the properties stated in Section (12e). We omit the proofs, which are straightforward. • Change of sign. Let ˇ ! M be the trivial line bundle with the nontrivial Z=2-action. Obviously, HF  .L#0;C ; L#1;C ˝ ˇjL1;C / D HF  .L#0;C ; L#1;C / for all pre-equivariant branes L#k;C , but these isomorphisms anti-commute with (14.8). In particular, if L#0 and L#1 are actual equivariant branes, then tensoring one of them with ˇ reverses the sign of the Z=2-action on HF  .L#0 ; L#1 /, hence

14 Symplectic involutions

203

(if our ground field has char.K/ ¤ 2) exchanges the invariant and anti-invariant parts. • PSS isomorphism. If L# is an equivariant brane, then the PSS isomorphism HF  .L# ; L# / Š H  .LI K/ is compatible with the Z=2-actions on both sides. • Isotopy invariance. Take an isotopy of exact pre-equivariant Lagrangian branes, # which means a family f.L#s;C ; Ps; ; fs;C ; fs; /g, 0  s  1. As usual, the # fLs;C g determine an element of HF 0 .L#0;C ; L#1;C /, and their shadows one of HF 0 .L#0; ; L#1; /. These two elements are in fact images of each other under the Z=2-action . In particular, it follows that if two exact equivariant branes are isotopic within that class, there is an invariant isomorphism between them in H F .M /strict . (14c) Equivariant transversality. Our next issue is to realize the Z=2-symmetry by a naive group action on the chain level. Rather than aiming at maximal generality, we choose to impose a range of geometric constraints which simplify the solution. Assumption 14.3. Of our given exact symplectic manifold with corners M and involution M , we require that the fixed point set M fix is of real codimension two, and that its normal bundle  is trivial. We actually want to fix a complex trivialization of that bundle,  ! C, and add the following partial integrability condition: the trivialization should be the derivative of a map  W U ! C defined on some neighbourhood N U  M of the fixed point set, which satisfies  ı M D  and is such that @ fix (taken with respect to IM ) vanishes to first order along M . Suppose that we are given IM and the trivialization DjM fix . If N.IM / is the Nijenhuis tensor, the existence of any  with the properties described above implies that D ı N.IM / vanishes on M fix , which explains the name “partial integrability”. Assumption 14.4. Among all M -invariant exact Lagrangian submanifolds L, we consider only those which either do not meet the fixed point set at all, or else approach it from a constant direction. This means that there is some  2 R, the normal angle of L, such that D.T Lx / D e i R  C for all x 2 Lfix . There is a Z worth of possible choices for , but we will assume that a preferred one has been selected for each relevant Lagrangian submanifold. Before going on to the next part, we need to introduce some terminology. Let H be a M -invariant function on M . At every point x 2 M fix , the Hessian D 2 H in

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II Fukaya categories

normal direction is a well-defined quadratic form on x (because the first derivatives have to vanish). We say that H infinitesimally rotates x with speed h 2 R if .D 2 H /.Z/ D h jZj2

(14.9)

for Z 2 x , where j  j is the hermitian metric obtained by identifying  with the anti-invariant part of TM jM fix . Equivalently, take the Hamiltonian vector field XH , and project it by D to get a function D.XH / W U ! C. Then (14.9) says that D.XH / D ih at first order around x. Assumption 14.5. Take .L0 ; L1 / to be a pair of invariant exact Lagrangian submanifolds as in Assumption 14.4, both of which meet the fixed point set. Let 0 ; 1 be their normal angles. When considering Z=2-invariant Floer data .H; J /, we want to impose the following additional requirements. First, just like the assumption on IM , N taken with respect to any J t to vanish at first order along we want the derivative @ fix M . For the second part, let  be the “flow” of H . Then, there should be an h 2 R satisfying 1  0  h 2 .I 0/ (14.10) and an open subset V  Œ0I 1  M fix which contains ftg   t .Lfix 0 / for all t 2 Œ0I 1, such that H.t;  / infinitesimally rotates x with speed h for all .t; x/ 2 V . We have settled for controlling the normal Hessian only on a sufficiently large open subset, since by definition of Floer datum H.t;  / must vanish near @M , hence cannot satisfy (14.9) on the whole of M fix . What matters is that for x 2 Lfix 0 , Dxt W x !  t .x/ is rotation by e iht . In particular,  1 .L0 / approaches the fixed point set from the direction e i.0 Ch/ R. Again by definition of Floer datum, this must be transverse to L1 , which by itself already implies that 1  0  h 2 R n Z (unless of course  1 .L0 / \ L1 \ M fix D ;). The additional requirement (14.10) picks out one “chamber” or connected component of R n Z. This may seem rather vacuous, since one can always change 0 or 1 to achieve the desired result, but it becomes meaningful when considering more than one Floer datum at the same time. Functions with these properties are still plentiful. In particular, we can arbitrarily prescribe H both on the fixed point set and away from an arbitrarily small neighbourhood of that set (subject to being zero near the boundary, and of course M -invariant). The same holds for the almost complex structures, by the following argument. First, make the neighbourhood U of M fix smaller if necessary, so that jU becomes a submersion with symplectic fibres. The tangent bundle TM jU splits into TM v D ker.D/ and its symplectic orthogonal complement TM h . The latter is trivialized by D, and its restriction to M fix is precisely . Because IM is compatible with the trivialization DjM fix , We know that along the fixed point set, the .hv/ component of IM vanishes and the .hh/ component equals i. The additional

14 Symplectic involutions

205

N says that these two conditions should extend to the first derivatives requirement on @ (actually, for the .hh/ component this already follows from M -symmetry). When finding other almost complex structures with the same property, one is clearly free to choose the .vv/ component along M fix as desired. Lemma 14.6. Let L0 ; L1 be invariant exact Lagrangian submanifolds, and .H; J / an invariant Floer datum as in Assumption 14.5. Take some u 2 MZ .y0 ; y1 / such that the image of u is entirely contained in M fix . Then, u is regular in M iff it is regular in M fix . Proof. The linearized operator is Z=2-equivariant, hence splits into C  ˚ DZ;u : DZ;u D DZ;u

The invariant part is readily identified. Namely, if we write ufix for u considered as a solutions of Floer’s equation in M fix , with respect to the Lagrangian submanifolds fix fix fix .Lfix 0 ; L1 / and the Floer datum .H jM ; J jM /, then C DZ;ufix D DZ;u :

(14.11)

Concerning the anti-invariant part, we can use D to identify the relevant vector bundle with E D Z  C ! C, and then the subbundle F  Ej@Z becomes Fs;0 D e i 0 R, Fs;1 D e i1 R. Up to first order along M fix , we know that D ı J t D i ı D. Moreover, for jsj  0 we know that x D u.s; t/ lies close to  t .L0 /, hence .t; x/ 2 V , which means that the Hamiltonian vector field X.t;  / of H.t;  / satisfies D.X.t; x// D ih .x/ to first order at x. This means that on the strip-like ends,  DZ;u D @s C i@ t C h:

(14.12)

One can deform the operator so that (14.12) holds everywhere, and this shows that  index.DZ;u / D 0, see for instance Lemma 11.9. On the other hand, Lemma 11.5  ( D 0, j† j D 1) says that DZ;u is injective, hence must be invertible. In view of (14.11), this implies the desired result.  With this at hand, it is routine to prove that Floer data within the rather special class defined in Assumption 14.5 are generically regular. Given one such datum, one first looks only at the fixed point set M fix , and perturbs it slightly to make it regular there. By Lemma 14.6, this ensures regularity of all invariant Floer trajectories in M . The remaining ones can be made regular by a version of the standard transversality argument; see [85, Proposition 5.13]. Let .L0 ; : : : ; Ld /, d  2, be M -invariant exact Lagrangian submanifolds as in Assumption 14.4, each of which meets the fixed point set. Let 0 ; : : : ; d be their normal angles. Suppose that we have chosen Floer data for the pairs .L0 ; Ld / and

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II Fukaya categories

.Lk1 ; Lk /, 1  k  d , as in Assumption 14.5. Consider the universal family S d C1 ! Rd C1 of .d C 1/-pointed discs, with the Lagrangian labels .L0 ; : : : ; Ld /, and equip it with some M -invariant perturbation datum .K; J /, compatible with the previous choices of Floer data. Lemma 14.7. Let .r; u/ 2 M d C1 .y0 ; : : : ; yd / D MS d C1 .y0 ; : : : ; yd / be an inhomogeneous pseudo-holomorphic polygon such that the image of u is entirely contained in M fix . Then, u is regular in M iff it is regular in M fix . Proof. As before we have to show invertibility of the anti-invariant part of the operator  DS d C1 ;r;u . The first step is again to write that part as DS;u W W 1;p .S; E; F / ! p d C1 ik L .S; E/ for S D Sr , E D S  C, and F jCk D e R. On each strip-like end, the behaviour of this operator is prescribed by the Floer data, which means that   S;k DS;u D @s C i@ t C hk as in (14.12), where the hk satisfy suitable conditions (14.10). One then combines Proposition 11.13, which shows vanishing of the index, with Lemma 11.5.  In parallel with the discussion following Lemma 14.6, this implies that a generic choice of M -invariant perturbation datum .K; J / makes all the moduli spaces MS d C1 regular (in fact, the argument is somewhat simpler this time around, because .d C 1/pointed discs have no automorphisms). Remark 14.8. As mentioned in Remark 11.6, one can prove a version of Lemma 11.5 which does not require as much control over the behaviour of the relevant operators on the strip-like ends. That in turn would enable one to weaken some of the conditions imposed here, replacing them by purely topological counterparts which are just strong enough to ensure that the anti-invariant linearized operators have index zero. In a different direction, one could consider Z=2-actions where the fixed point set has higher codimension, but then the explicit control over the anti-invariant operators needs to be maintained, and in fact extended to the entire Riemann surfaces, since degree arguments are no longer available. (14d) Grading and signs. We remain in the situation from Assumption 14.3, but require additionally that dim.M fix / D dim.M /  2 > 0 to avoid making trivial exceptions. Assume that M carries a (not necessarily M -invariant) quadratic complex 2 volume form M . Because of the triviality of the normal bundle, this induces a similar form on the fixed point set; in terms of local square roots, M jM fix D ˙  .dz/ ^ M fix : Let L be an invariant exact Lagrangian submanifold as in Assumption 14.4, which meets the fixed point set. Any brane structure .˛ # ; P # / on L induces one on the fixed

14 Symplectic involutions

207

point set Lfix  M fix . The grading is given by ˛ fix;# D ˛ # jLfix  1 , where  is the normal angle. For the Pin structure, we take the frame bundles P ! L and P fix ! Lfix associated to the tangent bundles, and embed the second into the first by mapping an Lfix -frame .X1 ; : : : ; Xn1 / to .e i ; X1 ; : : : ; Xn1 /, where the first entry is a vector in normal direction. The pullback of the double cover P # ! P by this embedding, together with its inherited action of Pinn1  Pinn , defines the desired Pin-structure P fix;# . Now let .L#0 ; L#1 / be two such Lagrangian submanifolds with brane structures, and .H; J / a Floer datum as in Assumption 14.5. As mentioned before, the restriction fix of this to the fixed point set defines a Floer datum for .Lfix 0 ; L1 /. Take some y 2 fix fix C.L0 ; L1 / which is contained in M , and write y for the same object considered fix as an element of C.Lfix 0 ; L1 /. Then, with the induced brane structures on the fixed point sets as defined above, we have Lemma 14.9. i.y fix / D i.y/ and o.y fix / Š o.y/. Proof. Following (11.39), we think of i.y/ and o.y/ as the index and determinant # line of an orientation operator for the pair of linear Lagrangian branes ƒk;y given by # the tangent spaces of the Lk at y. The claim is that one can choose the orientation operator to be Z=2-equivariant, with the invariant part being the orientation operator for y fix , and the anti-invariant part invertible. This is similar to the proofs of Lemmas 14.6 and 14.7. To keep the notation simple, we will write down the details only for H D 0. In that case, y is an invariant intersection point of L0 and L1 , and one can construct an orientation operator DH;y as follows. Take the vector bundle E D H  TMy , with a non-constant complex structure JE which at infinity is determined by our Floer datum (this means that if  is the single strip-like end of H , then JE . .s; t// D J t for s  0). We equip E with the trivial connection r. The subbundle F D F  Ej@H is given by a path in Gr.TMy / which joins .0/ to

.1/. Finally, to make this into an actual orientation operator, one needs to choose a brane structure .˛# ; P# / for , which of course constrains the homotopy class of the path. Within that class, one can choose JE to be invariant under DM jTMy , and the path to consist of invariant Lagrangian subspaces, so that DH;y D @N r becomes Z=2equivariant. Moreover, the anti-invariant part of , which is a family  .s/ of lines in y Š C, can be taken of the form  .s/ D e i .s/ R for some function  W Œ0I 1 ! R going from .0/ D 0 to .1/ D 1 . Given that, the invariant part inherits an induced brane structure, following the same procedure as for the Lagrangian submanifolds C above, which makes DH;y into an orientation operator for y fix . On the other hand, the condition (14.10), which in our case just says that 1  0 2 .I 0/, implies that   index.DH;y / D 0. Again by Lemma 11.5, it follows that DH;y is invertible. 

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Remark 14.10. It is maybe interesting to rephrase this argument in terms which avoid index theory, and instead rely on the elementary definitions of index and orientation space from (11.25). Namely, choose so that, in addition to the conditions imposed above, it lies in P  Gr.V /. Then the index and determinant line depend only on the intersections .s/ \ .1/. However, we can choose the function  to be strictly decreasing, and then the fact that 1  0 2 .I 0/ means that  .s/ \  .1/ D 0 for all s < 1. Hence, the intersections are all concentrated in the invariant part.  Suppose that our Floer datum is regular. Take a u 2 MZ .y0 ; y1 / which lies in M and is an isolated point of the moduli space. As in the proof of Lemma 14.6,  .y0fix ; y1fix /. Our next we write ufix for the same map considered as an element of MZ claim is that with respect to the isomorphisms from Lemma 14.9, the contributions of u and ufix to the Floer differentials have the same sign. In the notation from Section (12f), this means that the following diagram commutes: fix

oy1 

cu

/ oy0

Š

oy fix

cufix

1

(14.13)

Š



/ oy fix . 0

To see that, one needs to recall from (12.2) that cu is obtained by gluing together the linearized operator DZ;u and the orientation operator DH;y1 , and deforming the outcome to DH;y0 . In our case, all these operators will be Z=2-equivariant; their invariant parts can be identified with the corresponding operators in M fix , which give rise to cufix ; and their anti-invariant parts are invertible, hence contribute trivially to the determinant lines. Moreover, the homotopy from DZ;u #DH;y1 to DH;y0 goes through operators of the same form, which means that the anti-invariant part remains invertible. The same argument applies to the contributions cr;u of invariant isolated pseudo-holomorphic polygons, in the situation of Lemma 14.7. (14e) The equivariant Fukaya category. We now combine the two strands of our discussion. Let M be an exact symplectic manifold with corners, equipped with an involution M which obeys Assumption 14.3. Moreover, it should carry an invariant 2 quadratic complex volume form M . Consider pre-equivariant exact Lagrangian branes on M , with the additional condition that if they are equivariant, then Assumption 14.4 must be satisfied (which, unless they are disjoint from the fixed point set, entails a choice of normal angle). Choose a Floer datum for any pair of such branes. This should be natural with respect to the Z=2-action , in the sense that .HL#

# 0; ;L1;

; JL#

# 0 ;L1;

 / D .M HL#

# 0;C ;L1;C

 ; M JL#

# 0;C ;L1;C

/:

(14.14)

14 Symplectic involutions

209

If both branes involved are equivariant ones, this means that the Floer datum must be M -invariant; in that case, we further constrain the choice as in Assumption 14.5. We also want all the Floer data to be regular. For pairs of equivariant branes, we saw after Lemma 14.6 that this can be achieved, and in all other situations, there is no problem since (14.14) relates one Floer datum to another one: choose one of the two to be regular, and then the other will obviously have the same property. The symmetry in the choice of Floer data means that we have canonical isomorphisms  W CF  .L#0;C ; L#1;C / ! CF  .L#0; ; L#1; /;

(14.15)

which strictly commute with the differential and whose square is the identity. Next, choose perturbation data for all .d C 1/-tuples of objects, subject to the usual consistency requirements, and to the analogue of (14.14). Regularity can be achieved within that class thanks to Lemma 14.7, and the resulting composition maps d strictly commute with (14.15). The outcome is that we have an A1 -category F .M /naive , the equivariant Fukaya category, whose objects are pre-equivariant branes subject to the condition stated above, with morphisms spaces CF  .L#0;C ; L#1;C /. This A1 -category supports a naive Z=2-action given by (14.15). To compare it to the previous construction, let us pass to cohomology, where there is an obvious full and faithful embedding H F .M /naive ! H F .M /strict compatible with the strict Z=2-actions. This is in fact an equivalence: invariant Lagrangian submanifolds which violate the conditions in Assumption 14.4 cannot be realized as equivariant branes in H F .M /naive , but one can admit them as pre-equivariant branes, which are isomorphic objects on the cohomology level. In other words, such submanifolds exist as objects of H F .M /naive , but they can not be made into fixed points of the Z=2-action. The advantage gained is that H F .M /naive is indeed the cohomological category of an A1 -category with a naive Z=2-action (unlike H F .M /strict , which was a purely cohomology level construction). To describe (14.15) in a little more detail, let us pick some yC 2 C.L#0;C ; L#1;C / (we add the brane structures to the notation, since the Hamiltonian H and hence the set C now depend on them) and consider the associated pair of linear Lagrangian branes ƒ#k;C , which are the tangent spaces of the L#k;C at yC .k/. For the corresponding element of C .L#0; ; L#1; /, which is y D M .yC /, the relevant linear Lagrangian branes ƒ#k; are the tangent spaces of the L#k; , by definition of the Z=2-action on objects. As part of our equivariant brane structure, we have isomorphisms .fk; /y .k/ W ƒ#k; ! ƒ#k;C ˝ top .T Lk;C /yC .k/

(14.16)

covering the action of DM W .T Lk; /y .k/ ! .T Lk;C /yC .k/ . In view of Remark 11.18 we have preferred isomorphisms top .T Lk;C /yC .k/ Š yC .k/ , and can further use parallel transport along y to identify yC .0/ Š yC .1/ . Tensoring both linear Lagrangian branes involved with a common real line does not affect the absolute index or (up to canonical isomorphism) the orientation space. Hence, (14.16)

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II Fukaya categories

induces an isomorphism o.y / ! o.yC /. The K-normalizations of the inverses of these isomorphisms form (14.15). The fact that this is a Z=2-action comes down to the condition on (14.2) and the fact that, if one acts by the nontrivial element e 2 ker.Pinn ! On / on both linear Lagrangian branes ƒ#k;C simultaneously, the induced automorphism of o.y/ is the identity. One case which may deserve special attention is when we have two equivariant branes L#k , and y is an element of C.L#0 ; L#1 / which is invariant under the action of M . In that case, the construction above yields an automorphism of o.y/, whose sign is determined by the local geometric data. We make the same simplifying assumption H D 0 as in the proof of Lemma 14.9, so y is an invariant intersection point. Recall that by Lemma 14.2, the isomorphisms fk determine preferred trivializations of the normal bundles to the fixed parts Lfix . Under D, these trivializations project to give k ik orientations of the lines e R. Set wk D C1 if this orientation agrees with the one given by the generator e ik , and wk D 1 otherwise. Then, Lemma 14.11. jo.y/jK belongs to the -invariant part of CF  .L#0 ; L1# / if w0 w1 D C1, and to the anti-invariant part otherwise. Proof. We follow the strategy from Remark 14.10, choosing a path 2 P  Gr.V / from .0/ D .T L0 /y to .1/ D .T L1 /y which consists entirely of Z=2-invariant Lagrangian subspaces. Denote by F the associated vector bundle over Œ0I 1. This time, however, we want to choose the homotopy class of the anti-invariant summand in a slightly different way, namely so that  .s/ D e i .s/ R with .0/ D 0 , .1/ D 1 C k for some k 2 Z which satisfies .1/k D w0 w1 . Choose a brane structure on F , consisting of a real-valued phase ˛# and Pin structure P# , which over the endpoints restricts to our given linear Lagrangian branes. To realize (14.15), we also want to have an isomorphism f W P# ! P# ˝ top .F / which covers the action of DM on F . Lemma 14.2 tells us that such isomorphisms correspond bijectively to orientations of  . We can certainly adjust f so that it agrees with (14.16) at s D 0, and by following the induced orientations and using our assumption on the homotopy class of  , it then follows that the same thing with hold at s D 1. As a consequence, one can read off the induced automorphism of o.y/ by taking (11.25) and looking at the action of DM on (11.20). In our case, this yields a .1/ factor for each nonzero intersection .s/ \ .1/ , s < 1. The number of such points will be  w0 C w1 modulo 2, which yields the desired statement (equivalently, the argument can be translated back into index-theoretic terms, as in the proof of Lemma 14.9). 

III Picard–Lefschetz theory

The third ingredient in our mix is the symplectic geometry of Lefschetz fibrations. This is a remarkably rich topic, and we cannot even remotely do it justice. The definition used here is basically the same as in [131]. In fact, much of our discussion amplifies or comments on that paper (this means that our treatment is somewhat onesided; readers interested in getting a more comprehensive view may find [40], [64], [11], [17] to be possible starting points). The first two sections review some elementary aspects. The main addition compared to [131] is the notion of matching cycle, due to Donaldson (see also [16]). Roughly speaking, matching cycles arise when one has a Lefschetz fibration whose generic fibres are in their turn total spaces of secondary Lefschetz fibrations. We will call such a setup a bifibration; in the lowest nontrivial dimension, it is a less sophisticated version of the symplectic branched covers considered in [12], [13]. After that, we consider an extension of the Floer TQFT based on holomorphic sections of Lefschetz fibrations, and use that to derive a basic comparison theorem between the algebraic theory of Section 5 and the symplectic geometry of Section 16. This theorem, which equates (on the level of objects) the action of the twist functor T with that of a symplectic Dehn twist , is in fact a reformulation of the long exact sequence from [131]. We will not reproduce the entire proof of that sequence, but we will explain in some detail how to transform the result into the form needed here. Given a Lefschetz fibration whose base is a disc,  W E ! D, one can take the double cover branched along one fibre Ez (in algebraic geometry, this would be called “base extension”); in fact, by varying z one gets a family of such covers, and this is a particularly simple example of a bifibration. The double cover obviously has a Z=2action, to which one can apply the equivariant techniques from Sections 6 and 14. We will use this idea to define an A1 -category F ./, called the Fukaya category of the Lefschetz fibration, which serves as an intermediate between the Fukaya categories of the fibre Ez and total space E. Admittedly, the definition via the double cover trick is not the most natural one, but it allows one to reuse a lot of machinery. We then look at Lefschetz pencils in the classical algebro-geometric sense, and prove the results stated in the Introduction. Finally, the last section is devoted to a sequence of examples arising from a slightly different part of algebraic geometry, namely the Milnor fibres of simple singularities of type .Am /. These are primarily of historical interest, being one of the first cases in which explicit algebraic models of derived Fukaya categories were proposed [85] (as it turns out, these models were only correct in real dimension  4). At that point, the discussion becomes somewhat less self-contained than in the rest of the book, but it hopefully demonstrates the kind of geometric and algebraic tricks used in concrete computations. For more complex

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III Picard–Lefschetz theory

applications of the same general machinery, we refer the reader to the literature, for instance [14], [15], [130], [59].

15 First notions (15a) Fibrations with singularities. We will be looking at maps  W E ! B where E and B are exact symplectic manifolds with corners, with B connected and dim.E/  dim.B/, and where  is .IE ; IB /-holomorphic. We make three additional assumptions, which together ensure that the maps have reasonable behaviour along the boundary. • Transversality to @B. Let C  @B be any boundary stratum. At every point x 2 E such that y D .x/ 2 C , we have D.TEx / C T Cy D TBy . This obviously implies that  1 .C / is a boundary stratum of E of the same codimension as C . The union of such strata is @v E D  1 .@B/, which we call the vertical boundary of E. The union of all the boundary faces not contained in @v E is called the horizontal boundary of E, and denoted by @h E. • Regularity along @h E. If F is a boundary face of E which is not contained in @v E, then jF W F ! B is a smooth fibration. This implies that any fibre Ey is smooth near its boundary @Ey D Ey \ @h E. So far, the discussion has focused on the differential topology, but we also need to take care of the symplectic aspect. The tangent space at any point x 2 E splits into TEx D TExh ˚ TExv

(15.1)

where the vertical part is TExv D ker.Dx /, and the horizontal part TExh is its orthogonal complement with respect to øE (or equivalently for the metric obtained from øE and IE ; we have already encountered such a splitting briefly in Section (14c), but in a purely technical role). Our final requirement is • Horizontality of @h E. If x lies in some boundary face F which is not contained in @v E, then TExh  TFx . A map  W E ! B which satisfies the three conditions above is called an (exact symplectic) fibration with singularities. Let us explore some easy consequences. Write Crit./  E for the set of critical points, and Critv./ D .Crit.//  B for that of critical values. As already mentioned above, regularity along @h E means that Crit./  E n @h E. In the case where dim.B/ D 2, transversality to @B implies that Crit./ \ @v E D ; as well, but this is no longer true for higher-dimensional base spaces.

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15 First notions

Lemma 15.1. If y … Critv./, the fibre Ey (equipped with the restrictions øEy D øE jEy , Ey D E jEy , IEy D IE jEy ) is an exact symplectic manifold with corners. Proof. The only nontrivial thing to check is that the negative Liouville vector field points inwards along @Ey D Ey \ @h E. We know that the negative Liouville vector field of E points strictly inwards along every boundary face F . Now suppose that F does not belong to @v E. Because TE h is parallel to F , the TE v component of our vector field also points strictly inwards along F . But that component is just the  negative Liouville vector field for Ey , which yields the desired result. The splitting (15.1) defines a connection over B n Critv./. Since TE h is parallel to @h E, this connection has well-defined parallel transport maps, which means that for every path ˇ W Œ0I 1 ! B n Critv./ we get a distinguished diffeomorphism (depending smoothly on the choice of path) hˇ W Eˇ.0/ ! Eˇ.1/ :

(15.2)

A standard computation shows that hˇ is actually an isomorphism of exact symplectic manifolds with corners. Fix a regular value of , write M D E for the corresponding fibre, and let B 0  B n Critv./ be a small neighbourhood of . By using a suitable family of parallel transport maps, one gets a local trivialization E 0 D B 0  M ! E over B 0 . The differential forms E 0 ; øE 0 obtained by pulling back E ; !E are of the form E 0 D M C C d (some function);

!E 0 D øM C d ;

(15.3)

where is a one-form which vanishes in .TE 0 /v -direction. This can be thought of as a function-valued one-form on the base, 2 1 .B 0 ; C 1 .M; R//. In fact, if one passes from functions to Hamiltonian vector fields, then is precisely the connection one-form of .TE 0 /h . To see that, take some vector field Y on B 0 , and let Y h D .Y; X/ be its unique horizontal lift to E 0 . In the Cartan formula iY h .d / D LY h d.iY h /, the Lie derivative is again a one-form which vanishes in fibre direction. The defining property of Y h can therefore be written as follows: iX øM  d.iY / D 0

on each fibre fyg  M .

Hence, X is fibrewise the Hamiltonian vector field of the function  .Y /, which establishes the claimed property of . To summarize, these considerations show that over the regular part of B, what we have is an Aut.M /-fibre bundle with a preferred connection. Note that the symplectic structure on B plays no role in this. Remark 15.2. The definition of fibration with singularities used in [131] replaces our third condition (horizontality of TE h ) with the following stricter version:

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III Picard–Lefschetz theory

• Triviality near @h E. Let M D E be some fibre. Then there is an open neighbourhood V 0  B  M of B  @M , an open neighbourhood V  E of @h E, and a fibrewise diffeomorphism V 0 ! V which is the identity on V 0 \ .f g  M /, such that the pullback of E is B C M , and the pullback of IE is the product almost complex structure IB  IM . The condition on E means that the connection TE h is trivial near @h E, and (over the regular part of the base) provides a canonical reduction of the structure group to Aut.M; @M /. This property makes fibrations a little easier to deal with, but on the other hand, there are natural constructions which fail to satisfy it. (15b) Isomorphisms, deformations. Two fibrations with singularities are isomorphic if there are isomorphisms of the total spaces and bases (as exact symplectic manifolds with corners) which are compatible with the fibration maps. It is equally obvious how to define a general notion of deformation of such fibrations, namely as a one-parameter family of all the geometric data involved. Instead of doing that, we concentrate on two more special kinds of deformations, which will appear later on in the formulation of several technical results. The principal feature of the first kind is that the isomorphism class of the regular fibres is preserved; while in the second kind, the structure of the fibres does not change at all. Fix manifolds E and B as well as a map  between them. Take smooth families s .øsE ; Es ; IEs /, .øB ; Bs ; IBs /, s 2 Œ0I 1, of exact symplectic structures on them, which for each s make  into a fibration with singularities. We say that this deformation is well-behaved along @h E if there are smooth functions Rs on E, such that the vector fields Y s defined by @ s øsE .; Y s / D @s E  dRs (15.4) are parallel to @h E. For an equivalent formulation, take the product Q D Id W Ez D Œ0I 1  E ! BQ D Œ0I 1  B, and equip the total space with øEz D d.Es C Rs ds/. This is obviously symplectic on ker.D /, Q hence we get a splitting of T Ez as in (15.1); what (15.4) says is that the horizontal part T Ez h of that splitting is parallel to @h Ez D Œ0I 1  @h E. This allows one to define parallel transport maps for paths in Œ0I 1  .B n Critv.//. In particular, by going along Œ0I s  fyg one gets exact symplectic isomorphisms kys W .Ey ; ø0Ey ; E0 y / ! .Ey ; øsEy ; Es y /:

(15.5)

More concretely, these are defined by integrating the vertical (with respect to øsE ) component of Y s jEy , which would be the vector field in Moser’s argument if one used @s Es y  dRs jEy as a primitive for @s øsEy . The kys depend smoothly on .s; y/, and commute with parallel transports along paths in B n Critv./ up to Hamiltonian isotopy. In other words, the fibre bundle structure of the regular part of E is unchanged under deformations which are well-behaved along @h E.

15 First notions

215

The other, more restrictive, notion is that of a deformation which is constant along s ; Bs ; IBs / as the fibres. This is given by one-parameter families .øsE ; Es ; IEs / and .øB before, but where we now require that the one-forms @s Es  dRs should vanish on TE v . Clearly, this means that the maps (15.5) are the identity. In addition to that, we also require that in a neighbourhood of @h E, IEs jTE v should be independent of s. Take a fibration with singularities  W E ! S, where S is a surface with boundary. Let be a one-form on E which vanishes in TE v -direction, and which is zero in a neighbourhood of @E [ Crit./. For s 2 Œ0I 1 and some constant c  0, consider Ss D .cs C 1/S ; øsS D dSs ; ISs D IS ; Es D E C s  C cs    S ; øsE D dEs ;

(15.6)

s jTE v D !E jTE v is nondegenerate, we get an s-dependent splitting TEx D Since !E TExh;s ˚ TExv which deforms (15.1). By choosing c sufficiently large, we can ensure that øsE jTExh;s is positive (with respect to the orientation induced from the base), so that øsE is symplectic. Take IEs to be the unique øsE -compatible almost complex structure which equals IE in TE v -direction, and agrees with the pullback of IS on TE h;s . Because TE h;s D TE h near @E, IEs will agree with IE there, which implies that the boundary is weakly IEs -convex. Next, the negative Liouville vector field associated to Es and øsE points strictly inwards along @h E, since that property depends only on the TE v summand; see the proof of Lemma 15.1. Now consider a boundary circle C  @S , and the corresponding F D  1 .C /  @v E. For x 2 F , TExh contains a one-dimensional subspace TFx \ TExh , which moreover has a canonical orientation by projecting to S and using the boundary orientation there. The fact that the negative Liouville vector field points inwards along F is equivalent to

E j .TFx \ TExh / > 0:

(15.7)

Because S itself makes S into an exact symplectic manifold with corners, one has   S j .TFx \ TExh / > 0. This, together with the fact that Es D E C cs    S near @v E, implies that the analogue of (15.7) continues to hold throughout the deformation. We have now verified that .E; øsE ; Es ; IEs / is a exact symplectic manifold with corners for each s, hence that (15.6) is a deformation which is constant along the fibres. From a fibre bundle point of view, the effect is to change the symplectic connection by . For future use, we record one specific application: Lemma 15.3. Let  W E ! S be a fibration with singularities, with S being a surface with boundary. Let ˇ W Œ0I 1 ! int.S/ n Critv./ be an embedded path, and . s / an isotopy in Aut.Ec.0/ ; @Ec.0/ / starting with  0 D id. Then there is a deformation which is constant along the fibres, starting at s D 0 with the given fibration structure, such that the resulting family of parallel transport maps along ˇ satisfies hsˇ D hˇ ı  s :

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III Picard–Lefschetz theory

Additionally, if U  S is an open subset containing some part of ˇ, one can arrange the deformation to be such that @s .Es    Ss / vanishes outside  1 .U /.  (15c) Relative quadratic volume forms. Let  W E ! B be a fibration with singularities. The bundle of relative quadratic volume forms is 2 KE=B D   C .TB/˝2 ˝ C .TE/˝2 : top

top

Suppose that we have a nowhere zero section 2E=B of this (the simplest way to get 2 one would be as the quotient 2E =B of quadratic volume forms on the total space and base, but this is not the general situation). If b is a regular value of , we have the short exact sequence 0 ! T .Eb / ! TEjEb !   TBb ! 0, which induces a canonical isomorphism 2 2 KE=B jEb Š KE : b Hence, 2E=B determines quadratic complex volume forms 2Eb on the smooth fibres. Similarly, we get a relative squared phase map ˛E=B W Gr.TE v / ! S 1 , defined away from the critical points, and whose restriction to each smooth fibre agrees with the respective ˛Eb . In this situation, the parallel transport maps (15.2) acquire natural gradings, which means functions h# ˇ W Gr.Eˇ.0/ / ! R such that exp.2 i h# ˇ .ƒ// D

˛Eˇ.1/ .Dhˇ .ƒ// : ˛Eˇ.0/ .ƒ/

(15.8)

These are obtained by starting with the zero function for IdEˇ .0/ , and then extending continuously to ˇjŒ0I s for all s. In the special case of a closed path ˇ, the outcome is that hˇ becomes a graded symplectic automorphism, as defined in Section (12i). (15d) Lefschetz fibrations. Let S be a connected compact Riemann surface with nonempty boundary. We make this into an exact symplectic manifold with corners by choosing a one-form S such that S j@S > 0 and øS D dS > 0. An (exact) Lefschetz fibration is a fibration with singularities  W E 2nC2 ! S, whose critical points are generic (also called nondegenerate) and locally integrable. This means that IE is integrable in a neighbourhood of Crit./, and that D (seen as a section of the bundle HomC .TE;   T S / of complex linear maps) is transverse to the zero-section. The second condition is equivalent to saying that the complex Hessian D 2  at every critical point is nondegenerate. In addition, we will assume that there is at most one critical point in each fibre, so that the projection Crit./ ! Critv./ is bijective; this last assumption is for convenience only, and could easily be removed, at the cost of making notations somewhat more awkward later on. Nondegeneracy implies that the critical points are isolated, so Crit./ is a finite subset of int.E/, and similarly Critv./ a finite subset of int.S/. Locally near each

15 First notions

217

critical point and its value, one has holomorphic coordinates in which  becomes the standard quadratic map 2 Q.x/ D x12 C    C xnC1 (15.9) Generally øE will not be standard in these coordinates. However, one can find a deformation of the fibration which is well-behaved along @h E (and which in fact is local near the critical point), such that at the other end of the deformation the Kähler form becomes the standard form in a given holomorphic Morse chart. This increases the importance of the following basic model: Example 15.4. Let Q W CnC1 ! C be as in (15.9), and k W CnC1 ! RC the function k.x/ D .jxj4  jQ.x/j2 /=4. For some fixed r; s > 0, define E D fx 2 CnC1 W jQ.x/j  r; k.x/  sg;

(15.10)

and equip it with the restriction of the standard symplectic form on CnC1 , its standard primitive .i=4/.z d zN  zN dz/, the given complex structure IE D i, and the map  W E ! S D rD obtained by restricting Q. The boundary faces are @v E D fx 2 E W j.x/j D rg;

@h E D fx 2 E W k.x/ D sg:

The cutoff function k is chosen so as to make TE h parallel to @h E. To see that, one notes that TExh is generated over C by .rQ/x D 2x, N and checks that d kx .x/ N D 0, d kx .i x/ N D 0. Each nonsingular fibre Ez , z ¤ 0, is symplectically isomorphic to the subset Bs S n  T S n consisting of cotangent vectors of length (in the standard metric)  s. Explicitly, Bs S n D f.u; v/ 2 RnC1  S n W hu; vi D 0g with the symplectic form du ^ dv, and an isomorphism Ez ! Bs S n for z > 0 is given by z .x/ D .im.x/jre.x/j; re.x/jre.x/j1 /: Unfortunately, while the negative Liouville (negative radial) vector field does point inwards along @E, it is not true that E is weakly IE -convex (the fibres are, but not the total space). Hence, this is not quite an example of an exact Lefschetz fibration as defined here, even though from a purely symplectic viewpoint, it has all the desired features. Of course, one could change IE to improve the situation, but there is no real point in doing that, since ultimately E will serve only as a local model. Remarks 15.5. For other definitions of symplectic Lefschetz fibrations, see e.g. [39] or [64]. Usually, both base and fibres are assumed to be closed, which makes things technically a little simpler. The existence of a local normal form (15.9) is a consequence of the holomorphic Morse Lemma [8] (more precisely, the statement is that for any choice of holomorphic coordinates on the base, one can find coordinates on the total space in which  D Q). The deformation which allows one to make the symplectic structure standard in such coordinates is constructed in [131, Lemma 1.6]. The geometry of Example 15.4 is discussed exhaustively in [131, Section 1.2].

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III Picard–Lefschetz theory

(15e) Bifibrations. We will now look at diagrams of the type ‰D ı$

E

$

/S

/) W :

(15.11)

Here ‰ is a Lefschetz fibration whose total space is of dimension 2n C 4  6. The intermediate space S is an exact symplectic four-manifold with corners, and both $ and are fibrations with singularities. Of , we assume that it has no critical points, and that its fibres are connected surfaces with boundary (in particular, S has only corners of codimension  2). Of $, we require only that it should have no critical values on @h S, where this part of the boundary is defined with respect to . Take a point w 2 W and restrict $ to the fibres lying over that point. The outcome, denoted by $w (15.12) Ew ! Sw ; is a fibration with singularities, unless w is one of the critical values of ‰ (all other necessary properties follow from the assumptions about $, and ‰). One can therefore think of E ! W as a Lefschetz fibration whose smooth fibres are in turn fibered over the surfaces Sw . We call (15.11) an (exact) Lefschetz bifibration, provided it satisfies the following additional conditions: • The critical points of $ are generic, in the sense that D$ is transverse in HomC .T E; $  T S/ to the zero-section and to the stratum of linear maps of complex rank one. If x is a critical point of $ but a regular point of ‰, then the restriction of IE to the fibre Ew , w D ‰.x/, is integrable locally near x. Moreover, (15.12) is a Lefschetz fibration for each w 2 @W . • Take w 2 Critv.‰/, and let x be the unique critical point in Ew . Then we require that the restriction of D 2 ‰x to ker.D$x /  T Ex is nondegenerate. The almost complex structure IS is integrable in a neighbourhood of $.x/. Moreover, all critical points of $w lying on Ew n fxg must be nondegenerate, and their $-values pairwise different and also different from $.x/. We will now gradually unravel the meaning of this. Genericity and dimension counting show that D$ never vanishes, and that the set Crit.$/ where it has C-rank one is a smooth two-dimensional submanifold. In fact, Lemma 15.6. Crit.$/ is a smooth IE -holomorphic curve. Proof. For x 2 Crit.$/ and X 2 T Ex , consider the partial second derivative x 2 $x;X W ker.D$x / ! coker.D$x /: D

(15.13)

The tangent space of Crit.$/ consists of those X for which this vanishes. An easy x 2 $x;I X D IS D x 2 $x;X . computation in local coordinates shows that D  E

219

15 First notions

Lemma 15.7. Let x be a point of Crit.$/ n Crit.‰/, with ‰.x/ D w. Then x is a critical point of $w ; moreover, the intersection Crit.$/ \ Ew is transverse at x iff it is nondegenerate as critical point of $w . Proof. Because x is not a critical point of ‰, the image of D$x is transverse to the tangent space of Sw . This implies that ker.D$x / D T .Ew /x , and that the map T .Sw /x ! coker.D$x / induced by inclusion is an isomorphism. It follows that D.$w /x D 0, and that for X 2 T .Ew /x one can identify (15.13) with the ordinary second derivative D 2 .$w /x;X . If the intersection Crit.$/\Ew is transverse, (15.13) must be nonzero for any nonzero X 2 T .Ew /x , which therefore means that the critical point is nondegenerate. The same argument yields the converse.  Lemma 15.8. At every critical point x of ‰, the map $jCrit.$/ W Crit.$/ ! S is a local embedding, and ‰jCrit.$/ W Crit.$/ ! W has a branch point of order 2. Proof. Since D$x has rank one but D‰x vanishes, D $.x/ maps coker.D$x / isomorphically to T Ww , w D ‰.x/. Under this identification (15.13) becomes the ordinary second derivative of ‰ at x, which means that the tangent space to Crit.$/ is generated over C by a nonzero X such that D 2 ‰x .X; Y / D 0 for all Y 2 ker.D$x /. We assumed that D 2 ‰x is nondegenerate on ker.D$x /; hence D$x .X/ ¤ 0, which proves our first assertion. The nondegeneracy assumption also implies that D 2 ‰x .X; X/ ¤ 0, which shows that ‰jCrit.$/ has nonvanishing second derivative at x.  Lemma 15.9. Around each critical point of ‰ and its values in S, W , there are local holomorphic coordinates in which 2 $.x1 ; : : : ; xnC2 / D .x12 C x22 C    C xnC2 ; x1 /;

.y1 ; y2 / D y1 :

Proof. This is a consequence of the parametrized version of the holomorphic Morse Lemma (whose proof is the same as in the unparametrized case). First introduce holomorphic local coordinates on S and W , centered around the $-value and ‰-value of our critical point, in which has the required form .y1 ; y2 / D y1 (this is possible because IS is locally integrable by assumption). Writing $ D .$1 ; $2 / W E ! C2 locally in these coordinates, one finds that D$2 is nonzero at the critical point, so that the level set f$2 D 0g is smooth, and moreover that $1 D ‰ restricted to this level set has a nondegenerate critical point at x. As a consequence, nearby level sets f$2 D y2 g are also smooth, and $1 restricted to each has a single nondegenerate critical point, depending holomorphically on y2 . One can therefore find local coordinates x2 ; : : : ; xnC2 on each level set, depending holomorphically on y2 , in which $1 restricted to the level set has the usual quadratic normal form. Adding x1 D $2 yields 2 a coordinate set on E in which $.x1 ; : : : ; xnC2 / D .f .x1 / C x22 C    C xnC2 ; x1 /

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III Picard–Lefschetz theory

for some holomorphic function f . By assumption x D 0 is a nondegenerate critical point of $1 D ‰, so f .t / D ct 2 C O.t 3 / for some c ¤ 0. As a last step, replace x1 by f 1=2 .x1 /, and y2 with f 1=2 .y2 /.  We will now discuss the implications of these technical results for the geometry of the critical curve and its images, ‰jCrit.$/

Crit.$/

$ jCrit.$/

/ Critv.$/

jCritv.$/

/) W :

Because we have assumed that Critv.$/ \ @h S D ;, the boundary of Crit.$/ equals its intersection with ‰ 1 .@W /; and since øE is exact, each connected component of Crit.$/ must have a boundary. This implies that no component can be contained in a fibre of ‰, and therefore that ‰jCrit.$/ is a finite branched covering. In view of Lemma 15.7, the requirement that $w is a Lefschetz fibration for w 2 @W means that in some neighbourhood of @Crit.$/, the map $jCrit.$/ is an embedding, and ‰jCrit.$/ a local isomorphism. Applying some basic pseudo-holomorphic curve theory one sees that $jCrit.$/ is almost everywhere injective, so that its image Critv.$/ is an IS -holomorphic curve without multiply covered components. Moreover, near its boundary @Critv.$/ D Critv.$/ \ @v S the curve is smooth, and jCritv.$/ a local isomorphism. Lemmas 15.7 and 15.8 combined show that Critv.$/ is also smooth in a neighbourhood of Sw for w 2 Crit.‰/, and that each such Sw contains a single branch point of jCritv.$/, which moreover has order 2. Figure 15.1 summarizes the situation. We define the fake critical value set Fakev.‰; $/ to be the set of those w 2 W nCritv.‰/ such that Critv.$/ either has a singularity lying on Sw , or is smooth near Sw but intersects it in a non-transverse way. Equivalently, these are the w such that Ew is smooth but (15.12) fails to be a Lefschetz fibration. By our previous discussion, Fakev.‰; $/ is a finite subset of int.W /, in particular W n.Critv.‰/[Fakev.‰; $// is still connected.

16 Vanishing cycles and matching cycles (16a) Framings. Let .M 2n ; !/ be any symplectic manifold, and V  M a Lagrangian sphere. A framing of V is a diffeomorphism v W S ! V , where S is the unit sphere in some .n C 1/-dimensional inner product space (S depends on V ; note that this is not the same concept as the framed manifolds appearing in, say, the Pontryagin–Thom construction). Similarly, if we have two framed Lagrangian spheres vk W Sk ! Vk (k D 0; 1), a framed isotopy I D .i; j0 ; j1 / between then

221

16 Vanishing cycles and matching cycles

E

S

$

Critv.$ / Crit.$ /

‰

W Critv.‰/

Fakev.‰; $ /

Figure 15.1

consists of a family of Lagrangian embeddings i s W S ! M , 0  s  1, where S is again the unit sphere in some inner product space, and two isometries jk W S ! Sk such that vk ı jk D i k . Remark 16.1. Suppose that we have the same V0 D V1 D V and S0 D S1 D S, but with two different maps v0 ; v1 . Obviously, if the vk are isotopic in the space of diffeomorphisms Diff.S; V /, then our two framed Lagrangian spheres are isotopic. On the other hand, if the vk differ by some orientation-reversing isometry of S, we get an isotopy of framed Lagrangian spheres by setting i s D v0 for all s, and j0 D Id, j1 D v01 ı v1 . Therefore, as long as one is only interested in isotopy classes (which is basically true of all our uses of the concept), the framing information reduces to a class in 0 .Diff.S; V /=O.S//. It is known that Diff.S n /=OnC1 is contractible for n  3, so framings are inessential in those dimensions. However, for high n, the existence of exotic spheres means that 0 .Diff.S n /=OnC1 / is often nontrivial. (16b) Vanishing cycles. Let  W E 2nC2 ! S be an (exact, as usual) Lefschetz fibration. A vanishing path is an embedded path  W Œ0I 1 ! S running into a critical point, more precisely such that  1 .Critv.// D f1g. To each such path one can associate its Lefschetz thimble  , which is the unique embedded Lagrangian .n C 1/-ball in E satisfying . / D .Œ0I 1/, .@ / D f.0/g. The boundary V D @ , which is a Lagrangian sphere in E .0/ , is called the vanishing cycle of  . Since it bounds a Lagrangian disc in E, any vanishing cycle is automatically exact (of course, this is nontrivial only for n D 1). Before continuing, we note an elementary fact:

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III Picard–Lefschetz theory

Lemma 16.2. For any Lagrangian submanifold F  E, a point x 2 F is a critical point of  iff it is a critical point of jF .  In the case of a disc  with the properties described above, the function p D  1 ı j W  ! Œ0I 1 must have a maximum somewhere in the interior, and by Lemma 16.2, this point is necessarily the unique singular point x1 2 E .1/ . A similar argument shows that the Hessian of p at that point is nondegenerate, so p is a Morse function with a single critical point. After equipping  with a metric which is standard in local Morse coordinates near the critical point, one can use the flow of rp to provide a diffeomorphism B D B.Tx1  / ! 

(16.1)

from the unit ball in the tangent space at x1 to the whole Lefschetz thimble. In particular, the restriction to the boundary yields a framing v W S D @B ! V , which is independent of the choice of metric up to isotopy. We take v to be part of the structure of the vanishing cycle, which therefore will always be considered as a framed exact Lagrangian sphere. We have drawn some consequences of the defining property of the Lefschetz thimble, but its existence and uniqueness still need to be verified, and we will do so now, starting with uniqueness. Here is a useful fact: Lemma 16.3. Let ˇ be an embedded path in the regular part of the base S, and F  E a submanifold which is fibered over ˇ, such that each Fˇ.s/  Eˇ.s/ is Lagrangian. Then the whole of F is Lagrangian iff parallel transport along any piece of ˇ maps the Fˇ.s/ into each other.  For a Lefschetz thimble, this means that parallel transport along  carries the level sets  \ E .t/ D p 1 .t/, 0  t < 1, into each other. As t ! 1, these level sets shrink to fx1 g. Hence,  is necessarily contained in the set 0 D fx 2 E .t0 / ; 0  t0 < 1 W lim h jŒt0 It1  .x/ D x1 g [ fx1 g: t1 !1

The structure of 0 may not be immediately obvious, because the parallel transport vector field has a pole at x1 . To avoid that difficulty, choose a smooth function h W S ! R with h..t// D 0, and with the property that . 0 .t/; rh .t / / is a positively oriented basis of tangent vectors for any t . Let XH be the Hamiltonian vector field of H D h ı . It is easy to see that on  1 ..Œ0I 1///, this points in the same direction as the parallel transport vector field, while having a stationary point at x1 . Hence 0

is contained in s 00 D fx 2 E .t/ ; 0  t  1 W lim H .x/ D x1 g: s!1

(16.2)

16 Vanishing cycles and matching cycles

223

Since the stationary point x1 is hyperbolic, one can apply the stable manifold theorem to prove that 00 is an embedded .nC1/-ball, whose boundary is the sphere 00 \E .0/ . Because the same holds for  by assumption, the inclusions   0  00

are necessarily equalities, which proves the uniqueness of the Lefschetz thimble. Existence is dealt with by observing that (16.2) is Lagrangian (because the symplectic flow H is contracting on it) and has all the other desired properties. Remark 16.4. As a byproduct of the discussion above, we see that the vanishing cycle of any piece jŒt0 I 1 is related to that of the whole by parallel transport: V D h1

jŒ0It0  .V jŒt0 I1 /:

(16.3)

This observation can be used to define Lefschetz thimbles and vanishing cycles for paths  which may not be embedded, but which still satisfy  1 .Critv.// D f1g and  0 .1/ ¤ 0. Namely, for t0 sufficiently close to 1, jŒt0 I 1 will be an embedded path, and one can extend  jŒt0 I1 by parallel transport. The Lefschetz thimbles defined in this way are immersed, but not necessarily embedded, Lagrangian balls; but their boundaries, which are precisely (16.3), will have the same properties as in the embedded case. Example 16.5. Take the toy model  W E ! S as defined in Example 15.4. This is not quite a Lefschetz fibration, but the missing condition (lack of holomorphic convexity) is irrelevant for the present purpose. The only critical value is 0, and for any vanishing path , the Lefschetz thimble can be explicitly determined: [ p  D .t/S n I (16.4) p

0t1

p here zS D fx 2 C W x D ˙ zy for some y 2 S n  RnC1 g. To see that (16.4) holds, one uses the function k from Example 15.4, which is unchangedp under parallel transport, and observes that k 1 .0/ is precisely the union of the subsets zS n for all z 2 S . n

nC1

(16c) Dehn twists. Let M be an exact symplectic manifold with corners. To any framed exact Lagrangian sphere V  M one can associate its Dehn twist (also called Picard–Lefschetz transformation) V , which is an element of Aut.M; @M / unique up to isotopy (it would be more precise to say that the space of additional choices made in constructing V is contractible). To define this, one uses the Lagrangian tubular neighbourhood theorem, which extends the framing v W S ! V to a symplectic embedding W Bs S ! int.M /, where Bs S is the subset of cotangent vectors on S of length  s, for some s > 0. Now take a function b W R ! R satisfying ( b.r/ D 0 for r  s; and (16.5) b.r/  b.r/ D r everywhere:

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III Picard–Lefschetz theory

Define H.x/ D b.j 1 .x/j/ on .Bs S/ n V , and extend it to zero on the rest of M n V . Then V is the time 2 flow of this Hamiltonian function, extended in the unique possible way over V itself. To get a more concrete picture, recall that the Hamiltonian flow of the length function jj on T  S n S can be identified with the normalized geodesic flow (moving any nonzero tangent vector with unit speed along the geodesic emanating from it, irrespective of the length of this vector). Hence, the flow of b.jj/ moves tangent vectors with speeds b 0 .jj/. As the length approaches zero, the derivative goes to 1=2, which in the limit yields the antipodal map on the zero-section. To see that the extension over the zero-section is smooth (and not just continuous), write b.r/ D ˇ.r 2 /  r=2, where ˇ is smooth (this is possible because b.r/ C r=2 is even). The flow of ˇ.jj2 / is smooth on the whole cotangent bundle, while that of jj2 =2 becomes the antipodal map at time 2 (moreover, the two Hamiltonians Poisson-commute, since both are functions of jj). With this in mind, one can write the time 2 map of b.jj/ as composition of two symplectic automorphisms, each of which is obviously smooth near the zero-section. By looking a little closer at this construction, one can see that V is actually exact (this is nontrivial only in the lowest dimension, where M is a surface, so one can also prove it by inspecting the standard picture of a Dehn twist). By definition, the class ŒV  2 0 .Aut.M; @M // is invariant under isotopies of V as a framed exact Lagrangian sphere. In particular, it does not depend on an orientation of V . Remark 16.6. The question of whether ŒV  depends on the framing is somewhat subtle. For the induced auto-equivalence of Fukaya categories, it is known that the framing does not matter (up to isomorphism of A1 -functors). We will not prove this statement, but a weak version (concerning the action on objects only) follows from Corollary 17.17. Take an exact Lefschetz fibration  W E ! S, and a vanishing path . Let V be the associated vanishing cycle, lying in M D E .0/ . Take a loop in S n Critv./ which doubles , winding anticlockwise around .1/ (Figure 16.1). Then the monodromy around is essentially a Dehn twist along V , more precisely h ' V

(16.6)

where ' is isotopy in Aut.M / (if one assumes that the fibration is trivial near @h E in the sense of Remark 15.2, the statement can be made slightly sharper: the isotopy will take place inside the group Aut.M; @M /). This is the symplectic Picard–Lefschetz theorem. We will not give the proof here, but it is maybe worth while summarizing the basic strategy. First, note that deforming the structure of the Lefschetz fibration (in the sense of deformations well-behaved along @h E, see Section (15b)) does not affect the validity of the statement, since it changes both sides only by an isotopy. As mentioned in Section (15d), one can find such a deformation which makes the

16 Vanishing cycles and matching cycles

225

symplectic structure on E standard in a local holomorphic Morse chart around the critical point. In such local coordinates, the vanishing cycle of a short path going into the critical point can be written down explicitly, taking Example 16.5 as the model. Similarly, the monodromy along a sufficiently small circle around the critical point can be determined by integrating the relevant vector field, and turns out to be the Dehn twist along the vanishing cycle. An easy parallel transport argument then allows one to deduce the corresponding statement for the original paths  and .



Figure 16.1

Remark 16.7. Obviously, the deformation used to prove (16.6) is not quite unique, but the space of possible choices is contractible. This shows that there is in fact a unique homotopy class of isotopies (rel endpoints) between the two symplectic automorphisms. (16d) Distinguished bases. We now concentrate on exact Lefschetz fibrations where the base is a disc, S D D. Choose a base point 2 S n Critv./. We want to think of this as lying on or close to the boundary. This means that if … @S , we additionally choose an embedded base path W Œ0I 1 ! S n Critv./ leading from .0/ D to some point .1/ 2 @S. A distinguished basis of vanishing paths is an ordered family  D .1 ; : : : ; m /, m D jCritv./j, of vanishing paths starting at , such that the following conditions hold: • Different k intersect only at , and their starting directions RC  k0 .0/  T S should be pairwise distinct. • If 2 @S , we can think of the RC  k0 .0/ as radial half-lines in the upper half-plane H , so they have a natural clockwise ordering. The rule is that this should coincide with how the k are ordered in . In the other case … @S , we

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III Picard–Lefschetz theory

ask that each k should intersect only at , and that RC  k0 .0/ ¤ RC  0 .0/. Then, the RC  k0 .0/ are radial half-lines in a slit plane C n RC , where the missing direction is that of 0 .0/. This leads to an ordering, and we impose the same condition as before. The set of all possible isotopy classes of bases carries an action of the braid group Br m , by so-called Hurwitz moves: the k-th standard generator ˇk (1  k < m) corresponds to the k-th elementary Hurwitz move  7! ı D Lk . This leaves almost all paths unchanged, ıj D j for j ¤ k; k C 1, and transforms the remaining two as shown in Figure 16.2. It is easy to verify that the braid relations hold.

ık Lk

k

kC1

ıkC1 D k

Figure 16.2

A related but slightly different perspective is the following one. Let D  Diff C .S/ be the subgroup of those oriented diffeomorphisms of S which are the identity on (and on the base path , if appropriate), and which map the subset Critv./ to itself. Obviously, 0 .D/ acts on the set of isotopy classes of distinguished bases; using the contractibility of Diff.S; @S/, one sees easily that this action is simply transitive. The same kind of argument shows that 0 .D/ is isomorphic to Br m ; however, this isomorphism is not canonical, so a little more care needs to be exercised. First, recall that to any embedded path  W Œ1I 1 ! S which avoids (and ) and satisfies 1 .Critv.// D f1I 1g, we can associate its half-twist t 2 D, which is the identity outside a small neighbourhood of .Œ1I 1/, and whose local behaviour is characterized by Figure 16.3. Obviously, there are some choices involved, but the resulting class Œt  2 0 .D/ is unique, and is invariant under isotopies of  within the relevant class of paths. Given a distinguished basis , one can define paths k (1  k < m) by concatenating the inverse of k with kC1 , and smoothing out the corner. Then, mapping the standard generators ˇk to half-twists along the k gives an isomorphism  W Br m ! 0 .D/. With respect to this isomorphism, an elementary move is simply expressed by Lk  D t1 ./ D  .ˇk1 /./: k

16 Vanishing cycles and matching cycles

227

  t . /

Figure 16.3

On the other hand, such a move also affects  , changing it by a conjugation: ı  .ˇ/ ı tk D  .ˇk1 ˇˇk /: Lk  .ˇ/ D t1 k By combining these two observations, one sees that for all k1 ; : : : ; kr 2 f1; : : : ; m1g and 1 ; : : : ; r 2 f˙1g, 1 r : : : ˇk /./: Lkrr : : : Lk11  D  .ˇk r 1

In words, the action of an element ˇ 2 Br m on a distinguished basis , defined as a sequence of elementary moves, agrees with the image of  under the diffeomorphism  .ˇ 1 /. This shows in particular that any two bases are related by Hurwitz moves. We will now introduce the corresponding concepts for vanishing cycles. Inside a fixed exact symplectic manifold with corners, consider ordered collections of m framed exact Lagrangian spheres, V D .V1 ; : : : ; Vm /. We will not distinguish between collections obtained by applying (framed exact) isotopies to the Lagrangian spheres involved. In this context, the elementary Hurwitz moves are transformations V 7! W D Lk V for some 1  k < m, given by setting Wj D Vj for all j ¤ k; k C 1, and Wk D Vk .VkC1 /; WkC1 D Vk : (16.7) As usual, a general Hurwitz move is any composition of these transformations and their inverses. The connection to the previous topic should be clear: any distinguished basis of vanishing paths  determines an ordered collection V D V  , consisting of the vanishing cycles Vk D V k ; this is called the associated distinguished basis of vanishing cycles. Remark 16.4 and (16.6) show that the elementary Hurwitz move Lk on vanishing paths induces the corresponding move of the distinguished basis (up to isotopy): V L k  ' Lk V :

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III Picard–Lefschetz theory

Define Hurwitz equivalence of collections V to be the relation generated by Hurwitz moves and isotopies. In those terms, the outcome is that the Hurwitz equivalence class of V is an invariant of the Lefschetz fibration (strictly speaking, of the fibration together with the choice of base point and if necessary base path). In a suitable sense, it also remains invariant under deformations which are well-behaved along the horizontal boundary, as defined in Section (15b). (16e) Constructing Lefschetz fibrations. When defining Hurwitz moves on collections V , we did not a priori ask that these should arise as vanishing cycles. As it turns out, there is no difference, meaning that any collection can be realized as distinguished basis for some fibration. We will now make this more precise. Lemma 16.8. Let V be a framed exact Lagrangian sphere in an exact symplectic manifold with corners M . Take the disc S D D, and an embedded path from some point .0/ 2 @S to .1/ D 0. Then there is an exact Lefschetz fibration  W E ! S having Critv./ D f0g, which comes with an identification E .0/ D M under which the vanishing cycle of  corresponds to V (compatibly with framings). Moreover, this fibration is trivial near @h E. The construction is easy. One starts with the model from Example 15.4, whose fibre is a compact piece Bs S n of the cotangent bundle of the sphere, for sufficiently small s. As in the definition of Dehn twist, one then uses the Lagrangian tubular neighbourhood theorem to find an embedding W Bs S n ! M extending the framing v W S n ! V . This is used to glue together the local model with the trivial fibration D  .M n V /. The symplectic forms on the fibres agree, but on the total space, they have to be patched together using suitable cutoff functions. All one needs to do is to preserve the two-form of the local model in a neighbourhood of the relevant Lefschetz thimble (16.4); then, all vanishing paths will yield the desired Lagrangian sphere. Once one can construct Lefschetz fibrations with a single prescribed vanishing cycle, it is straightforward to glue them together with a nonsingular part to get the general result: Lemma 16.9. Let V D .V1 ; : : : ; Vm / be a collection of framed exact Lagrangian spheres in an exact symplectic manifold with corners M . On the disc S D D, choose a base point (and if necessary base path ), and a distinguished basis of vanishing paths .1 ; : : : ; m /. Then there is an exact Lefschetz fibration  W E ! S D D, whose critical values are precisely the endpoints 1 .1/; : : : ; m .1/; this comes with an identification E D M , under which the (framed) vanishing cycles V k correspond to the Vk . Moreover, this fibration is trivial near @h E.  Lemma 16.8 has a uniqueness counterpart. Namely, suppose that we have a Lefschetz fibration over D which has a unique critical point lying in the fibre over 0,

16 Vanishing cycles and matching cycles

229

and which is trivial near @h E (this last requirement is technical and of marginal importance; it arises from the condition that @E should be weakly IE -convex, which behaves awkwardly with respect to deformations of the symplectic connection). Then this fibration is completely determined up to deformation (in the sense of deformations well-behaved along @h E) by its fibre over 1 and the vanishing cycle in it, considered as a framed Lagrangian sphere up to isotopy. A corresponding statement holds in the more complicated situation of Lemma 16.9. (16f) Grading issues. Let M 2n be an exact symplectic manifold with corners, car2 rying a quadratic complex volume form M . If V  M is a framed exact Lagrangian sphere which admits a grading (no particular choice of grading is necessary), then the Dehn twist V is naturally a graded symplectic automorphism, in the sense of Section (12i); the real-valued grading function #V is determined by the condition that it is identically zero outside a neighbourhood of V (where V is the identity). The proof is not difficult: a cohomology check shows that there are no obstructions in dimensions n > 1, and the remaining case is elementary. On the other hand, consider a Lefschetz fibration equipped with a relative quadratic complex volume form, 2E=S . This induces forms on the smooth fibres, and with respect to these, any vanishing cycle admits a grading (but there is no naturally distinguished choice of one). One could argue as before, namely that all cases are trivial except for the lowest-dimensional one, which can be solved by hand. However, there is a more elegant argument, which is similar to the one proving exactness of vanishing cycles: let  be the vanishing path. In a neighbourhood of  1 ./  E, write 2E=S D 2E =2S (one can actually do this everywhere, since by assumption T S is trivial, but we will not need that). Since  is contractible, the squared phase function ˛ W  ! S 1 associated to 2E certainly admits a lift to R. The restriction of that function to the boundary agrees with the squared phase function of V up to a constant (which depends on  0 .0/), so the desired fact follows. Finally, inspection of the proof of (16.6) sketched above shows that the isotopy constructed there lifts to the group Aut gr .M / of graded symplectic automorphisms, if the monodromy is given the grading defined by (15.8), and the Dehn twist is graded in the way we have just described. 2 Finally, if we are given M , an M , and a collection of framed exact Lagrangian spheres which all admit a grading, then the associated Lefschetz fibration as constructed in Lemma 16.9 will admit a relative quadratic complex volume form, unique up to homotopy relative to E D M . Again, there is no reason to make a specific choice of grading (only the existence is relevant). Remarks 16.10. The fact that higher-dimensional Dehn twists are symplectic was discovered by Arnol’d [3]. The corresponding Lagrangian nature of vanishing cycles

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III Picard–Lefschetz theory

was pointed out to the author by Donaldson, even though it may well have been known earlier. As for the Picard–Lefschetz theorem, it is classical on the topological (monodromy as diffeomorphism) level. Two of the many possible references are [92], [5]. The symplectic version can be found in [131, Proposition 1.15]. Distinguished bases of vanishing cycles are also a classical notion, particularly so in the local cases considered in singularity theory [5], [4]. The observation that, on the symplectic level, the total space can be reconstructed from such a basis is more recent (see the papers quoted at the start of this chapter). Gradings of Dehn twists are discussed in more detail in [125]. (16g) Matching cycles. Let  W E ! S be an (exact, as always) Lefschetz fibration. Take an embedded path  W Œ1I 1 ! int.S/ such that 1 .Critv.// D f1I 1g. We can split this into a pair of vanishing paths with the same starting point, ˙ .t/ D .˙t/ for t 2 Œ0I 1, hence get a pair of vanishing cycles V ˙  M D E.0/ . Let us begin by looking at a particularly simple situation, namely when these two are equal. In that case, † D   [  C (16.8) is a smooth Lagrangian submanifold of the total space E (by definition of Lefschetz thimble, parallel transport along  maps the intersections † \ 1 .t/ to each other for all 1 < t < 1, which gives a local chart .1I 1/  V ˙ around the overlap   \  C D V ˙ ). Being the result of gluing two balls along their boundaries, † is necessarily a homotopy sphere. If we assume that the framings of the V ˙ are isotopic, then it is a standard sphere differentiably. In fact, given a choice of isotopy between the two framings, one can combine the diffeomorphisms from (16.1) to obtain a framing of † . We will refer to (16.8) as the naive matching cycle construction. Requiring two vanishing cycles to strictly agree seems hardly realistic, but one can relax the assumption somewhat, at the cost of losing control over the position of the resulting sphere in E. Namely, suppose that the V ˙ are isotopic as framed exact Lagrangian spheres in M . When that is the case, we say that  is a matching path. Any framed exact Lagrangian isotopy I D .i; j0 ; j1 / between the two vanishing cycles will be similarly called a matching isotopy. Elementary symplectic geometry tells us that there is an isotopy . s / in Aut.M; @M / such that  0 D Id,  s ı i 0 D i s . By Lemma 15.3 there is a deformation of the given exact Lefschetz fibration, which is constant along the fibres and is concentrated in a neighbourhood of E.1=4/ , such that the resulting deformation of the parallel transport map along jŒ1=2I 0 is hsjŒ1=2I0 D  s ı hjŒ1=2I0 I here, the parallel transport on the right-hand side is that of the initial (s D 0) structure. The effect on the vanishing cycles is that V s D  s .V  /, while V C remains unchanged. At the other endpoint s D 1 of the deformation, we have an exact Lefschetz fibration structure for which the two vanishing cycles agree, hence can define a

16 Vanishing cycles and matching cycles

231

Lagrangian sphere in the total space by the naive matching cycle construction. Since the two framings of the vanishing cycles are also the same (up to the identifications i 0 , i 1 between the relevant vector spaces), the naive matching cycle comes with a framing. Lemma 7.1 applied to the reverse deformation yields a conformally sym1 0 / into .E; !E D !E /, and we use that to map the naive plectic embedding of .E; !E matching cycle back to the original symplectic form. Up to isotopy of framed exact Lagrangian spheres, the outcome is independent of the choices of . s /, of the deformation of Lefschetz fibration, and of . We call it the matching cycle associated to  and I , and denote it by †;I  E. To keep the notation light, we will continue to write † in situations where there is an obvious choice of isotopy, or where we do not care which choice is adopted. Remark 16.11. The role of the isotopy I in the construction is not easy to estimate. Deforming it rel endpoints does not affect the isotopy class of the matching cycle. In the lowest dimension n D 1, the space of exact embedded circles in the surface M has trivial 1 , hence †;I is independent of I . The corresponding question in higher dimensions seems hard to answer. However, we will see later on, from Proposition 18.21 combined with Proposition 18.13, that the isomorphism class of †;I as an object of the Donaldson–Fukaya category is independent of I . In general, there is no guarantee that a given Lagrangian sphere in E can be represented as a matching cycle (however, see [16] for an asymptotic statement of that kind in the context of Lefschetz pencils). Moreover, different isotopy classes of matching paths can give rise to isotopic matching cycles. This is not a sporadic phenomenon, but rather something one can expect to happen in any sufficiently complex Lefschetz fibration. The following example is due to Auroux: Example 16.12. For simplicity, consider the lowest dimension dim.E/ D 4. Suppose that we have two embedded paths  and  in the base S joining pairwise distinct critical values, which intersect each other exactly once (at some point in the interior of the paths, obviously) and do so transversally. Suppose that  is a matching path; and that the vanishing cycles associated to the two halves of  intersect each other transversally and in a single point. Let t be the half-twist along . Because Dehn twists along circles intersecting once satisfy a braid relation, 0 D t3 ./ is again a matching path, and it turns out that the associated Lagrangian two-sphere †0 is isotopic to † (as a direct if somewhat unintuitive approach, one could: take a neighbourhood of all the Lefschetz thimbles involved; look at it from the point of view of cell attachment; realize that it is symplectically isomorphic to a compact piece of T S 2 ; and then apply [74] to prove uniqueness of Lagrangian S 2 s up to isotopy). There is also a slightly less interesting variation on this construction, where the vanishing cycles associated to the two halves of  are assumed to be disjoint, and one takes 0 D t2 ./.

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(16h) Concatenation of paths. Let 0 ; 1 be two matching paths, which meet only once, at a common endpoint. We assume that the paths are oriented in such a way that 0 .1/ D 1 .1/ (this is for convenience only; the orientations play no role in the following construction). Consider the path  D 1 #0 obtained by joining the k together, and then moving them to the right of the common endpoint (Figure 16.4; this is an unsymmetric operation, depending on how one orders our two paths). Quite clearly,  is again a matching path. The result we want to explain says that the associated matching cycle † is obtained from †0 by a Dehn twist along †1 . To

0

1



Figure 16.4

make the statement precise, framings and isotopies need to be taken into consideration. Each k splits into a pair of vanishing paths k˙ , so we have four associated vanishing cycles. It is convenient to assume that jŒ1I 1=2 agrees with 0 jŒ1I 0 up to the obvious reparametrization, and similarly jŒ1=2I 1 with 1 jŒ0I 1. The concatenated paths .jŒ1=2I 0/1  0C and .jŒ0I 1=2/  1 are homotopic rel endpoints; from that one gets a framed isotopy in E1 .0/ ,  hjŒ1=2I0 .V C / ' h1 jŒ0I1=2 .V 1 /:

(16.9)

0

Choose isotopies Ik as necessary to define the matching cycles †k ;Ik . Combined with (16.9) these give rise to an isotopy I between the vanishing cycles for the two halves  ˙ of , I

V  D hjŒ1=2I0 .V 0 / ' h1 jŒ0I1=2 .V C / D V C ; 1

and we use that to define another matching cycle †;I . Lemma 16.13. There is an isotopy of framed Lagrangian spheres in E, †;I ' † 1 ;I1 .†0 ;I0 /:

(16.10)

Proof. This is a typical argument by deformation to a simpler special case. If the result is true for some exact Lefschetz fibration, then it also holds for any deformation of it, in the sense of deformations well-behaved along @h E; that is because once one has found an isotopy for the deformed Lefschetz fibration structure, one can

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233

combine that with a family of conformally symplectic embeddings as in Lemma 7.1 to obtain an isotopy for the original structure. In view of Lemma 15.3 and of the trick used to prove the Picard–Lefschetz formula, it is therefore permitted to make the following assumptions: the pairs of vanishing cycles associated to the two halves of 0 coincide; the same for 1 ; the isotopies I0 ; I1 are constant; and near the unique critical point in E0 .1/ , there are holomorphic coordinates in which  becomes the standard quadratic form (15.9), and !E the standard Kähler form on CnC1 . Again without affecting the statement, we may deform the paths involved in such a way that: in the local coordinate near 0 .1/, 0 and 1 are the negative and positive real half-axes, respectively; and  agrees with 0 [ 1 outside a small neighbourhood of that point. Then †;I D †0 ;I0 [ †1 ;I1 outside a small neighbourhood of the critical point in E1 .1/ , while near that point we have the following local picture, coming from Example 16.5: [p †0 ;I0 D iRnC1 ; †1 ;I1 D RnC1 ; †;I D .t/S n (16.11) t

p

where  is a smooth embedded path in C lying in the closed lower right quadrant, p p which is such that .t / D it for t <  and .t/ D t for t > , where  is some small number. The third submanifold in (16.11) is the Lagrangian handle which appears in the Lagrangian connected sum or Lagrangian surgery construction. Denoting that sum equally by #, and using the conventions of [125, Section 2e], we have †;I D †1 ;I1 # †0 ;I0 : (16.12) But the right-hand sides of (16.10) and (16.12) are the same up to Lagrangian isotopy [124, Appendix]. The isotopies I0 ; I1 and I are all constant, so one checks easily that this is compatible with framings.  Remark 16.14. Given that we have used the notation  D 1 #0 , (16.10) can be written more intuitively (ignoring homotopies) as †1 #0 D †1 #†0 . There is also another important interpretation. Namely, let t1 be the half-twist along 1 . Since  is isotopic to t1 .0 /, what we have proved is that † t 1 .0 / ' † 1 .†0 /: In this form, the statement generalizes to matching paths 0 ; 1 intersecting arbitrarily. However, the proof is quite different from the special case treated above, and we will not present it here, since we have no real use for it (note however that Lemma 18.3 can be viewed as another special case). (16i) Vanishing cycles as matching cycles. Take an exact Lefschetz bifibration, with the notation as in (15.11). Our first observation is a simple consequence of

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the topology of the critical value curve Critv.$/  S and its projection to W . For each w 2 W n .Critv.‰/ [ Fakev.‰; $// we consider the surface Sw with its finite configuration of points Critv.$/ \ Sw D Critv.$w / in it. Since the intersection is transverse, this configuration varies smoothly with w. By following its motion along any path ˇ W Œ0I 1 ! W n .Critv.‰/ [ Fakev.‰; $//, one can define a braid parallel transport map, which is an oriented diffeomorphism kˇ W .Sˇ.0/ ; Critv.$ˇ.0/ // ! .Sˇ.1/ ; Critv.$ˇ.1/ // unique up to isotopy (one can make this symplectic, but because of the low dimension, no additional information would be gained by that). Now suppose that  W Œ0I 1 ! W is a vanishing path for ‰ which also avoids Fakev.‰; $/. Lemma 15.8 tells us that as t ! 1, two of the points of Critv.$/ \ S .t/ come together at a branch point of the covering jCritv.$/ W Critv.$/ ! W . Take the short path (unique up to unoriented isotopy) connecting these two points in S .s/ , for s close to 1. Its preimage under k jŒ0Is is an embedded path  W Œ1I 1 ! S .0/ such that 1 .Critv.$ .0/ // D f1I 1g. We call it the matching path associated to  , for the following reason: Lemma 16.15.  is a matching path for $.0/ , and for an appropriate choice of isotopy I , the associated matching cycle †;I is isotopic, as a framed Lagrangian sphere in E .0/ , to the vanishing cycle V . Proof. It is sufficient to show that, in E .s/ with s close to 1, the short path between the two points of Critv.$ .s/ / which are close to each other is a matching path, and that the resulting matching cycle is the vanishing cycle for jŒsI 1. The original statement then follows by transporting everything back from E .s/ to E .0/ . Consider the unique critical point of ‰ which lies in E .1/ . We introduce local holomorphic coordinates near that point and its images, with the properties stated in Lemma 15.9. As in the proof of (16.6) or of Lemma 16.13, we can locally modify the symplectic form on E so that it becomes standard in these coordinates. We may also deform the path  so that in the local coordinates around ‰.x/ 2 W , it is .t/ D 1  t for t close to 1. Consider the short vanishing path jŒsI 1, for s close to 1. The associated vanishing cycle is p V jŒsI1 D 1  s S n  CnC1 as in Example 16.5. On the other hand, the same sphere is also the matching cycle associated p to the straight line in S .s/ joining the two critical values .y1 D 1  s, y2 D ˙ 1  s/ of $ .s/ ; there is no difficulty in checking this, since in this case the matching cycle can be defined by the naive procedure (16.8). Finally, as in the proof of Lemma 16.13, one uses Lemma 7.1 to move back to the original symplectic form. Of course, the exact coincidence of matching cycle and vanishing cycle no longer holds for that form, but the argument provides a framed isotopy between them. 

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235

Remark 16.16. The isotopy class of V depends only on that of  within the space of all vanishing paths; in particular, it does not change if we move  over a point of Fakev.‰; $/. However, the braid parallel transport will usually change, hence so does the associated matching path . There is no contradiction in that, because different matching paths can give rise to the same matching cycle. In fact, the instances of that phenomenon mentioned in Example 16.12 arise in this way from the simplest (generic) fake critical values, where the curve Critv./ has cusps or nodes. Remark 16.17. As already mentioned, matching cycles were introduced by Donaldson (unpublished). The Lagrangian connected sum, especially in two dimensions, is studied in [109]. More recently, it has acquired some importance in mirror symmetry, see [140] and [139]; the last-mentioned paper contains a version of the formula (16.12). The contents of Section (16i) will be obvious to readers familiar with braid monodromy techniques, for instance with the relation between Auroux’ branched covers [11] and Donaldson’s pencils.

17 Pseudo-holomorphic sections (17a) Counting sections. One way to generalize the standard setup of Floer TQFT is to replace the target space with a Lefschetz fibration, and consider pseudo-holomorphic sections rather than maps. We begin by explaining this generalization in the simplest possible case, which is when the base surface is compact, and then enlarge that framework to allow strip-like ends. Our intent in doing that is to relate the algebraic twist construction T with the geometric Dehn twist  (Corollary 17.17). We devote special care to establishing this identification in a reasonably canonical form, which then carries over easily to the Z=2-equivariant context. Let  W E 2nC2 ! S be an (exact, as always) Lefschetz fibration. A Lagrangian boundary condition is a submanifold F nC1  E which is contained in @v E and disjoint from @h E, with the property that there are ˛F 2 1 .@S/ and hF 2 C 1 .F; R/ such that E jF D .jF / .˛F C S j@S/ C dhF : (17.1) Let us decode this a little. F is obviously Lagrangian in E. Since  has no critical points in @v E, jF W F ! @S must be a fibre bundle (Lemma 16.2). Each Fz  Ez , z 2 @S , is in fact an exact Lagrangian submanifold, and these submanifolds are carried into each other by parallel transport along the boundary (Lemma 16.3). To a large extent, the converse is also true: suppose that Fz  Ez , z 2 @S , is a family of exact Lagrangian submanifolds which are carried into each other by boundary parallel transport, and define F to be their union. Then E jF is closed, and one can find a function hF such that E jF  dhF vanishes when restricted to each Fz . If

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one additionally assumes that the Fz are connected, then E jF  dhF is necessarily the pullback of some one-form on @S , so F is a Lagrangian boundary condition. Fix such an F . A relative perturbation datum is a pair .K; J /, where: K is a one-form on E which is zero on vectors lying in TE v , which vanishes altogether in some neighbourhood of @h E [ Crit./, and which satisfies KjF D 0 2 1 .F /; and J is an almost complex structure on E which is øE -compatible, makes  pseudoholomorphic, and agrees with IE in some neighbourhood of @h E [ Crit./. Given that, an inhomogeneous pseudo-holomorphic section with boundary in F is a map u W S ! E satisfying

 .u.z// D z;

Du.z/ C J.u/ ı Du.z/ ı IS D Y.u/ C J.u/ ı Y.u/ ı IS ; u.@S/  F:

(17.2)

Here, Y is a section of the bundle Hom.  T S; TE v / determined by K in the following way: given any  2 T Sz , z D .x/, one can take its lift to a tangent vector X on the regular part of Ez ; the function K.X/ is independent of the choice of lift, and one defines Y ./x to be value at x of the associated Hamiltonian vector field on Ez . Since K is required to vanish near the critical points, so does Y , which therefore extends trivially over those points. Y vanishes near the critical points (hence extends continuously over those points) as well as near @h E. Remark 17.1. Relative perturbation data have a simple geometric interpretation. Consider the modified (still closed and fibrewise symplectic) two-form ønew E D øE  dK

(17.3)

and its associated connection, which is TExh;new D fX D X h C X v 2 TEx D TExh ˚ TExv W X v D Y.D.X//g: (17.4) Concerning the almost complex structure, note that by assumption J preserves the splitting TE h ˚ TE v , and is equal to IE on the first summand. Hence, the choice here is essentially that of an almost complex structure on TE v . There is unique J new new with the following properties: ønew  / is symmetric;  is J new -holomorphic; E .; J v new and on TE , J D J . In these terms, (17.2) is just the equation for a section to be new J -holomorphic in the standard sense. The connection viewpoint is also useful in understanding some of the basic properties of solutions u. For instance, the a priori estimate for the energy is Z Z Z v 2 1 j.Du/  Y.u/j D ˛F C  S  u R; E.u/ D 2 S

@S

S

17 Pseudo-holomorphic sections

237

where .  /v is the vertical component with respect to TE D TE h ˚ TE v , and the curvature term is given in a local trivialization (15.3) by   R.@s ; @ t / D @s .K  /.@ t /  @ t .K  /.@s / C f.K  /.@s /; .K  /.@ t /g ds ^ dt: (17.5) R 1 v;new 2 j , where the modified connecEquivalently, one can write E.u/ D S 2 j.Du/ tion (17.4) is used, and then (17.5) is indeed the curvature of that connection. Remark 17.2. There is also an analogue of moving boundary conditions in this context. Such a boundary condition simply consists of an arbitrary family of exact Lagrangian submanifolds Fz  Ez , z 2 @S. The appropriate notion of relative perturbation datum is a pair .K; J / where K, instead of vanishing when restricted to F , has the following property: for any X 2 TF jFz , z 2 @S , such that D.X/ is parallel to the boundary, we have d.iX K/jFz D ø.; X/jFz :

(17.6)

Otherwise, the definition of inhomogeneous pseudo-holomorphic sections remains the same. The meaning should be clear: the Fz are not necessarily preserved by parallel transport for the given symplectic connection on E, but passing to the modified form (17.3) with respect to some K satisfying (17.6) precisely restores that compatibility. Write ME=S for the moduli space of all solutions of (17.2). The linearization of N this equation at any point u of that space is given by a @-operator DE=S;u defined on the vector bundle u .TE v /, with boundary values given by u .TF v / over @S . As usual, we say that .K; J / is regular if DE=S;u is onto for all u, in which case the moduli space is a closed smooth manifold. This property holds for generic choices of .K; J /. Taking any such choice, one can define a toy model invariant by a mod 2 count of isolated points: ˆE=S D #ME=S 2 Z=2: (17.7) Example 17.3. Let  W E ! S be the local model from Examples 15.4 and 16.5. We put a Lagrangian boundary condition F on this which is the union of the standard p vanishing cycles Fz D zS n for jzj D r. On F we have xj D ..x/=r/xNj , hence X   E jF D 4i xj d xNj  xNj dxj D   8ri .z d zN  zN dz/ : j

Defining ˛F so that ˛F C S j@S agrees with the right-hand side of this, and taking hF D 0, shows that F is a Lagrangian p boundary condition (of course, we already knew this would be true, since the zS n are preserved by parallel transport in any

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direction). We take the standard complex structure J D IE and trivial inhomogeneous term K D 0. The resulting moduli space ME=S can be determined explicitly: it consists of the maps ua W S ! E  CnC1 ;

ua .z/ D r 1=2 az C r 1=2 aN

(17.8)

2 D 0 and jjajj2 D 1=2 [131, Lemma 2.16]. where a 2 CnC1 satisfies a12 C    C anC1 Moreover, all these are regular points of the moduli space. Strictly speaking, a little care should be exercised, namely s in (15.10) should be large enough so that the image of all these sections is indeed contained in E. This is related to the fact that @h E is not holomorphically convex, which keeps E from being a fully valid example of an exact Lefschetz fibration. Temporarily overlooking this deficiency, we find that the invariant (17.7) would be trivial for dimension reasons (no isolated points in the moduli space).

This computation serves as the model for a more general vanishing result. Let  W E ! S be a Lefschetz fibration whose base S is a disc, and which has a single critical point. Choose a point z on @S, and a vanishing path  from z to the critical value. We write M D Ez and V D V . By (16.6), the monodromy h around the boundary loop is isotopic to V (in a way which is unique up to homotopy rel endpoints). This means that we can put a moving Lagrangian boundary condition F on our fibration in the following way: start with Fz D V and move this almost all the way around the boundary by parallel transport, then use the isotopy h .V / ' V .V / D V to connect the outcome back to Fz . We will refer to this as a standard boundary condition. Lemma 17.4. ˆE=S D 0. We will only sketch the proof; for details see [131, Section 2.3]. To simplify the notation, assume that S D D is the unit disc, and that Critv./ D f0g. After a local deformation (which does not affect the invariant), one can assume that there are holomorphic Morse coordinates around the critical point in which the symplectic form is standard. Consider the family  r W E r ! S r , r 2 .0I 1, of Lefschetz fibrations obtained by restricting the given one to smaller discs S r D rD. Starting with F , one can equip these with a smooth family of boundary conditions F r , which for small r equals the one described in the local model in Example 17.3. Take a family of perturbation data for .E r ; F r / which is trivial when r is small. In that case, a compactness argument [131, Lemma 2.15] shows that in the limit r ! 0, all elements of ME r =S r will be contained in an arbitrarily small prescribed neighbourhood of the critical point, so we are in the same situation as Example 17.3. Finally, a routine cobordism argument carries the result back to the original .E; F /.

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239

As a complement to that, we would like to mention a more elementary but less general proof, which relies only on the dimension of the moduli spaces. First we need a basic fact about the topology of .E; F /: Lemma 17.5. If the relative first Chern class 2c1 .M; V / 2 H 2 .M; V / vanishes, its counterpart 2c1 .E; F / 2 H 2 .E; F / is .n C 1/ times the pullback of the standard generator of H 2 .S; @S/. From Section (15c) we know that E admits a quadratic complex volume form, hence 2c1 .E/ D 0, which implies that 2c1 .E; F / comes from an element of H 1 .F /. For n > 1, this means that it is the image of an element of H 2 .S; @S/ under   . For n D 1, one reaches the same conclusion after noticing that the restriction of 2c1 .E; F / to H 2 .; V /, where V is the Lefschetz thimble, necessarily vanishes because of the Lagrangian nature of . With that in mind, the argument reduces to a computation which can be done in the local model, and which we omit. The expected dimension of the moduli space ME=S at a point u is given by the index of the operator DE=S;u . Applying the relative Riemann–Roch formula (Lemma 11.7, in a slightly different formulation) and Lemma 17.5, one finds that R index.DE=S;u / D n C S 2c1 .u .TE v /; u .TF v // D 2n  1; which of course agrees with what we have seen in the local model. In particular, there are never any isolated points in ME=S , independently of the choice of (regular) relative perturbation datum. This contrasts with the outcome of the previous (more general) cobordism argument, which only ensured cancellation between such isolated points. (17b) Fibrations with strip-like ends. Let M be an exact symplectic manifold with corners, and O W Ey ! SO a Lefschetz fibration. Suppose that we have a finite set of marked boundary points † D † t †C on the base, and at each such point, an identification of the fibre Ey with M . We will also assume that the fibration is trivial locally near the marked points. This means that each  has a neighbourhood Uy D Uy  SO such that y Uy ! Uy ; ø y ;  y ; I y / D .Uy  M ! Uy ; ø O C øM ;  O C M ; I O  IM /: .j O Uy W Ej E E E S S S As usual take S D SO n †; and choose, for each , a strip-like end  whose image lies inside U D Uy n fg. Then, the restriction  W E ! S of O comes with canonical trivializations Z ˙  M ! E over the strip-like ends. We will call such objects Lefschetz fibrations with strip-like ends. Remark 17.6. In many cases, local triviality near the marked points does not hold a priori, but one can remedy that by a deformation, as follows. Let O W Ey ! SO

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y Choose some  2 @SO and set be a Lefschetz fibration which is trivial near @h E. M D Ey . One can then achieve local triviality near  by changing the symplectic and almost complex structures near that fibre (more precisely, in the terminology of Section (15b), this will be a deformation which is constant along the fibres). This is y Uy , we have similar to Lemma 15.3: in a local trivialization Uy  M ! Ej Ey D M C SO C C dR;

(17.9)

where both and the function R vanish near Uy  @M . We wish to replace these two terms in (17.9) by . C dR/, where is a suitable cutoff function on Uy vanishing close to . The exterior derivative of the modified one-form may no longer be symplectic on the total space, but that is easily remedied by adding the pullback of a suitable term on the base. In the same way, if we look at the almost complex structure IEy in our local trivialization, it becomes equal to ISO  IM near Uy  @M , and one can easily modify it so that it has the same product form close to the entire fibre Ey . We need to extend the notions of Lagrangian boundary condition and relative perturbation datum to the strip-like end context. For the first one, one takes an F  E which projects properly to the base S and satisfies (17.1). The family Fz is then automatically locally constant over each strip-like end, which means that there is some pair of Lagrangian submanifolds .L ;0 ; L ;1 / such that F  .s;k/ D L ;k for all s. Additionally, we then necessarily have ˛F D 0 on the strip-like ends. Now assume that a Floer datum .H ; J / has been fixed for each .L ;0 ; L ;1 /. Then, a relative perturbation datum is a pair .K; J / as before, with the additional restriction that over each strip-like end, K.s; t; x/ D H .t; x/dt and J.s; t; x/ D i  J .t; x/. Take a solution u of (17.2), and use the trivialization over each end to write .u ı  /.s; t/ D .s; t; u .s; t// with u W Z ˙ ! M . Then (17.2) reduces to Floer’s equation for u , hence it makes sense to impose a condition of asymptotic convergence to some y 2 C .L ;0 ; L ;1 /. We denote the resulting moduli spaces by ME=S .fy g/. Supposing that suitable regularity assumptions are satisfied, one can define a chain homomorphism O O C ˆE=S W CF pr .L C ;0 ; L C ;1 / ! CF pr .L  ;0 ; L  ;1 / (17.10)

C 2†C

 2†

by a mod 2 count of isolated points in ME=S .fy g/, as in (8.20). The induced maps ˆE=S on Floer cohomology are independent of the choice of perturbation data, and obey a straightforward generalization of the TQFT axioms. In particular, one has a composition law, where two Lefschetz fibrations k W Ek ! Sk are glued together by identifying an outgoing end of S1 with an incoming one of S2 (of course, the symplectic structures on the Sk will not usually agree, but that can be dealt with by locally patching things together, in a way similar to Remark 17.6).

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241

This framework contains both the previously considered case of a compact S and the theory from Section 8 (which corresponds to allowing only trivial fibrations E D S  M ). By borrowing ideas from those cases, one can extend it further in several ways: • Moving boundary conditions. As discussed in the compact case, with the additional proviso that the Fz need to be locally constant in z over each striplike end. • Parametrized moduli spaces. Families of Lefschetz fibrations, and the associated moduli spaces, already appear in the basic construction, where one uses them to provide chain homotopies between the maps (17.10) for different choices of relative perturbation data. The general structure is that one looks at a family E r ! S r with Lagrangian boundary conditions F r and perturbation data .K r ; J r / depending on some parameter space r 2 R, where the structure of the strip-like ends is constant in r. Assume for simplicity that R is compact. The associated maps between Floer cochain groups will satisfy an equation of the form @ ı (map) C (map) ı @ C (boundary terms) D 0, where the boundary terms come from restricting our family to @R. In particular, for R D Œ0I 1 we get a chain homotopy between two maps, corresponding to the endpoints. We will see several two-parameter examples later (looking beyond that, one can build a general theory along the lines of Section 9, but the meaning of the resulting algebraic structures is only partially understood). • Evaluation maps. In addition to the strip-like ends, one can fix a finite set of points … D … t …C on @S, and consider the evaluation maps evz˙ .u/ D u.z˙ / at those points. This gives rise to an invariant which, as in (8.24), has additional factors H  .Fz˙ I K/ in the tensor products. For instance, in the local model from Example 17.3, the evaluation map at any point can be identified with projection S.T S n / ! S n from the tangent sphere bundle to the base. Fundamentally, introducing marked points does not make the theory more general, since (modulo technicalities, and using the PSS isomorphism) they can be replaced with strip-like ends; but it is often more convenient for computational purposes. There is also a variation of the composition law for this situation, where one takes boundary connected sums of the base surfaces along marked points. (17c) A vanishing theorem. The last-mentioned composition law will be used several times in our subsequent discussion, hence it deserves to be stated a little more precisely. For simplicity, let k W Ek ! Sk (k D 1; 2) be two Lefschetz fibrations, the first one with strip-like ends and the second one with a compact base, each equipped with (possibly moving) Lagrangian boundary conditions Fk . In addition, we want to have marked boundary points z1 2 @S1 , z2 2 @S2 , such that the fibres .Ek /zk can

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both be identified with a common M , and the .Fk /zk become the same submanifold L  M . We also assume that both fibrations are locally trivial, and the boundary conditions locally constant, near the marked points. Choose relative perturbation data .Kk ; Jk /, which are trivial (Kk D 0, Jk D IEk / near .Ek /zk . Now form the boundary connected sum of S1 and S2 at our marked points. This can be thought of in one of two ways: either one first removes the zk , then introduces strip-like ends on the resulting punctured surfaces, and carries out a standard gluing process for those ends as in (8.5), for some gluing length l; or one considers local coordinates modelled on a half-disc DC D D \ H around zk , and then uses a transition map of the form z 7! z 1 for some  2 .1I 0/. These two pictures are completely equivalent (with the usual relation l D  log./=). We have mentioned them because each is intuitive in its own way, and because they suggest two slightly different, but equally viable, analytic frameworks: the first one results in a Morse–Bott version of the usual Floer theory, while the second one is close to gluing procedure for holomorphic discs in (relative) Gromov–Witten theory. In any case, the outcome carries a Lefschetz fibration  W E ! S , which comes with an induced boundary condition F and relative perturbation datum .K; J /; and the basic gluing theorem [131, Equation (2.28)] says that if suitable regularity conditions are satisfied, and if l is large (or  close to 0), then a q ME=S .fy g/0 Š ME1 =S1 .fy g/p L ME : (17.11) 2 =S2 pCqDn

Here, the superscripts are dimensions: on the right-hand side, these are adjusted in such a way that the fibre product, taken with respect to the evaluation maps evz1 W ME1 =S1 .fy g/ ! .F1 /z1 Š L and evz2 W ME2 =S2 ! .F2 /z2 Š L, is zerodimensional (the regularity assumptions mentioned above include the property that evz1 , evz2 are mutually in general position). We now translate this into a statement about the associated invariants. For E2 , consider the homology class represented by evz2 , and denote this by ˆE2 =S2 ;z2 2 H .LI K/. For E we just take (17.10). Lemma 17.7. If ˆE2 =S2 ;z2 vanishes, so does ˆE=S . The proof is straightforward: take a (pseudo-)cycle b W B ! L which bounds evz2 . Then the zero-dimensional part of ME1 =S1 .fy g/ L B gives rise to a map of Floer cochain groups which is a nullhomotopy for ˆE=S . Remark 17.8. In all our applications of this lemma, the compact piece will be as in Lemma 17.4. One can then impose the additional assumption that 2c1 .M; L/ D 0, and obtain a slightly sharper result. If n > 1, the space ME2 =S2 is of dimension 2n  1 > n, so the fibre product (17.11) is empty. In the remaining case n D 1, the only nontrivial term is the one with p D 0, q D 1. We already know from Lemma 17.4 that the evaluation map 1 evz2 W ME ! L 2 =S2

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has even degree. For each isolated point u1 2 ME1 =S1 .fy g/0 , evz1 .u1 / is a regular value of evz2 (due to the transversality assumption), hence there is an even number of maps u2 such that evz2 .u2 / D evz1 .u1 /. This shows that the fibre product again contains an even number of points. The outcome is that for this particular choice of perturbation datum, the cochain level invariant C ˆE=S will itself be zero. (17d) The basic morphism. Let M be an exact symplectic manifold with corners, and V a framed exact Lagrangian sphere in it. It will be convenient for us to take advantage of the cochain level vanishing result from Remark 17.8, so we assume that 2c1 .M; V / vanishes. Set L0 D V , and take two other exact Lagrangian submanifolds L1 ; L2 , with the property that L2 is isotopic to V .L1 /; more precisely, we want to specify a choice of isotopy. We also want to fix Floer data for the pairs .L0 ; L1 /, .L0 ; L2 / and .L1 ; L2 /, as well as the perturbation datum required to define the product 2 W CF pr .L1 ; L2 / ˝ CF pr .L0 ; L1 / ! CF pr .L0 ; L2 /. For better comparison with the algebraic theory, we write Y0 , Y1 , Y2 for the objects of F pr .M / given by L0 , L1 , L2 . Fix a Lefschetz fibration  W E ! S over a disc with fibre M , having a single critical point whose vanishing cycle is V . More precisely, we want this fibration to be equipped with a marked point 2 @S and an identification of the fibre E over that point with M ; it should also come with a vanishing path  starting at , such that V D V . For technical simplicity, we further assume that the fibration is locally trivial near . E will be the essential building block in all the subsequent constructions, which basically consist of gluing various trivial pieces onto it. Remark 17.9. Recall that such fibrations exist for any given M and V , and can in fact be constructed “by hand” (see Section (16e) for an outline, and [131, Lemma 1.10] for a comprehensive account). By paying close attention to the details of the construction, one can obtain somewhat more control: the symplectic connection will be flat outside a small neighbourhood of the critical point; the vanishing cycles of any vanishing paths are all equal to V ; and the monodromy for any loop winding once around the critical value is a Dehn twist V (not merely isotopic to one). However, none of these properties is essential for our construction. As a first, and most elementary, application we will define a Floer cocycle c 2 CF pr .L1 ; L2 /; 1 .c/ D 0:

(17.12)

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For that we take our given E ! S and remove . The outcome is a Lefschetz fibration E c ! S c with one point at infinity, which we consider to be an incoming one. Denote this by  c; and choose a strip-like end  c; . There is a canonical (up to fibrewise isotopy) moving Lagrangian boundary condition F c , which is defined as follows. Start with L1 in the fibres over  c; .s; 0/, s  0. Then carry this around @S c by parallel transport, almost all the way. The outcome is isotopic to V .L1 /, hence also to L2 , and we use that isotopy to extend it continuously so that it becomes equal to L2 in the fibres over  c; .s; 1/, s  0. The construction is shown schematically in Figure 17.1. There, the graphical notation is to remove the singular fibres and then cut open the base along the dotted line, to get a trivial symplectic fibration. The monodromy is then thought of as an identification between the two sides of the cut. In these kinds of pictures, we usually do not distinguish between isotopic objects, hence write the monodromy simply as V . L2 V

V .L1 / L1

L1 Figure 17.1

Then, (17.12) is the associated invariant C ˆE c =S c , for some (regular) relative perturbation datum which is compatible with the existing choice of Floer datum. The second piece of data will be a chain homotopy between 2 .c;  / and the zero map, namely k W CF pr .L0 ; L1 / ! CF pr .L0 ; L2 /; (17.13) 1 .k.  // C k.1 .  // C 2 .c;  / D 0: As one would expect, this is constructed from a family of Lefschetz fibrations with strip-like ends, depending on a parameter r 2 Œ0I 1. At one end, for r D 0 one takes the trivial fibration over the three-pointed disc, with Lagrangian boundary conditions .L0 ; L1 ; L2 /. With an appropriate perturbation datum, the invariant associated to this just gives back 2 . Now take one of the two outgoing ends, namely the one with labels .L1 ; L2 /, and glue that to the incoming end of E c ! S c . This should be done with a large gluing length, so that the gluing formula for the zero-dimensional moduli spaces of pseudo-holomorphic sections holds on the cochain level. The outcome, denoted by E k;0 ! S k;0 , is a fibration equipped with a moving boundary condition F k;0

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17 Pseudo-holomorphic sections

and a relative perturbation datum, whose associated invariant is precisely 2 .c;  /. Two pictures of this, which differ only in the way we have mentally cut open the base, are shown in Figure 17.2. At the other endpoint r D 1, we proceed as follows. L2 L2

V .L1 / V L1

L1 L0

L2 L2

V

V .L1 /

L0 V .L0 / D L0 Figure 17.2

Take the trivial fibration over Z. Equip this with the moving Lagrangian boundary condition which is equal to L0 over R  f0g, and which realizes the isotopy from L2 to V .L1 / over R  f1g. This means that the pairs of Lagrangian submanifolds associated to two ends are .L0 ; L2 / and .L0 ; V .L1 //, respectively. For the latter pair, we take the Floer datum which is the image of that for .L0 ; L1 / under V , keeping in mind that V .L0 / D V .V / D V D L0 . We also want to mark a boundary point z C 2 R  f0g. Now take our original  W E ! S and equip that with a standard boundary condition F , which should be equal to L0 in the fibre over , and should also be locally constant near that point. Finally, glue together these two fibrations by taking the boundary connected sum along z C and z  D , and denote the result by E k;1 ! S k;1 ; see Figure 17.3. This is an instance of the process discussed in Lemma 17.7, and in fact Remark 17.8 applies here; hence, a suitably

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III Picard–Lefschetz theory

L2

V .L1 /

L0

L0

V

Figure 17.3

small choice of gluing parameter ensures that all zero-dimensional moduli spaces of pseudo-holomorphic sections of E k;1 are empty (n > 1) or consist of an even number of points (n D 1). Direct comparison of Figures 17.2 and 17.3 shows that there is indeed a one-parameter family of fibrations E k;r ! S k;r and boundary conditions F k;r interpolating between the ones we have constructed for r 2 f0; 1g. The structure at infinity (over the strip-like ends) of these fibrations is the same for all r. Choose a family of relative perturbation data, which for r D 0; 1 agree with the previously constructed ones, and consider the resulting parametrized moduli spaces of pseudoholomorphic sections. By counting isolated points in these spaces, one defines a homomorphism k W CF pr .L0 ; L1 / Š CF pr .L0 ; V .L1 // ! CF pr .L0 ; L2 / which is a chain homotopy between C ˆE k;0 =S k;0 and C ˆE k;1 =S k;1 , hence has exactly the desired property (17.13). In constructing c, some choices had to be made, since neither the boundary condition nor the perturbation datum were entirely canonical. Even after one has specified these, there are further choices in defining the family E k;r ! S k;r , which will in turn affect k. However, in both cases, the parameter space is path-connected, so one can move smoothly between any two possibilities. Concretely, this means that if .c; k/ Q are defined using two different choices, one can consider moduli spaces and .c; Q k/ with one more parameter, and thereby find  2 CF pr .L1 ; L2 /;

W CF pr .L0 ; L1 / ! CF pr .L0 ; L2 /; 1 ./ C c C cQ D 0;

(17.14)

1 . .  // C .1 .  // C 2 .;  / C k C kQ D 0: This is a straightforward extension of our previous considerations, so we will not

17 Pseudo-holomorphic sections

247

explain the details. Instead, we will now interpret the outcome in terms of the algebraic theory from Section 5. Remark 17.10. Note that our Floer-theoretic framework has not yet acquired gradings, and we are working in char.K/ D 2 where signs are trivial (the absence of both is clearly noticeable in the formulae above). However, Remark 9.7 applies here as well: first of all, it would be quite trivial to rework the algebraic theory for ungraded A1 -categories; and secondly, the present context is just provisional, and will later on be modified to include proper gradings and signs. With this in mind, we appeal to results from Section 5 without any further hesitations. Namely, Lemma 5.3 (i) together with (17.12) and (17.13) says that .c; k/ give rise to a morphism TY0 .Y1 / ! Y2 in the cohomological category of A1 -modules. Similarly, Lemma 5.3 (ii) and (17.14) say that this morphism is canonical; which means, independent of the auxiliary choices made in the definition of the Lefschetz fibrations and moduli spaces used to define c and k. (17e) An auxiliary chain complex. Take another exact Lagrangian submanifold Q. In addition to the previous choices, fix some Floer data for the pairs .Q; Lk /, (k D 0; 1; 2), and also the perturbation data needed to define the triangle products 2 W CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ! CF pr .Q; L1 / and 2 W CF pr .L1 ; L2 / ˝ CF pr .Q; L1 / ! CF pr .Q; L2 /, as well as (consistently with those) the datum on the universal family of four-pointed discs which gives rise to 3 W CF pr .L1 ; L2 / ˝ CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ! CF pr .Q; L2 /. Using a variation of the ideas above, one can define maps d W CF pr .Q; L1 / ! CF pr .Q; L2 /; l W CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ! CF pr .Q; L2 /; 1 .d.  // C d.1 .  // D 0;

(17.15)

d.2 .  ;  // C 1 .l.  ;  // C l.1 .  /;  / C l.; 1 .  // D 0: To encode these relations in a more compact form, define D D CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ˚ CF pr .Q; L1 / ˚ CF pr .Q; L2 /; 0 1 1  ˝ Id C Id ˝ 1 A 2 End.D/: 2 1 @D@ 1 l d 

(17.16)

Then (17.15) is equivalent to @2 D 0. The maps d and l are actually the ones used to construct the Floer cohomology long exact sequence, see [131, Sections 3.3 and 3.4],

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and (17.16) plays a key role in its proof [131, Lemma 2.32]. Nevertheless, because of the notational (and minor technical) differences between here and [131], it still makes sense to quickly review the definitions. For d , we start with  W E ! S and remove not only  d; D , but also another boundary point  d;C (let us say, a point obtained by moving in positive direction by a small amount; this has the advantage that the fibration is still locally trivial near that point). The result is a Lefschetz fibration E d ! S d Š Z with one incoming and one outgoing strip-like end. To define the relevant boundary condition F d , one proceeds as follows: over the “short” boundary component, this will be constant and equal to Q; while over the “long” one, we have L1 in the fibre over  d;C .s; 1/ for s  0, and L2 over  d; .s; 1/ for s  0, interpolating between the two by using parallel transport and the isotopy V .L1 / ' L2 . For two pictures of this fibration (differing only in how the base is drawn) see Figure 17.4. One equips it with a relative perturbation datum, and defines d D C ˆE d =S d to be the resulting invariant. L2 L2

L1 V

V

Q

Q

L1 Figure 17.4

The construction of l closely parallels that of k. Take the trivial fibration over the three-pointed disc with boundary conditions .Q; L0 ; L1 /, and glue its incoming end to the outgoing end of E d ! S d , to get a fibration E l;1 ! S l;1 which comes equipped with a boundary condition F l;1 and a relative perturbation datum. On the other hand, we can take another trivial fibration over the three-pointed disc, this time with a boundary condition which is constant and equal Q, respectively to L0 , over the first two boundary components, and which realizes the isotopy from V .L1 / to L2 over the third one. Mark a boundary point on the L0 component, and take the boundary connected sum with the fibration E ! S, carrying its standard boundary condition F , at the point . This yields another fibration E l;0 ! S l;0 , to which Remark 17.8 applies as before. There is an obvious family E l;r ! S l;r interpolating between the two, and one takes l to be the invariant obtained from the resulting parametrized moduli spaces (see Figure 17.5). (17f) Additional remarks. As before, one can show that the choices involved in Q are defined defining .d; l/ are essentially irrelevant. More precisely, if .d; l/ and .dQ ; l/

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17 Pseudo-holomorphic sections

V .L1 / L0

L2

V

L0

Q

L1 L2

L1 V L0 Q

Q

Figure 17.5

using different choices, then one can consider additional moduli spaces interpolating between the two versions of the construction, and thereby obtain ı W CF pr .Q; L1 / ! CF pr .Q; L2 /;

W CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ! CF pr .Q; L2 /; 1 .ı.  // C ı.1 .  // C d C dQ D 0; ı.2 .;  // C 1 . .;  // C .1 .  /;  / C .; 1 .  // C l C lQ D 0:

The algebraic meaning of this is that we get an isomorphism between the chain complexes (17.16): 1 0 Id ˝ Id z A W D ! D: @ 0 Id (17.17)

ı Id More generally, one can look at how .d; l/ depends on L1 . Suppose that we have Q of .d; l/. Take Q 1 isotopic to L1 , and associated versions .dQ ; l/ another submanifold L

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III Picard–Lefschetz theory

the continuation maps Q 1 /; a W CF pr .Q; L1 / ! CF pr .Q; L b W CF pr .L0 ; L1 / ! CF pr .L0 ; LQ 1 /I these are compatible with the triangle product up to chain homotopy, which means that there is an h W CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ! CF pr .L0 ; LQ 1 / such that 1 .h.  // C h.1 .  /;  / C h.; 1 .  // C 2 .b.  /;  / C a.2 .;  // D 0: From our general point of view, the continuation maps come from the trivial fibrations M  Z ! Z, where the boundary condition is constant and equal to Q or L0 over R  f0g, and realizes the isotopy from LQ 1 to L1 over R  f1g (this was already pointed out in Section (8j)). One can take the trivial fibration over a three-punctured disc, with boundary conditions .Q; L0 ; L1 /, and glue its incoming end to the outgoing end of the fibration defining a. On the other hand, one can take boundary conditions Q 1 / for the tree-punctured disc, and glue the second outgoing end to the .Q; L0 ; L incoming end of the fibration defining b. The two fibrations constructed in this way can be deformed into each other, and the resulting family gives rise to h. One can use the same idea to relate d and dQ , as follows: glue the incoming end Q

Q

of the fibration defining a to the outgoing one of E d ! S d , and deform the outcome back to the original E d ! S d . The result is a one-parameter family of fibrations E ˛;r ! S ˛;r , shown in Figure 17.6, whose associated invariant has the following form: ˛ W CF pr .Q; L1 / ! CF pr .Q; L2 /; 1 .˛.  // C ˛.1 .  // C dQ .a.  // C d.  / D 0:

Q1 L

L2

Q1 L

L1

V

L2

L1 V

Q

Q

Q Figure 17.6

17 Pseudo-holomorphic sections

251

Q the outcome A slightly more complicated argument of the same kind works for l and l, taking the form ˇ W CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ! CF pr .Q; L2 /; 1 .ˇ.;  // C ˇ.1 .  /;  / C ˇ.; 1 .  // Q  /;  / C l.;  / C ˛.2 .;  // C dQ .h.;  // D 0: C l.b. Together, these give rise to a chain map 1 0 b ˝ Id z A W D ! D: @ h a ˇ ˛ Id

(17.18)

The diagonal terms in (17.18) are continuation maps, hence quasi-isomorphisms, so the whole will again be a quasi-isomorphism. Finally, a parallel argument shows that D remains the same, up to quasi-isomorphism, under isotopies of L2 ; to avoid repetition, we omit this part. (17g) Relating the two constructions. Given .c; k/, one can consider dQ D 2 .c;  / and lQ D 2 .k.  /;  / C 3 .c;  ;  /. These satisfy (17.15), and one can relate them to .d; l/. This can be viewed as a limiting case of the previous uniqueness statement for solutions of (17.15), but since it is important for our application, we use separate notation and give a fully detailed account. Namely, we will construct  W CF pr .Q; L1 / ! CF pr .Q; L2 /; W CF pr .L0 ; L1 / ˝ CF pr .Q; L0 / ! CF pr .Q; L2 /; 1 ..  // C .1 .  // C d C 2 .c;  / D 0;  . .;  // C

. .  /;  / C

1

1

(17.19)

.;  .  // 1

C . .;  // C l.;  / C 2 .k.  /;  / C 3 .c; ;  / D 0: 2

For , start with E ;0 ! S ;0 being E d ! S d with the previously used boundary condition and relative perturbation datum. At the other endpoint of the deformation, we take the trivial fibration over the three-pointed disc with boundary conditions Q; L1 ; L2 over the three boundary components, and glue the second outgoing end of that to the incoming end of E c ! S c . The family that interpolates between the two is obvious; see Figure 17.7. The definition of begins by writing down five one-parameter families of Lefschetz fibrations, equipped with boundary conditions and perturbation data, which

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III Picard–Lefschetz theory

V

L2

L1

L1

L2

V

Q

Q Figure 17.7

together form a closed chain (the endpoint of the first is the starting point of the second, and so on; see Figure 17.8 for a graphical account). The first family is E l;r ! S l;r as defined above. To get the second one, which will be denoted by 2 2 E . /;r ! S . /;r , one takes the E ;r ! S ;r and glues the single outgoing end to the incoming end of a trivial fibration over a three-pointed disc; the latter piece comes equipped with boundary conditions Q; L0 ; L1 , and with the perturbation datum that defines the 2 product. Hence, provided that the gluing length is sufficiently large, counting pseudo-holomorphic sections of this family of fibrations precisely gives .2 .;  //. Recall that E l;1 ! S l;1 was obtained by taking E d ! S d and gluing in a trivial fibration over a three-pointed disc; on the other hand, we had .E ;0 ! S ;0 / D .E d ! S d / by definition. What this means is that if the gluing parameters are chosen appropriately, we have .E .

2 /;0

! S .

2 /;0

/ D .E l;1 ! S l;1 /;

(17.20)

where the equality also includes boundary conditions and perturbation data. At the 2 2 other end, for r D 1, we have a fibration E . /;1 ! S . /;1 which gives rise to the expression 2 .c; 2 .;  //. This immediately suggests the next step: take a sufficiently large closed interval Œ0I 1  R4 , and consider the universal family of four-pointed discs over that, with boundary components labeled by .Q; L0 ; L1 ; L2 /. Turn this into a family of trivial Lefschetz fibrations with their appropriate boundary conditions and perturbation data, and then glue the third outgoing end of each disc 3 to the incoming end of E c ! S c . We denote the outcome by E  .c˝Id/;r ! 3 S  .c˝Id/;r , since counting pseudo-holomorphic sections in this family just yields 3 .c;  ;  /. By an appropriate choice of gluing parameters, one can certainly achieve

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17 Pseudo-holomorphic sections

0 (vanishes)

d.2 / l

.2 /

2 .c; 2 /

0 (vanishes) 3 .c ˝ Id/

2 .k ˝ Id/

2 .0 ˝ Id/ (vanishes)

2 .2 .c ˝ Id/ ˝ Id/ Figure 17.8

that .E 

3 .c˝Id/;0

! S

3 .c˝Id/;0

/ D .E .

2 /;1

! S .

2 /;1

/

in the same sense as in (17.20). At the other extreme r D 1 we get a Lefschetz fibration whose sections define the homomorphism 2 .2 .c;  /;  /. For the fourth 2 2 family E  .k˝Id/;r ! S  .k˝Id/;r , we take the three-pointed disc with boundary components labeled by .Q; L0 ; L2 /, make it into a trivial Lefschetz fibration, and then

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III Picard–Lefschetz theory

glue its second outgoing end with the incoming end of the fibrations E k;r ! S k;r . By definition of k, .E 

2 .k˝Id/;0

! S

2 .k˝Id/;0

/ D .E 

3 .c˝Id/;1

! S

3 .c˝Id/;1

/:

2

On the other hand, for r D 1 one can think of E  .k˝Id/;1 as being constructed as follows: start with the trivial fibration over a three-pointed disc, where the boundary conditions are constant and equal to Q; L0 over two boundary components, and yield the isotopy from V .L1 / to L2 over the remaining one. Mark a boundary point on the L0 component, and take the boundary connected sum with E (carrying its standard boundary condition), as in the definition of k. Note that E l;0 ! S l;0 was defined in essentially the same way. Details (such as the position of the marked point) may vary, but one can clearly interpolate between the two with a fifth and final family, denoted by E 0;r ! S 0;r ; this has the property that each E 0;r is obtained by a boundary connected sum with E, where the gluing parameter remains small throughout. By definition, this completes our chain: .E 0;0 ! S 0;0 / D .E 

2 .k˝Id/;1

! S

2 .k˝Id/;1

/;

.E 0;1 ! S 0;1 / D .E l;0 ! S l;0 /: The vanishing results from Remark 17.8 will apply to E 0;r ! S 0;r in an appropriately parametrized version, so the count of associated zero-dimensional moduli spaces of pseudo-holomorphic sections will be trivial. Finally, the entire chain forms a family of Lefschetz fibrations parametrized by S 1 , which can obviously be filled in by a family over D 2 . We use the resulting parametrized moduli spaces to define , and then (17.19) will hold by construction. Remark 17.11. It is instructive to try to fit this procedure into a more general and standardized framework. Let R3;1 be the moduli space of 3-pointed discs with an additional interior marked point. Equivalently, and more appropriately for our purpose, one can fix the interior point and the first boundary point, which breaks the Aut.D/-symmetry, and consider R3;1 as the configuration space of the other two boundary points. Take the disc D to be the base of our standard Lefschetz fibration E ! S , with the interior marked point being the unique critical value, and the first boundary point being . For any configuration .z1 ; z2 / 2 R3;1 , one can restrict E to S n f ; z1 ; z2 g and make that restriction into a Lefschetz fibration with strip-like ends. Equip this family of fibrations with Lagrangian boundary conditions consisting of Q; L0 ; L1 , and the isotopy from V .L1 / to L2 ; and consider the invariant obtained by looking at the resulting parametrized moduli spaces. To understand the relation which this satisfies, one needs to look at the Deligne– x 3;1 , which is a hexagon (just like R x 5 , this can be Mumford style compactification R x realized as real locus of the complex Deligne–Mumford space M0;5 , with respect

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17 Pseudo-holomorphic sections

to an anti-holomorphic involution which combines complex conjugation with the exchange of two marked points). The degenerations corresponding to the various strata are shown in Figure 17.9. For the edges (i) and (ii), the resulting one-parameter families are limits (letting the gluing parameter to go zero) of the ones in Figure 17.8, giving rise to the terms 3 .c ˝ Id/ and 2 .k ˝ Id/ in (17.19). The rest of the picture is slightly more complicated: the family giving rise to our .2 / term corresponds to part of edge (vi), and E 0;r ! S 0;r similarly covers part of edge (iii); the remainder of those two edges, as well (iv) and (v), have no direct counterparts in Figure 17.8, but their union can be deformed rel endpoints to the one-parameter family giving rise to l. So, graphically speaking, the five families from Figure 17.8 can be thought of as the dotted edges in 17.9. For practical purposes, what this means that is that x 3;1 itself, the outcome would be slightly more complicated versions if one used R (containing more terms) of (17.15), (17.19).

(ii)

(iii)

(i)

V L2

L1

Q

L0

(iv)

(vi) (v)

Figure 17.9

Remark 17.12. In our later applications, the condition 2c1 .M; V / D 0 will always hold, as a consequence of the existence of a grading (see the discussion in Section (12a)). Nevertheless, one can do without it if necessary, by adding correction

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III Picard–Lefschetz theory

terms involving suitable bounding chains, as in [131, Lemma 3.12] or Lemma 17.7 above. (17h) Acyclicity. The algebraic arguments from the proof of [131, Lemma 2.32], combined with the geometric ones from [131, Chapter 3], prove that the complex D defined in (17.16) is acyclic; this is important for us in view of Lemma 5.3 (iii). Strictly speaking, [131] establishes acyclicity only under a number of additional technical conditions, namely: (i) Q; L0 ; L2 are in general position. The Floer data for the pairs .Q; L0 /, .L0 ; L2 / and .Q; L2 / should have zero Hamiltonian terms. Moreover, there is a condition on the values of the action functionals (8.11) at intersection points: namely, the numbers A.x0 / for x0 2 C.Q; L2 /, and A.x1 / C A.x2 / for .x1 ; x2 / 2 C.Q; L0 /  C .L0 ; L2 /, must all be pairwise different. (ii) One needs to be able to extend the framing L0 D V Š S n to a symplectic embedding of the cotangent ball bundle, Bs S n ! int.M / for some s > 0, in such a way that the pullback of M is the standard one-form on T  S n . Moreover, in this tubular neigbourhood, both Q and L2 should be unions of cotangent fibres. (iii) L1 D V1 .L2 /, where the Dehn twist is defined using the previously chosen tubular neighbourhood. The function b determining the speed of the twist (16.5) must satisfy certain conditions; these depend on the minimal difference between the action values in (i), as well as the minimal distance between points x1 ; x2 with respect to the spherical metric on V . (iv) The Floer data for the pairs .L0 ; L1 / and .Q; L1 / should have zero Hamiltonian term; similarly, the perturbation datum for the triple .Q; L0 ; L1 / should have K D 0. (v) Finally, the auxiliary geometric data (Lefschetz fibrations, Lagrangian boundary conditions, relative perturbation data) entering into the definition of .d; l/ must be chosen in a precisely controlled way. As a preliminary observation, all these conditions can be fulfilled if one is willing to make the following changes: Lagrangian isotopies of the Lk ; passing from M to M C dh, for some function h supported in int.M /; and changing the Floer data and perturbation data used to define the Fukaya category. Since none of this affects the isomorphism classes of objects on the cohomology level (either in the Fukaya category, or its category of A1 -modules), we can apply Lemma 5.3 (iii) and already conclude that TY0 .Y1 / Š Y2 . This is good enough for most uses, but not entirely satisfactory, and we will therefore embark on a more detailed analysis, using the results of Section (17f). First of all, (v) is clearly unnecessary, since the isomorphism class of D is independent of those choices, as we have seen in (17.17).

17 Pseudo-holomorphic sections

257

Similarly, (17.18) shows independence of L1 , which allows one to remove (iii); since the tubular neighbourhood from (ii) was only used to define L1 D V1 .L2 /, this becomes redundant as well; moreover, passing to an isotopic LQ 1 also entails complete freedom in the choice of Floer data associated to that submanifold, which was previously constrained by (iv). An argument analogous to that for L1 shows that we may move L2 within its isotopy class, without affecting the quasi-isomorphism class of D; and finally, as already mentioned in Lemma 5.3 (iii), the fact that .d; l/ are related to .c; k/ via (17.19) means that acyclicity of (17.16) for some Q implies z Together, this makes (i) and the remaining part of (iv) the same for all isotopic Q. unnecessary. We have therefore shown that the complex D from (17.16) is acyclic, for all .L0 ; L1 ; L2 ; Q/, and irrespective of the choices made in defining .d; l/. In terms of Lemma 5.3, this means that the morphism TY0 .Y1 / ! Y2 defined by .c; k/ is always an isomorphism. Remark 17.13. To round off this discussion, we want to outline roughly how one can produce additional algebraic data .g; p; q/ satisfying the conditions from Lemma 5.4. On the cohomology level, Œg is Poincaré dual to the continuation isomorphism HF pr .L0 ; L2 / ! HF pr .L0 ; V .L1 // Š HF pr .L0 ; L1 /. We can simplify our task by a trick, which is to choose the Floer datum for .L2 ; L0 / in such a way that the cochain group becomes naturally dual to that for .L0 ; L1 /. If L2 was equal to V .L1 /, it would be obvious how to achieve that: one would take the Floer datum .H; J / for .L0 ; L1 /, replace it by .H.t/; J.t//, and then apply V to it. In general, L2 is only isotopic to V .L1 /, but one can take the isotopy into account by building it into a modified version of the Floer datum defined above. Now, take g to be the dual of the identity map CF pr .L2 ; L0 /_ ! CF pr .L0 ; L1 /. With that in mind, the other terms and their required properties can be written as p W CF pr .L2 ; L0 / ! CF pr .L1 ; L0 /; q W CF pr .L2 ; L0 / ˝ CF pr .L0 ; L1 / ! CF pr .L0 ; L0 /; 1 .p.  // C p.1 .  // C 2 .; c/ D 0; 1 .q.;  // C q.1 .  /;  / C q.; 1 .  //

(17.21)

C 3 .; c;  / C 2 .; k.  // C 2 .p.  /;  / C e h; i D 0I in the last line of this, h; i is the dual pairing, and e 2 CF pr .L0 ; L0 / is a representative for the cohomological identity. From the first of these equations, it is clear how to construct p by adapting the definition of k above. For q, one considers a two-parameter family over R3;1 as in Figure (17.9), but now setting Q D L0 . The contributions from the boundary sides (i), (ii) and (vi) give rise to the terms in the second-to-last line of (17.21), while the ones from (iii) and (v) vanish. This leaves

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(iv) as the last and most interesting case. The relevant geometric setup consists of two Lefschetz fibrations with an identification of the fibres over boundary marked points (Figure 17.10). On the left-hand piece, which we denote by E e ! S e , the position of the marked point is variable, so we get an evaluation map with an additional parameter. Suppose that the Floer datum for .L0 ; L0 / is such that the Hamiltonian is a standard Morse function on L0 D S n , so that CF pr .L0 ; L0 / has two generators e D y0 and yn . Then the moduli spaces ME e =S e .y0 / and ME e =S e .yn / are of dimension n  1 and 2n  1, respectively. Take the first of these spaces, and consider the parametrized evaluation map, where we use parallel transport to identify the Lagrangian boundary condition at any point of @S e with L0 : ev W R  ME e =S e .y0 / ! L0 : One can show that ev is of odd degree over L0 n fy0 ; V .y0 /g (the two missing points correspond to the limits where z goes to infinity). This is done by reducing to the case of a closed disc, and then to the standard local model; compare [131, Remark 2.17 (iii)]. Once one has that, it is easy to see that the contribution from (iv) is indeed eh; i. L0 L0 V L1

L0 L0 Figure 17.10

(17i) Relative brane structures. Let  W E ! S be a Lefschetz fibration (we will discuss the case where S is compact first, and do the version for strip-like ends later). Suppose that E carries a relative quadratic complex volume form 2E=S , and let ˛E=S be the relative squared phase map. Now suppose that F is a Lagrangian boundary condition on E. A relative grading of F is a lift ˛ # W F ! R of x 7! ˛E=S .TFxv /. A relative brane structure on F is a pair .˛ # ; P # / consisting of a relative grading and a relative Pin structure, where the latter is just a Pin structure P # on the vector

17 Pseudo-holomorphic sections

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bundle TF v ! F . The obstructions to existence of such structures can be viewed as follows: pick a point z on each boundary circle C . Then, each Fzj  Ez must admit a grading with respect to 2Ez ; additionally, if hC is the monodromy around the boundary, equipped with its natural grading as in Section (15c), then the image hC .Fz / should be equal to Fz as a graded Lagrangian submanifold. Similarly, each Fz needs to have a Pin structure Pz# , with the property that .hC jFz / .Pz# / is isomorphic to Pz# (a choice of isomorphism determines a relative Pin structure for F jC ). Example 17.14. Take the local model from Example 17.3, setting r D 1 for simplicity. The standard relative quadratic complex volume form is the squared quotient of E D dx1 ^    ^ dxnC1 and S D dz. For each x 2 Fz , ˛E=S .TFxv / D

E .v1 ; : : : ; vnC1 /2 ; S .iz@z /2

the whole of F . where fv1 ; : : : ; vnC1 g is an orthonormal basis of the tangent p space top Concretely, v1 ;p : : : ; vn can be chosen to be tangent to zS n  zRnC1 , while vnC1 D ix 2 i zRnC1 , and then ˛E=S .TFxv / D z n1 . Hence, F admits a relative grading when n D 1, and does not admit one for n > 1. This is a reflection of the fact that any Dehn twist V , when equipped with its natural grading, shifts the grading of the central sphere V by 1  n [125, Lemma 5.7]. In contrast, relative Pin structures always exist, in fact there are four different choices if n D 1, and two if n > 1. The presence of a grading ensures that for any section u with boundary values on F , the pair .u .TE v /; u .TF v // has zero relative first Chern class, so the dimension of the moduli space (assuming regularity) will be n .S/. As before, pick a point on each component of @S, and denote these points by zj . Given a relative Pin structure, one gets a preferred orientation of ME=S relative to the product of evaluation maps at the zj . This is a consequence of Proposition 11.13 (translated from the language of abstract brane structures into the present one). The evaluation maps, with the preferred relative orientation, then represent a cohomology class (by Poincaré duality) Y  ˆE=S;z1 ;:::;zk 2 H n.k.S// Fzj : (17.22) j

Take some real line bundle ˇ ! @S . One can modify the relative Pin structure by P # 7! P # ˝ .jF / ˇ; the effect on (17.22) is to multiply the invariant by .1/i , where i is the number of components where ˇ is nontrivial. Finally, this whole discussion extends without any difficulties to moving boundary conditions. For our purpose, the most important special case of (17.22) is when S is a disc, and the fibre dimension is 2n D 2. The latter assumption means that the linearized N operators DE=S;u are @-operators on line bundles, so the arguments from Section (11d)

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apply. In particular, from (11.13) we see that for every u 2 ME=S , ker.DE=S;u / is one-dimensional and is generated by a nowhere vanishing section X of u .TE v /. N In the trivialization given by X, DE=S;u becomes the standard @-operator on the trivial line bundle, with constant boundary conditions. Looking back at the proof of Lemma 11.12, one sees that the relative orientation ker.DE=S;u / Š .TFz /u.z/ is given by the map X 7! X.z/ if the Pin structure on u .TF v / is trivial; otherwise, the orientation is the opposite one. This means that the locally constant function ˆE=S;z 2 H 0 .Fz / can be computed as follows: pick a point x 2 Fz . Then, count the sections u 2 ME=S with u.z/ D x, assigning C1 multiplicity if the Pin structure on u .TF v / is trivial, and 1 otherwise. Example 17.15. We return to Example 17.3, assuming n D 1. It is possible to explicitly parametrize the boundary condition by S 1  S 1 ! F , .t1 ; t2 / 7! .e i t1 =2 cos.t1 =2 C t2 /; e it1 =2 sin.t1 =2 C t2 //. The moduli space of sections is the disjoint union of two circles ME=S;˙ , given by taking a D 12 .e it ; ˙ie it / in (17.8). In these coordinates, the evaluation maps along the boundary become @S  ME=S;C ! F; @S  ME=S; ! F;

.s; t/ 7! .t1 ; t2 / D .s; t  s/ .s; t/ 7! .t1 ; t2 / D .s; t/:

If we fix a point in F , say .t1 ; t2 / D .0; 0/, there is one section u˙ in each family going through that point, namely the one with t D 0. Note that in homology, the circles u˙ .@S/  F differ by one copy of the fibre Fz . In view of the general observations made above, this has the following consequence: if the relative Pin structure is chosen to be nontrivial on Fz , then the contributions of uC and u cancel out, so the invariant vanishes. Otherwise ˆE=S;z D ˙2 (both signs occur, for two Pin structures which differ by the pullback of the nontrivial line bundle over @S ). Fix an exact symplectic manifold with corners M , and a quadratic complex volume 2 form M on it. Take a Lefschetz fibration with strip-like ends, where the trivialization over each end is modelled on Z ˙  M . Equip it with a (possibly moving) Lagrangian boundary condition F , and let .L ;0 ; L ;1 / be the pair of Lagrangian submanifolds associated to each end . Now suppose that these carry brane structures .L# ;0 ; L# ;1 /, and that E comes with a relative quadratic complex volume form 2E=S , which over 2 each end agrees with M . Then, a compatible relative brane structure on F consists # of a relative grading ˛ which, over each end, equals those of the L ;k ; and a relative Pin structure P # , together with a specified isomorphism between that structure and those of the L ;k over the ends. Given this kind of structure, we can carry out an analysis of determinant line bundles similar to that in Sections (12c). The result is easiest to state in the case where each boundary circle contains exactly one incoming point at infinity (this will be the case in all our applications); then, a signed count

17 Pseudo-holomorphic sections

261

of isolated points in ME=S .fy g/ gives rise to a refined version of (17.10), namely a cochain map O O C ˆE=S W CF  .L# C ;0 ; L# C ;1 / ! CF  .L#  ;0 ; L#  ;1 /Œn. .S/j† j/;

C 2†C

 2†

where CF  is the Z-graded Floer cochain complex with an arbitrary coefficient field K. More general surfaces can be allowed, but then one either needs to orient some of the Fz , see (12.5), or change some of the brane structures associated to the ends, as in (12.7). (17j) Algebraic versus geometric twisting. Suppose that we have a Lefschetz fibration over a disc, with fibre M , having a single critical point. If the fibre dimension is 2n > 2, such fibrations always admit a relative quadratic complex volume form, inherited from that of M ; for 2n D 2, this will be the case if and only if the vanishing cycle V admits a grading. With this in mind, the entire argument from Sections (17d)–(17h) goes through as before, and all the resulting signs agree with those expected in Lemma 5.3; this is a straightforward but somewhat tedious verification, which we omit. As in Example 17.14, standard boundary conditions do not admit relative gradings if 2n > 2, but that is not a problem, since in those dimensions, the vanishing argument from Remark 17.8 does not rely on any cancellation. In the remaining case, Example 17.14 applies, so if one wishes to work in char.K/ ¤ 2, it is necessary to choose the nontrivial Pin structure on the vanishing cycle V D L0 . The main result, taken from Section (17h) but now equipped with its proper grading and signs, is the following. Theorem 17.16. Let V D L0 be an exact framed Lagrangian sphere in M , equipped with a brane structure L#0 (in the case n D 1 and char.K/ ¤ 2, the Pin structure belonging to this should be nontrivial). Let L1 , L2 be two other exact Lagrangian submanifolds, with a fixed isotopy L2 ' V .L1 /, and suppose that they too carry brane structures L#k , which are compatible with the isotopy (giving V its natural grading). Denote by Yk the objects of F .M / given by L#k , and by Yk theirYoneda A1 modules. Then, by counting sections of the fibrations E c ! S c and E k;r ! S k;r , one obtains c 2 CF 0 .L#1 ; L#2 / and k W CF  .L#0 ; L#1 / ! CF  .L#0 ; L#2 /Œ1, such that the resulting Œt  W TY0 .Y1 / ! Y2 , defined as in Lemma 5.3, is an isomorphism in H 0 .mod.F .M ///.  To get a more concrete statement, we now replace A1 -modules by twisted complexes, as in (5.6). We also assume that L2 D V .L1 /, and omit some of the finer details. This turns Theorem 17.16 into the following “T D ” statement: Corollary 17.17. In the derived category DF .M /, TL# .L#1 / Š L0 .L#1 /. 0



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TL# .L#1 / fits into an exact triangle (5.4), and one may replace it with L0 .L#1 / 0 by general properties of exact triangles. Note that one of the morphisms in the triangle is the (purely algebraic) evaluation morphism Œf , defined in Section (5b). By Lemma 5.3, the second morphism is Œc, so we have: Corollary 17.18. In DF .M / there is an exact triangle HF  .L#0 ; L#1 / ˝ L#0

Œf 

hQQQ QQQŒ1 QQQ QQQ

/ L# w 1 w ww ww w {ww Œc

L0 .L#1 /:

(17.23) 

Remark 17.19. The third map in (17.23) is an element of HomDF .M / .L0 .L#1 /; HF  .L#0 ; L#1 / ˝ L#0 Œ1/  1 Š HF  .L#0 ; L#1 / ˝ HF  .L0 .L#1 /; L#0 /  n Š HF  .L#0 ; L#1 / ˝ HF  .L#1 ; L#0 / ; where the last isomorphism uses the equality L0 .L#0 / D L#0 Œ1  n mentioned in Example 17.14. It is natural to expect this to be the element given by the Poincaré dual pair of bases; equivalently, one can think of this as a dual evaluation map of the kind used in (5.20). It seems that in principle, one could prove that this is indeed the case, by combining Remarks 5.5 and 17.13. However, we have not checked all the details; in particular, the sign issues are tricky, and will affect the result in a nontrivial way (compare Remark 3.12). Finally, we have a more substantial application. Let V # D .V1# ; : : : ; Vm# / be a collection of framed exact Lagrangian spheres in M , each equipped with a brane structure (for n D 1 and char.K/ ¤ 2, we require that all the Pin structures be nontrivial). Let W # D Lk V # be the collection obtained from this one by an elementary Hurwitz move (16.7), with the natural induced brane structures (for the grading, this uses the natural grading of Vk ). Let F ! .V # /, F ! .W # / be the associated directed A1 -subcategories of F .M /. Corollary 17.17 says that the objects of W # are isomorphic in D b .F .M // to those of the collection obtained by applying a mutation (5.24) to V # . Passing to isomorphic objects does not change the quasi-isomorphism class of the directed A1 -category, by Lemma 5.21. Hence, Lemma 5.23 applies and shows that Tw F ! .V # /, Tw F ! .W # / are quasi-equivalent. A far more elementary fact is that Tw F ! .V # / does not change if we shift the grading of one or several Vj . We summarize the outcome:

17 Pseudo-holomorphic sections

263

Theorem 17.20. Up to quasi-equivalence, Tw F ! .V # / is an invariant of the Hurwitz equivalence class of V .  When applied to distinguished bases of vanishing cycles, this immediately yields the following: Corollary 17.21. Let  W E ! S be a Lefschetz fibration whose base S is a disc, carrying a relative quadratic complex volume form 2E=S . Choose a base point 2 , and equip M D E with the associated form M . Let  be a distinguished basis of vanishing paths starting at , and V  the associated collection of vanishing cycles in M . Equip each of these cycles with a grading and Pin structure (choosing nontrivial Pin structures if n D 1 and char.K/ ¤ 2), and denote the outcome by V # . Let F ! .V # / be the associated directed A1 -subcategory of F .M /. Up to quasi-equivalence, the category Tw F ! .V # / of twisted complexes is independent of the choice of  and of the gradings, hence is an invariant of the Lefschetz fibration. Hence, the same holds for the derived category DF ! .V # /, up to equivalence of triangulated categories.  Remark 17.22. Corollary 17.17 was originally conjectured by Kontsevich. Obviously, one would expect a stronger version to hold, namely, the automorphism induced by  should be canonically isomorphic to T in H 0 . fun.F .M /; F .M ///. We will not prove that here; but it is expected to be a consequence of ongoing work of Wehrheim–Woodward on Lagrangian correspondences. (17k) The equivariant case. We begin with some simple geometric preliminaries. Let M be an exact symplectic manifold with a Z=2-action M . An invariant framed exact Lagrangian sphere in M is a M -invariant exact Lagrangian submanifold V together with an equivariant diffeomorphism v W S ! V , where S is the unit sphere in some inner product space equipped with a linear Z=2-action. Since the Lagrangian tubular neighbourhood theorem can be made compatible with finite group actions, and the local model for a Dehn twist is equivariant with respect to OnC1 , the Dehn twist L along such a sphere is naturally an equivariant exact symplectic automorphism. Given an exact Lefschetz fibration which carries a fibrewise involution, the associated parallel transport maps are equivariant, the vanishing cycles are equivariant framed spheres, and an appropriate equivariant version of the Picard–Lefschetz formula (16.6) holds. Conversely, given one or several equivariant framed spheres in M , one can arrange that the Lefschetz fibrations from Lemmas 16.8 and 16.9 carry fibrewise involutions. Now assume that we are in the situation of Section (14e), so that the equivariant Fukaya category can be defined. We denote this by A D F .M /naive , and the Z=2-action on it by A . Assume that we have an equivariant framed exact Lagrangian

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III Picard–Lefschetz theory

sphere V D L0 , which can be made into an equivariant brane L#0 D .L0 ; ˛0# ; P0# ; f0 /. We also want to have two other equivariant branes L#k D .Lk ; ˛k# ; Pk# ; fk /, k D 1; 2, together with an isotopy V .L1 / ' L2 through invariant exact Lagrangian submanifolds, and which can be made compatible with the equivariant brane structures. In this situation, we can consider the A -invariant objects Yk 2 ObA given by the L#k , as well as their associated equivariant Yoneda modules Yk 2 Obmod eq .A/. As pointed out in Section (6c), the twisted A1 -module TY0 .Y1 / is again equivariant. The equivariant analogue of Theorem 17.16 says that .c; k/ give rise to a morphism of equivariant modules Œt eq  2 HomH 0 .mod eq .A// .TY0 .Y1 /; Y2 /;

(17.24)

which is an isomorphism in that category. Even though there is nothing new here, a brief explanation may still be helpful. First of all, t eq is defined by the same formulae as in Lemma 5.3 (i). Now recall that if we forget the Z=2-action, A becomes quasiequivalent to F .M /. More precisely, by making each Lagrangian brane into a preequivariant one (this is always possible, see Remark 14.1), one can identify F .M / with a full A1 -subcategory of A which is quasi-equivalent to the whole thing; and the resulting functor mod eq .A/ ! mod.A/ ! mod.F .M //

(17.25)

is cohomologically full and faithful (the first part, forgetting equivariance, is full and faithful on the chain level; and the second part, restriction to F .M /  A, is a quasi-equivalence by Lemma 2.8). Moreover, one can arrange the embedding F .M /  F .M /naive in such a way that the Lagrangian branes L#k get mapped to their equivariant counterparts. Then, the image of t eq under (17.25) is the previously considered module homomorphism t, which shows that Œt eq  is an isomorphism, as stated above. Lemma 17.23. Œt eq  belongs to the Z=2-invariant part of the Hom group in (17.24). One can fairly reasonably claim that this is trivial: we have previously shown that Œt is canonical, hence it should be invariant under any symmetries which are inherent in our geometric setup, and the same would apply to its equivariant version. We still want to take a look at the details, to make sure that no unwelcome surprises are lurking there. One can arrange that the Lefschetz fibrations E c ! S c and E k;r ! S k;r , as well as their boundary conditions, carry fibrewise Z=2-actions compatible with the one on M . All the Lagrangian submanifolds involved are equivariant branes, which come with Z=2-invariant Floer data by construction of F .M /naive . The remaining choice is that of relative perturbation data. At this point, we prefer to avoid equivariant transversality questions, and instead argue as follows. Suppose that L#1 ; L2# were more generally pre-equivariant branes. Consider first E c ! S c with its fibrewise

18 The Fukaya category of a Lefschetz fibration

265

involution. Denote by F the Lagrangian boundary condition used to define c, by F # its relative brane structure, and by .K; J / the relevant relative perturbation datum. One can apply the fibrewise involution to both .K; J / and F # , and then tensor the Pin structure on the latter by the line bundle top .TF v /. The outcome of counting inhomogeneous pseudo-holomorphic sections in this modified situation is a class CF 0 .M .L#1 ˝ top .T L1 //; M .L#2 ˝ top .T L2 ///, which by definition is just A .c/. Now, in the case where both L#1 and L#2 are equivariant branes, the isomorphisms fk yield a natural identification   CF 0 M .L#1 ˝ top .T L1 //; M .L#2 ˝ top .T L2 // D CF 0 .L#1 ; L#2 /  .Pk# ˝ top .T Lk // which comes from the isomorphisms of Pin structures Pk# ! M for k D 1; 2. In our case, these isomorphisms are compatible with the isotopy V .L1 / ' L2 , which means that they give rise to an isomorphism of relative Pin structures for the Lagrangian boundary condition F . With this in mind, it is clear that A .c/ differs from c only by the choice of relative perturbation datum. The situation for k is entirely parallel. Therefore, the construction from (17.14) applies (with suitable signs inserted), yielding .; / such that

1A ./ C c  A .c/ D 0;   .1/jc0 j1 1A . .c0 //  .1A .c0 // C 2A .; c0 /  A . .A .c0 /// C .c0 / D 0 for all c0 2 CF  .L#0 ; L#1 /. With this at hand, the argument from Lemma 5.3 (ii) produces a homotopy between t eq and its image under the Z=2-action, thereby proving that on the cohomological level, Œt eq  is indeed invariant.

18 The Fukaya category of a Lefschetz fibration (18a) The double branched cover. Fix an exact Lefschetz fibration  W E 2nC2 ! S, where S D D is the unit disc in C (with the standard complex structure). Choose a small ı > 0, and write S shrink  S for the disc of radius 14ı, as well as E shrink  E for its preimage in the total space. We assume that all critical values of  lie in the interior of S shrink , and moreover, that the negative Liouville flow on E, for some positive time, will push the whole of E into E shrink . Both conditions are certainly satisfied for sufficiently small ı. We will take D 1  3ı 2 int.S/ as a base point, with M D E being the fibre at that point. Whenever a choice of base path becomes necessary, we will use the straight line from to 1 2 @S . Let Sz be the double cover of S branched at , and similarly, Ez the double cover of E branched along M D E . Of course Sz is again isomorphic to D, but we prefer

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III Picard–Lefschetz theory

not to make that identification, and instead write Sz D fy 2 C W y 2 C 2 S g; Ez D f.y; x/ 2 Sz  E W ty 2 C D .x/g:

(18.1)

Ez comes with an induced almost complex structure (it is an almost complex submanifold of the product Sz  E), and this makes @Ez weakly convex. With respect to this structure, we have pseudo-holomorphic maps ) Sz .y/ D y covering involutions; Ez .y; x/ D .y; x/ ) ˇSz .y/ D y 2 C W Sz ! S quotient maps; ˇ z .y; x/ D x W Ez ! E E

z .y; Q x/ D y W Ez ! S;

lift of .

The critical points of Q are the preimages of those of , and since the local structure remains the same, they are still nondegenerate. To complete the construction, we need to introduce suitable exact symplectic structures. Recall that, as part of the data defining our fibration, the base S carries some S 2 1p.S/ and øS D dS 2 2 .S/. From (18.1) one sees that Sz contains the disc of radius p2ı around the origin. Let be a one-form supported inside the smaller disc of radius ı, which is Sz -invariant, and has the property that d is positive at the origin. We want to think of as small, hence reserve the right to multiply our first choice by some positive small constant if necessary. Set Sz D ˇSz S C ;

øSz D dSz D ˇSz øS C d :

(18.2)

Since d is small, and is positive at the origin, øSz is everywhere positive; and since is supported in the interior, Sz is positive along the boundary. These are the conditions for (18.2) to make Sz into an exact symplectic manifold with boundary. In parallel with this, we equip Ez with  Ez D ˇE Q  ; z E C 

 øEz D dEz D ˇE Q  d : z øE C 

(18.3)

The first term ˇ z øE fails to be symplectic only in normal direction to the branch fibre E fy D 0g, so by the same argument as before, adding a sufficiently small Q  d term makes it symplectic everywhere. Note that the associated symplectic connection on the regular part of Q is the pullback of that for , since the second summand in (18.3) comes from the base. This shows that @h Ez is horizontal, and also (using the same argument as in Lemma 15.1) that the negative Liouville flow points inwards along z the negative Liouville vector field is the pullback that part of the boundary. Near @v E,

18 The Fukaya category of a Lefschetz fibration

267

of the one on E, hence again points inwards. With that in mind, we have verified that Q W Ez ! Sz is again an exact Lefschetz fibration. Two classes of exact Lagrangian submanifolds in Ez will figure prominently in the following discussion. We denote them by (U) for unbranched, and (B) for branched: • Type (U). The restriction of ˇSz to S shrink is an unbranched covering, hence trivial, which means that the total space consists of two copies of the base. The same applies to the restriction of ˇEz to E shrink . We write shrink shrink Szshrink D ˇS1 / D SzC t Szshrink ; z .S 1 shrink shrink / D EzC t Ezshrink : Ez shrink D ˇE z .E

(18.4)

More concretely, points y which satisfy jy 2 C j  1  4ı necessarily have im.y/ ¤ 0, and the sign of that imaginary part p distinguishes the two components in (18.4). Such y also have jyj  ı, which means that on the restricted versions of each double branched cover, the summand in (18.2), (18.3) becomes zero. As an obvious consequence, if L is an exact Lagrangian submanifold of E which is contained in E shrink , its preimage in Ez consists of two copies of L, which are again exact Lagrangians. In analogy with the above, we use the notation Q D ˇ 1 .L/ D L Q C t LQ  : L z E • Type (B). Let  W Œ0I 1 ! S be a vanishing path with .0/ D , and   E its Lefschetz thimble. Then Q D ˇ 1 . /  z E

(18.5)

z One way to see this is to represent it is a smooth Lagrangian sphere in E. as a matching cycle, as follows. Up to orientation-reversal, there is a unique smooth path Q W Œ1I 1 ! Sz such that ˇSz ..t// Q D .t 2 /. The endpoints of Q are the preimages of the critical value .1/, hence the two halves Q˙ .t/ D .˙t/ Q W Œ0I 1 ! Sz are vanishing paths for . Q As pointed out above, the symplectic connection is the pullback of the one for , and from that it follows that each Lefschetz thimble for Q˙ projects down to the Lefschetz thimble for  . This means that the vanishing cycles of Q˙ coincide, being equal to the vanishing cycle of  inside M , and that (18.5) is just the matching cycle for , Q taken in the naive sense; see the start of Section (16g). Instead of D 1  3ı, one could use another regular value w 2 S n Critv./ as a branch point. The technicalities of proving that the double branched cover is a Lefschetz fibration require that jwj  1  2ı. For the construction of type (U) Lagrangian submanifolds, one would have to assume that L lies inside the region

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III Picard–Lefschetz theory

Q Q where j.x/j p  jwj  ı. Moreover, the distinction between LC and L depends on a choice of w. This of course means that if one moves w around, say, the circle of radius 1  3ı, the two components get interchanged. We will make this more rigorous below, see Lemma 18.1. (18b) A family of double covers. We will now construct a Lefschetz bifibration which contains Q W Ez ! Sz together with its analogues for other branch points, including degenerate (singular) ones. Let W  C be the disc of radius 1  2ı. Consider the double cover of W  S branched along the diagonal, and the double cover of W  E branched along the graph of  (or more precisely, those parts of the diagonal resp. the graph where one coordinate lies in W  S): SQ D f.w; y/ 2 W  C W y 2 C w 2 Sg; EQ D f.w; y; x/ 2 SQ  E W y 2 C w D .x/g: SQ comes with a natural complex structure, and EQ with an almost complex structure; both times, the boundaries are weakly convex. We have pseudo-holomorphic maps ) SQ .w; y/ D .w; y/ W SQ ! SQ covering involutionsI EQ .w; y; x/ D .w; y; x W EQ ! EQ ) ˇSQ .w; y/ D y 2 C w W SQ ! S quotient maps, forgetting wI ˇ Q .w; y; x/ D x W EQ ! E E

Q $.w; y; x/ D .w; y/ W EQ ! S; ) .w; y/ D w W SQ ! W ‰.w; y; x/ D w W EQ ! W

lift of , for each w; projections to the parameter w:

Note that ‰ D ı $ . The singularities of $ are a a Crit.$/ D fy 2 C w D .x/g; Critv.$/ D fy 2 C w D zg: (18.6) x2Crit./

z2Critv./

Among the first set, the critical points of ‰ are those of the form .w D .x/; 0; x/, which means that Critv.‰/ D Crit./. If we choose local holomorphic Morse coordinates near a point x 2 Crit./ with .x/ D z, the defining equation for EQ turns into 2 I (18.7) w D z  y 2 C x12 C    C xnC1 2 ; y/. after eliminating the w variable, one has $.x; y/ D .z  y 2 C x12 C    C xnC1 From that, one sees easily that both $ and ‰ have generic critical points. Moreover, the Hessian of ‰ at a critical point .z; 0; x/ remains nondegenerate if we restrict to

18 The Fukaya category of a Lefschetz fibration

269

ker.D$/, which is the subspace tangent to y D 0. These computations show that as far as almost complex geometry is concerned, ‰D ı$

EQ

$

/ SQ

/) W

(18.8)

satisfies all the properties of a Lefschetz bifibration (actually, this is a particularly simple example of a bifibration, since there are no fake critical points). In spite of the elementary nature of this whole discussion, it may be worth while to rephrase the main points in a more intuitive form. For each w, the restriction $w W EQ w D ‰ 1 .w/ ! SQ w D

1

.w/

(18.9)

is the pullback of  along the double branched cover ˇSQ jSQ w W SQ w ! S (in particular, z Whenever w … Critv./, for  D we get back the previously defined Q W Ez ! S). EQ w is smooth and (18.9) has generic critical points, which come in pairs corresponding to the critical points of . Morepprecisely, if we fix w, then to each x 2 Crit./ correspond the two points .w; ˙ .x/  w; x/. As w approaches a critical value z D .x/, the two square roots come together, at what is the unique singular point of EQ z . Take W ; øW which make W into an exact symplectic manifold with boundary. These will be assumed to be sufficiently large. Consider the one-form SQ obtained by adding the pullbacks of W (by ), (by the second projection), and S (by ˇSQ ), and set øSQ D dSQ . We have already seen that W jSQ  D øSz is symplectic, and the same holds for the restriction to any other fibre. Therefore, because we have added the pullback of the large symplectic form øW from the base, øSQ itself is symplectic. By the same argument as in Lemma 15.3, see in particular (15.7), the Liouville vector field will point inwards along the boundary face @v SQ which lies over @W . To deal with the other face, consider the partial trivialization p Q W  Sz W  Uz ! S; .w; y/ 7! .w; y 2 C  w/ (18.10) z one chooses the unique defined on the subset Uz D fjy 2 C j  1  ıg  S; square root which gives the identity map for w D . p The image of (18.10) p is h Q 2 a neighbourhood of the fibrewise boundary @ S. Since j y C  wj  ı on W  Uz , the partial trivialization is compatible with the given symplectic structures and one-form primitives, in the sense that the pullback of SQ by (18.10) is W C .Sz jUz /. From this, it follows easily that SQ is an exact symplectic manifold with corners, and Q that the horizontal tangent spaces for run parallel to @h S.   Correspondingly, set EQ D $ .W C / C ˇ Q E . The same argument as before E shows that if øW is sufficiently large, øEQ D dEQ will be symplectic, except possibly

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near the critical points of ‰. If we now look at a critical point in local coordinates .x; y/, as in (18.7), then øEQ equals d (in the y-coordinate) plus øE (in the x2 coordinate) plus the pullback of øW by ‰.y; x/ D z  y 2 C x12 C    C xnC1 ; which Q is obviously a Kähler form. Next, we will look at the various boundary faces of E. (i) On the face defined by jwj D 12ı, which is ‰ 1 .@W /, the argument used before for @v SQ applies, and shows that the negative Liouville vector field points inwards. (ii) In a neighbourhood of the face jy 2 C wj D 1 (which is where x 2 @v E), one has a partial trivialization similar to (18.10). The behaviour of the Liouville field therefore z Note that this face is part of the horizontal boundary of EQ reduces to that of W  E. taken with respect to ‰; the trivialization also shows that the symplectic connection for ‰ is trivial near it. (iii) The remaining face consists of those .w; y; x/ where x 2 @h E. At such a point, consider DˇEQ W T EQ D T EQ h ˚ T EQ v ! TE D TE h ˚ TE v :

(18.11)

The second summand T EQ v is the tangent space along the fibre of $, hence maps h isomorphically to TE v . The first summand T EQ is its orthogonal complement with respect to øEQ . This is actually the same as the orthogonal complement with respect to ˇ Q øE , since the difference between the two forms is pulled back from the base of $. E Therefore, DˇEQ preserves the splittings (18.11). Since the map ˇEQ is transverse to @h E, the tangent space along our boundary face is the preimage of T .@h E/ under h (18.11), hence contains the whole of T EQ . This shows that the horizontal tangent spaces of the symplectic connection associated to $ run parallel to the boundary at .w; y; x/; for obvious reasons, this implies the same property for ‰. Moreover, the negative Liouville vector in T EQ differs from the corresponding vector field on TE h only by a T EQ component, and since that component is tangent to the boundary, the Liouville vector must point inwards. We have now shown that EQ is an exact symplectic manifold with corners, and that $, ‰ are exact fibrations with singularities; this verifies that (18.8) satisfies the symplectic aspects of the definition of Lefschetz bifibration. For future use, we record one simple observation concerning the geometry of ‰: Lemma 18.1. Let be the closed path going once (positively) around the circle of radius j j D 1  3ı in W . Consider the associated monodromy automorphism of z On the subset Ez shrink , this map agrees with the involution  z . EQ  D E. E Proof. At any point p .w; x; y/ where jwj D 13ı and j.x/j  14ı, we have jyj D p j .x/  wj  ı, hence the d -contribution to the symplectic form vanishes. As a consequence, the restriction of the symplectic connection for ‰ equals the pullback of the trivial connection by projection to W  E. Hence, parallel transport along

18 The Fukaya category of a Lefschetz fibration

271

.t/ D .1  3ı/e 2 it becomes q   .x/.t/ y 2 C .0/  .t/; x/ D .t/; y .x/.0/ ;x ; (18.12) where the square roots are chosen so that (18.12) is the identity for t D 0. As t goes from 0 to 1, .t/  .x/ describes a loop around the origin, hence the square root changes sign.  hjŒ0It  W . ; y; x/ 7! . .t/;

p

(18c) Vanishing cycles and matching cycles in the double cover. Take a vanishing path  W Œ0I 1 ! S for , starting at .0/ D , and which remains inside the smaller disc W  S. Since ‰ W EQ ! W has the same critical values as , one can also think of  as a vanishing path for that Lefschetz fibration, hence there is an associated vanishing cycle VQ , which is a Lagrangian sphere in Ez D EQ  . On the other hand, we already saw in Section (18a) that the preimage of , suitably parametrized as z is a matching path for , Q . Q W Œ1I 1 ! S, Q with associated matching cycle  z  Q is isotopic to VQ . Lemma 18.2. As a framed Lagrangian sphere in E, Proof. Most of this can be derived from the general theory of bifibrations, more Q specifically Lemma 16.15. Namely, by looking at the structure of Critv.$/  S, see (18.6), one sees that the matching path associated to  is isotopic to Q . The only problem with this argument is that Lemma 16.15 does not characterize the isotopy I which occurs. Of course, in the present case the isotopy is somewhat redundant, since the matching cycle can be formed in the naive way; but still, it is maybe worth while to check that all the ingredients come together properly. As usual, we first apply a local deformation near the unique critical point x 2 Ez , z D .1/, in order to make the Kähler form øE standard in local holomorphic Morse coordinates. We may also assume that .t/ D z C .1  t / for t close to 1. Both these changes are permitted because they do not affect the isotopy classes of the two framed Lagrangian spheres under consideration. Take the short vanishing path jŒsI 1, for some s sufficiently close to 1, and its preimage under S .s/ ! S, which is a matching path for $ .s/ . By assumption, this preimage is actually the straight line connecting the two square roots of z  .s/. Take local coordinates .y; x/ as in (18.7), so that øEQ D d C øE C ‰  øW , where now the second summand is the standard form in the x-coordinates. This is not quite the standard form on CnC2 , but it is still true that the subspace defined by y 2 iR, x 2 RnC1 is øEQ -Lagrangian. From our characterization of Lefschetz thimbles, it is easy to see that a suitable piece of this subspace is necessarily the Lefschetz thimble for jŒsI 1; hence, the vanishing cycle is VQ jŒsI1 D fy 2 iR; x 2 RnC1 ; jyj2 C jxj2 D .s/  zg;

(18.13)

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III Picard–Lefschetz theory

with the obvious framing. Until now, we have considered jŒsI 1 as a vanishing path for ‰. If we now go back temporarily to looking at it as a vanishing path for , the associated Lefschetz thimble is of course fx 2 RnC1 ; jxj2  .s/  zg; hence, the Q jŒsI1 , coincides preimage of this thimble under ˇEQ jE .s/ , which by definition is  with (18.13). As in Lemma 16.15, one now transports this fact back to the entire path  by using suitable parallel transport maps.  For future reference, we will look at two other geometric situations, still involving the classes of Lagrangian submanifolds defined above. Suppose that we are given three vanishing paths k , kC1 , ık for  as in Figure 16.2 (of course, we do not need a whole distinguished basis at this point, but we keep the notation for convenience). Q k,  Q kC1 , … z k the associated type (B) Lagrangian submanifolds of E. z Denote by  Q kC1 /. z … z k is isotopic to  Q . Lemma 18.3. As a framed Lagrangian sphere in E, k Proof. Without loss of generality, we may assume that our paths are contained in W . Q k is the vanishing cycle of k considered as a vanishing path Then, by Lemma 18.2,  for ‰. Therefore, the associated Dehn twist is the monodromy of ‰ along a loop

which doubles around that path, which means that Q kC1 / ' h . Q kC1 /: Q k . By definition, ık is the result of taking kC1 and moving its starting point along . This means that there is a smooth family of vanishing paths  s for , with  s .0/ D .s/ and  s .1/ D kC1 .1/, which starts at  0 D kC1 and ends at  1 D ık . These Q s  EQ .s/ interpolating give rise to a smooth family of framed Lagrangian spheres  Q0 D  Q kC1 and  Q1 D … z k . By looking at the image of these spheres between  Q kC1 / under parallel transport along jŒsI 1, one obtains an isotopy between h . z k. and …  Finally, let  be a matching path for , and †  E its matching cycle. By assumption, the negative Liouville flow moves each Lagrangian submanifold into E shrink , and since † is defined up to isotopy only, we may just as well assume z D † z ;C [ † z ; is a type (U) that it lies inside E shrink , so that its preimage † z Lagrangian submanifold, consisting of two disjoint framed Lagrangian spheres in E. Choose vanishing paths k ; kC1 , both starting at and ending at the two endpoints of , as in Figure 18.1. The key requirement is that the triangle described by these three paths should contain no critical values of  in its interior, so that the composite 0 kC1 D #k (formed in the same way as in Section (16h)) is a vanishing path Q k;  Q kC1 be the type (B) submanifolds associated to k ; kC1 . isotopic to kC1 . Let  Q Q kC1 is isotopic to  z  z Lemma 18.4. As a framed Lagrangian sphere,  † ; † ;C .k /.

18 The Fukaya category of a Lefschetz fibration

Q C

 0 kC1

k

273

Qk QkC1 kC1 0 QkC1

Q  Figure 18.1

0 Proof. Denote by Q D Q C [ Q  and Qk ; QkC1 ; QkC1 the preimages of the respective z paths in S. All of these are matching paths for , Q and moreover 0 QkC1 ' QkC1 D Q  #.Q C #Qk /:

Applying Lemma 16.13 twice yields the desired result, except for potential framing issues, but these can be resolved easily: deform the symplectic connection on our original Lefschetz fibration E such that the matching cycle of  is constructed in the naive sense (with a constant isotopy I ). For the matching paths Qk ; QkC1 , the isotopies are constant too, and this is compatible with the Lagrangian connected sum argument used to prove Lemma 16.13.  Remark 18.5. All three lemmas above admit improvements which take into account the involution Ez : • The isotopy from Lemma 18.2 is naturally equivariant, because equality holds in the local model, and the deformation used to reduce to that model case is compatible with EQ . • For Lemma 18.3, one simply uses the equivariant version of the Picard– Lefschetz formula. We want to point out a useful additional detail. The Q kC1 yields an equivariant diffeomorcomposition of the isotopy with Q k j phism Q kC1 ! … z k;  (18.14) so it makes sense to compare co-orientations of the fixed point sets inside the two spheres. Now, such co-orientations naturally correspond to orientations of the paths ıQk , QkC1 in Sz. The statement is that two co-orientations are compatible with (18.14) iff the intersection number ıQk  QkC1 at the unique common point (y D 0) is 1. This follows by taking the loop from the proof of Lemma 18.3, and comparing the associated monodromy map on Ez and braid

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III Picard–Lefschetz theory

z Generally the braid monodromy is a diffeomorphism monodromy map on S. of Sz preserving Critv./, Q and in the present instance this is simply a half-twist along Qk . If one chooses an orientation of QkC1 and equips its image ıQk D t Qk .QkC1 / with the induced orientation, then the local intersection number is indeed 1, see Figure 18.2.

ıQk D t Qk .QkC1 /

QkC1 Qk

Figure 18.2

• Our proof of Lemma 18.4 is less ideal for this purpose, since we break the symmetry by doing one Dehn twist at a time. However, this is easily remedied by an appropriate variation of the argument, which is to first provide an isotopy kC1 ' † .k / of Lagrangian balls (in E rel M ), and then to lift that to the double branched cover. (18d) Gradings. Throughout the rest of this section, we will assume that E comes with a quadratic complex volume form 2E . Division by the form dz 2 on the base S yields a relative form 2E=S D 2E =dz 2 . The pullback of 2E to Ez is nonzero away from the branch locus M D E . To see what happens at that locus, write 2E D 2E=S ^dz 2 , which makes sense near M since that fibre is nonsingular. The pullback acquires a 4y 2 dy 2 factor, which means that we can define a quadratic complex volume form on z and a corresponding relative form, by setting E,  2 2 2Ez D .ˇE z E /=4y ;

2Ez =Sz D 2Ez =dy 2 :

(18.15)

Note that this is arranged in such a way that the induced forms on the nonsingular fibres Ezy agree with those on Ez , z D ˇSz .y/. One can apply the same argument to EQ considered as a double cover of W  E branched along fw D .x/g. The outcome

18 The Fukaya category of a Lefschetz fibration

275

is that the Lefschetz fibration ‰ W EQ ! W carries forms 2EQ D ˇEQ 2E ^ dw 2 =4y 2 ;

2E=W D 2EQ =dw 2 Q

with the property that the induced forms on the regular fibres EQ w are equal to .ˇEQ jEQ w / 2E =4y 2 , hence in particular give back (18.15) for w D . Recall that in such a situation, the monodromy map along any loop in the regular part of the base W naturally becomes a graded symplectic automorphism. Lemma 18.6. Take the monodromy from Lemma 18.1, and let  # be its natural grading. Then, for all .y; x/ 2 Ez shrink , we have  # .y; x/ D 1. Proof. By definition of 2Q

E=W

and (18.12), one has

  .x/  .0/ hjŒ0It 2E=W jEQ .t/ D 2Ez  Q .x/  .t/

(18.16)

at every point .y; x/ satisfying our assumption. As t goes from 0 to 1, the fraction in (18.16) turns once around the origin in clockwise direction, hence so do all the phase functions in (15.8), which implies the desired statement.  Q k the We also need to take a second look at Figure 18.1. Denote by Vk ; k ;  vanishing cycle, Lefschetz thimble, and type (B) Lagrangian submanifold associated 0 0 Q0 ; 0kC1 ;  for the same data obtained from kC1 . Vk is to k ; and write VkC1 kC1 0 isotopic to the vanishing cycle of kC1 , hence also to VkC1 (more precisely, the isotopy I used to construct the matching cycle for  induces an isotopy between them). Since 0 k and kC1 are equal near the base point, this induces a (fibrewise) isotopy between k \ U and 0kC1 \ U , where U  E is a suitable small neighbourhood of E . Q k \ Uz and  Q 0 \ Uz in Lifting this to the double cover, we get an isotopy between  kC1 the preimage Uz of U . Choose gradings which are compatible with this local isotopy, Q kC1 . and then deform the latter to a grading of  Q kC1 , and with the natural Q k and  Lemma 18.7. With these choices of gradings of  gradings of the Dehn twists involved, the isotopy in Lemma 18.4 becomes one of graded Lagrangian submanifolds. Proof. The usual deformation trick reduces the question to the case where the matching cycle † can be formed in the naive way, and where the symplectic form on E is standard in local holomorphic Morse coordinates near the critical point in k .1/. We have imposed the second requirement in order to take advantage of the fact that, in the local model from Example 16.5, parallel transport along any path ˇ in the base p p n maps the vanishing cycle ˇ.0/S to ˇ.1/S n .

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III Picard–Lefschetz theory

0 Choose kC1 carefully, so that it agrees with k or  outside a small neighbourhood of the critical value k .1/. In that case, one sees by following the par0 allel transport maps that Vk D VkC1 . Therefore, taking Uz as before, we have 0 Q k \ Uz D  Q  \ Uz . By definition, the gradings of these two submanifolds kC1 z ;˙ , hence trivial will be the same. Because the Dehn twists are concentrated near † z on U , Q k / \ Uz D  Q0 z †z ; †z ;C . (18.17) kC1 \ U ;

and the gradings induced from the canonical gradings of the Dehn twist will still agree. Now, the isotopy between the two Lagrangian submanifolds in (18.17) provided by z ;˙ , so the equality of gradings is preserved.  Lemma 16.13 is again local near † (18e) Some (pre-)equivariant branes. We now review the two kinds of Lagrangian submanifolds of Ez introduced in Section (18a), with the aim of turning them into z naive . Recall that, in order to define objects of the equivariant Fukaya category F .E/ that category, we had to impose some technical conditions on the Z=2-action, see Assumption 14.3. Those are obviously satisfied in the present case: the fixed point set of Ez is fy D 0g Š M , and we have a natural trivialization of its normal bundle, z given by D jM Q W T EjM ! T Sz0 D C. • Type (U). Let L# D .L; ˛ # ; P # / be an exact Lagrangian brane in E, which is Q be the preimage of L in E. z In view contained in the interior of E shrink . Let L Q are related by of (18.15), the squared phase maps for L and L ˛Ez .T LQ .y;x/ / D

jyj2 y2

˛E .T Lx /:

Q  , where the connected component L Q ˙ lies in the QC [ L Recall that LQ D L subset ˙im.y/ > 0. Since im.y/ is never zero, we can take the argument function arg W C n RC ! .0I 2/, and define a grading of LQ by setting ˛Q # .y; x/ D ˛ # .x/ 

1 2

arg.y 2 /:

(18.18)

# Q ˙ . As for the Pin structures, our for L Restriction then yields gradings ˛Q ˙ Q  with the convention is to equip LQ C with the pullback of the given P # , and L # # top Q Q pullback of P ˝ .T L/. Denote these by P˙ ! L˙ , and their union by Q PQ # ! L.

With this in mind, it is easy to make LQ C into a pre-equivariant brane in the Q # , the shadow is L Q # , sense of Section (14a). The principal component is L C  Q C // is the identity, and one .PQC# ˝ top .T L the isomorphism fQ W PQ# ! M adjusts fQC so that (14.1) comes out right. The same procedure, with C and Q  ; by taking the  exchanged, yields a pre-equivariant brane structure for L Q disjoint union of the two, L becomes an equivariant brane.

18 The Fukaya category of a Lefschetz fibration

277

We can already draw some first consequences. Let L#k D .Lk ; ˛k# ; Pk# / (k D 0; 1) be two branes in E of the kind considered above. When choosing a Floer datum .H; J / for this pair, we may require that outside E shrink , H should vanish and J should agree with IE . This does not interfere with the regularity of moduli spaces MZ .y0 ; y1 /, since the choice is still free in a neighbourhood of L0 [ L1 . For the Q 0;C ; LQ 1;C /, take the Floer datum .HQ ; JQ / obtained by lifting .H; J /; preimages .L this is unproblematic because our Floer datum is trivial near the branch locus. Every Q 0;C ; LQ 1;C / remains inside Ez shrink , and projects to a y 2 C.L0 ; L1 /, which yQC 2 C.L C establishes a bijection between the two sets. This correspondence preserves absolute indices and (up to canonical isomorphism) orientation spaces: i.yQC / D i.y/;

o.yQC / Š o.y/:

(18.19)

shrink There is a minor point of concern, which can be addressed as follows: while 2z jEzC E is not equal to the pullback of 2E , one can be deformed into the other (up to an overall sign) by choosing a deformation retraction of C n RC onto the point 1, and applying that to the term jyj2 =y 2 in (18.15). This homotopy of quadratic complex volume # Qk of L forms induces a homotopy of gradings, which in fact deforms the grading ˛Q k;C # to the pullback of ˛k 1=2. The Pin structures of course agree anyway, so (18.19) follows easily. For each u 2 MZ .y0 ; y1 /, the map v D  ı u W Z ! S is holomorphic x on  D v 1 .S nS shrink /, which means that the maximum principle applies to jvj on . Since  is relatively compact and does not intersect @Z, the conclusion is that  D ;, shrink so u again remains inside EC . This means that projection establishes a bijection of between such u and elements uQ C 2 MZ .yQ0;C ; yQ1;C /. This comes with an identification of the corresponding linearized operators, so .HQ ; JQ / is again regular. Finally, this identification is compatible with the gluing procedures used to define the contribution (12.17) to the Floer differential @. This defines an isomorphism of Floer cochain complexes. A parallel argument applies to the  parts of the preimages; the only Q k; , difference is that the Pin structures are twisted by top .T Lk / Š E jLk Š Ez jL but since this twist applies equally to both Lagrangian submanifolds involved, the Floer complex is not affected. To summarize, one has a canonical isomorphism

Q #0; ; LQ #1; / CF  .LQ #0 ; LQ #1 / Š CF  .LQ #0;C ; LQ #1;C / ˚ CF  .L Š CF  .L#0 ; L#1 / ˝K KŒC; ;

(18.20)

where by KŒC;  we mean the two-dimensional vector space with two generators Q # , the Floer complex carries a C and . In view of the equivariant nature of the L k canonical involution; with respect to (18.20), this is just the map exchanging the C and  generators. More generally, given any .d C 1/-tuple of Lagrangian branes of the same kind, one can choose the perturbation data to have the same property as the Floer data considered above. Then, perturbed holomorphic .d C 1/-gons are also confined to E shrink , which means that (18.20) is compatible with the A1 -structure.

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III Picard–Lefschetz theory

Remark 18.8. Suppose that all our quadratic complex volume forms admit continuous square roots, so that branes are naturally oriented; see Remark 11.18. More Q .y;x/ at some point is posiexplicitly, a basis .v1 ; : : : ; vnC1 / of the tangent space T L tively oriented iff Ez .v1 ^    ^ vnC1 / : jEz .v1 ^    ^ vnC1 /j

exp. i˛ # .y; x// D

If we pass from .y; x/ to .y; x/, the sign of Ez D ˇ z E =2y changes, while ˛ # E Q C, L Q  carry opposite remains unchanged by definition. Hence, the two components L orientations. • Type (B). Let  W Œ0I 1 ! D be a vanishing path starting at .0/ D . We require that  0 .0/ 2 C n RC , and that j.0I 1 be disjoint from our base path ; from now on, these will be standing assumptions whenever vanishing paths Q be the preimage in Ez of the associated Lefschetz for  are considered. Let  thimble  . We equip this with an arbitrary grading ˛Q # , and with the unique (up to isomorphism) Pin structure PQ # . To make our brane into an equivariant one, it remains to choose an isomorQ //. By Lemma 14.2, such isomorphisms phism f W PQ # ! z .PQ # ˝ top .T  E correspond canonically to co-orientations of the fixed point set, which in our Q . Since  Q is fibered over the preimage Q of , we have case is V   p Q jV / D RQ 0 .0/ D R  0 .0/; (18.21) D .T Q  so co-orientations naturally correspond to choices of square roots. Our convention is to always take the square root with positive imaginary part. Equation Q satisfies Assumption 14.4. Finally, the definition (18.21) also shows that  naive z Q , and of F .E/ requires a choice of real-valued normal angle for each  0 we take this to be arg. .0//=2 2 .0I /. Q induces As pointed out in Section (14d), an equivariant brane structure on  a brane structure on its fixed point set, namely V . Concretely, the grading is # ˛V#  .x/ D ˛ Q .0; x/  

1 arg. 0 .0//; 2

and the Pin structure is the unique one (up to isomorphism) that bounds a Pin structure on  . This means that for n D 1, the vanishing cycle will carry the nontrivial Pin structure rather than the trivial one (for any compact connected surface S, the relative second Stiefel–Whitney class in H 2 .S; @S I Z=2/ Š Z=2 is the mod 2 Euler characteristic; since this is nonzero for S D D, there can be no Pin structure which is trivial on the boundary).

18 The Fukaya category of a Lefschetz fibration

279

Q # , HF  . Q #;  Q #/ Š Q# D  Lemma 18.9. (i) For any type (B) equivariant brane  H  .S nC1 I K/. The Z=2-action is Id on HF 0 and Id on HF nC1 . Q # be the associ(ii) Take a pair of vanishing paths k (k D 0; 1), and let  k Q #; Q#/ Š ated type (B) equivariant branes. Then the Poincaré duality map HF  . 1 0 nC1 Q # Q # _ HF .0 ; 1 / is anti-compatible with the Z=2-actions (which means, it is equivariant up to an overall 1 sign). Proof. These are purely cohomology level statements, and follow from the general Q properties of equivariant Floer cohomology listed in Section (14b). Indeed, Ez j Q is an orientation-reversing automorphism of , and since the PSS isomorphism is compatible with the Z=2-action, (i) holds. For (ii), we recall that the Poincaré duality isomorphism comes from the triangle product Q #1 ;  Q #0 / ˝ HF nC1 . Q #0 ;  Q #1 / ! HF nC1 . Q #0 ;  Q #0 / Š H nC1 . Q 0 I K/ Š K; HF  . Q #; Q # / Š H nC1 . Q 0 I K/ ˝ hence should be written more canonically as HF  . 1 0 nC1 Q # Q # _ .0 ; 1 / . In this form, it is again compatible with the Z=2-actions; HF Q 0 I K/ factor introduces a sign, because of (i). removing the H nC1 .  As was the case for type (U) branes, special choices of Floer data and perturbation data can help to make the Z=2-action more transparent on the cochain level. Take Q # and vanishing cycles V # . two vanishing paths k , with associated type (B) branes  k k Suppose that arg.00 .0// > arg.10 .0//; (18.22) 0 .Œ0I 1/ \ 1 .Œ0I 1/ D f g; and V0 t V1 :



Q 0;  Q 1 /, we want to impose a considerably When choosing a Floer datum .H; J / for . strengthened version of Assumption 14.5, namely: each J t should be such that Q is J t -holomorphic, and H D 0. The second part is compatible with Assumption 14.5 because, by (18.22), the normal angles satisfy 1  0 2 .I 0/, so that one can Q 0;  Q 1/ take h D 0 in (14.10). Under , Q any intersection point yQ D .y; x/ 2 C. necessarily projects to the unique intersection point of Q0 and Q1 . Hence x D 0, which means that yQ lies in Ez fix D M . By definition, the equivariant Pin structure Q k corresponds to the trivialization of the normal bundle to  Q fix which on each  k q 0 projects to the square root k .0/ with positive imaginary part. This square root is a positive real multiple of exp.ik / D exp.i arg.k0 .0//=2/. Therefore, the signs wk in Lemma 14.11 are both C1, which means that the Z=2-action on jo.y/j Q K is trivial. Q #; Q # / is Z=2-invariant. This shows that the entire chain group CF  . 0 1 Similarly, any uQ 2 MZ .yQ0 ; yQ1 / projects to a holomorphic map vQ D Q ı uQ W Z ! Sz with v.R Q  fkg/  Qk .Œ1I 1/, lims!˙1 v.s; Q  / D 0. The energy of any such vQ is

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III Picard–Lefschetz theory

necessarily zero, so it must be constant, which means that uQ is again contained in M . Assuming that .H; J / was chosen in such a way that the Floer moduli spaces on M z Moreover, Lemma 14.9 shows that are regular, Lemma 14.6 implies regularity in E. z the absolute indices and orientation spaces are the same whether we work in M or E, and (14.13) says the same about the sign with which each isolated uQ contributes to the Floer differential. To summarize, we have an isomorphism of complexes Q #0 ;  Q #1 / Š CF  .V0# ; V1# /; CF  .

(18.23)

where the right-hand side uses the brane structures on the vanishing cycles induced Q # . Moreover, the canonical Z=2-action on the left-hand side is trivial. More from  k generally, suppose that we have vanishing paths 0 ; : : : ; d (d  2) such that

 arg. .0// > arg. 0 k

0 .0// kC1

for all k; j .Œ0I 1/ \ k .Œ0I 1/ D f g for j ¤ k, and the Vk are in general position.

(18.24)

Then, after choosing a perturbation datum .K; J / with K D 0 and Q J -holomorphic, one finds that all holomorphic .d C 1/-gons remain within M . Assuming regularity (see Lemma 14.7), they also contribute in the same way to the A1 -structures on M z and E. Remark 18.10. In view of Lemma 18.9 (ii), (18.23) implies that the canonical Z=2Q #; Q # / is Id. In principle, one can prove this on the action on the dual group HF  . 1 0 cochain level, by taking a Floer datum with zero inhomogeneous term. In order for this to satisfy (14.10), where necessarily h D 0, one would have to replace the normal angle 1 by 1 C , which would mean that w1 D 1 in Lemma 14.11, making the Z=2-action on the relevant orientation spaces nontrivial. Unfortunately, the change in the normal angle makes this incompatible with the previously used Floer datum Q 0;  Q 1 /. Concretely, the attempt to use both at the same time runs into potential for . problems with equivariant transversality for pseudo-holomorphic polygons, and we z naive , the cochain will therefore abandon it. This means that in our version of F .E/ Q 1;  Q 0 / is true only on the level identity (18.23) holds, while its counterpart for . cohomological level. This choice is dictated by Proposition 18.14 below. Remark 18.11. A result similar to (18.20) holds in the mixed case, where one subQ D ˇ 1 .L/ is of type (U), and the other  Q D ˇ 1 ./ of type (B). First, manifold L z z E E even though  has boundary, the Floer cohomology of .L; / can be easily defined, as follows. Take a Floer datum .H; J / which is trivial outside E shrink , so that for any u 2 MZ .y0 ; y1 /, v D  ı u is holomorphic outside v 1 .S shrink /. Then, the image of v does not contain , for if it did, then by openness it must also contain nearby points z … .Œ0I 1/, which means that the intersection number u  ŒEz  would be

18 The Fukaya category of a Lefschetz fibration

281

positive. Since S n .S shrink [ v.Œ0I 1// is connected, the same positivity would hold for z 2 @S , which is a contradiction. The upshot is that, since u does not meet @, the existence of this boundary can be ignored in Floer-theoretic considerations. With this in mind, one can use isotopy invariance to reduce our question to the case where .t/ D  t for t 2 Œ0I ı, and where the rest of  remains inside S shrink . In this case, the maximum principle applies to jvj as before, yielding Q # / Š HF  .L# ; # / ˝K KŒC; : HF  .LQ # ;  On the right-hand side of this equation,  carries the unique (trivial) Pin structure Q reversing the process in (18.18). and the grading descended from that of , (18f) Two full embeddings. At this point, it is convenient to introduce a technical condition.  W E ! S has generic parallel transport if, within each homotopy class of distinguished bases of vanishing paths, there is one whose associated vanishing cycles are in general position. Given any Lefschetz fibration, one can find a small perturbation of .E ; øE ; IE /, which is constant near @E and also near M D E , so that the perturbed structure has generic parallel transport (this follows easily from Lemma 15.3 and the fact that there are countably many homotopy classes of distinguished bases). Such a perturbation does not change the Fukaya categories of M or E (in the second case, this is up to quasi-isomorphism, and uses Moser’s Lemma). With this in mind, we will assume from now on that our given Lefschetz fibration already has generic parallel transport. We require from now on that char.K/ ¤ 2; this allows one to apply results from Section 6, and even more importantly, it makes Id different from Id in Lemma 18.9. z naive to be the full A1 -subcategory whose objects are the preTake AQ  F .E/ Q # and the equivariant branes LQ # ,  Q # . Even though this largely equivariant branes L ˙ repeats the previous discussion, we still want to be explicit about what kinds of Floer Q For any pair of type (U) data and perturbation data are used in constructing A. equivariant or pre-equivariant branes, the Floer datum should be chosen in such a way that (18.20) holds. This in particular implies that each such brane is a free Q in the sense of Section (6a). Similarly, for any .d C 1/-tuple object of H 0 .A/, of type (U) branes, d  2, the perturbation data are taken to be of the same kind, to make the isomorphism compatible with the A1 -structure maps. Next, for any pair of type (B) equivariant branes, the Floer datum should be an invariant datum satisfying Assumption 14.5. However, in the special case where the branes arise from paths 0 ; 1 such that (18.22) holds, we want to adopt the more restrictive choice leading to the isomorphism (18.23). The same will apply to choices of perturbation data whenever a collection .0 ; : : : ; d / of vanishing paths satisfy (18.24). This is compatible with the consistency conditions for global choices, because the class of such .d C 1/-tuples is closed under taking ordered subsets.

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Definition 18.12. The Fukaya category of the Lefschetz fibration, denoted by F ./, inv is the invariant part AQ . Proposition 18.13. There is a cohomologically full and faithful A1 -functor F .E/ ! F ./; Q #. which sends each exact Lagrangian brane L# lying in E shrink to its preimage L Proof. By assumption, every exact Lagrangian brane on E is isotopic to one that lies inside E shrink . Therefore, the inclusion of the full A1 -subcategory consisting of such branes into F .E/ is a quasi-equivalence, which can be inverted by Theorem 2.9 (in fact, looking at the argument immediately preceding that theorem, one can find an inverse quasi-equivalence which is the identity on the subcategory itself). On the other hand, the same subcategory can be identified with the full A1 -subcategory of inv Q # , by (18.20). AQ consisting of objects L  Now let  D .1 ; : : : ; m / be a distinguished basis of vanishing paths. For each k D 1; : : : ; m, choose a grading so as to make the associated type (B) Lagrangian Q k into an equivariant brane, and equip the corresponding vanishing submanifold  Q # and V # the resulting colcycle Vk with the induced brane structure. Denote by  lections of objects in F ./ and F .M /, respectively. These come with directed Q # / and F ! .V # /. A1 -subcategories F ! . Q # / and Q # is an exceptional collection in F ./. Moreover, F ! . Proposition 18.14.  F ! .V # / are quasi-isomorphic, through a strictly unital A1 -functor which preserves the ordering of objects. As a consequence, there is a cohomologically full and faithful A1 -functor F ! .V # / ! F ./; Q#. which sends each Vk# to  k Proof. Suppose first that the vanishing cycles are in general position. From Lemma 18.9 (i) we know that Q # ; Q # / D HF  . Q # ; Q # /Z=2 Š H  .S nC1 I K/Z=2 D K; HomH F ./ . k k k k Q # is an exceptional object. Moreover, as already mentioned in Remark 18.10, so each  k part (ii) of the same lemma together with (18.23) implies that the Z=2-action on Q #; Q # / is Id for j > k, so that the morphism group in F ./ is acyclic, HF  . j k proving that we indeed have an exceptional collection. Lemma 5.20 then provides a Q # / to F ./. On the other hand, cohomologically full and faithful functor from F ! . from (18.23) one knows that that directed subcategory is identical to F ! .V # /.

18 The Fukaya category of a Lefschetz fibration

283

For the general case, we use the assumption that the Lefschetz fibration has generic parallel transport, which implies that for any  there is a homotopic  0 to which the previous argument applies. Since each vanishing cycle Vk0 is isotopic to Vk , these become isomorphic objects of H 0 F .M /. Lemma 5.21 shows that the two resulting Q 0 is isotopic to directed A1 -subcategories are quasi-isomorphic. Similarly, each  k Q k through Z=2-invariant Lagrangian submanifolds, so the associated objects in  H 0 F ./ are isomorphic, which yields a quasi-isomorphism between the associated  directed (and also full) A1 -subcategories. (18g) Type (U) branes and global monodromy. As before, let  be a distinguished Q # in basis of vanishing paths. To each k associate the type (B) equivariant brane  k z where the grading is chosen arbitrarily. In addition, let L# be an exact Lagrangian E, Q# D L Q # [ LQ # the corresponding type (U) brane in E, contained in E shrink , and L C equivariant brane. Q Q D H 0 .Tw A/, Lemma 18.15. In the derived category D.A/ Q # /: TQ # : : : TQ #m .LQ #C / Š S.L 1

(18.25)

Proof. In a simplified version, which ignores brane structures and the shift S, one can explain this as follows. By Corollary 17.17, the object on the left-hand side of Q C /. (18.25) is isomorphic to that given by the Lagrangian submanifold Q 1 : : : Q m .L Q k are the vanishing cycles Lemma 18.2 tells us that (up to framed isotopy) the  Q associated to k viewed as a vanishing path for ‰ W E ! W , so their Dehn twists are monodromies for ‰. It follows that the composition of all the Q k is isotopic z to the monodromy of ‰ around the circle of radius 1  3ı in W . By in Aut.E/ Lemma 18.1, the restriction of that monodromy to the subset Ez shrink , which by definition contains LQ ˙ , is equal to Ez , hence maps LQ C to LQ  . To get the full statement, one argues a little more carefully as follows. The action of TQ # on objects corresponds to that of Q k equipped with its canonical grading, k which by the graded version of the Picard–Lefschetz formula is also the grading naturally associated to the corresponding monodromy of ‰. By Lemma 18.6, the Q C ; ˛Q # ; P # / under this monodromy is . z .L Q C /;  ˛Q #  1;  P # /. image of LQ #C D .L C C E z C z C E E Q # by a constant 1 in the grading, as well a ˝ top .T L / in the This differs from L Pin structure. But that difference is precisely the action of the shift operation on Lagrangian branes, as defined in Section (12a); we checked in Section (12h) that this indeed quasi-represents the abstract (algebraic) shift.  Remark 18.16. The appearance of S in (18.25) may seem puzzling if one thinks of it as an operation reversing the orientation of Lagrangian submanifolds. In fact,

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opposite orientations are built into the definition of LQ #˙ (see Remark 18.8), and the shift merely restores a common choice. Proposition 18.17. LQ # lies in the triangulated subcategory of Tw F ./ generated by Q #;:::; Q #m .  1 Proof. This is a straightforward application of Lemma 6.3. LQ # is an equivariant brane, hence an object of AQ which is fixed under the involution AQ . By definition, it is Q # /. Moreover, HF  .L Q # ; LQ # / D isomorphic to the direct sum of LQ #C and LQ # D AQ .L C C Q # are again 0, because the underlying Lagrangian submanifolds are disjoint. The  k Q # ; Q # / is Z=2-invariant for all k < l, AQ -fixed objects, with the property that HF  . k l by (18.23). Lemma (18.15) is just the required relation (6.2), and by definition inv AQ D F ./.  (18h) Matching cycles. We will now consider more closely an important special case of the previous situation, namely when L D †;I is the matching cycle associated to some matching path  in S, whose position relative to the k is as indicated in Q k;  Q kC1 with equivariant brane structures, in such a way that the Figure 18.1. Equip  induced brane structures on Vk ; VkC1 become isotopic (this is just a reformulation of the way in which gradings were fixed for Lemma 18.7). Similarly, we pick a grading for L, and equip LQ with the corresponding equivariant brane structure. Q # ; Q # / Š H  .S n I K/, and the Z=2-action on it is trivial. Lemma 18.18. HF  . k kC1 Q # ; Q # / Š HF  .V # ; V # / Proof. This is just a special case of (18.23): HF  . k kC1 k kC1 with trivial Z=2-action, and the latter group is isomorphic H  .Vk I K/ because the two branes are isotopic.  Q # / are one-dimensional, and concentrated in Lemma 18.19. The groups HF  .LQ #˙ ;  k the same degree. Proof. This should be clear once one has thought through the construction of matching cycles. Using Lemma 15.3, one finds a deformation which is constant along the fibres and which, starting with the given fibration structure, arrives at a Lefschetz fibration for which the matching cycle can be defined in the naive way, with a constant isotopy I . We can require that this deformation be concentrated near the preimage of  itself, hence lifts to a deformation of Ez which is of the same type. Working in this deformed structure, one finds that the analogues of the Lagrangian submaniQ  , intersect transversally and in folds under consideration, which we call LQ ˙ and  k a single point (which is one of the preimages in Ez of the critical point in E k .1/ ). Now by Lemma 7.1, the deformed double branched cover comes with a conformally

285

18 The Fukaya category of a Lefschetz fibration

Q ˙ and Q  to L symplectic embedding into the original Ez which, up to isotopy, maps L ˙  Q Q k to k . This is obvious in the first case, since such isotopies are part of the construction of matching cycles; in the second case, it follows by looking at the family of vanishing cycles produced continuously throughout the deformation. The outcome is that there is an isotopy of our Lagrangian submanifolds after which they intersect transversally in a single point, which of course implies one-dimensionality of Floer cohomology. To see that the nontrivial pieces lie in the same degree, one simply uses the Z=2-action , which by definition exchanges the two Floer groups.  For simplicity, assume that the grading of L has been chosen in such a way that Q # / are concentrated in degree 1. the groups HF  .LQ #˙ ;  k Q of the form Lemma 18.20. There is an exact triangle in H.A/ Q# L `BB

Œ1

BB BB BB

/ Q# k {{ { { {{ {} {

(18.26)

Q# .  kC1

Q Proof. From Lemma 18.7 and Corollary 17.17 we know that in D.A/, Q# Q#  Q # TL Q # .k /: kC1 Š TL 

(18.27)

C

Moreover, the LQ ˙ are disjoint, so there are no morphisms between them, which means that Q # / Š T Q # Q # . Q # /; TLQ # TLQ # . k k L ;L 

C



C

where the right-hand side is as defined in Remarks 5.1, 5.2. With this in mind, (18.27) Q of the form leads to an exact triangle in H.Tw A/ Q#/˝L Q # ˚ HF  .LQ # ;  Q # / ˝ LQ # Q # ;  HF  .L C C k k

jUUUU UUUU UUUU UUUU Œ1 UUUU

/ Q# k { {{ { {{ {} {

Q# .  kC1

Q # / ˚ S 1 .L Q # / Š S 1 .L Q # /. Now By Lemma 18.19, the upper left vertex is S 1 .L C rotate the triangle (Corollary 3.17).  Q# ! Q # /, where the Proposition 18.21. In DF ./, LQ # is isomorphic to Cone. k kC1 cone is formed over any nontrivial degree zero morphism.

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III Picard–Lefschetz theory

Q Q # Š Cone. Q# !  Q # / in D.A/. Proof. From (18.26), we get an isomorphism L k kC1 Up to isomorphism, the cone of any morphism is invariant under multiplication of that morphism by some element of K . In our case, the morphism group is onedimensional, hence all that remains to be shown is that the morphism used is not zero. Q#,  Q# , If it were zero, the two summands LQ #˙ would have to be isomorphic to  k kC1 but that is immediately contradicted by our previous Floer cohomology computations Q # is free, we can apply Lemma 6.2 to (Lemma 18.19, for instance). Finally, since L inv get an isomorphism in D.AQ / D DF ./.  Remark 18.22. An alternative approach would be to treat this situation strictly as a special case of Proposition 18.17. Suppose for simplicity that k D 1. One would need to prove (by looking at a suitable local model) that the product Q #2 ; LQ #C / ˝ HF n . Q #1 ;  Q #2 / ! HF Cn . Q #1 ; LQ #C / HF  . is an isomorphism (all spaces involved are actually one-dimensional). From this and Q # ; LQ # / D 0 for j > 2, it follows that the object U from the proof the fact that HF  . C j Q# ! Q # / in this particular case. of Lemma 6.3 reduces to Cone. 1 2 (18i) Hurwitz moves. Let k , kC1 , ık be vanishing paths as in Figure 16.2, and Q#,  Q# , … z # the associated type (B) branes. More precisely, the grading of  Q#  k kC1 k k is arbitrary, but the other two gradings should be adjusted in such a way that (when combined with the canonical grading of the Dehn twist) they are compatible with the isotopy from Lemma 18.3. To restate the result, we now have an isotopy of branes z # '  Q . Q # /: … k kC1 k

(18.28)

As pointed out in Remark 18.5, this isotopy is in fact through Ez -invariant Lagrangian submanifolds. To make it into an isotopy of equivariant branes, it remains to compare the equivariant Pin structures. By Lemma 14.2, we know that a choice of equivariant Pin structure corresponds to a co-orientation of the fixed point set. Our convention was Q kC1 and … z k so that, under D , to choose these co-orientations for  Q they project to the 0 0 square roots of kC1 .0/ and ık .0/, with positive imaginary parts. The computation in Remark 18.5 shows that these choices are indeed compatible with (18.28). We can now apply the considerations from Section (17k), and in particular Lemma 17.23. The algebraic data provided by that plugs into Lemma 6.4, the outcome being: z # is isomorphic to T Q #  Q# . Proposition 18.23. In DF ./, …  k kC1 k



(18j) First consequences. The most immediate application of Proposition 18.23 is to relate Hurwitz moves and mutations. Take a distinguished basis of vanishing paths

18 The Fukaya category of a Lefschetz fibration

287

Q # of type (B) equivari D .1 ; : : : ; m /, and consider the associated collection  ant branes, where the gradings are chosen arbitrarily, as well as the corresponding collection of vanishing cycles V # , with the induced brane structures. We already Q # is an exceptional collection. Proposition 18.23 know from Proposition 18.14 that  says that if we apply an elementary Hurwitz move to our basis, as defined in (16.7), the outcome is an elementary mutation of the exceptional collection, in the sense of (5.24). We have already implicitly recognized this correspondence in choosing the relevant notation, so one can write Q # : Q #L  Š Lk   k

(18.29)

Strictly speaking, in order to get (18.29) to be an actual isomorphism of exceptional collections in DF ./, the gradings must be chosen specifically as in Proposition 18.23 (otherwise, it will only be an isomorphism up to shifts, which is perfectly fine for our purposes). Of course, just by changing the notation for the collections , and by composition for any mutation. In involved, the same will hold inverses L1 k other words, the algebraically defined action of the braid group Br m on exceptional collections up to shift and isomorphism (see Section (5j)) can be identified with the geometric action on distinguished bases of vanishing paths (see Section (16d)). Fix some . Temporarily, write T for the triangulated subcategory of Tw F ./ Q # is a full exceptional Q # ; : : : ;  Q # . Tautologically,  generated by the objects  m 1 collection in that subcategory, and this continues to hold for all mutations. Using (18.29), this means that if we apply a Hurwitz move to our basis and consider the collection associated to that, then all its objects again lie in T , and generate that category. Since Hurwitz moves act transitively on exceptional bases, it follows that T is independent of . In fact, Proposition 18.17 tells us that any type (U) equivariant brane lies in T , so by definition of F ./ we must have T D Tw F ./. On the other hand, Lemma 5.22 says that T is quasi-equivalent to the category of twisted Q # , which in turn by complexes over the directed A1 -subcategory associated to  # ! Proposition 18.14 is quasi-isomorphic to F .V  /. Therefore, Theorem 18.24. Tw F ! .V # / is quasi-equivalent to Tw F ./.



As an immediate consequence, one gets an alternative proof of Corollary 17.21 (and in fact Theorem 17.20 as well, since any collection of Lagrangian spheres can be realized by a basis of vanishing cycles for an appropriately constructed Lefschetz fibration). This actually represents an improvement over the original proof, in that Tw F ! .V # / is shown to be independent of  by identifying it with Tw F ./, which manifestly does not involve any choice of basis. As a concrete payoff, one can construct quasi-equivalences between the categories Tw F ! .V # / for different  which are canonical, at least up to isomorphism in the relevant category of functors H 0 . fun/.

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The only price to pay for this is the technical assumption char.K/ ¤ 2, which has been forced on us by using the double branched cover trick. Next, note that Proposition 18.13 and Lemma 3.23 yield a cohomologically full and faithful functor Tw F .E/ ! Tw F ./. In view of Theorem 18.24, this yields: Corollary 18.25. For any , there is a cohomologically full and faithful embedding of F .E/ into Tw F ! .V # /.  The point is that, since the directed Fukaya category is formed inside F .M /, Floer cohomology computations in the total space E can, abstractly speaking, be fully reduced to ones carried out in the fibre. We will now try to give a more concrete shape to these considerations. (18k) Matching cycles revisited. Take a quasi-equivalence inverse to the one from Theorem 18.24, and denote that by G W Tw F ./ ! Tw F ! .V # /: Our first observation is that one can determine where each type (B) equivariant brane gets mapped under G . By construction, for the specific basis used to define G we have Q # / D V # ; G . j j where the right-hand side is the vanishing cycle seen as the j -th object of F ! .V # / itself. Now consider any collection ı and its k-th elementary Hurwitz move  D Lk ı. Then, using Lemma 5.6 and Proposition 18.23, we find that

‚ G .Q

Q /Š G . j #

for j ¤ k; k C 1;

#

/ ıj # Q G .ık / TG .Q #

ık

for j D k C 1;

Q# / G .ıkC1 /

(18.30)

for j D k:

where the Š is in DF ! .V # /, and in fact may only hold up to shift (unless one makes the extra effort to keep track of the correct gradings, which we will not insist on). By substituting, one establishes a similar formula for the inverse Hurwitz move  D L1 ı, namely k

‚ G .Q

Q /Š G . j #

for j ¤ k; k C 1;

#

/ ıj # Q G .ıkC1 / T_ Q# G .ı

kC1

/

for j D k; Q # // .G . ık

for j D k C 1:

Each vanishing path is part of a distinguished basis, which finishes our explanation of the first step.

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289

Next, let  be a matching path whose position relative to ı is as shown in FigQ# Q # to  in DF ./ ure 18.3. By Lemma 18.18, the space of morphisms from  ık ıkC1 has two generators, which lie in degrees differing by n. The precise absolute degrees

ım ıkC1  ık ı1 Figure 18.3

depend on the choices of gradings, and we will shift those until the degrees become 0 and n. Take the matching cycle † and the associated type (U) equivariant brane z # . By Corollary 3.9 and Proposition 18.21, we know that †     Q# ! Q# Q# Q# z # / Š G Cone. G .† ık ıkC1 / Š Cone G .ık / ! G .ıkC1 / ; where the cone is formed with respect to any nontrivial degree 0 morphism, and the isomorphism is in the same sense as in (18.30). Note that for any given matching path  one may find a distinguished basis ı so that this recipe applies, which means that the images under G of all matching cycles can be explicitly determined. By applying Proposition 18.13, we conclude that: Corollary 18.26. Let 1 ; : : : ; l be a finite number of matching paths for . Then, from knowing F ! .V # / up to quasi-isomorphism for just one distinguished basis , one can compute the full A1 -subcategory of F .E/ formed by the matching cycles †#j , up to quasi-isomorphism and possibly shift of the objects.  (18l) A spectral sequence. Let  be a distinguished basis of vanishing paths, and ı the one obtained from  by applying Hurwitz moves corresponding to the element 1=2 2 Br m from (5.25) (note that, even though this is a positive element of the braid group, the discussion from Section (16d) means that geometrically, the associated diffeomorphism of the disc is the square root of the negative Dehn twist along a boundary parallel circle). In addition, we want to consider the collection of paths  Š obtained from ı by moving the base point from to  in a clockwise semicircle; see Figure 18.4.

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III Picard–Lefschetz theory

1Š

m

1 ' ım

Š m

ı1 Figure 18.4

Q # the resulting Let Lj# (j D 0; 1) be two branes contained inside E shrink , and L j type (U) equivariant branes. We will apply Proposition 5.17 to these branes viewed as objects of DF ./. The result is a spectral sequence converging to HF  .LQ #0 ; LQ #1 /Z=2 Š HF  .L#0 ; L#1 /; where the isomorphism is part of Proposition 18.13. The E1 page of the spectral sequence consists of the tensor products of two kinds of Floer groups. One is Q # ; LQ #1 /Z=2 Š HF  .# ; L#1 /; HF  . k k

(18.31)

Q k its preimage in E. z where k is the Lefschetz thimble associated to k , and  We make the latter into a type (B) equivariant brane as usual, choosing the grading arbitrarily, so that the left-hand side of (18.31) is the space of morphisms in H F ./, and then apply Remark 18.11 to identify that with a suitable Floer cohomology group in E itself, which is the right-hand side. The other Floer groups which occur can be written similarly as z # /Z=2 Š HF  .L#0 ; …# / Š HF  .L#0 ; Š;# /; HF  .LQ #0 ; … k k k where …k ; Šk are the Lefschetz thimbles for ık ; kŠ . The second isomorphism is given by the obvious isotopy between these thimbles (the isotopy moves the boundary, but it still preserves Floer cohomology groups, since the relevant solutions of Floer’s equation never reach the boundary). To be precise, we want to use the grading of Šk which is compatible with the mutation picture: one can characterize this by the fact that the unique point of mC1k \ Šk has Maslov index zero, since that is one of the general properties of Koszul dual collections from Section (5k). With this in mind, we have:

19 Algebraic varieties

291

Corollary 18.27. There is a spectral sequence converging to HF  .L#0 ; L#1 /, which starts with  rCs # E1rs D HF  .L#0 ; Š;rC1 / ˝ HF  .#mr ; L#1 / : (18.32)  This is interesting even in the case L#0 D L#1 D L# , when the target of the spectral sequence reduces to H  .LI K/. Note that Proposition 5.17 also describes the shape of the first differential on (18.32): it is given by multiplication with elements of the mutually dual morphism groups between subsequent objects in our two collections; moreover, by Proposition 18.14, those groups can be written in terms of vanishing Š;# Š;# # / and HF  .Vmk ; VmC1k /. cycles as HF  .Vk# ; VkC1 Remark 18.28. The existence of the spectral sequence (18.32) was originally suggested to the author by Donaldson. His viewpoint was based on a geometric decomposition of the diagonal: suppose for simplicity that no two critical values of  have the same imaginary part. Take the diagonal   E  E, where the sign of the symplectic form on the first factor is reversed, and deform it by applying the Hamiltonian flow of im./ on the first factor. Part of the diagonal will flow out of E, but the rest will approximate the product of stable and unstable manifolds (compare for instance [73]). These manifolds are the Lefschetz thimbles for paths going to ˙1 in horizontal directions, which should be viewed as deformed versions of the paths k , kŠ from Figure 18.4. In our proof, this idea has been replaced by an algebraic decomposition of the diagonal, in the sense of Beilinson.

19 Algebraic varieties (19a) Plurisubharmonic functions. A Stein manifold is a complex manifold .X; IX / which admits an exhausting (which means, proper and bounded below) plurisubharmonic function  W X ! R. Plurisubharmonicity says that if we set  D d c  D d ı IX , then ø D d D dd c  is a Kähler form. Note that d d c  is always of type (1,1); the nontrivial condition is that it should be negative on complex tangent lines (real IX -invariant two-dimensional subspaces of TX, with the natural orientation). As an immediate consequence, given any holomorphic map u W S ! X, the function  D  ı u satisfies   0 as in (7.4), hence obeys the maximum principle. Due to the convex nature of the plurisubharmonicity condition, such functions can be manipulated relatively easily. We need two elementary instances of this (see [42] or [22] for much more along these lines).

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Lemma 19.1. Let  be an exhausting plurisubharmonic function, and h W R ! R a function such that h0 > 0, h00  0 everywhere. Then h./ is again an exhausting plurisubharmonic function. Proof. We have dd c h./ D h0 ./dd c  C h00 ./d ^ d c . On a complex tangent line, the first summand is negative, and the second one nonpositive.  Lemma 19.2. Let ; be exhausting plurisubharmonic functions on .X; IX /. Suppose that grows faster than , in the sense that   is still exhausting. Fix some compact subset of X. Then there is another exhausting plurisubharmonic function Q such that Q D  on that compact subset, and Q D C (constant) at infinity. Proof. We choose a function l such that l.t/ D 0 for t  0, l.t / D t for t  0, and l 00 .t/  0 everywhere. Take  D  C l.    b/ for some constant b. This is plurisubharmonic for the same reason as before: dd c  D .1l 0 . b//.dd c /C l 0 .    b/.dd c / C l 00 .    b/d.  / ^ d c .  /. By choosing the constant b large enough, we can ensure that l.    b/ D 0 on our chosen compact subset.  We say that  is of finite type if its critical point set is compact. In general, it may be difficult to decide whether a Stein manifold admits such a function (for four-manifolds, there are subtle obstructions coming from the differentiable structure [63]). However, there are large classes of cases where it can be done easily. Examples 19.3. (i) Let Xx be a smooth complex projective variety, equipped with an ample line bundle L. Ampleness means that there is a metric j  j on that line bundle, x Let s be a holomorphic section of L such whose curvature is a Kähler form on X. that s 1 .0/ is a divisor with normal crossings. Then .x/ D  log js.x/j is an exhausting strictly plurisubharmonic function of finite type on X D Xx n s 1 .0/. The associated Kähler form ø is the restriction of the previously given Kähler form to X . The finite type property is proved by a straightforward local computation [133, Lemma 8]. (ii) Let X  CN be a smooth affine algebraic variety, and  W X ! R an exhausting plurisubharmonic function which is (the restriction of) a real polynomial. Then  is automatically of finite type, because its set of critical points is itself a real algebraic variety (hence has finitely many connected components, each of which corresponds to a single critical value). (iii) Along the same lines, suppose that q W CN ! C is a complex polynomial with an isolated critical point at the origin, where q.0/ D 0, Dq.0/ D 0. Then, for

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all 0 < jj  ı, the Milnor fibre X D fx 2 CN W q.x/ D ; jxj < ıg is a Stein manifold, and .x/ D .ı 2  jxj2 /1 is a plurisubharmonic function of finite type on it. Note that Milnor fibres are not generally (biholomorphic to) algebraic varieties. From now on, fix .X; IX /, which additionally should come with a quadratic com2 and a finite type exhausting plurisubharmonic function . plex volume form X Choose c to be larger than all critical values of , and consider the sublevel set M D X ;c D  1 ..1I c/ with the restrictions of X ; !X . This is an exact symplectic manifold with contact type boundary (and the restriction of IX is an almost complex structure of contact 2 type near the boundary). By also using the restriction of X , we have all the data necessary to define the Fukaya category F .M /. Proposition 19.4. Up to quasi-equivalence, F .M / is independent of c and . Proof. The entire discussion will be somewhat analogous to that in Section (10d). The first step is to show that for a fixed , the choice of c does not matter. Take some other d > c. When defining F .X ;d / one can arrange that, for exact Lagrangian submanifolds which are contained inside the smaller subset M D X ;c , the Floer data and perturbation data use only almost complex structures which are equal to IX on a neighbourhood of X ;d nint.X ;c /, and Hamiltonian functions (inhomogeneous terms) supported inside int.X ;c /. The maximum principle shows that inhomogeneous pseudo-holomorphic polygons remain inside X ;c , which means that F .X ;c / is contained in F .X ;d / as a full A1 -subcategory. On the other hand, every exact Lagrangian submanifold of X ;d can be pushed inside X ;c by the negative Liouville flow, which in the Stein case is the same as the negative gradient flow of . This shows that the inclusion F .X ;c / ! F .X ;d / is a quasi-equivalence. Secondly, if  D outside a compact subset, the associated Fukaya categories are the same. One can interpolate between the two functions and apply a standard Moser-type argument, to show that suitably large sublevel sets are symplectically isomorphic, by a symplectic isomorphism which is exact and equals the identity near the boundary. Third, Suppose that we pass from  to some h./, as in Lemma 19.1. Then the Fukaya category remains the same. We can choose an auxiliary function hQ with the Q same properties, which agrees with h for r  0, and such that h.r/ D ar C b for Q r  c with constants a > 0, b 2 R. Passing from  to h./ changes the symplectic geometry of the sublevel set X ;c only by a constant rescaling, which can be easily taken into account by adjusting the Floer data and perturbation data. Since neither function has a critical point outside that sublevel set, the associated Fukaya categories Q agree. On the other hand, passing from h./ to h./ is a compactly supported change, hence the argument given above applies.

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After these preliminaries, we come to the main argument. Suppose that we are given ; which are exhausting plurisubharmonic functions of finite type. Appealing to Lemma 19.1, we find some function h./ which grows faster than . Fix some d which is larger than all critical values of . Lemma 19.2 then yields a Q which, at infinity, equals h./ C (constant), and which on X ;d agrees with . We shall now run the same argument again: fix some c which is larger than all critical values of Q Q and such that X ;d  int.X ;c , /. Again by Lemma 19.1, we can find some k. / Q Apply Lemma 19.2 again which on X ;d equals , and which grows faster than . to find another function Q which, at infinity, equals k. / C (constant), and which on Q Q X ;c equals . By the discussion above, the Fukaya category associated to Q is the same as that for . The same holds for Q and , but in that case, we have an additional observation to make. Consider the symplectic isomorphism between .X; ø Q ;  Q / and .X; øk. / ; k. / / which arises by linearly interpolating between the two functions, and applying Moser’s technique. This isomorphism will be the identity on X ;d , Q where Q , , and k. / all agree up to adding constants. The negative Liouville flow of k. / will push every compact subset of .X; øk. / ; k. / / into that sublevel set. Hence, applying the Moser symplectomorphism, every exact Lagrangian submanifold of .X; ! Q ;  Q / is isotopic to one lying inside X ;d . Q Q Q Choose dQ  c. The inclusion .X ;c ; ø Q ;  Q / ,! .X ;d ; ø Q ;  Q / induces a cohomologically full and faithful functor on Fukaya categories, as in Proposition 10.5. On the other hand, that inclusion contains X ;d in its image. It therefore follows from the previous observation that the functor is a quasi-equivalence.  Remark 19.5. There are other possible ways of cutting down a Stein manifold to a compact subset. For instance, if X  CN is an affine variety, one can consider the intersection M D X \ P with a polyball P D fjx1 j2 C    C jxN1 j2  c1 ; : : : ; jxN1 CCNr1 C1 j2 C    C jxN j2  cr g; for N1 C    C Nr D N and sufficiently large c1 ; : : : ; cr . Choose a plurisubharmonic function whose negative Liouville flow points inwards with respect to any variable xk (for instance, the standard function 14 jxj2 , or the Fubini–Study Kähler potential log.1Cjxj2 /). Then, the restriction of the associated one-form and two-form make P into an exact symplectic manifold with corners. As a consequence of Proposition 10.5, the Fukaya category F .M / will be the same as the one obtained from our previous construction. As yet another equivalent approach, one could forget about compactness and work with the whole of X, using almost complex structures that are equal to IX at infinity. This does not quite fit into the framework of the present book, but it is maybe the most natural idea, and it also extends to the infinite type case.

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(19b) Lefschetz pencils. Let Xx be an .nC1/-dimensional smooth projective variety equipped with an ample line bundle L. Take two linearly independent sections s0 ; s1 2 H 0 .L/, and consider the rational function pN D s0 =s1 , whose fibres are the hypersurfaces fs0 .y/ D zs1 .y/g for z 2 CP1 . The indeterminacy locus of pN is the base locus B, defined by the vanishing of both s0 and s1 . We say that pN is a Lefschetz pencil if B is smooth, and pN has only nondegenerate critical points, at 1 most one in each fibre. We will assume in addition that s1 .0/ itself is smooth, and consider the restriction of pN to its complement, denoted by 1 .0/ ! C: p W X D Xx n s1

(19.1)

Choose a metric on L which gives rise to a Kähler form. As in Example 19.3 (i), the log norm of s1 defines a finite type plurisubharmonic function . The resulting setup is closely related to, but not quite the same as, the notion of exact Lefschetz fibration introduced in Section 15. There are two ways of dealing with the discrepancy: one can argue that the geometry of (19.1) is sufficiently well-behaved, allowing one to carry over the necessary arguments without major problems; or alternatively, one can modify the given situation, to make it fit squarely into the formalism set up previously. In the first approach, the main point is to show that the connection on (19.1) given by ø has well-behaved parallel transport maps (in spite of the noncompactness of the fibres). To see that, blow up the base locus to get the graph of p, N which is Xy D f.z; y/ 2 CP1  Xx W s0 .y/ D zs1 .y/g: x We take the pullback of the This comes with projections pO W Xy ! CP1 and Xy ! X. given Kähler form by the second of these maps, and denote the result by øXy . This is no longer Kähler, since it degenerates along the exceptional divisor By D CP1  B, but it does restrict to a Kähler form on the smooth fibres, hence defines a symplectic y so its parallel transport on the connection. The connection is in fact trivial on B, complement Xy n By is well-defined away from the singularities. On the other hand, if we remove the fibre at 1 from pO W Xy n By ! CP1 , we get back p W X ! C with the connection given by ! . This argument gives a little bit more information, which is particularly interesting when considered in a graded context. Namely, suppose that KXx Š L˝d

(19.2)

˝d for some d 2 Z. The preimage of s1 under this isomorphism is a rational holo1 morphic .n C 1/-form, which may have a zero (d < 0) or pole (d > 0) along s1 .0/. Hence, it will restrict to a holomorphic volume form X on X. In the presence of 2 that, or rather of the associated relative quadratic form X =dz 2 , all parallel transport maps naturally become graded, as in Section (15c). Consider the case where our path is a sufficiently large circle. The fibration extends smoothly over 1, but the

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relative quadratic form does not. Hence, the (graded) monodromy around infinity is isotopic to the shift S k (which means the identity map equipped with constant grading k), for some k 2 2Z. Clearly, k depends only on the pole order of X =dz along the fibre at infinity, which is d 2. For simplicity, take instead a trivial fibration C  Y ! C , with relative volume form z d 2 Y . If .t/ D e 2 it is the unit circle, parallel transport along jŒ0I t  is IdY equipped with the grading .2d  4/t, hence after going all the way around the circle, we get k D 4  2d . We will now discuss one possible implementation of the alternative approach, which is to turn (19.1) into an exact Lefschetz fibration in the sense of Section 15. This requires a choice of base point 2 C; for notational convenience, we take D 0, which is therefore assumed to be a regular value of p. Fix the standard symplectic form øC on C, and its standard primitive C . Later on, we will want to r deform the given øX to øX D øX C rp  øC , and similarly X by Xr D X C rp  C , for some r  0. In spite of the lack of compactness, Moser’s argument applies here, because the vector fields Y r satisfying ør .; Y r / D @Xr =@r D p  C are everywhere (positive) proportional to the horizontal lift of the inwards pointing radial vector field, z@z , on the base C. The outcome is a family of exact symplectic embeddings r W X ! X;

r .r / øX D øX ; .r / Xr D X C d.some function/:

We find it useful to remember one particular property of these embeddings, which follows directly from our description of the Y r : Lemma 19.6. Take a compact subset K  X. Then there is another compact subset Q KQ  X such that for all r  0, r .K/  K.  The next step is to use parallel transport in radial direction, for the given symplectic connection, to construct a trivialization at infinity of (19.1). This consists of an open subset U  X such that p j .X n U / is proper, and a fibrewise diffeomorphism between U and C  U0 , where U0 D U \ X0 . In this trivialization X D X0 C C dR;

øX D øX0 C d ;

where X0 ; øX0 are the restrictions of X ; øX to X0 ; is some one-form which vanishes on vectors tangent to the fibres, and R a function (both actually vanish at all points .z; x/ 2 C  U0 with z D 0). We now want to modify parallel transport, to make it trivial at infinity in our given trivialization. For that, take a compactly supported function l on X0 which is equal to 1 outside U0 , and consider the modified forms QX D X0 C l C d.lR/ C rp  C ;

øQX D d QX

(19.3)

for r  0, with the understanding that outside the region where the trivialization r applies, these are extended to be equal to Xr ; øX . Take a large disc S  C, and

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equip it with the restrictions S D C jS, øS D øC jS. Large means first of all that the interior of S contains all critical values of p; more conditions will be imposed later on. Take a compact subset E0  X0 which is a large sublevel set of some plurisubharmonic function, and consider the piece E  p 1 .S/ which looks like S  E0 at infinity. Here, the understanding is that E will contain all the part of p 1 .S/ not visible in the trivialization at infinity. Moreover, we want to choose E0 so that l is supported in its interior. Set øE D øQX jE, E D QX jE, and consider the restriction  D pjE W E ! S. Finally, take an almost complex structure IE which is compatible with øE , agrees with the given IX near the critical points, makes  pseudo-holomorphic, and looks like the natural product structure i  IX0 in the trivialization near infinity. We claim that the outcome is an exact Lefschetz fibration. Because øQX D øX when restricted to the fibres, and r is large, it is clear that øQX is again symplectic, and moreover, that the negative Liouville vector field points inwards along @v E D  1 .@S/; we have seen this argument before, when considering (15.6). On the other hand, we know that the negative Liouville field points inwards along the boundary of E0 , hence by construction, the same will hold along all of @h E. In fact, we have arranged things so that the fibration is trivial near @h E, in the sense of Remark 15.2. Weak IE -convexity of @h E again follows from the corresponding property of E0 , and for @v E it holds by projecting pseudo-holomorphic curves to the base. Finally, suppose that Xx comes with an isomorphism (19.2); by construction, IE is homotopic to IX jE as an almost complex structure, and one can then follow X jE through that homotopy to get a complex volume form E for .E; IE /. Having verified the required properties, it remains to reflect on the changes (19.3) made to the symplectic structure, and how this affects the Fukaya category. The natural plurisubharmonic function  on X, given as in Example 19.3 (i), is of finite type. Take the sublevel set X ;c for some c which is larger than all critical values. Set Q In the construction, this to be K in Lemma 19.6, and consider the corresponding K. r Q so that there is an exact symplectic we may arrange that øQX D øX and QX D Xr on K, embedding of X ;c into E. By Proposition 10.5, F .E/ therefore contains a full A1 subcategory quasi-equivalent to F .X ;c /; this is entirely sufficient for our purpose, which is to use the Lefschetz fibration to make statements about X (any other objects of F .E/ can be considered as spurious, and simply ignored). If, instead of considering the total space E, we look at the geometry of the fibration, things become even simpler. The fibre E0 is, by definition, a large sublevel set of a plurisubharmonic function on X0 . The parallel transport maps for  W E ! S are not, of course, restrictions of those of p W X ! C, but they will agree with those restrictions on a large compact subset of E n@h E (the part not touched by our trivialization at infinity). One can arrange things in such a way that every Lagrangian submanifold in E0 can be pushed into that part by the negative Liouville flow. This, in particular, allows us to carry over the previous results about the global monodromy: if is a loop in S n Critv./ running parallel to the boundary in positive sense, and L an exact Lagrangian submanifold in the fibre

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E.0/ , then the monodromy image h .L/ will be isotopic to L. In the presence of (19.2), there is again an improved version of this, which says that the image of any exact Lagrangian brane L# under the natural graded version of the monodromy h satisfies (19.4) h .L# / ' S 42d L# : Proposition 19.7. Let  W E 2nC2 ! S be an exact Lefschetz fibration coming from a Lefschetz pencil. Suppose that this carries a relative complex volume form obtained from (19.2). Fix a base point 2 S, and let V # D .V1# ; : : : ; Vm# / be a basis of vanishing cycles in the fibre M D E , made into Lagrangian branes by an arbitrary choice of gradings (and, for n D 1 and char.K/ ¤ 2, taking nontrivial Pin structures). Let A D ….Tw F .M // be the split-closure of the A1 -category of twisted complexes, so that by definition H 0 .A/ D D  F .M /. Then the objects in V # split-generate A, provided that d ¤ 2. Proof. Combine (19.4), Corollary 17.17, and Corollary 5.8.



Remark 19.8. To see that the condition d ¤ 2 is indeed necessary, take Xx D CP1  CP1 , with L D OXx .1; 1/. In that case X0 D C , and both vanishing cycles are isotopic to the same circle L D S 1  C , carrying the nontrivial Pin structure. Denote this by L# , and its counterpart with the trivial Pin structure by L# ˝ ˇ. If one chooses the coefficient field K to have characteristic ¤ 2, then HF  .L# ; L# ˝ ˇ/ D H  .LI jˇjK / D 0 by (12.14). This implies that L# ˝ ˇ does not lie in the subcategory split-generated by L# . In spite of this, one can try to push the general theory for d D 2 a little further. Write Lk D VkC1 : : : Vm .L/, and equip it with the induced brane structure. The composition of all the ! maps in the exact triangles / L# L#k gN k1 NNN o o NNN oo o o NNN ooo NN wooo Œ1 HF  .Vk# ; L#k / ˝ Vk#

is a number cL# 2 HF 0 .L#m ; L#0 / Š HF 0 .L# ; L# / Š H 0 .L# I K/ D K. By inspecting the proof of Corollary 5.8, one sees that cL# D 0 is a sufficient condition for L# to be contained in the subcategory split-generated by the vanishing cycles. On the other hand, thinking back to how these exact triangles were constructed in Corollary 17.18, one finds that each ! map is an invariant counting pseudo-holomorphic sections of a small piece of our given Lefschetz fibration (containing only one critical point). Using standard gluing theorems, one can write cL# as an invariant ˆE=S; associated to the entire fibration, for the moving Lagrangian boundary condition F which realizes the

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isotopy (19.4). Equivalently, in terms of the original algebro-geometric data, one can think of F as a Lagrangian submanifold of X which is fibered over a large circle in the base of  W X ! C, hence lies close to the fibre at infinity. cL# then counts holomorphic discs in X with boundary in F . One can try to further analyze the situation by a degeneration to the normal cone (or symplectic cut). This degenerates 1 X to the union of two pieces, Xx and the total space of L1 ! s1 .0/, glued together 1 along s1 .0/. The Lagrangian submanifold F moves into the second piece, and techniques similar to [79] can be applied to the count of holomorphic discs. We will not attempt to make this more explicit here. (19c) Picard–Lefschetz data. We consider again an .n C 1/-dimensional smooth projective variety, but this time explicitly embedded into projective space, Xx  CPN . As before, this should satisfy KXx Š OXx .d /

(19.5)

for some d 2 Z. The projective embedding will be assumed to be in general position with respect to the standard homogeneous coordinates .y0 W    W yN /. The precise conditions will be specified below, but the main point is that they will always hold after applying a suitable element of PGLN C1 , or equivalently, passing to a different homogeneous coordinate system. Take the hyperplane at infinity to be CPN 1 D fyN D 0g, and identify its complement with CN , with coordinates .x0 D y0 =yN ; : : : ; xN 1 D yN 1 =yN /. For 0  k  n C 1, take the subspace H k D fx0 D    D xk1 D 0g  CN , and x k  CPN . Define its closure H x k; Xx k D Xx \ H Xk D X \ H k; ‰ k D xk W X k ! C; $ k D .xk ; xkC1 / W X k ! C2 : Generically, the following will hold: x k \ CPN 1 . This means x k and H • (For 0  k  n C 1) Xx is transverse to all H k that each X is smooth, as well as smooth at infinity. Note that by definition, X kC1 D .‰ k /1 .0/ and ‰ kC1 D $ k jX kC1 . Moreover, from (19.5) we get KXx k Š OXx k .k  d /. Continuing with our list of generic properties, • (For 0  k  n) Each of the functions ‰ k has only nondegenerate critical points, no two of which lie in the same fibre.

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• (For 0  k  n  1) D$ k is generic (transverse to the subvarieties of Hom.TX k ; C2 / consisting of linear maps of lower rank). If w 2 C is a critical value of ‰ k , and x the unique critical point lying in the fibre .‰ k /1 .w/, then the following additional conditions hold: .D 2 ‰ k /x remains nondegenerate when restricted to the kernel of .D$ k /x ; the function $k

.‰ k /1 .w/ n fxg ! fwg  C D C has nondegenerate critical points; and the values of any two such points are pairwise different, as well as different from $ k .x/ itself. The fact that these properties are satisfied generically is classical, but for convenience, we still sketch the proof. The first part is a straightforward application of the Bertini–Sard theorem. The second part can be reduced to the same theorem by the following standard argument. Consider P the section of the pullback bundle T  X k ! CN k  X k given by . ; x/ 7! d. ik i xi /. This always intersects the zero-section transversally, hence its zero-set Z  CN k  X k is smooth. If we take P a regular value of the projection Z ! CN k , then the function i i xi has only nondegenerate zeros. Clearly, any such function can be transformed to xk by a linear x we can change its embedding by coordinate change. This shows that given any X, an element of PGLN C1 close to the identity, such that the function ‰ k D xk on the transformed variety has only nondegenerate critical points. A further perturbation of the embedding will make the critical points lie in different fibres. The same argument applies to $ k . It will not have escaped the reader that these conditions are algebrogeometric counterparts of those in Section (15e). The implications are similar: if x is a critical point of ‰ k , with value w, then Critv.$ k / is tangent to the line fwg  C at $ k .x/, and is transverse to that line everywhere else. Define C k D Critv.‰ k /, 0  k  n, and F k D Fakev.‰ k ; $ k /, 0  k  n  1, as in Section (15e). To simplify the notation, we set F n D ;. Note that by assumption, 0 … C k [ F k . Fix 0  k  n, and write C k D fcjk g. Choose for each j an embedded path jk W Œ0I 1 ! C satisfying jk .0/ D 0, jk .1/ D cjk , jk .t/ … C k for t < 1, jk .t/ … F k for all t. These paths should form a basis in the sense of Section (16d). In the lowest-dimensional case k D n, these are just paths going from 0 to one of the branch points of the branched covering ‰ n W X n ! C; one can associate to each of them an unordered pair of points jnC1  X nC1 , which marks the two sheets of the covering which come together along the path. In higher dimensions, one proceeds as in Section (16i), by following how the subsets Critv.$ k / \ .fjk .t/g  C/  C move for t 2 Œ0I 1/. This gives rise to an embedded path jkC1 W Œ1I 1 ! C satisfying .jkC1 /1 .C kC1 / D f1I 1g, unique up to isotopy and orientation-reversal. The outcome of this process is the following information, called the Picard–Lefschetz data of X :

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• the finite set X nC1 , together with the collection of subsets fjnC1 g; • for each 0  k  n: in the plane C with marked points f0g [ C k [ F k , the collections of paths fjk g and (assuming k > 0) fjk g; x ¤ 0, an additional homotopy datum, to be specified • in the case where H 1 .X/ presently. Note that these are intended to be topological pictures (one can move the marked points, and accompany this by a suitable isotopy of the relevant paths, without changing the data in any essential way). Hence, each path can be encoded by specifying how to obtain it from a standard path by a diffeomorphism of the punctured plane, which leaves us mainly with a collection of words in the braid group (this is a weak version of Moishezon’s braid factorization technique). Let us see how much information about X these data contain. First of all, the branching data allow one to reconstruct ‰ n W X n ! C topologically. In order to get a well-defined Fukaya category for X n , one also needs to know the trivialization of KX n , at least topologically (as a C 1 -trivialization considered up to homotopy). By construction, this trivialization comes from a nowhere vanishing complex one-form with a .d  n/-fold pole at every point at infinity. To specify its homotopy class, we should write down the “rotation numbers” (Maslov indices) for a collection of immersed curves which span H1 .Xx n /. If a curve is a vanishing cycle for ‰ n1 , we already know that the rotation number is zero (because the curve admits a grading). x D 0. Then H 1 .Xx n1 / D 0 by weak Lefschetz, and the Suppose that H 1 .X/ vanishing cycles span H 1 .Xx n /, which means that the trivialization of KX n is uniquely determined. Otherwise, one may need to remember some additional rotation numbers, which is the last piece of data mentioned above. Remark 19.9. Alternatively (going a little against the general philosophy), one can assume precise knowledge of the branch point set C n . In that case, X n is determined up to holomorphic isomorphism, and it carries a unique holomorphic one-form with no zeros and with the required pole order around the points at infinity. On the next level, there are unique simple closed curves in X n whose image under ‰ is jn . Moreover, none of these curves is nullhomologous (because they arose as vanishing cycles, hence are exact for a suitable choice of one-form). Following Donaldson and Gompf, this collection of curves, together with the corresponding paths jn1 , determines Xx n1 up to symplectic isomorphism (relative to the divisor at infinity, hence inducing an exact symplectic isomorphism of X n1 ). Finally, the trivialization of KX n induces one of KX n1 , in a way which is unique up to homotopy. For k < n, it seems doubtful whether the Picard–Lefschetz data themselves suffice to determine the symplectic isomorphism type of X k or Xx k . This is because we are remembering the matching paths, but not the isotopies required to build the actual n

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matching cycles (when compactifying X k to Xx k , there is actually another potential problem, which is the way in which the divisor at infinity is attached). One can try to overcome these issues by specifying additional data, but that is not the direction we choose to take. (19d) Conclusions. Instead of trying to recover the X k geometrically from the Picard–Lefschetz data, we will perform an analogous construction algebraically in the framework of Fukaya categories. It turns out that the difficulties encountered above disappear at this level (compare Remark 16.11). To simplify the exposition, we treat ‰ k and $ k as if they were actually exact Lefschetz fibrations and bifibrations, respectively. This is unproblematic, as explained in Section (19b) for the case of ‰ k (the discussion for $ k would be entirely analogous). Hence, we have two sequences of Fukaya categories F .X k / and F .‰ k /. For each 0 < k  n, let Vjk  X k be the vanishing cycle associated to jkC1 , which by construction is also the matching cycle associated to jk (see Lemma 16.15). We turn these into Lagrangian branes as usual, and denote the resulting collection by V k;# . Write Ak D F .V k;# /; Ak;! D F ! .V k;# /I in words, Ak is the full A1 -subcategory of F .X k / formed by our collection of vanishing cycles, and Ak;! the associated directed A1 -subcategory. The start of our algorithm is the same as before, namely to reconstruct topologically X n together with the trivialization of KX n , and the vanishing cycles V n . Given that, An;! can be computed combinatorially, as explained in Section 13. The iterative step is as follows. Suppose that we are given a directed A1 -category B kC1;! quasi-isomorphic to AkC1;! . From Proposition 18.13 and Theorem 18.24, we get a cohomologically full and faithful A1 -functor F .X k / ! F .‰ k / ! Tw AkC1;! :

(19.6)

As in the proof of Corollary 18.26, we can determine the image of any matching cycle under (19.6), as well as the corresponding object in Tw B kC1;! , possibly up to a shift in the grading (that ambiguity is ultimately irrelevant; for simplicity, let us assume that the gradings have been chosen correctly). Hence, Ak is quasi-isomorphic to an explicitly given full A1 -subcategory B k  Tw B kC1;! . If k > 1, we throw away part of the information by retaining only B k;! , which is quasi-isomorphic to Ak;! , and continue with the next higher dimension. Arriving in this way to k D 1, what we have is an A1 -category B 1 quasi-isomorphic to A1 . Finally, we could go half a step further and apply (19.6) again, which shows that F .X/ itself is quasi-equivalent to some (a priori unknown) full A1 -subcategory of Tw B 1;! .

19 Algebraic varieties

303

Proof of Theorem B, for n  1. In view of the argument above, this is now obvious,  with the A1 -category under consideration being Tw B 1;! . Proof of Corollary C, for n  1. By Corollary 5.27, we can replace B 1;! by a quasiisomorphic directed A1 -category C ! which is actually a dg category. Take the cohomologically full and faithful A1 -functor F .X/ ! Tw.C ! /. After possibly introducing duplicate copies of existing objects, we may assume that this functor is injective on objects, hence F .X/ becomes quasi-isomorphic to its image, which is a dg category with finite-dimensional hom spaces.  Proof of Theorem A. By Proposition 19.7, we know that the vanishing cycles splitgenerate … Tw F .X 1 /. By Corollary 4.9 and Lemma 4.7, … Tw F .X 1 / is quasi equivalent to … Tw A1 , hence also to … Tw B 1 . Attentive readers will have noticed that the algorithm does not yield full information in the lowest dimension (we can easily determine any directed A1 -subcategory of F .X n /, but this does not a priori work for F .X n / itself). In part, this is because we have not started the induction process with the genuinely trivial zero-dimensional case, and that can be remedied as follows. Take ‰ n W X n ! C, and define F .‰ n / following the procedure from Section 18, which means taking the double cover Xz n ! X n branched over a smooth fibre, and considering Z=2-invariant Lagrangian submanifolds. In particular, we have the type (B) equivariant branes in Xz n which are the preimages of the Lefschetz thimbles associated to the vanishing paths jn . By inspecting the proof of Proposition 18.14, one sees that these form an exceptional collection in F .‰ n /. Hence, the associated full A1 -subcategories and directed A1 subcategories are quasi-isomorphic. On the other hand, the directed A1 -subcategory is amenable to the combinatorial methods from Section 13, since in defining the relevant Floer cochain groups and d operations, we may use the standard complex structure and zero inhomogeneous terms (Assumption 14.5 will still be satisfied). In this particular situation, all pseudo-holomorphic polygons will actually be constant; their regularity follows from Lemma 11.5, as in the proof of Lemmas 14.6 and 14.7. Denote the resulting directed A1 -category by B nC1;! . This comes with a cohomologically full and faithful functor F .X n / ! Tw B nC1;! :

(19.7)

Using that, one can extend the proofs of Theorem B and Corollary C to the case n D 0, where X n D X. Remark 19.10. In order to similarly extend Theorem A, one would need to determine the image of the vanishing cycles of ‰ n1 under (19.7). Unfortunately, there is a minor but tricky point, which did not occur in higher dimensions: in the analogue of Proposition 18.21, there is now a two-dimensional space of degree zero morphisms,

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III Picard–Lefschetz theory

and the cone needs to be formed with respect to the correct element of that space (which corresponds to the nontrivial Pin structure on the vanishing cycle).

20 .Am / type Milnor fibres (20a) Generators. For any n  1 and m  2, consider the function ‰.x/ D p.x0 / C x12 C    C xn2 W CnC1 ! C, where p is some polynomial of degree m C 1. We shall be interested in n D ‰ 1 .w/; Xm where w is a regular value of ‰. This is a smooth hypersurface, hence in particular an affine algebraic n-fold with trivial canonical bundle. Instead of taking a suitable n compact piece of it, we prefer (in the spirit of Remark 19.5) to work with Xm itself, nC1 and its standard equipping it with the restriction of the flat Kähler form on C one-form primitive, as well as with the quadratic complex volume form 2 which n is satisfies  ^ d ‰ D dx0 ^    ^ dxn . Up to exact symplectic isomorphism, Xm n independent of the choice of p and w, hence so is F .Xm /. Suppose that p is generic, which means that its critical points are nondegenerate. In that case, after making suitable technical adjustments, ‰ can be considered as a Lefschetz fibration. Choosing a basis of vanishing paths in an appropriate way, one gets a basis of vanishing cycles .V1 ; : : : ; Vm / whose (mutually transverse) intersections are: ( 1 for ji  j j D 1; jVi \ Vj j D (20.1) 0 for ji  j j  2: Lift these to Lagrangian branes .V1# ; : : : ; Vm# /, and take the associated full A1 n subcategory Fmn  F .Xm /. The cohomology level (Donaldson–Fukaya) category H.Fmn / can be computed easily, using only (20.1) and general properties of Floer cohomology. Following [85] we formulate the outcome in terms of quiver algebras. Consider the graded quiver 1



r

n 0

2

2



r

n 0

2  r

n 0

1m :

(20.2)

Generally speaking, one can associate to any such quiver a graded associative algebra, called the path algebra. In the example under discussion, elements of the path algebra are linear combinations of the basic expressions .il ; : : : ; i0 /, l  0, with ij 2 f1; : : : ; mg and jij C1  ij j D 1. The degree of such a basis element is n times the number of those j such that ij C1 D ij 1. Multiplication is given by composition of paths, for instance .i C 2; i C 1/ .i C 1; i/ D .i C 2; i C 1; i/. In particular, the

20 .Am / type Milnor fibres

305

length zero paths .i/ are mutually orthogonal idempotents. Their presence makes the path algebra into an algebra over the semisimple ring Rm Š Km . Let Anm be the quotient of the path algebra by the relations .i C 2; i C 1; i/ D 0;

.i  2; i  1; i/ D 0;

.i; i C 1; i/ C .i; i  1; i/ D 0:

This still a graded Rm -algebra, so one can equivalently consider it as a graded linear category with m objects; the morphism spaces .j /Anm .i/ in that category are represented by paths from the i-th to the j -th vertex. With that in mind, the statement is that, for a suitable choice of grading of our vanishing cycles, Anm Š H.Fmn /. To make that more explicit, note that a basis of Anm as a K-vector space is given by the paths .i/ and ˙.i; i ˙ 1; i/ for each i, corresponding to the standard generators of H  .Vi I K/, together with .i; i C 1/ and .i C 1; i/, which are the Floer elements corresponding to the unique point in Vi \ ViC1 . The product  D V1 : : : Vm is a symplectic automorphism of X, which in classical terms is the monodromy map of the .Am / type hypersurface singularity. Since that singularity is weighted homogeneous, one can analyze the geometry of  explicitly as in [125, Section 4c]. Taking into account the natural gradings of Dehn twists, the outcome is that for any exact Lagrangian brane L# ,  2mC2 .L# / ' S .42n/.mC1/4 L# : The amount of shift is always nonzero. By the same argument as in Proposition 19.7, this implies that the Vj# split-generate the Fukaya category, hence that we have a quasi-equivalence n … Tw.F .Xm // ' … Tw.Fmn /: (20.3) n // fully, it remains to characterize Fmn among all In order to determine … Tw.F .Xm quasi-isomorphism classes of A1 -categories with cohomology Anm .

(20b) The high-dimensional case. Assume that n  2. One can then use Lemma 1.9, together with an appropriate vanishing result for Hochschild cohomology groups, to prove that any A1 -category with cohomology Anm is necessarily formal (quasi-isomorphic to the trivial A1 -structure on Anm , which is the one with vanishing composition maps of order > 2). This was carried out in [134]. Here, we take a more concrete approach to the last-mentioned result, which avoids abstract classification theory. On the geometric side, this relies on material from Section 18, hence requires the technical assumption that char.K/ ¤ 2. Take the function  W X ! C, .x/ D x0 . Again after some technical adjustments, one can regard this as a Lefschetz fibration, and the pair .‰; / as a bifibration. Moreover, the fibres and parallel transport maps of  can be analyzed explicitly [85]: every smooth fibre is symplectically isomorphic to the cotangent bundle T S n1 ; every vanishing path gives rise to the same vanishing cycle, which is the zero-section

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III Picard–Lefschetz theory

S n1  T S n1 ; and therefore, every path connecting two critical values is trivially a matching path. In particular, the Vj arise as matching cycles for a chain of matching paths j , shown in Figure 20.1. V3

V2

V1



2

1

3

Figure 20.1

From now on fix m and n  2, and write F D Fmn , A D Anm . If we consider just the zero-section S n1 inside its cotangent bundle, the associated Fukaya A1 -algebra is quasi-isomorphic to K  e ˚ K  f , where jej D 0; jf j D n  1; 2 .e; e/ D e; 2 .e; f / D .1/n1 f; 2 .f; e/ D f;

(20.4)

with all other products vanishing. This is obvious on the cohomology level, where we are just looking at HF  .S n1 ; S n1 / D H  .S n1 I K/. On the cochain level, the necessary vanishing result follows from degree considerations, at least once one has made the A1 -structure strictly unital (alternatively, one could use the results from [58] and the fact that S n1 is formal in the traditional algebro-topological sense). The next step is to write down the directed A1 -category B ! associated to a basis of vanishing cycles for , which of course just consists of m C 1 copies of S n1 . This is straightforward: denoting objects by .Z0 ; : : : ; Zm /, the morphisms are

K  e

homB ! .Zi ; Zj / D

j i ˚ K  fj i ; i < j; K  ei i ; i D j; 0; i > j;

(20.5)

with the same degrees and compositions as in (20.4). Take the objects .C1 ; : : : ; Cm / of Tw B ! given by Cj D Cone.ej;j 1 /: (20.6)

20 .Am / type Milnor fibres

307

Proposition 18.21 says that the full A1 -subcategory consisting of those objects is quasi-isomorphic to F . It is easy to check that on the cohomology level, this reproduces A. More importantly, this process yields an explicit chain level model. Finally, it turns out that this model admits a helpful additional symmetry, which is not a priori of geometric origin. To simplify the notation, we use the version of twisted complexes from Remark 3.26. Equip B ! with another grading by length, which is such that the ej i have length zero, and the fj i length two (the name will be justified later). One can then introduce the homogeneous additive completion †ho B ! , whose objects are finite sums M S i ƒi Yi ; (20.7) i

where ƒ is a formal operation decreasing the length grading by one. Morphisms between any two such objects then carry a bigrading, by degree and length. Along the same lines, we have the A1 -category Twho B ! of homogeneous twisted complexes, whose objects are expressions (20.7) equipped with a differential which preserves the length grading. For instance, (20.6) can be turned into homogeneous twisted complex as follows:   Cj D ƒj Cone.ej;j 1 / D Sƒj Zj 1 ˚ ƒj Zj ; ıCj D ej;j01 00 : As before, take the full A1 -subcategory of Twho B ! consisting of those objects, which is a bigraded A1 -category. On the level of cohomology, the additional grading indeed reproduces the length grading that A inherits as the quotient of the path algebra of (20.2). Let us apply Proposition 1.12 to the previously constructed full A1 -subcategory, so as to kill 1 . This can be done in a way which preserves the bigrading, and the result is an A1 -structure A on A which is homogeneous with respect to the length grading. Inspection shows that the higher order composition maps dA , d > 2, are then necessarily trivial. On the other hand, our A1 -structure is quasi-isomorphic to F by construction, so F is formal. (20c) The surface case. The remaining case n D 1 is more tricky, and we will approach it in a less direct way, using a mixture of explicit computations and abstract classification theory. We assume throughout that the coefficient field satisfies char.K/ D 0 (this is used repeatedly, even in the purely algebraic parts of the argument; for one thing, if one took char.K/ D 2, Lemma 20.2 would be false). Our first task is to look at a potential situation where an A1 -category A with cohomology A D A1m might not be formal. Without loss of generality, we may suppose that 1A D 0, so that A is an A1 -structure on A itself. We may also suppose that A is strictly unital.

308

III Picard–Lefschetz theory

Lemma 20.1. Suppose in addition that 3A D 0, and that m  5. Consider 4A ..3; 2/; .2; 1/; .1; 2/; .2; 3//  4A ..3; 2/; .2; 3/; .3; 2/; .2; 3// C 4A ..3; 4/; .4; 3/; .3; 2/; .2; 3// C 4A ..3; 2/; .2; 3/; .3; 4/; .4; 3// 

4A ..3; 4/; .4; 3/; .3; 4/; .4; 3//

C

4A ..3; 4/; .4; 5/; .5; 4/; .4; 3//

(20.8) D q.3/;

where q 2 K. If q ¤ 0, then A is not formal. Proof. Let A; AQ be two A1 -structures of this kind, and denote by q, qQ the resulting coefficients as in (20.8). Suppose that there is a formal diffeomorphism ˆ W A ! AQ with linear order term ˆ1 D Id, and which is strictly unital. A straightforward computation shows that  2 .qQ  q/.3/ D ˆ2 ..3; 2/; .2; 3//  ˆ2 ..3; 4/; .4; 3// :

(20.9)

Consider also the inverse formal diffeomorphism ‰, which has ‰ 2 D ˆ2 , and look at the corresponding equation for q  q. Q Comparing the two shows that the right-hand side in (20.9) vanishes, hence that q D q. Q To apply this, take some given A, and assume that it is formal. One can then find a formal diffeomorphism ˆ of the kind introduced above, where AQ is the trivial A1 -structure, and therefore q must be zero.  (20d) A Hochschild cohomology computation. Instead of trying to classify A1 N structures on A directly, we prefer to temporarily pass to a slightly larger algebra A. This is the quotient of the extended (one more vertex than before) quiver 0



r

1 0

2

1



r

1 0

2

2



r

1 0

2  r

1 0

2m 

(20.10)

by the relations .i C 2; i C 1; i/ D 0; .i; i C 1; i/ C .i; i  1; i/ D 0;

.i  2; i  1; i/ D 0; .0; 1; 0/ D 0:

x D KmC1 . Maybe surprisingly, the enlarged By construction, AN is linear over R algebra is better-behaved than A. It has finite cohomological dimension, and moreover is a Koszul algebra. We will now explain the meaning of the latter property. Since it is a quotient of a path algebra by an ideal which is homogeneous with respect to degree and pathlength, AN again admits a bigrading. Hence, the Ext algebra x R/ x ExtAN .R;

(20.11)

20 .Am / type Milnor fibres

309

is a priori trigraded (by the cohomological degree; by the internal grading derived from N and finally by the length grading, again derived from the degrees of elements of A; N the corresponding one in A). If one takes the reduced bar construction underlying (20.11), it is obvious that the cohomological grading of the cochains is always less or equal than their length grading. Moreover, the piece of the cochain complex where those two gradings agree can be described explicitly. The outcome of this is as N This has generators .i; 1 ˙ 1/ dual to follows. Let AN be the quadratic dual of A. .i ˙ 1; i/, with degree j.i; i ˙ 1/ j D 1  j.i ˙ 1; i/j, and relations .i; i C 1; i/ C .i; i  1; i/ D 0;

.m; m  1; m/ D 0:

Then, by taking bar cochains and restricting them to the generators .i ˙ 1; i/, one defines a canonical homomorphism x R/ x ! AN : ExtAN .R;

(20.12)

In our case, it follows from the projective resolutions of simple modules given in [85, Proposition 2.1] that the trigrading of the Ext algebra is partially diagonal, meaning that all nonzero pieces have the property that their cohomological and length gradings agree. This is precisely the property of being a Koszul algebra, in the sense of [110] x as a base, rather than a field). (mildly generalized, since we use the semisimple ring R It is equivalent to (20.12) being an isomorphism. N which consists Next, we consider the reduced Hochschild cochain complex of A, N R x to A. N One can filter this complex by pathlengths on of multilinear maps from A= the target of the maps. The result is a spectral sequence whose E1 page can be written as x R/ x ˝ x x AN Š AN ˝ x x AN : Ext AN .R; (20.13) R˝R R˝R Because of the finite-dimensionality of both AN and its dual, this spectral sequence N A/. N Graphically, generators on the E1 page can be clearly converges to HH.A; thought of as closed paths consisting of two pieces, lying in the quiver (20.10) and its dual, respectively. Concretely, they can be written as expressions .il ; : : : ; i0 / ˝ .j0 ; : : : ; jr /

with jr D il , j0 D i0 .

(20.14)

Grading issues can be untangled as follows. Suppose that we are interested in N A/ N s , which is the group represented by r-linear Hochschild cocycles of HH rCs .A; internal degree s (this notation is the same as in Section (1f)). Then, the contributing terms on (20.13) consist of those expressions (20.14) with the given r, and with j.j0 ; : : : ; jr / j C j.il ;    ; i0 /j D r C s. Because of the way in which our quivers are graded, we then get l D 2s C r. N A/ N 2r is zero for r  5 and also for r D 3, and is K for Lemma 20.2. HH 2 .A; r D 4.

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III Picard–Lefschetz theory

Sketch of proof. This is straightforward for r  5, since contributing terms (20.14) would need to have l D 4  r < 0. The cases r D 3; 4 are slightly more involved; we omit the details but record one consequence, for later use. The edge homomorphism of (20.13) yields a map N A/ N 2 ! R x ˝ x x .AN /2 : HH 2 .A; R˝R

(20.15)

Here, the target space consists of elements in AN of degree two, on which the idemx act diagonally (in the same way from the left and right). From the potents of R computation, one finds that (20.15) is injective, and that its image consists of multiples of (the central element of AN ) .0; 1; 2; 1; 0/ C .1; 2; 3; 2; 1/ C    C .m  2; m  1; m; m  1; m  2/ : (20.16)  In view of general classification theory, Lemma 20.2 implies that, up to quasiisomorphism fixing the cohomology, there is at most a one-parameter family of A1 N Š A. N Denote the parameter by q 2 K (q D 0 corresponds to categories AN with H.A/ the formal A1 -structure). Lemma 20.2 by itself does not provide a proof of existence of non-formal structures, which would correspond to values q ¤ 0 (actually, different nonzero values are related by a rescaling of the various graded pieces, so they are not substantially different). However, assuming for the moment that such structures do exist, one can arrange that their first nontrivial higher order product is 4 . Moreover, it follows from (20.16) that for m  5, this product will satisfy the analogue of (20.8) with nonzero q. In particular, if one then drops the object corresponding to the extra vertex, the resulting full A1 -subcategory will still be non-formal by Lemma 20.1. (20e) Application. We follow the same strategy as in the higher-dimensional case. 1 Take a basis of vanishing paths for the double branched covering  W X D Xm ! C, and consider the associated full A1 -subcategory of F ./, which we denote by B ! . Explicitly, this has objects .Z0 ; : : : ; Zm / with

K  c

ji

homB ! .Zi ; Zj / D

K  ei i 0

˚ K  dj i

i < j; i D j; i > j;

where all degrees are zero. The ei i are strict units, and otherwise we have 2B ! .ckj ; cj i / D cki ;

2B ! .dkj ; dj i / D dki ;

2B ! .dkj ; cj i / D 0;

2B ! .ckj ; dj i / D 0:

20 .Am / type Milnor fibres

311

The discrepancy between this and its higher-dimensional analogue (20.5) eventually comes down to the fact that the cohomology ring of S 0 behaves differently from that of higher-dimensional spheres. Take the twisted complexes .C0 ; : : : ; Cm / given by C 0 D Z0 ;

Cj D Cone.cj;j 1 ˙ dj;j 1 / for j > 0,

(20.17)

where the sign can be chosen freely for each j (see Remark 20.5 below for a discussion of this ambiguity), and let AN be the associated full A1 -subcategory of Tw B ! . One N Š A. N computes easily that H.A/ Lemma 20.3. AN is not formal. Proof. It is clearly enough to prove this for m D 2, since all other AN contain that as a full A1 -subcategory. Since the collection (20.17) generates Tw.B ! /, the A1 categories of twisted complexes over AN and B ! are quasi-equivalent, which in turn implies that they have isomorphic Hochschild cohomology groups (results of this kind go back to [71] for algebras, and the proofs go through in the A1 -context with minor adjustments). It is straightforward to compute that HH.B ! ; B ! / D K (in degree zero) ˚ K2 (in degree one): This disagrees with the Hochschild cohomology of AN as (partially) computed in Lemma 20.2, and the desired non-formality result follows from that.  Let A be the full A1 -subcategory of AN with objects .C1 ; : : : ; Cm /. As a consequence of our previous discussion, this will again be non-formal, at least for m  5. On the other hand, Proposition 18.21 (appropriately extended, see Remark 20.5) shows that A is quasi-isomorphic to F . Hence, Corollary 20.4. F D Fm1 is not formal for any m  5.



Remark 20.5. We have used Proposition 18.21 in the two-dimensional case, where the Lefschetz fibration reduces to a double branched cover. Even though this case was not covered in our original discussion, the proof adapts easily, or one could use a more elementary argument instead. However, when formulating the precise statement, some caution needs to be exercised (this point was already mentioned briefly in Remark 19.10). The morphisms cj;j 1 and dj;j 1 correspond to intersection points between Lefschetz thimbles, and the statement is that the associated matching cycle is the cone over cj;j 1 ˙ dj;j 1 , where the choice of sign determines which Pin structure the matching cycle carries. In our case, the matching cycles are the curves forming the .Am / configuration on X, and we originally assumed that they carry the nontrivial Pin structure (because we have applied Corollary 17.17 to prove that they are split-generators). Luckily these curves happen to be linearly independent in

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III Picard–Lefschetz theory

H1 .XI Z=2/, which means that any two choices of Pin structures can be mapped to each other by twisting with a real line bundle on X (a brane automorphism, in the language of Section (12i)). Algebraically, this is reflected in the fact that B ! admits automorphisms which change the signs of cj;j 1 ˙ dj;j 1 . This is the reason why the choice of sign in (20.17) is irrelevant. (20f) The braid group and the mapping class group. We now specialize to the simplest non-formal case n D 1, m D 5, and look at the situation from a different point of view, which explains the geometric significance of the higher order composition maps. By combining the equivalence (20.3) with the Yoneda embedding, one gets a full and faithful A1 -functor F .X/ ! Q D mod.F /:

(20.18)

This takes an exact brane L# to an A1 -module M whose L cohomology is LLagrangian  # M D H.M/ D i HF .Vi ; L# /, seen as a module over A D i;j HF  .Vi# ; Vj# /. We will be especially interested in the curve L drawn in Figure 20.2. This has zero Maslov number and can be made exact, hence becomes an exact Lagrangian brane. The associated module, for a suitable choice of grading, is M D U3 ˚ SU3 ; where the notation is that Ui D .i/R Š K are the elementary simple modules (the direct summands of R as a right module over A), and S is the shift as usual.

L

V2

V4

V3

V1

V5

Figure 20.2

We will now examine the homological algebra of the simple module U3 in a little more detail. Denote by Pi D .i/A the elementary projective modules. There is an acyclic complex 0 ! S 3 U3 ! S 2 P3 ! S 1 P2 ˚ S 2 P4 ! P1 ˚ S 1 P3 ˚ S 2 P5 ! P2 ˚ S 1 P4 ! P3 ! U3 ! 0;

313

20 .Am / type Milnor fibres

which can be looped together to form an infinite projective resolution of U3 . Using this, one computes that (with a notation for the grading which is analogous to the one for Hochschild cohomology groups) ExtArCs .U3 ; U3 /s D 0 ExtA0 .U3 ; U3 /0 ExtA1 .U3 ; U3 /1

whenever r C s < 0;

D K; and ExtA0 .U3 ; U3 /s D 0

for all s ¤ 0I

D K; and ExtA1 .U3 ; U3 /s D 0

for all s ¤ 1:

(20.19)

Finally, knowing that and using either the bar resolution, or else a comparison with N one also sees that the product the analogous results for the extended algebra A, ExtA1 .U3 ; U3 /1 ˝ ExtA1 .U3 ; U3 /1 ! ExtA2 .U3 ; U3 /2

(20.20)

is nonzero. The significance of these computations is as follows. Let AQ be the trivial z D mod.A/ Q its category of A1 -modules. There is a general A1 -structure on A, and Q Q z with fixed cohomology module M , governed approach to classifying A-modules M by ExtA .M; M / in the sense of Kodaira–Spencer deformation theory. In this context, (20.19) and (20.20) describe a theory with one-dimensional first order deformations, all of whom are obstructed at second order. It follows that up to isomorphism in z there is only one way (the trivial one) to make U3 into an A-module Q z 3. H 0 .Q/, U A slightly more complicated argument using the same formalism, whose details we omit, yields the following: z be an A1 -module over A, Q whose underlying cohomology Lemma 20.6. Let M z module is H.M/ Š U3 ˚ SU3 . Then z z Š Cone.t/ in H 0 .Q/, M z 3; S S U z 3 / represents an element of Ext 2 .U3 ; U3 /. Hence, where t 2 hom0z .SS 1 U A Q

z M// z ¤ 0: H 1 .homQz .M; Returning to geometry, let us assume momentarily that F was formal, in which z are quasi-equivalent. Then, the curve L from Figure 20.2 would give rise case Q and Q to an A1 -module to which Lemma 20.6 applies. But on the other hand, since (20.18) is faithful, the endomorphisms of this module should reproduce HF  .L# ; L# / Š H  .LI K/, which is zero in degree 1, a contradiction. This provides an alternative proof of Corollary 20.3 (for m D 5, but the general case follows straightforwardly from that). z carry actions of the braid group Br 6 , given by the twist functors Both Q and Q associated to the basic (spherical) objects. However, in the geometric case of Q Š mod.F /, this embeds into an action of a larger group, namely the group of graded

314

III Picard–Lefschetz theory

exact symplectic automorphisms of X up to isotopy. An example of an element of this larger group which does not belong to Br 6 is given by the Dehn twist along L (this is because the isotopy class of L itself is not invariant under the covering involution of  W X ! C). Intuitively, one can think of the non-formality of F as being tied to this larger symmetry group. Remark 20.7. The relation between braid groups and mapping class groups, which appeared in our discussion above, is a well-understood subject of classical lowdimensional topology [23]. More recently, it has been shown that the situation for generalized braid groups is somewhat more complicated [142]. It is worth while to take a brief look at this from the point of view of Fukaya categories. Consider the Milnor fibre of the .E6 / singularity, which is fx03 C x14 D 1g  C2 . Topologically, this is a once-punctured genus three surface, and it carries a basis of vanishing cycles which form an .E6 / configuration. We denote these by .V1 ; : : : ; V6 /, following the ordering indicated in [142, Fig. 2], even though that is not the natural order in which they appear in the basis. Let A be the full A1 -subcategory of the Fukaya category of the Milnor fibre with objects Vi# . On the cohomology level, A D H.A/ can be expressed as a quotient of the path algebra of (an appropriately graded version of) the .E6 / Dynkin quiver, in the same sense as before. Set L D V4 V3 V5 V4 V2 V6 V3 V5 V4 .V1 /: After a suitable isotopy, L will intersect V1 in a single point, and that gives rise to a braid relation between the Dehn twists associated to the two curves. As pointed out in [142], no corresponding holds relation in the .E6 / type generalized braid group, and therefore the natural homomorphism from that braid group to the mapping class group of the Milnor fibre is not injective. Let AQ be the trivial A1 -structure on A. In Tw AQ consider the twisted complex C D TY4 TY3 TY5 TY4 TY2 TY6 TY3 TY5 TY4 .Y1 /; where Yi are the objects of AQ itself. A straightforward computation shows that H.homTw AQ .Y1 ; C // is three-dimensional (with generators in degrees 2; 1; 0). Q one could use Corollary 17.17 to identify this If A was quasi-isomorphic to A, H.hom/ group with the Floer cohomology of .V1 ; L/. But we know that to be onedimensional, hence A cannot be formal. The upshot is that the higher order products in A lead to cancellations between homomorphisms of twisted complexes, and those in turn appear to be responsible for the failure of the .E6 / braid group action to be faithful (we say “appear to be” since it is not known whether that group acts faithfully Q on Tw A).

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Symbols

zn Aut.M / y/ Aut.M

Aut gr .M; @M / Aut # .M; @M /

frequently occurring term in sign conventions, 8 group of exact symplectic automorphisms, 96 symplectic automorphism group after attaching a cone, 98 group of exact symplectic automorphisms rel boundary, 96 group of graded symplectic automorphisms, 188 group of brane automorphisms, 188

CF pr .L0 ; L1 / CF  .L#0 ; L#1 / Crit./ Critv./

Floer cochain group (preliminary version), 109 Floer cochain group, 181 critical point set, 212 critical value set, 212

D.A/, D  .A/ DS;u DS;r;u @h E @v E

derived category, 42 linearized operator, 107 extended linearized operator, 122 horizontal boundary, 212 vertical boundary, 212

Fakev.‰; $/ F .M /pr F ./

set of fake critical values, 220 Fukaya category (preliminary version), 126 Fukaya category of a Lefschetz fibration, 282

H.A/, H 0 .A/ H F .M / H F .M /pr

cohomological category, 8 Donaldson–Fukaya category, 181 Donaldson–Fukaya category (preliminary version), 101 Floer cohomology (preliminary version), 101, 109 Floer cohomology, 182

Aut.M; @M /

HF pr HF  .L#0 ; L#1 / M d C1 .y0 ; : : : ; yd / ME=S .fy g/ MS .fy g/, MS .fy g/  MZ .y0 ; y1 /

moduli space of pseudo-holomorphic polygons, 126 moduli space of inhomogeneous pseudo-holomorphic sections, 240 moduli spaces of pseudo-holomorphic maps, 105, 122 moduli space of Floer trajectories, 105

nu-fun.A; B/, fun.A; B/ nu-mod.A/, mod.A/

categories of A1 -functors, 10, 28 categories of A1 -modules, 19, 30

324

Symbols

x d C1 Rd C1 , R

moduli space of pointed discs and its compactification, 115, 118

TE v ; TE h TY0 .Y1 /

tangent space along the fibres and its complement, 212 twisting of objects, 63

Z Z˙

infinite strip, 100 half-infinite strips, 103

Index

absolute index, 164, 175 A1 -category, 8 A1 -functor, 9 A1 -module, 19 associativity equations, 8, 132, 185 Beilinson spectral sequence, 80, 290 braid parallel transport, 234 c-finite, A1 -category, 64 cohomological category, 8 cohomologically full and faithful A1 -functor, 9 composition law, 101 consistent, choice of strip-like ends or perturbation data, 120, 124 continuation map, 110, 137 crossing form, 160 c-unital, A1 -category, functor, or module, 22, 23, 30 Dehn twist, 223 derived category, 42, 60 determinant line, 149 directed A1 -category, 80 distinguished basis of vanishing cycles, 227 of vanishing paths, 225 Donaldson–Fukaya category, 101, 181 energy of a pseudo-holomorphic map, 105 of a pseudo-holomorphic section, 236 exact Lagrangian submanifold, 96 exact symplectic manifold with contact type boundary, 97

with corners, 96 exact triangle, 36 exceptional collection, 75 fake critical value, 220 family of Riemann surfaces, 113 fibration with singularities, 212 Floer cohomology, 101, 109, 182 Floer datum, 104 formal diffeomorphism, 10 formality, of an A1 -category, 305 framed Lagrangian sphere, 220 free object of a category with Z=2-action, 88 Fukaya category, 126 equivariant, 209 of a Lefschetz fibration, 282 generators, of an A1 -category, 42 grading of a Lagrangian submanifold, 174 of a Lagrangian subspace, 169 of a symplectic automorphism, 187 Gromov compactification, 130 group action on a category, 87, 135 Hochschild cohomology, 13 homotopy of A1 -functors, 15 Hurwitz equivalence, 228 Hurwitz moves, for bases of vanishing paths, 226 Hurwitz moves, for bases of vanishing cycles, 227 Lagrangian boundary condition, 235 standard, 238 Lagrangian brane, 174 abstract, 163

326 equivariant, 199 linear, 169 relative, 258 Lefschetz bifibration, 218 Lefschetz fibration, 216 with strip-like ends, 239 Lefschetz pencil, 295 Lefschetz thimble, 221 linearized operator, 107 extended, 122 mapping cone, 35, 48 Massey product, 9 matching isotopy, 230 matching path, 230 mutation, of an exceptional collection, 77 natural transformation between A1 -functors, 11 normal angle, of a Z=2-invariant Lagrangian submanifold, 203 octahedral axiom, 41 orientation operator, 174 orientation space, 164, 175 perturbation datum, 104 relative, 236 Perturbation Lemma, 17 Picard–Lefschetz data, 300 Picard–Lefschetz theorem, 224 Pin structure, 167 Piunikhin–Salamon–Schwarz (PSS) isomorphism, 102, 180 pointed disc, 114 Postnikov decomposition, 76

Index

pseudo-holomorphic map, 105 pseudo-holomorphic polygon, 126 pseudo-holomorphic section, 236 quadratic complex volume form, 169 relative, 216 quasi-equivalence, of A1 -categories, 28 quasi-representing object, 32 shift algebraic, 34, 48 for Lagrangian branes, 171, 174, 186 spherical object, 72 split-closed, category or A1 -category, 55, 59 split-generators, of an A1 -category, 60 Stein manifold, 291 strictly unital A1 -category, 22 strip-like ends, 103 symplectic automorphism exact, 96 graded, 187 tree, 115 triangle product, 101, 180 triangulated A1 -category, 39 twisted complex, 44 twisting of objects, 63, 71 Type (B), Lagrangian submanifold or brane of, 267, 278 Type (U), Lagrangian submanifold or brane of, 267, 276 vanishing cycle, 221 vanishing path, 221 Yoneda embedding, 21, 31