Deformation theory, homological algebra, and mirror symmetry

Kenji Fukaya

Department of Mathematics Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto Japan.

January 30, 2002

Contents

Contents 5 0.1 Introduction. 6

1 Classical Deformation theory 11 1.1 Holomorphic structure on vector bundles. 11 1.2 Family of holomorphic structures on vector bundle. 13 1.3 Cohomology and Deformation. 16 1.4 Bundle valued Harmonic forms. 20 1.5 Construction of versal family and Feynman diagram. 21 1.6 Kuranishi family. 25 1.7 Formal deformation. 31

2 Homological algebra and Deformation theory 38 2.1 Homotopy theory of A_{∞} and L_{∞} algebras. 38 2.2 Maurer-Cartan equation and moduli functor. 45 2.3 Canonical model, Kuranishi map, and moduli space. 50 2.4 Super space and odd vector field - an alternative formulation

of L_{∞} algebra - 58

3 Application to Mirror symmetry 61 3.1 Novikov ring and filtered A_{∞}, L_{∞} algebras. 61 3.2 Review of a part of global symplectic geometry. 64 3.3 From Lagrangian submanifold to A_{∞} algebra. 70 3.4 Maurer-Cartan equation for filtered A_{∞} algebra. 78 3.5 Homological mirror symmetry. 86

0.1 Introduction.

In this article the author would like to explain a relation of deformation theory to mirror symmetry. Deformation theory or theory of moduli is related to mirror symmetry in many ways. We discuss only one part of it. The part we want to explain here is related to rather abstract and formal point of the theory of moduli, which was much studied in 50’s and 60’s. They are related to the definition of scheme, stack and its complex analytic analogue, and also to various parts of homological and homotopical algebra. Recently those topics again call attention of several people working in areas closely related to mirror symmetry.

The author met them twice. He first met them when he was working [34] with K.Ono on the construction of Gromov-Witten invariant of general symplectic manifolds and the study of periodic orbit of periodic Hamiltonian system. There we found that a C^{∞} analogue of the notion of scheme and stack is appropriate to attach transversality problem. The transversality problem we met was one on the moduli space of holomorphic maps from Riemann surface. Later the author learned that in the algebraic geometry side, the same problem is studied by using stacks [72, 8]. This point however is not our main concern in this article. Our main focus is the next point.

The author met the relation of homological algebra to the theory of moduli while he, together with Y.G.Oh, H.Ohta, K.Ono [33], was trying to find a good formulation of Floer homology of the Lagrangian submanifold. There we first met a trouble that Floer homology of Lagrangian submanifold is not defined in general. So studying the condition when it is defined becomes an interesting problem. For this purpose, we developed an obstruction theory for the Floer homology to be defined. We next found that the Floer homology, even when it is defined, is not independent of the various choices involved. ^{1}

An example of this phenomenon is as follows. Let us consider a Lagrangian submanifold L in a symplectic manifold M. A problem, which is related to the definition of Floer homology, is to count the number of holomorphic maps φ : D^{2} → M such that φ(∂D^{2}) ⊂ L.^{2} Then the trouble is the number thus defined depends on the various choices involved. For example it is not independent of the deformation of (almost) complex structure of M. So unless clarifying in which sense the number of holomorphic disks is invariant of various choices, it does not make mathematical sense to count it. It is this essential point where we need deformation theory

1 This problem is quite similar to the case of Donaldson’s Gauge theory invariant of 4 manifolds with b_{2}^{+} = 1 [17]. (Here b_{2}^{+} = 1 is the number of positive eigenvalue of the intersection matrix on the second homology.) 2 Definitions of several notions we need in symplectic geometry will be given at the beginning of §3.2.

and homological algebra. Namely we construct an algebraic structure using the number of disks and homotopy types of this algebraic structure is independent of the perturbation.

We thus developed a moduli theory of the deformations of Floer homology. Namely we defined a moduli space M(L) for each Lagrangian submanifold L and Floer homology is defined as a family of graded vector spaces parametrized by M(L_{1}) × M(L_{2}). The moduli space M(L) is related to the actual deformation of Lagrangian submanifolds but the relation is rather delicate. The algebraic machinery to construct such moduli space is one of A_{∞} algebra and Maurer-Cartan equation.

The A_{∞} algebra we use there is a version of one found by the author in [23]^{3}. Using this A_{∞} structure, M.Kontsevich [64, 68] discovered a very interesting version of mirror symmetry conjecture which he called homological mirror symmetry conjecture. There it is conjectured that Lagrangian submanifold corresponds to a coherent sheaf on the mirror bundle.

After developing the theory of deformation of and obstruction to Floer homology of Lagrangian submanifold, we could make homological mirror symmetry conjecture more precise. For example, we now conjecture that the moduli space M(L) will become a moduli space of holomorphic vector bundles on the mirror. It is the purpose of this article to explain the formulation of homological mirror symmetry based on homological algebra and deformation theory.

During the conference “Geometry and Physics of Branes”, the author leaned that recently there are several works by Physicists which seems to be closely related to the story we had been developed. For example, the obstruction phenomenon seems to be rediscovered. The fact that our Maurer-Cartan equation which control the deformation of Floer homology (see §3.4) is inhomogeneous and 0 is not its solution seems to be related to what is called “Tachyon condensation”^{4}. The author does not quote references of the papers by Physicists on those points, since he expects that it is included in other parts of this book and since it is hard for the author to find correct choice of the papers to be quoted. It seems interesting to find a good dictionary between Physics side and Mathematics side of the works. The author hopes that this volume is helpful for this purpose.

As we already mentioned, the main purpose of this article is to describe a version of homological mirror symmetry precisely. For example, we want

3 The original motivation of the author to introduce A_{∞} structure on Floer homology was to use it to study gauge theory Floer homology of 3-manifolds with boundary. (The author was inspired by Segal and Donaldson to use category theory for this purpose.) The research toward this original direction is still on progress ([25, 26]) and the author believes that the relation of A_{∞} structure of Floer homology of Lagrangian submanifolds to gauge theory of 3-manifolds with boundary, would be related to some kind of duality in future. 4 The author does not yet understand precise relation of our story to ones developed by Physicists.

to state precisely the conjectured coincidence of the moduli spaces mentioned above. For this purpose, we need to review various basic aspects of moduli theory (especially its local version, the deformation theory). Hence the classical theory of deformation of the holomorphic structures of vector bundles on complex manifolds (together with proofs of various parts of it) is included in this article.

Chapter 1 of this article thus is devoted to theory of deformations. Deformation theory or local theory of moduli is a classical topic initiated by Riemann in the case of moduli space of complex structures of Riemann surface. Kodaira-Spencer [61, 63], generalized it to higher dimension and studied the deformation theory of complex structures of complex manifolds of higher dimension. It was further amplified by many people, for example, [71, 19, 105, 85]. There are many versions of the deformation theory, that is deformation theory of holomorphic vector bundles, deformation theory of complex submanifolds, deformation theory of holomorphic maps etc.. But, as far as the points mentioned in Chapter 1 concern, the difference among them are rather minor. So we mostly restrict ourselves to the case of vector bundles. In this article, we are taking analytic point of view and use (nonlinear) partial differential equation. There is algebraic theory of deformation (and of moduli). Some of basic references of it are [20, 45, 4, 93, 79]. The author tried to make Chapter 1 selfcontained. Especially he tried to explain several abstract notions which are popular among algebraic geometers but not so much popular among researchers in the other fields. For example, we explain a notion of analytic space (complex analytic analogue of scheme), the relation of category theory to the problem of moduli, especially notion of “Functor from Artin ring”, which is basic to study formal moduli. (Here formal moduli means that we consider formal power series solutions of defining equation of moduli space.) The proof of several of the main results of the local theory of moduli (existence of Kuranishi family, its completeness, versality, etc.) are postponed to Chapter 2, where we give a proof of them based on homotopy theory of A_{∞} algebra developed there. The contents of Chapter 1 is classical and no new points of view is introduced. We include them here since most of the reference the author found requires much background on algebraic geometry etc.

In Chapter 2, we explain systematically how the points of view of homological algebra of A_{∞} or L_{∞} algebra can be applied to the problem of moduli. (A_{∞} and L_{∞} algebra are generalization of differential graded algebra and of differential graded Lie algebra, respectively.) In §2.1, we give a definition of them and define A_{∞} and L_{∞} homomorphisms. Also we developed homotopy theory of them. Namely we define homotopy between two A_{∞} or L_{∞} homomorphisms and homotopy equivalence between two A_{∞} or L_{∞} algebras.

We next study a Maurer-Cartan equation from the point of view of “Functor from Artin ring” by [93] which we explained in §1.7.

We then sketch an important theorem which says that the gauge equivalence class of solutions of Maurer-Cartan equation is invariant of homotopy types of the A_{∞} or L_{∞} algebra. In the case of differential graded algebra and differential graded Lie algebra, this result is due to [37, 38].

We then construct a Kuranishi family of the solutions of Maurer-Cartan equation, as a quotient ring of appropriate formal power series ring. We use a technique sum over trees (calculation of tree amplitude by Feynman diagram) for this purpose. Several basic results postponed from Chapter 1 (together with its generalization to A_{∞} or L_{∞} algebra) follows. In §2.4, we briefly explain a translation of the story of Chapter 2 into the language of formal super geometry.

The theory developed in Chapter 2, seems to be studied by various people by now. Let us quote here a few related papers [54, 46, 92, 99, 78, 77, 14, 7, 69, 49, 50, 97, 56] which the authors found. (The author is very sorry for the authors of the other papers on the subject which he did not quote. The author does not have enough knowledge to quote all important papers.) (Y. Soibelman informed me that he and M. Kontsevitch is preparing a book which has overlap with this article.)

In Chapter 3, we study the application of the discussion in Chapters 1 and 2, to homological mirror symmetry. We need to introduce a kind of formal power series ring which we call the universal Novikov ring to study instanton (or quantum) effect (in symplectic geometry side of the story). Our A_{∞} algebra is a module over universal Novikov ring, and we need a slight modification of the definition of A_{∞} algebra which is explained in §3.1.

In §3.2 and §3.3, Floer homology is explained. In this Chapter 3 we do not assume the reader to be familiar with global symplectic geometry. So we include §3.2, which is an introduction of a part of global symplectic geometry related to §3.3. Especially we explain Floer’s original construction [22] of Floer homology. In §3.4, we explain the main construction of [33] which associates A_{∞} algebra to Lagrangian submanifolds. Our discussion in §3.2 and §3.3, are rather brief especially in the geometric and analytic points we need for the construction, since the main purpose of this article is to explain algebraic formalism rather than basic geometric-analytic construction which is essential to give examples of the algebraic formalism. Detail of the construction is in [33]. [28, 30, 84] are other surveys.

§3.5 is devoted to the definition of the moduli space M(L). To define it, we explain the modifications of the argument of Chapter 2 which are necessary to apply it to the case when the coefficient ring is not ℂ but is universal Novikov ring. §3.6 is devoted to the explanation of the complex geometry side of the story. An important point to be explained is what the Novikov ring in the complex geomegry side corresponds. Roughly speaking Novikov ring will become the ring of functions on the disk which parametrize maximal degenerate family of mirror manifolds. We then discuss that a mirror of a Lagrangian submanifold is a family of vector bundles (or more generally of object of the derived category of coherent sheaves) over maximally degenerate family of Calabi-Yau manifolds. A version of homological mirror symmetry conjecture is then stated that two A_{∞} algebra over Novikov rings, one for Floer homology the other for sheaf cohomology, coincide up to homotopy equivalence. There are some points which is not so clear for the author yet, which are related to various deep problems in algebraic geometry. We conclude Chapter 3, with giving an example.

The original plan of the author was to include several other deformation theories related to mirror symmetry in this article. For example extended deformation of Calabi-Yau manifold due to [7], deformation quantization due to [69], and contact homology announced in [21] ^{5}. They all can be treated by using the formalism of Chapter 2. However this article becomes already too thick and the author would like to postpone them to other occasion^{6}.

Parts of this article are announcement of our joint paper [33] with Oh,Ohta,Ono. (A preliminary version of [33] is completed in December 2000 and is available from author’s home page at the time of writing this article. We are adding several new materials to it, some of which is included in this article. The final version of [33] is now being completed.)

The author would like to thank to the organizers of the conference “Geometry and Physics of Branes” to give him an opportunity to communicate with various researchers in a comfortable atmosphere and to write this article.

5 Probably there is another example related to the Period of Primitive form due to K.Saito [91]. The author is unable to explain it at the time of writing this article. Some explanation from the point of view of mirror symmetry is found in [74]. 6 The reader who speaks Japanese can find them in [29]

Chapter 1

Classical Deformation theory

1.1 Holomorphic structure on vector bundles.

We start with describing deformation theory (that is a local theory of moduli) of holomorphic structures of complex vector bundles on complex manifolds. It is a classical theory, and is a direct analogue of Kodaira-Spencer [61, 63], who studied the case of deformation of complex structures of complex manifold itself. We present it here since it consists a prototype of the discussion which will appear later in less classical situation.

Let M be a complex manifold and Λ_{ℂ}^{k}M = ⊕_{p+q=k}Λ^{p,q}M be the decomposition of the set of the complex valued k forms according to their types. We denote by Ω^{p,q}(M) the set of all smooth sections of Λ^{p,q}M. The complex structure of M is characterized by the Dolbault differential \overline{∂} : Ω^{p,q}(M) → Ω^{p,q+1}(M). (A standard text book of complex manifold written from the transcendtal point of view is [40].) We remark that (Ω(M), ∧, \overline{∂}) is a differential graded algebra, which we define below. Hereafter we denote by R a commutative ring with unit.

Definition 1.1.1. A differential graded algebra or DGA over R is a triple (A^{∗}, ·, d) with the following properties. (1) For each k ∈ ℤ_{≥0}, A^{k} is an R module. We write deg a = k if a ∈ A^{k}. (2) · : A^{k} ⊗ A^{ℓ} → A^{k+ℓ} is an R module homomorphism, which is associative. Namely (a · b) · c = a · (b · c).

(3) d : A^{k} → A^{k+1} is an R module homomorphism such that d ∘ d = 0.

(4) d(a · b) = d(a) · b + (−1)^{deg a}a · d(b).

We may take either (⊕Ω^{0,k}(M), ∧, \overline{∂}) where k is the degree, or (∑_{p,q} Ω^{p,q}(M), ∧, \overline{∂}) where the degree is the (total) degree of differential form. Let π_{E} : E → M be a complex vector bundle. (A standard text book of differential geometry of holomorphic vector bundle is [58].) We put Ω^{p,q}(M; E) = Γ(M, Λ^{p,q}(M; Λ^{p,q}M ⊗ E)). (We omit M in case no confusion occur.) We define a wedge product : ∧ : Ω^{p,q}(M) ⊗ Ω^{p′,q′}(M; E) → Ω^{p+p′,q+q′}(M; E) in an obvious way. We define holomorphic structure on our complex vector bundle as follows.

Definition 1.1.2. A holomorphic structure on E is a sequence of operators \overline{∂}_{E} : Ω^{0,q}(M; E) → Ω^{p,q+1}(M; E) (p, q ∈ \{1, ⋯, n\}), such that (1) \overline{∂}_{E} ∘ \overline{∂}_{E} = 0. (2) \overline{∂}_{E}(u ∧ α) = \overline{∂}(u) ∧ α + (−1)^{p+q}u ∧ \overline{∂}_{E}(α), for u ∈ Ω^{p,q}(M), α ∈ Ω^{∗}(M; E).

In other words, the holomorphic structure on E is a structure on Ω^{∗}(M; E) of left graded differential graded module over (Ω(M), ∧, \overline{∂}), which we define below.

Definition 1.1.3. A differential graded module on a differential graded algebra (A^{∗}, ·, d) is, by definition, a triple (C^{∗}, ·, d) such that : (1) For each k ∈ ℤ, M^{k} is an R module. We write deg a = k if a ∈ M^{k}. (2) · : A^{k} ⊗ M^{ℓ} → M^{k+ℓ} is an R module homomorphism, which is associative. Namely (a·b)·x = a·(b·x), for a, b ∈ A, x ∈ M. (3) d : M^{k} → M^{k+1} is an R module homomorphism such that d∘d = 0. (4) d(a·x) = d(a)·x+(−1)^{deg a}a·d(x), for a ∈ A, x ∈ M.

We remark that Definition 1.1.2 coincides with another definition of holomorphic vector bundle, the one which uses local chart (see [58]).

From now on we choose one holomorphic structure \overline{∂}_{E} on E and write E = (E, \overline{∂}_{E}). Once we fix \overline{∂}_{E}, other holomorphic structures can be identified with elements of an affine space satisfying a differential equations, as we describe below. We consider the vector bundle End(E) whose fiber at p is Hom(E_{p}, E_{p}). (We omit M in case no confusion can occur.) Let Ω^{p,q}(M; End(E)) be the set of all smooth sections of Λ^{p,q} ⊗ End(E). We define operators

\begin{aligned}∘ : Ω^{p,q}(M; End(E)) ⊗ Ω^{p′,q′}(M; E) & → Ω^{p+p′,q+q′}(M; E) \\ ∘ : Ω^{p,q}(M; End(E)) ⊗ Ω^{p′,q′}(M; End(E)) & → Ω^{p+p′,q+q′}(M; End(E)) \end{aligned}

by using End(E) ⊗ E → E, End(E) ⊗ End(E) → End(E) and the wedge product in an obvious way. A holomorphic structure \overline{∂}_{E} on E induces a holomorphic structure, (still denoted by \overline{∂}_{E}), on End(E) by

\overline{∂}_{E}(B) = \overline{∂}_{E} ∘ B − (−1)^{deg B}B ∘ \overline{∂}_{E}.  (1.1)

Theorem 1.1.1. Let \overline{∂}_{E′} be another holomorphic structure on π_{E} : E → M. Then, there exists a section B ∈ Ω^{0,1}(M; End(E)), such that

\overline{∂}_{E′}(α) = \overline{∂}_{E}(α) + B ∘ α  (1.2)

B satisfies the differential equation

\overline{∂}_{E}B + B ∘ B = 0.  (1.3)

On the other hand, let B ∈ Ω^{0,1}(M; End(E)) be a section satisfying (1.3). We define \overline{∂}_{E′} by (1.2). Then \overline{∂}_{E′} defines a holomorphic structure on π_{E} : E → M.

Proof. By definition, we find (\overline{∂}_{E′} −\overline{∂}_{E})(u∧α) = (−1)^{deg u}u∧(\overline{∂}_{E′} −\overline{∂}_{E})(α). It implies that \overline{∂}_{E′} − \overline{∂}_{E} is induced by a section of Ω^{0,1}(M; End(E)), which iwe denote by B. To show (1.3) we calculate

\begin{aligned}\overline{∂}_{E′}(\overline{∂}_{E′}(α)) & = \overline{∂}_{E′}(\overline{∂}_{E}(α) + B ∘ α) \\ & = \overline{∂}_{E}(\overline{∂}_{E}(α) + B ∘ α) + B ∘ (\overline{∂}_{E}(α) + B ∘ α) \\ & = \overline{∂}_{E}B ∘ α − B ∘ \overline{∂}_{E}(α) + B ∘ \overline{∂}_{E}(α) + B ∘ B ∘ α \\ & = (\overline{∂}_{E}B + B ∘ B) ∘ α. \end{aligned} (1.4)

Since (1.4) holds for any α, we have (1.3). The converse can be proved in the same way.

Equation (1.3) is an example of Maurer-Cartan equation, whose study is one of the main theme of this article.

1.2 Family of holomorphic structures on vector bundle.

We study holomorphic vector bundles which is sufficiently close to E. In other words, we are going to discuss a local theory of moduli.

We first define a family of complex structures. Let U ⊂ ℂ^{n} be an open set. Let πM̂ : M̂ → U be a fiber bundle whose fibers are diffeomorphic to M.

Definition 1.2.1. A smooth (complex analytic) family of complex structures on M parametrized by U is a complex structure JM̂ on M̂ such that πM̂ : (M̂, JM̂) → U is holomorphic.

We next define a family of holomorphic vector bundles. Let Ê → M̂ be a complex vector bundle. We assume that the restriction of Ê to π−1 ; M̂(x) ≅ M is isomorphic to E (as complex vector bundles).

Definition 1.2.2. A smooth (complex analytic) family of the holomorphic structures on E over (M̂, JM̂) is a holomorphic structure \overline{∂}Ê of the bundle Ê.

One important case is when (M̂, JM̂) is trivial, that is the case when (M̂, JM̂) is isomorphic to the direct product (M, J_{M}) × U. (However the case when the family (M̂, JM̂) is nontrivial also appears later in our story. §3.5.) In that case, we can use Theorem 1.1.1 to identify a family of holomorphic structures on E to a map U → Ω^{0,1}(M; End(E)) as follows. Let \overline{∂}Ê be a family of holomorphic structures on Ê = E × U → M̂ = M × U. Each x ∈ U determines a holomorphic structure \overline{∂}_{E_{x}} on E. Namely \overline{∂}_{E_{x}} is the restriction of \overline{∂}Ê to M × \{x\}. We put B_{x} = \overline{∂}_{E_{x}} − \overline{∂}_{E}. Theorem 1.1.1 implies

\overline{∂}_{E}B_{x} + B_{x} ∘ B_{x} = 0.  (1.5)

Using the fact that \overline{∂}Ê is a holomorphic structure on Ê we can prove that the map

B : U → Ω^{0,1}(M; End(E)), x ↦ B_{x}  (1.6)

is holomorphic. (Ω^{0,1}(M; End(E)) is a complex vector space (of infinite dimension). So it makes sense to say that the map B is holomorphic.) On the contrary, given a holomorphic map (1.6) satisfying (1.5), we can define \overline{∂}Ê by \overline{∂}Ê = \overline{∂}_{E×U} +B. Here \overline{∂}_{E×U} is a holomorphic structure on E ×U (the direct product) and B is regarded as a smooth section of Ω^{0,1}(M̂; End(Ê)).

Our next purpose is to define and study notions of completness, versality and universality of families. We define them only in the case when complex structure on M is fixed. The case of family of complex structures on a manifold M and the case of holomorphic structures of vector bundles over M̂ (with moving complex structures) are similar and are omitted.

We first need to define morphism of two families, for this purpose. We first define equivalence of holomorphic vector bundles. Let π_{E_{1}} : E_{1} → M_{1} and π_{E_{2}} : E_{2} → M_{2} be complex vector bundles. We consider a bundle homomorphism φ : E_{1} → E_{2} over a holomorphic map \overline{φ} : M_{1} → M_{2}. Namely π_{E_{2}} ∘ φ = \overline{φ} ∘ π_{E_{1}} and φ is complex linear on each fiber. A bundle homomorphism φ induces φ_{∗} : Ω^{p,q}(M; E_{1}) → Ω^{p,q}(M; E_{2}).

Definition 1.2.3. We say φ : (E_{1}, \overline{∂}_{E_{1}}) → (E_{2}, \overline{∂}_{E_{2}}) is holomorphic if \overline{∂}_{E_{2}} ∘ φ_{∗} = φ_{∗} ∘ \overline{∂}_{E_{1}}. We say that φ : (E_{1}, \overline{∂}_{E_{1}}) → (E_{2}, \overline{∂}_{E_{2}}) is an isomorphism if it is holomorphic and is a bundle isomorphism.

Let Ê_{i} = E × U_{i} and \overline{∂}_{E_{i}} be a holomorphic structure on it, that is a deformation of holomorphic structures on E_{i}. We put M̂_{i} = M × U_{i}.

Definition 1.2.4. A morphism from (M̂_{1}, \overline{∂}Ê_{1}) to (M̂_{2}, \overline{∂}Ê_{2}) is a pair (Φ, ϕ), where ϕ : U_{1} → U_{2} is a holomorphic map and Φ is a holomorphic bundle map Φ : (Ê_{1}, \overline{∂}Ê_{1}) → (Ê_{2}, \overline{∂}Ê_{2}) over id ×ϕ : M × U_{1} → M × U_{2}.

Let us define the notion of deformations of complex structure and of holomorphic vector bundle. It is a germ of a family and is defined as follows.

Definition 1.2.5. A deformation of a complex manifold (M, J) is a ∼ isomorphism class of a pair (((M̂, J), U), i) where (M̂, J) = M × U → U is a family of complex structures and i is a (biholomorphic) isomorphism π^{−1}(0) ≅ (M, J).

We say ((M × U, J), i) ∼ ((M × U′, J′), i′), if there exists an open neighborhood V of 0 such that V ⊂ U∩U′ and if there exists a biholomorphic map φ : (M × V, J) → (M × V, J′) which commutes with projection : M × V → V and which satisfies i ∘ φ = i.

Let E be a holomorphic vector bundle on a complex manifold M. A deformation of E is an equivalence class of maps B : U → Ω^{0,1}(M, End(E)) such that B(0) = 0 and \overline{∂}_{E}B(x) + B(x) ∘ B(x) = 0. We say that B : U → Ω^{0,1}(M, End(E)) is equivalence to B′ : U′ → Ω^{0,1}(M, End(E)) if there exists an open neighborhood V of 0 with V ⊂ U ∩ U′ and a holomorphic map Φ : V → Γ(M, End(E)) such that Φ(0) = id and Φ(x) ∘ (\overline{∂}_{E} + B(x)) = (\overline{∂}_{E} + B′(x)) ∘ Φ(x).

We can define a deformation of a pair of complex structure and vector bundle on it in a similar way. Hereafter we sometimes say B is a deformation of E or (B, U) is a deformation of E by abuse of notation.

Now we define completeness of a deformation. Roughly speaking, it means that all nearby holomorphic structures are contained in the family.

Definition 1.2.6. A deformation (B, U) of E is said to be complete if the following condition holds.

Let (B′, U′) be another deformation of E and Φ_{0} : E → E is an automorphism of E. Then, there exists a neighborhood V of 0 in U′ and a morphism

(Φ, ϕ) : (E × V, \overline{∂}_{E+B′}) → (E × U, \overline{∂}_{E+B})  (1.7)

of families in the sense of Definition 1.2.4 such that

Φ|_{M×{0}} = Φ_{0}.  (1.8)

The other important notion is universality and versality of a deformation, which we define below. Let (B, U) of E be a complete deformation.

Definition 1.2.7. We say that (B, U) is universal if for each (B′, U′) as in Definition 1.2.6 the morphism (Φ, ϕ) as in (1.7) satisfying (1.8) is unique.

We say that (B, U) is versal if the differential of ϕ at 0 is unique. Namely if (Φ′, ϕ′) as in (1.7) is another morphism satisfying (1.8) then d_{0}ϕ′ = d_{0}ϕ. (Note they both are linear maps : T_{0}V → T_{0}U).

The difference between versality and universality is related to the stability of bundles^{1}. We give an example of versal family which is not universal in §1.3.

1 See [79] for the definition of stability.

1.3 Cohomology and Deformation.

The Maurer-Cartan equation (1.3) is a nonlinear partial differential equation. In this section, we study its linearization. The solution of linearized equation is related to the cohomology group. Let E_{i} = (E_{i}, \overline{∂}_{E_{i}}) be holomorphic vector bundles on M. We consider Ω^{0,q}(Hom(E_{1}, E_{2})) = Γ(M; Λ^{0,q} ⊗ Hom(E_{1}, E_{2})), where Hom(E_{1}, E_{2}) is a bundle whose fiber at p is Hom((E_{1})_{p}, (E_{2})_{p}) Operations \overline{∂}_{E_{1}}, \overline{∂}_{E_{2}} defines an operation \overline{∂}_{E_{1},E_{2}} : Ω^{0,q}(Hom(E_{1}, E_{2})) → Ω^{0,q+1}(Hom(E_{1}, E_{2})) in the same way as (1.1). It is easy to see \overline{∂}_{E_{1},E_{2}} ∘ \overline{∂}_{E_{1},E_{2}} = 0. Namley (Hom(E_{1}, E_{2}), \overline{∂}_{E_{1},E_{2}}) is a holomorphic vector bundle.

Definition 1.3.1. The extension Ext^{q}(E, E′) is the qth cohomology of the chain complex (Ω^{0,∗}(Hom(E_{1}, E_{2})), \overline{∂}_{E_{1},E_{2}}).

Let (B, U) be a deformation of E. We are going to define a Kodaira-Spencer map T_{0}U → Ext^{1}(E, E). By Definition 1.2.5, we have

\overline{∂}_{E}B(x) + B(x) ∘ B(x) = 0.  (1.9)

We differentiate (1.9) at 0. Then, in view of B(0) = 0, we have

\overline{∂}_{E} (\frac{∂B}{∂x^{i}} (0)) = 0.

Here x = (x^{1}, ⋯, x^{n}) is a complex coordinate of U ⊆ ℂ^{n}.

Definition 1.3.2. We put

KS (\frac{∂}{∂x^{i}} ) = [\frac{∂B}{∂x^{i}} (0)] ∈ Ext^{1}(E, E).

KS is a linear map : T_{0}U → Ext^{1}(E, E), which we call the Kodaira-Spencer map of our deformation.

Kodaira-Spencer map is gauge equivariant. Namely it is independent of the choice of the representative (B, U) of the deformation. In other words, we have the following lemma . Let (B′, U′) be another representative of the deformation.

Lemma 1.3.1. If (B, U) is equivalent to (B′, U′) in the sense of Definition 1.2.5 then

[\frac{∂B}{∂x^{i}} (0)] = [\frac{∂B′}{∂x^{i}} (0)] ∈ Ext^{1}(E, E).

Proof. V ⊂ U ∩ U′, Φ : V → Γ(M; End(E)) in Definition 1.2.5. satisfies :

Φ(x) ∘ (\overline{∂}_{E} + B(x)) = (\overline{∂}_{E} + B′(x)) ∘ Φ(x)  (1.10)

We differentiate (1.10) at 0 and obtain

\frac{∂Φ}{∂x}(0) ∘ \overline{∂}_{E} + \frac{∂B′}{∂x^{i}} (0) = \frac{∂B}{∂x^{i}} (0) + \overline{∂}_{E} ∘ \frac{∂Φ}{∂x^{i}} (0).

Namely \frac{∂B}{∂x^{i}} (0) − \frac{∂B′}{∂x^{i}} (0) = \overline{∂}_{E} (\frac{∂Φ}{∂x^{i}} (0)) .

In a similar way, we can prove the following lemma.

Lemma 1.3.2. Let (B_{1}, U_{1}), (B_{2}, U_{2}) be deformations of E_{1}, E_{2} and Φ : (E_{1} × U_{1}, \overline{∂}_{E_{1}+B_{1}}) → (E_{2} × U_{2}, \overline{∂}_{E_{2}+B_{2}}), ϕ : U_{1} → U_{2} be a morphism of family of holomorphic structures in the sense of Definition 1.2.4. We assume ϕ(0) = 0. Then the following diagram commutes.

T_{0}U_{1} ⟶^{KS} Ext^{1}(E, E)
d_{0}ϕ↓  ∥
T_{0}U_{2} ⟶^{KS} Ext^{1}(E, E)

Diagram 1

The following result was proved by [62] in the case of deformation theory of complex structure.

Theorem 1.3.1. If Kodaira-Spencer map is surjective then the deformation is complete.

We will prove it in §2.3. Another main result of deformation theory is the following theorem which is due to Kodaira-Nirenberg-Spencer [60] in the case of deformation theory of complex structures.

Theorem 1.3.2. If Ext^{2}(E, E) = 0 then there exists a deformation of E such that Kodaira-Spencer map is an isomorphism.

We will prove it in §1.5. We also prove in §2.3 that the family obtained in Theorem 1.3.2 is unique up to isomorphism. Remark 1.3.1. The smooth family where Kodaira-Spencer map is surjective does not exist in general in the case when Ext^{2}(E, E) ≠ 0. (However there are cases where such family exists in the case Ext^{2}(E, E) ≠ 0. See Example 1.3.1 below.) Kuranishi [71] studied the case Ext^{2}(E, E) ≠ 0. It leads us to the notion Kuranishi map : Ext^{1}(E, E) → Ext^{2}(E, E). We will discuss it in §1.6.

We remark that :

Proposition 1.3.1. The deformation obtained in Theorem 1.3.2 is versal.

Proof. Completeness follows from Theorem 1.3.1. Let (Φ, ϕ) (Φ′, ϕ′) be morphisms as in (1.7) satisfying (1.8). By Lemma 1.3.2 we have the following commutative diagram.

T_{0}V ⟶^{KS} Ext^{1}(E, E) ⟵^{KS} T_{0}V
d_{0}ϕ↓  ∥  ↓^{d_{0}ϕ′}
T_{0}U ⟶^{KS} Ext^{1}(E, E) ⟵^{KS} T_{0}U

Diagram 2

d_{0}ψ′ = d_{0}ψ follows immediately.

We now give a few examples of deformations of holomorphic vector bundle.

Example 1.3.1. Let M be a complex manifold and let L → M be a complex line bundle. L has a holomorphic structure if its first Chern class is represented by a 1-1 form. (See [40, 58].) We fix a holomorphic structure of L and denote it by \overline{∂}_{L}. Other holomorphic structure is equal to \overline{∂}_{B} = \overline{∂}_{L}+B here B ∈ Ω^{0,1}(M; End(L)). Let us study the equation (1.3) in this case.

Using the fact that L is a line bundle, it is easy to see that End(L) is isomorphic to the trivial line bundle (as a holomorphic line bundle). Namely B ∈ Ω^{0,1}(M). Since B is 1 form we have B ∘ B = 0. Hence (1.3) reduces to a linear equation :

\overline{∂}B_{L} = 0.

We can find a vector subspace

U ⊂ Ker (\overline{∂} : Ω^{0,1}(M; End(L)) → Ω^{0,2}(M; End(L)))

such that the restriction U → Ext^{1}(L, L) of the natural projection is an isomorphism. Hence, in the case of line bundle, we have always a family whose Kodaira-Spencer map is an isomorphism. Note that in case dim M ≥ 2, the condition Ext^{2}(L, L) ≅ 0 of Theorem 1.3.2 may not be satisfied.

It is easy to see that our deformation is universal.

Example 1.3.2. Let L_{1}, L_{2} be line bundles on M such that

Ext^{1}(L_{2}, L_{1}) ≠ 0.
Ext^{0}(L_{2}, L_{1}) ≅ Ext^{1}(L_{1}, L_{1}) ≅ Ext^{1}(L_{2}, L_{2}) ≅ Ext^{1}(L_{1}, L_{2}) ≅ 0. (1.11)

(Let M = ℂP^{1} and c^{1}(L_{1}) = k_{2}[ℂP^{1}], c^{1}(L_{1}) = k_{2}[ℂP^{1}], with k_{1} > k_{2}. It is easy to check (1.11) in this case.) Let B(x) ∈ Ω^{0,1}(M; Hom(L_{2}, L_{1})) be a form representing non zero cohomology class x in Ext^{1}(L_{2}, L_{1}). We consider

𝔅(x) = (\begin{matrix}0 & B(x) \\ 0 & 0 \end{matrix}) ∈ Ω^{0,1}(M; End(L_{1} ⊕ L_{2})).

Then

\overline{∂}_{𝔅(x)} = \overline{∂}_{L_{1}⊕L_{2}} + 𝔅(x) = (\begin{matrix}\overline{∂}_{L_{1}} & B(x) \\ 0 & \overline{∂}_{L_{2}} \end{matrix})

satisfies \overline{∂}_{𝔅(x)} ∘ \overline{∂}_{𝔅(x)} = 0 and hence have a deformation of L_{1} ⊕ L_{2}. By (1.11) we can find easily Ext^{1}(L_{1} ⊕ L_{2}, L_{1} ⊕ L_{2}) ≅ Ext^{1}(L_{2}, L_{1}). The Kodaira-Spencer map of our deformation is the identity : Ext^{1}(L_{2}, L_{1}) → Ext^{1}(L_{1} ⊕L_{2}, L_{1} ⊕L_{2}) ≅ Ext^{1}(L_{2}, L_{1}). We thus have a versal deformation.

However this deformation is not universal.

In fact, let r : Ext^{1}(L_{2}, L_{1}) → ℂ\{0} be a holomorphic function with r(0) = 1. We define ϕ : Ext^{1}(L_{2}, L_{1}) → Ext^{1}(L_{2}, L_{1}) by ϕ(x) = r(x)x. We also define Ψ : Ext^{1}(L_{2}, L_{1}) → End(L_{1} ⊕ L_{2}), by Ψ(x)(v, w) = (r(x)v, w). Ψ defines a bundle homomorphism over ϕ. By definition it is easy to see that Ψ(x) : ((L_{1} ⊕ L_{2}), \overline{∂}_{L+𝔅(x)}) → ((L_{1} ⊕ L_{2}), \overline{∂}_{L+𝔅(x)}) is holomorphic. We thus find a morphism of family of holomorphic structures, which is identity on the fiber of 0 but is not identity at the fiber of other points. Hence our deformation is not universal.

We can explain this phenomenon as follows. We consider the group of (holomorphic) automorphisms of the bundles (L_{1} ⊕ L_{2}; \overline{∂}_{𝔅(x)}). In case x = 0 this bundle is a direct product. Hence we have Aut(L_{1}⊕L_{2}; \overline{∂}_{𝔅(0)}) ≅ Aut(L_{1}) × Aut(L_{2}) × Ext^{0}(L_{1}, L_{2}) ≅ ℂ_{∗}^{2} × Ext^{0}(L_{1}, L_{2}). On the other hand, for x ≠ 0 we have Aut(L_{1} ⊕ L_{2}; \overline{∂}_{𝔅(x)}) ≅ ℂ_{∗} × Ext^{0}(L_{1}, L_{2}) where c ∈ ℂ_{∗} acts on L_{1} ⊕ L_{2} by (v, w) ↦ (cv, cw). The difference Aut(L_{1} ⊕ L_{2}; \overline{∂}_{𝔅(0)})/ Aut(L_{1} ⊕ L_{2}; \overline{∂}_{𝔅(x)}) ≅ ℂ_{∗} acts on Ext^{1}(L_{2}, L_{1}) as a scaler multiplication. We can easily check that (L_{1} ⊕ L_{2}; \overline{∂}_{𝔅(x)}) is isomorphic to (L_{1} ⊕ L_{2}; \overline{∂}_{𝔅(x′)}) if and only if x′ = cx, namely in the case when x and x′ lie in the same orbit of ℂ_{∗}-action. Hence the part of the automorphism of (L_{1} ⊕ L_{2}, \overline{∂}_{B(x)}) : x = 0 which will be “lost” for x ≠ 0 will act on U = Ext^{1}(L_{2}, L_{1}). The set of isomorphism classes of nearby holomorphic structures will be the orbit space of this ℂ_{∗}-action. The quotient space of this action in our case is a union of ℂP^{N} and one point, where N + 1 = rank Ext^{1}(L_{2}, L_{1}). If we put quotient topology then the quotient space is not Hausdorff at the orbit of origin. This is because ℂ_{∗} is noncompact.

The phenomenon explained above that is the jump of the dimension of the automorphism group is related to versal but not universal family is rather general. Actually we can prove the following result. We recal that the Lie algebra of Aut(E) is identified with Ext^{0}(E, E)

Theorem 1.3.3. If the deformation (B, U) is versal and if the rank of Ext^{0}((E, \overline{∂}_{E+B(x)}), (E, \overline{∂}_{E+B(x)})) is independent of x in a neighborhood of 0, then (B, U) is universal.

We will prove it in §2.3.

1.4 Bundle valued Harmonic forms.

We prove Theorem 1.3.2 in the next section. To prove Theorem 1.3.2, we use Harmonic theory of vector bundle valued forms, which we review in this section (See [58] for detail). Let κ : Ω^{p,q}(M) → Ω^{q,p}(M) be the complex anti-linear homomorphism defined by

\begin{aligned}κ(u_{i_{1},··· ,i_{p};j_{1},··· ,j_{q}} & dz^{i_{1}} ∧ ⋯ dz^{i_{p}} ∧ d\overline{z}^{j_{1}} ∧ ⋯ d\overline{z}^{j_{q}}) \\ & = \overline{u}_{i_{1},··· ,i_{p};J_{1},··· ,j_{q}}d\overline{z}^{i_{1}} ∧ ⋯ d\overline{z}^{i_{p}} ∧ dz^{j_{1}} ∧ ⋯ dz^{j_{q}}. \end{aligned}

We next fix a hermitian metric g on M. Then g induces the Hodge ∗ operator ∗ : Λ^{k}(M) → Λ^{n−k}M by u ∧ κ(∗ v) = g(u, v) Vol_{g} where Vol_{g} ∈ Ω^{2n}(M) is the volume element, and g(u, v) in the inner product on Λ^{∗}(M) induced by g. We remark that ∗ is complex linear, κ ∘ ∗ = ∗ ∘ κ and ∗ : Λ^{p,q}(M) → Λ^{n−q,n−p}(M). We can also check

∗∗ = (−1)^{p+q} id  on Λ^{p,q}(M),  (1.12)

Definition 1.4.1. We define the operator \overline{∂}^{∗} is defined by \overline{∂}^{∗} = −κ ∘ ∗ ∘ \overline{∂} ∘ κ ∘ ∗ : Λ^{p,q}(M) → Λ^{p,q−1}(M).

\overline{∂}^{∗} is complex linear. Let us next include holomorphic vector bundle. Let E = (E, \overline{∂}_{E}) be a holomorphic vector bundle over M. We take and fix a hermitian inner product h on E. h induces an anti-complex linear homomorphism I_{h} : E → E^{∗}. I_{h} and κ induces an anti-complex linear homomorphism κ_{h} : Ω^{p,q}(M; E) → Ω^{q,p}(M; E^{∗}). We define ∧ : Ω^{p,q}(M; E) ⊗ Ω^{p′,q′}(M; E^{∗}) → Ω^{p+p′,q+q′}(M) by (u ⊗ a) ∧ (v ⊗ α) = α(a)u ∧ v. Then we define ∗ : Ω^{p,q}(M; E) → Ω^{n−q,n−p}(M; E^{∗}) by a ∧ κ_{h}(∗ B) = g_{h}(A, B)Ω_{g}, A, B ∈ Ω^{p,q}(M; E), where g_{h} is an inner product on Λ^{p,q}(M; E) induced by g and h. Formula (1.12) holds. We define

\overline{∂}_{E}^{∗} = −κ ∘ ∗ ∘ \overline{∂}_{E} ∘ κ ∘ ∗ : Λ^{p,q}(M; E) → Λ^{p,q−1}(M; E).

We define hermitian inner product ⟨·, ·⟩ on Ω^{p,q}(M) and Ω^{p,q}(E) by

⟨u, v⟩ = ∫_{M} g(u, v) Vol_{g}, ⟨A, B⟩ = ∫_{M} g_{h}(A, B) Vol_{g} .

Let L^{2}(M; Λ^{p,q}(M)), L^{2}(M; Λ^{p,q}(M; E)) be the completion. They are Hilbert spaces. We can prove ⟨\overline{∂}u, v⟩ = ⟨u, \overline{∂}^{∗}v⟩, ⟨\overline{∂}_{E}A, B⟩ = ⟨A, \overline{∂}_{E}^{∗}B⟩ by using stokes theorem and (1.12). We now define Laplace-Beltrami operator by Δ\overline{∂} = \overline{∂}\overline{∂}^{∗} + \overline{∂}^{∗}\overline{∂}, Δ_{\overline{∂}_{E}} = \overline{∂}_{E} \overline{∂}_{E}^{∗} + \overline{∂}_{E}^{∗} \overline{∂}_{E}, and space of Harmonic forms and sections by

H^{p,q}(M) = Ker(Δ\overline{∂} : Ω^{p,q}(M) → Ω^{p,q}(M)),
H^{p,q}(M; E) = Ker(Δ_{\overline{∂}_{E}} : Ω^{p,q}(M; E) → Ω^{p,q}(M; E)).

We can show that Δ\overline{∂}, Δ_{\overline{∂}_{E}} are elliptic and hence H^{p,q}(M) and H^{p,q}(M; E) are of finite dimensional if M is compact without boundary.

Let Π_{H} : L^{2}(M; Λ^{p,q}(M)) → H^{p,q}(M), Π_{H,E} : L^{2}(M; Λ^{p,q}(M; E)) → H^{p,q}(M; E) be orthonormal projections.

Now the basic results due to Hodge-Kodaira is :

Theorem 1.4.1. There exists an orthonormal decomposition :

L^{2}(M; Λ^{p,q}(M)) ≅ Im \overline{∂} ⊕ Im \overline{∂}^{∗} ⊕ H^{p,q}(M),
L^{2}(M; Λ^{p,q}(M; E)) ≅ Im \overline{∂}_{E} ⊕ Im \overline{∂}_{E}^{∗} ⊕ H^{p,q}(M; E).

There exist operators Q : L^{2}(M; Λ^{p,q}(M)) → L^{2}(M; Λ^{p,q}(M)), Q_{E} : L^{2}(M; Λ^{p,q}(M); E) → L^{2}(M; Λ^{p,q}(M); E) such that Δ\overline{∂} ∘ Q = id −Π_{H}, Δ_{\overline{∂}_{E}} ∘ Q_{E} = id −Π_{H,E}.

(For the proof see for example [106].) We remark that Q commutes with \overline{∂}, \overline{∂}^{∗}, Δ^{∗}. We put G = \overline{∂}^{∗} ∘ Q, G_{E} = \overline{∂}_{E}^{∗} ∘ Q_{E} and call them the propagators. We remark that G, G_{E} are chain homotopy between identity and the orthonormal projections Π_{H}, Π_{H,E}. Namely, we can prove easily that :

id −Π_{H} = G ∘ \overline{∂} + \overline{∂} ∘ G, id −Π_{H_{E}} = G_{E} ∘ \overline{∂}_{E} + \overline{∂}_{E} ∘ G_{E}.  (1.13)

1.5 Construction of versal family and Feynman diagram.

We prove Theorem 1.3.2 in this section. We are going to find a neighborhood U of origin in H^{p,q}(M; (End(E), \overline{∂}_{E})), construct a holomorphic map B : U → Ω^{0,1}(M; End(E)) such that

\overline{∂}_{E}B(b) + B(b) ∘ B(b) = 0,  (1.14)

and that d_{0}B : T_{0}U = H^{0,1}(M; (End(E), \overline{∂}_{E})) → Ω^{0,1}(M; End(E)) is the identity. As we discussed already, this is enough to show Theorem 1.3.2. We take a formal parameter T and put

B(b) = T b + ∑_{k=2}^{∞} T^{k}B_{k}(b).  (1.15)

We solve equation (1.14) inductively on k. Namely we solve

\overline{∂}_{E}B_{k}(b) = − ∑_{ℓ+m=k} B_{ℓ}(b) ∘ B_{m}(b)  (1.16)

inductively on k. The solution of (1.16) is given by using the operator G_{E}, the propagator, introduced in the last section. (Here we write G_{E} in place of G_{End(E)} for simplicity.)

Lemma 1.5.1. We define B_{1}(b) = b and

B_{k}(b) = ∑_{ℓ+m=k} G_{E}(B_{ℓ}(b) ∘ B_{m}(b))  (1.17)

inductively on k. Then it satisfies (1.16).

Proof. We remark that the harmonic projection Π_{H,End(E)} : L^{2}(M; Λ^{0,2}(M; End(E))) → H^{0,2}(M; End(E)) is zero because Ext^{2}(E, E) = 0 by assumption. Hence, by (1.4.1), we have :

\begin{aligned}\overline{∂}_{E}B_{k}(b) & = ∑_{ℓ+m=k} \overline{∂}_{E}G_{E}(B_{ℓ}(b) ∘ B_{m}(b)) \\ & = − ∑_{ℓ+m=k} B_{ℓ}(b) ∘ B_{m}(b) − ∑_{ℓ+m=k} G_{E} \overline{∂}_{E}(B_{ℓ}(b) ∘ B_{m}(b)). \end{aligned}

By using induction hypothesis, we have

∑_{ℓ+m=k}\overline{∂}_{E}(B_{ℓ}(b) ∘ B_{m}(b))

= ∑_{ℓ+m=k} \overline{∂}_{E}B_{ℓ}(b) ∘ B_{m}(b) − ∑_{ℓ+m=k} B_{ℓ}(b) ∘ \overline{∂}_{E}B_{m}(b)

= − ∑_{ℓ+m=k} ∑_{ℓ_{1}+ℓ_{2}=ℓ}(B_{ℓ_{1}}(b) ∘ B_{ℓ_{2}}(b)) ∘ B_{m}(b)

+ ∑_{ℓ+m=k} ∑_{m_{1}+m_{2}=m} B_{ℓ}(b) ∘ (B_{m_{1}}(b) ∘ B_{m_{2}}(b)) = 0.

(We remark that the associativity of ∘ plays an important role here.)

To complete the proof of Theorem 1.3.2, it suffices to show the following :

Lemma 1.5.2. There exists ϵ > 0 such that if |T |∥b∥ < ϵ then (1.15) converges.

(Here ∥b∥ is the Sobolev L_{k}^{2} norm of b with sufficiently large k, which is introduced below.) The proof of Lemma 1.5.2 is based on the standard result, in geometric analysis, which we briefly recall here. First for A ∈ Ω^{p,q}(End(E)) we define its Sobelev norm ∥A∥_{L_{d}^{2}} by

∥A∥_{L_{d}^{2}}^{2} = ∑_{i=0}^{k}⟨∇^{i}A, ∇^{i}A⟩

here ∇^{i}A is a ith covariant derivative of A and ⟨∇^{i}A, ∇^{i}A⟩ is its appropriate L^{2} inner product defined in a way similar to the last section. Let L_{d}^{2}(M; End(E)) be the completion of Ω^{p,q}(End(E)) with respect to ∥A∥_{L_{d}^{2}}. Then

(A) Q_{E} defines a bounded operator

L_{d}^{2}(M; Λ^{p,q}(M; End(E))) → L_{d+2}^{2}(M; Λ^{p,q}((M; End(E))).

(B) ∘ : Ω^{p,q}((M; End(E))⊗Ω^{p′,q′}((M; End(E)) → Ω^{p+p′,q+q′}((M; End(E)) can be extended to a continuous operator

\begin{aligned}L_{d+1}^{2}(M; Λ^{p,q}(M; End(E))) & ⊗ L_{d+1}^{2}(M; Λ^{p′,q′}(M; End(E))) \\ & → L_{d}^{2}(M; Λ^{p+p′,q+q′}(M; End(E))). \end{aligned}

if k is sufficiently large compared to 2n = dim_{ℝ} M.

The proof of (A) is given in many of the standard text books of Harmonic theory, for example [106]. The proof of (B) is given in many of the standard text books on Sobolev space, for example [36]. Now (A),(B) implies

∥G_{E}(B_{ℓ}(b) ∘ B_{m}(b))∥_{L_{k}^{2}} < C∥B_{ℓ}(b)∥_{L_{k}^{2}}∥B_{m}(b)∥_{L_{k}^{2}}

if k is large. Here C is independent of k. Therefore, we can show

∥B_{k}(b)∥_{L_{d}^{2}} < (C′∥b∥_{L_{d}^{2}})^{k}

inductively on k. Lemma 1.5.2 follows immediately.

We defined B_{k}(b) inductively on k. We can rewrite it and define B_{k}(b) as a sum over Feynman diagrams. In order to show the relation of Theorem 1.3.2 to quantum field theory, let us do it here.

Definition 1.5.1. A finite oriented graph Γ consists of the following date : (1) A finite set Vertex(Γ), the set of vertices. (2) A finite set Edge(Γ), the set of edges. (3) Maps ∂_{source} : Edge(Γ) → Vertex(Γ), ∂_{target} : Edge(Γ) → Vertex(Γ).

A ribbon structure of an oriented graph Γ is the cyclic ordering of the set ∂_{source}^{−1}(v) ∪ ∂_{target}^{−1}(v) for each v ∈ Vertex(Γ). A graph together with ribbon structure is called a ribbon graph.

We take copies of intervals [0, 1]_{e} corresponding to each element of e ∈ Edge(Γ) and copies of points v corresponding to each element v ∈ Vertex(Γ). We identify {0} ∈ [0, 1]_{e} with ∂_{source}(e) and {1} ∈ [0, 1]_{e} with ∂_{target}(e). We thus obtain a one dimensional complex, which we write |Γ|.

An embedding of |Γ| to an oriented surface (real 2 dimensional manifold) Σ induces a ribbon structure on Γ as in Figure 1.

Figure 1

In this article we only consider finite oriented graphs. So we call them graphs. Now we consider ribbon graphs Γ satisfying the following conditions.

Condition 1.5.1. (1) Vertex(Γ) is decomposed into a disjoint union of Vertex_{int}(Γ) and Vertex_{ext}(Γ). (2) If v ∈ Vertex_{int}(Γ) then ♯∂_{target}^{−1}(v) = 2, ♯∂_{source}^{−1}(v) = 1. (3) There exists v_{last} ∈ Vertex_{ext}(Γ), the last virtex, such that ♯∂_{target}^{−1}(v) = 1, ♯∂_{source}^{−1}(v) = 0. (4) If v ∈ Vertex_{ext}(Γ)\{v_{last}} then ♯∂_{target}^{−1}(v) = 0, ♯∂_{source}^{−1}(v) = 1. (5) |Γ| is connected and simply connected.

We say an element of Vertex_{int}(Γ) an interior virtex and an element of Vertex_{ext}(Γ) an exterior virtex. An edge e is called an exterior edge if {∂_{target}(e), ∂_{source}(e)} ∩ Vertex_{ext}(Γ) ≠ ∅. Otherwise it is called an interior edge.

Figure 2

Let Γ be a ribbon graph such that |Γ| is simply connected. It is then easy to see that there exists an embedding |Γ| → ℝ^{2} such that the ribbon structure is compatible with the orientation of ℝ^{2}.

We denote by RG_{k,2} the set of all ribbon graph Γ which satisfies Condition 1.5.1 and has exactly k + 1 exterior vertices.

Let Γ ∈ RG_{k,2}. Then, using the embedding |Γ| → ℝ^{2} compatible with ribbon structure, we obtain a cyclic order on Vertex_{ext}(Γ). By regarding the last vertex as 0th one, the cyclic order determine an order on Vertex_{ext}(Γ)\{v_{last}}. So we put Vertex_{ext}(Γ)\{v_{last}} = {v_{1}, ⋯, v_{k}}. We now are going to define B_{Γ} : H^{0,1}(M; (End(E), \overline{∂}_{E}))^{k⊗} → Ω^{0,1}(M; End(E)) for each Γ ∈ Gr_{k,2} such that

B_{k}(b) = ∑_{Γ∈Gr_{k,2}} B_{Γ}(b, ⋯, b).  (1.18)

We define B_{Γ} by induction on k. Let Γ ∈ RG_{k,2} and v_{last} be its last vertex. Let e_{last} be the unique edge such that ∂_{target}(e_{last}) = v_{last}. We remove [0, 1]_{e_{last}} together with its two vertices from |Γ|. Then |Γ|\[0, 1]_{e_{last}} is a union of two components which can be regarded as |Γ_{1}| and |Γ_{2}|, where Γ_{1} ∈ RG_{2,ℓ}, Γ_{2} ∈ RG_{2,m} with ℓ + m = k. We may number Γ_{1}, Γ_{2} so that v_{1}, ⋯, v_{ℓ} ∈ Γ_{1}.

Now we put

B_{Γ}(b_{1}, ⋯, b_{k}) = G_{E}(B_{Γ_{1}}(b_{1}, ⋯, b_{ℓ}) ∘ B_{Γ_{2}}(b_{ℓ+1}, ⋯, b_{k})).  (1.19)

(1.18) is quite obvious from definition.

The way we rewrite the induction process into sum over trees is a straightforward matter. However, rewriting the definition of B_{k} as in (1.18) leads us naturally to the following questions.

Question 1.5.1. (1) Can we generalize to the case when the interior vertex has more than two edges ?

(2) When we consider tree instead of ribbon tree, are there any corresponding story ?

(3) What happens when we include the graph which is not simply connected ?

Remarkably they all have good answers.

Answere 1.5.1. (1) We then study deformation of A_{∞} algebra in place of differential graded algebra.

(2) We then study deformation of differential graded Lie algebra or more generally of L_{∞} algebra.

(3) It corresponds to Reidemeister or Analytic torsion [88], (in the case H_{1}(|Γ|) = ℤ) and to Chern-Simons Perturbation theory [5, 6, 66, 24], or quantum Kodaira-Spencer theory [10], or pseudoholomorphic map from higher genus Riemann surface (with or without boundary).

We will explain the first two answers more in later sections. The detailed study on the third one is left to future, since Mathematical theory, in the case H_{1}(|Γ|) ≠ ℤ, is not yet enough developed.

1.6 Kuranishi family.

In this section, we remove the assumption Ext^{2}(E, E) = 0 from Theorem 1.3.1. We need to study deformations parametrized by a singular variety for this purpose. Let us start with briefly reviewing the notion of analytic variety. Since we discuss here only local theory of moduli it is enough to consider the case of analytic subspace of ℂ^{N}. (See [20] for more detail on analytic variety.)

Definition 1.6.1. Let X ⊂ ℂ^{N} be a locally closed subset. We say that X is a (complex) analytic subset if the following holds.

For each p ∈ X there exists its neighborhood U in ℂ^{N} and holomorphic functions f_{1}, ⋯, f_{m} such that X ∩ U = {z|f_{1}(z) = ⋯ = f_{m}(z) = 0}.

Definition 1.6.2. For p ∈ X, we put

ℑ_{X,p} = {f ∈ O_{p}|f ≡ 0 on X}.  (1.20)

Here O_{p} is a germ of holomorphic functions at p. (Namely the set of convergent power series in a neighborhood of p.)

The germ of holomorphic functions O_{X,p} of X at p is defined by O_{X,p} = O_{X,p}/ℑ_{X,p}. O_{X,p} is a local ring and its maximal ideal is O_{X,p,+} = {[f] ∈ O_{X,p} | f(p) = 0}. (Here and hereafter, we denote by R_{+} its maximal ideal of a local ring R.)

Let X ⊂ ℂ^{N}, X′ ⊂ ℂ^{N′} be analytic sets. A map F : X → X′ is said to be a holomorphic map if for each p ∈ X there exists a neighborhood U of p in ℂ^{N} such that the restriction of F to U ∩X is extended to a holomorphic map from U to ℂ^{N′}.

A germ of analytic subset at 0 ∈ ℂ^{N} is an equivalence class of analytic subset X containing 0, where X is equivalent to X′ if there exists a neighborhood V of 0 such that V ∩ X = V ∩ X′.

To study the problem of moduli, we need to consider the case when ℑ_{X,p} does not satisfy (1.20), that is a complex analytic analogue of scheme. The simplest example is X = {0} ⊂ ℂ and ℑ_{X,p} = (x^{2}), that is the set of holomorphic functions f(x) such that f(0) = f′(0) = 0. Let us define such objects. We need only its germ at 0 so we restrict ourselves to such case.

Definition 1.6.3. A germ at 0 of analytic subspace 𝔛 of ℂ^{N} is a germ of analytic subset X together with an ideals ℑ_{𝔛,0} ⊂ O_{X,0} such that the following holds. Let f_{1}, ⋯, f_{m} be a generator of ℑ_{𝔛,0}. (Since O_{0} is Noetherian it follows that we can choose such generator.) Let f_{i} are defined on U. Then

U ∩ X = {x | f_{i}(x) = 0, i = 1, 2, ⋯, m}.  (1.21)

Remark 1.6.1. By Hilbert’s Nullstellensatz, (1.21) is equivalent to ℑ_{X,0} = {f | f^{n} ∈ ℑ_{𝔛,0} for some n.}

One can define analytic variety as a ringed space^{2} which is locally isomorphic to (𝔛, ℑ_{𝔛,0}), in other words as the space obtained by glueing germs of analytic subspaces in ℂ^{N} defined in Definition 1.6.3. We do not try to do so since we do not use it. (See [20].)

Example 1.6.1. Let F = (f^{1}, ⋯, f^{k}) : U → ℂ^{k} be a holomorphic map, where U is an open neighborhood of 0 in ℂ^{N}. We assume f^{i}(0) = 0. We put X = F^{−1}(0). We let ℑ_{𝔛,0} be the ideal generated by f^{1}, ⋯, f^{k}. We then obtain a germ of analytic subspace. We write it by F^{−1}(0) by abuse of notation.

We put O_{𝔛,0} = O_{0}/ℑ_{𝔛,0}. O_{𝔛,0,+} = {[f] ∈ O_{𝔛,0} | f(0) = 0}.

Definition 1.6.4. Let 𝔛 = (X, ℑ_{𝔛,0}), (𝔛′, ℑ′_{0}(𝔛′)) be germs of analytic varieties. A morphism 𝔉 from 𝔛 to 𝔛′ is a ring homomorphism F^{∗} : O_{𝔛′,0} → O_{𝔛,0}.

2 We do not define this notion here since we do not use it. See [48].

(Here and hereafter ring all homomorphisms between commutative rings are assumed to preserve unit.) We remark that morphism of germ of analytic subspaces induce a map between analytic sets as follows.

Lemma 1.6.1. If 𝔉 : 𝔛 → 𝔛′ is a morphism as in Definition 1.6.4. Here 𝔛 = (X, ℑ_{𝔛,0}), X ⊂ ℂ^{N}, 𝔛′ = (X′, ℑ′_{𝔛′,0}), X′ ⊂ ℂ^{N′}. Then there exists a neighborhood U of 0 in ℂ^{N} and a holomorphic map F̃ : U → ℂ^{N′} with F̃(0) = 0, such that : (1) F̃(X ∩ U) ⊂ X′. (2) If f ∈ ℑ′ℑ′_{𝔛′,0} then f ∘ F ∈ ℑ_{𝔛,0}. (3) By (2) we have a ring homomorphism O_{𝔛′,0} → O_{𝔛,0} induced by f ↦ f ∘ F̃. This homomorphism coincides with F^{∗}.

Proof. Let x^{i}, i = 1, ⋯, N′ be the coordinate function on ℂ^{N′}. We have F^{∗}x_{i} ∈ O_{𝔛,0}. Let f̃^{i} ∈ O_{0} be any elements with represents F^{∗}x_{i}. It is easy to see that F̃ = (f̃^{1}, ⋯, f̃^{N′}) has required properties.

To define a deformation of complex structures parametrized by a germ of analytic subspace of ℂ^{N}, we need to define a fiber bundle over complex analytic variety etc. It is a straightforward analogue of the case of complex manifold. But since our main purpose is to study the case of vector bundle, we restrict ourselves to the case of deformation of holomorphic structures of complex vector bundle on a complex manifold M with fixed complex structure.

Let E be a holomorphic vector bundle on M. Let U ⊂ ℂ^{N} be an open neighborhood of origin and let X ⊂ U be a germ of complex analytic subset.

Definition 1.6.5. A deformation of E parametrized by X is a germ of a holomorphic map B : U → Ω^{0,1}(M; End(E)) such that B(0) = 0 and that

\overline{∂}_{E}B(x) + B(x) ∘ B(x) = 0.  (1.22)

holds for each x ∈ X.

Using (1.20), it is easy to see that (1.22) is equivalent to the following. There exists an open covering ∪U_{i} = M, and an open neighborhood

U of 0, and there exists a smooth sections e_{i}(x, q) of Λ^{0,1}(U × U_{i}, End(E)) such that

\overline{∂}_{E}B(x) + B(x) ∘ B(x) = ∑ f_{i}(x)e_{i}(x, q).  (1.23)

where f_{i} ∈ ℑ_{X,0}. Hereafter we say that (1.23) holds locally and do not mention U_{i}, U.

In view of (1.23), we can generalize Definition 1.6.5 to the case of a germ of analytic subspace as follows.

Definition 1.6.6. Let 𝔛 = (X, ℑ_{𝔛,0}) be a germ of analytic subspace and B : U → Ω^{0,1}(M; End(E)) be a deformation of holomorphic structures on E parametrized by X. We say that B is a deformation of E parametrized by 𝔛 if the following holds. Such that

\overline{∂}_{E}B(x) + B(x) ∘ B(x) = ∑ f_{i}(x)e_{i}(q, x).  (1.24)

holds locally with f_{i} ∈ ℑ_{𝔛,0}.

We way that B is the same deformation as B′ if

B(x) − B′(x) = ∑ f_{i}(x)e_{i}(q, x),

holds locally with f_{i} ∈ ℑ_{𝔛,0}.

Example 1.6.2. Let X = {0} ⊂ ℂ and ℑ_{𝔛,0} = (x^{2}). Let us write B(x) = xB_{1} + x^{2}B_{2} + · · · . (1.24), then implies \overline{∂}_{E}B_{1} = 0. Note any choice of B_{2}, ⋯ define the same family by definition. Hence the set of deformations parametrized by ({0}, (x^{2})) is identified with the kernel of \overline{∂}_{E} : Ω^{0,1}(End(E)) → Ω^{0,2}(End(E)).

To define completeness and universality of a deformation parametrized by a germ of analytic subspace, we define morphism between deformations. Let 𝔛 = (X, ℑ_{𝔛,0}), 𝔛′ = (X′, ℑ_{𝔛′,0}) be germs analytic subspaces and B : U → Ω^{0,1}(M; End(E)), B′ : U′ → Ω^{0,1}(M; End(E)) be deformations of E parametrized by 𝔛, 𝔛′ respectively.

Definition 1.6.7. A morphism from (𝔛, B) to (𝔛′, B′) is a pair (𝔉, Ψ), where 𝔉 : 𝔛 → 𝔛′ is a morphism of analytic subspace and Ψ : U → Γ(M; End(E)) is a holomorphic map such that

\overline{∂}_{E}(Ψ(x)) + B′(F̃(x)) ∘ Ψ(x) − Ψ(x) ∘ B(x) = ∑ f_{i}(x)e_{i}(q, x) (1.25)

holds locally with f_{i} ∈ ℑ_{𝔛,0}. Here F̃ is as in Lemma 1.6.1.

We regard (𝔉, Ψ) as the same morphism as (𝔉′, Ψ′) if 𝔉 = 𝔉′ in the sense of Definition 1.6.4 and if

Ψ(x) − Ψ′(x) = ∑ f_{i}(x)e_{i}(q, x)

locally with f_{i} ∈ ℑ_{𝔛,0}. Here e_{i}(q, x) is a local section of Λ^{0,1} ⊗ End(E).

Example 1.6.3. Let us consider a family on 𝔛 = ({0}, (x^{2})) as in Example 1.6.2. Let B = xB_{1} and B′ = xB′_{1} be two such families. Let (𝔉, Ψ) be a morphism from (𝔛, B) to (𝔛, B′). We may choose F̃ in Lemma 1.6.1 so that F̃(x) = xF_{1}, when F_{1} ∈ ℂ. We may write also Ψ = Ψ_{0} + xΨ_{1}. (1.25) then can be written as \overline{∂}_{E}Ψ_{0} = 0, F_{1}\overline{∂}_{E}Ψ_{1} + B′_{1} ∘ Ψ_{0} − Ψ_{0} ∘ B_{1} = 0. Hence B′_{1} − B_{1} = F_{1}\overline{∂}_{E}Ψ_{1}. Thus the set of the deformations of E parametrized by 𝔛 = ({0}, (x^{2})) is identified with Ext^{1}(E, E).

Morphism of deformations being defined, we can define completeness and universality in exactly the same way as the case of deformation parametrized by complex manifolds. We leave it to the reader.

To define versality and Kodaira-Spencer map we need to define Zariski tangent space of a germ of analytic subspace.

Definition 1.6.8. Let 𝔛 ⊂ ℂ^{N} be a germ of analytic subspace. The Zariski tangent space T_{0}𝔛 is defied by

T_{0}𝔛 = {V ∈ T_{p}ℂ^{N} | V (f) = 0 if f ∈ ℑ_{𝔛,0}.}

Let 𝔉 : 𝔛 → 𝔛′ be a morphism. Let F̃ : U → ℂ^{N′} be as in Lemma 1.6.1. Then it is easy to see that d_{0}F̃ : T_{0}ℂ^{N} → T_{0}ℂ^{N′} induces d_{0}𝔉 : T_{0}𝔛 → T_{0}𝔛′.

Now we can generalize the definition of versality in the same way as §1.2 by using Zariski tangent space in the same way as Definition 1.2.7.

We next generalize Kodaira-Spencer map. Let us consider the situation of Definition 1.6.5. Let f_{i}, e_{i} be as in (1.23). Let V = ∑ V^{i} \frac{∂}{∂x^{i}} ∈ T_{0}𝔛. By differentiating (1.23) we have

\begin{aligned}\overline{∂}_{E}V (B(x)) + V (B(x)) ∘ B(x) & + B(x) ∘ V (B(x)) \\ & = ∑ V (f_{i})(0)e_{i}(0) + f_{i}(0)V (e_{i})(0) = 0. \end{aligned}

It follows that

V (B(x)) ∈ Ker (\overline{∂}_{E} : Ω^{0,1}(M; End(E)) → Ω^{0,2}(M; End(E))).

Definition 1.6.9. We put

KS (V) = [V (B(x))] ∈ Ext^{1}(E, E).

KS is a linear map : T_{0}𝔛 → Ext^{1}(E, E), which we call the Kodaira-Spencer map of our family.

Lemmata 1.3.1, 1.3.2 hold in our case also. However Theorem 1.3.1 does not hold. In fact, we have the following counter example. Let us consider a universal family parametrized by an open neighborhood of 0 in ℂ^{2}. We restrict it to X = {(x, y)|xy = 0}. Since T_{0}X = ℂ^{2} it follows that this family still satisfies the assumption of Theorem 1.3.1. It is not complete however.

Now we have the following generalization of Theorem 1.3.2

Theorem 1.6.1. Let E be a holomorphic vector bundle on M. There exists a germ of complex analytic variety 𝔛 and a deformation B of E parametrized by 𝔛 such that : (1) B is complete. (2) The Kodaira-Spencer map, KS : T_{0}𝔛 → Ext^{1}(E, E) is an isomorphism.

Moreover there exists an open neighborhood U of origin in Ext^{1}(E, E) and a holomorphic map Kura : U → Ext^{2}(E, E) such that 𝔛 = Kura^{−1}(0). Here the Kura^{−1}(0) is as in Example 1.6.1.

Definition 1.6.10. We call the map Kura the Kuranishi map.

We will give a proof of Theorem 1.6.1 in §2.3. Theorem 1.6.1 is a vector bundle analogue of [71]. [71] was generalized to various other situations by many people. See for example [45, 93, 105, 85, 4]. Theorem 1.6.1 has also an analogue in the case of the moduli space of gauge equivalence classes of Yang-Mills equations [102, 16].

Proposition 1.3.1 can be generalized directly to our situation by the same proof. Namely the deformation obtained in Theorem 1.6.1 is versal. Theorem 1.3.1 still holds in the case of deformation parametrized by a germ of analytic subspace. (We prove it in §2.3.)

We close this section by giving an example where the Kuranishi map is nonzero.

Example 1.6.4. Let 𝕋 be an elliptic curve and L → 𝕋 be a line bundle with c^{1}(L) ∩ [𝕋] = −m < 0. We have

Ext^{k}(𝕋; ℂ, L) ≅ \begin{cases}ℂ^{m} & k = 1, \\ 0 & k = 0. \end{cases}

Let 𝔟_{i} : i = 1, ⋯, m be a generator of Ext^{1}(𝕋; ℂ, L) we may regard it as a L valued (0,1)-form.

By Serre duality, we have Ext^{0}(𝕋; L, ℂ) ≅ Ext^{1}(𝕋; ℂ, L)^{∗}. Let 𝔟^{i} : i = 1, ⋯, m be the dual basis.

We consider M = 𝕋 × 𝕋 and let pr_{i} : i = 1, 2 be projections to first and second factors. We put

E = pr_{1}^{∗} L ⊕ pr_{2}^{∗} L = (L ⊠ 1) ⊕ (1 ⊠ L),

where ⊠ is the exterior product and 1 is the trivial line bundle. We consider

Ext^{1}(M; E, E) ≅ Ext^{1}(M; L ⊠ 1, L ⊠ 1) ⊕ Ext^{1}(M; 1 ⊠ L, 1 ⊠ L)

⊕ Ext^{1}(M; 1 ⊠ L, L ⊠ 1) ⊕ Ext^{1}(M; L ⊠ 1, 1 ⊠ L).

The first two factors are isomorphic to H^{0,1}(M) ≅ ℂ^{2}. The third and fourth factors are isomorphic to ℂ^{m^{2}} and their generators are pr_{1}^{∗} 𝔟_{i} ∧ pr_{2}^{∗} 𝔟^{j} and pr_{1}^{∗} 𝔟^{i} ∧ pr_{2}^{∗} 𝔟_{j} respectively.

Using these basis we define coordinate a_{i}, b_{i}, x_{ij}, y_{ij} of Ext^{1}(M; E, E) as follows. Let B ∈ Ext^{1}(M; E, E), then we put :

\overline{∂}_{E+B} = \overline{∂}_{E} + (\begin{matrix}∑_{i} a_{i}d\overline{z}^{i} & ∑_{i,j} x_{ij}(pr_{1}^{∗} 𝔟_{i} ∧ pr_{2}^{∗} 𝔟^{j}) \\ ∑_{i,j} y_{ij}(pr_{1}^{∗} 𝔟^{i} ∧ pr_{2}^{∗} 𝔟_{j}) & ∑ b_{i}d\overline{z}^{i}. \end{matrix}) ,

where z_{1}, z_{2} are complex coordinate of the first and the second factor M = 𝕋 × 𝕋. We also have

Ext^{2}(M; E, E) ≅ Ext^{2}(M; L ⊠ 1, L ⊠ 1) ⊕ Ext^{2}(M; 1 ⊠ L, 1 ⊠ L) ≅ ℂ^{2}.

We find

(pr_{1}^{∗} 𝔟_{i} ∧ pr_{2}^{∗} 𝔟^{j}) ∧ (pr_{1}^{∗} 𝔟^{k} ∧ pr_{2}^{∗} 𝔟_{ℓ}) = \begin{cases}(1, 0) & (i, j) = (k, ℓ), \\ 0 & otherwise. \end{cases}

(pr_{1}^{∗} 𝔟^{i} ∧ pr_{2}^{∗} 𝔟_{j}) ∧ (pr_{1}^{∗} 𝔟_{k} ∧ pr_{2}^{∗} 𝔟^{ℓ}) = \begin{cases}(−1, 0) & (i, j) = (k, ℓ), \\ 0 & otherwise. \end{cases}

Other components of the products Ext^{1}(M; E, E)×Ext^{1}(M; E, E) → Ext^{2}(M; E, E) vanish. Therefore, the Kuranishi map is

(a_{1}, a_{2}; b_{1}, b_{2}; (x_{i,j}), (y_{i,j})) ↦ (∑_{i,j} x_{i,j}y_{i,j}, − ∑_{i,j} x_{i,j}y_{i,j}) .

Hence origin is a singular point of Kura^{−1}(0).

We remark that origin is not a stable bundle in the sense of [79]. We need to consider 3 dimensional case to obtain a stable bundle whose Kuranishi map is nontrivial. R.Thomas [103] found such an example for Calabi-Yau 3 fold. We will discuss a mirror of Example 1.6.4 in section §3.6.

1.7 Formal deformation.

In §1.4, we proved the convergence of the series (1.15). In less classical situation, which we will study in later chapters, the convergence of a similar series is not yet proved. So until the convergence will be proved (in future hopefully) we need to regard a series like (1.18) as a formal power series. This leads us to the theory of formal deformation developed in algebraic geometry. [45, 93, 4] seems to be a standard reference. We review formal deformation in this section.

We need to translate the notion of deformation or family of structures, into more algebraic language. Let R be a commutative ring with unit. Here we consider the case R is a field. We consider a local ring R such that R ≅ R/R_{+}. We assume that there exists an embedding R → R preserving unit. Hence the composition R → R → R/R_{+} ≅ R is the identity. Let (A, ·, d) be a differential graded algebra defined over R. Hereafter we assume A is free as R module. We simply write A in place of (A, ·, d) sometimes for simplicity.

Definition 1.7.1. A deformation of A over R is a pair (A_{R}, i) where A_{R} is a differential graded R algebra and i : A_{R}/R_{+}A_{R} ≅ A, is an isomorphism of differential graded R algebras.

In a similar way, we can define deformation of differential graded module as follows. Let (C, ·, d) be a differential graded module over (A, ·, d). We sometimes write C in place of (C, ·, d).

Definition 1.7.2. A deformation of C over A_{R} is a pair of (C_{R}, i) such that C_{R} is a differential graded A_{R} module and i : C_{R}/R_{+}C_{R} ≅ C is an isomorphism of differential graded A modules.

In case A_{R} = A ⊗_{R} R, (that is the case of trivial deformation of A), we say C_{R} is a deformation of C over R.

Let us explain a relation of Definitions 1.7.1 and 1.7.2 to the definitions in §1.2. Let M̂ → U be a family of complex structures on M. We consider a vector bundle Λ^{p,q}(M̂/U) whose fiber at x̂ ∈ M̂ is Λp,q ; x̂(M_{x}) where x = π(x̂) ∈ U.

Definition 1.7.3. A section ω of Λ^{p,q}(M̂/U) is said to be holomorphic in U direction if the following holds. Let x̂ ∈ M̂. We choose a complex coordinate w^{1}, ⋯, w^{N} of a neighborhood of x = π(x̂) ∈ U. We choose z^{1}, ⋯, z^{n} so that z^{1}, ⋯, z^{n}, w^{1}, ⋯, w^{N} is a complex coordinate of a neighborhood of x̂. (Here we identify w^{i} with w^{i} ∘ π). Now we may write

ω = ∑ ω_{i_{1},··· ,i_{p},j_{1},··· ,j_{q}}(z, w)dz^{i_{1}} ∧ ⋯ ∧ dz^{i_{p}} ∧ d\overline{z}^{j_{1}} ∧ ⋯ ∧ d\overline{z}^{j_{q}}. (1.26)

Now we say ω is holomorphic in base direction if ω_{i_{1},··· ,i_{p},j_{1},··· ,j_{q}}(z, w) is holomorphic with respect to w. We assume ω to be smooth in both directions. We denote by Ω^{p,q}(M̂/U) the set of fiberwise (p, q) form which is holomorphic in base direction.

We remark that the holomorphicity in base direction is independent of the choice of the coordinate z, w. It is also easy to see that Ω^{p,q}(M̂/U) is a module over O(U), the ring of holomorphic functions on U.

We define an operator \overline{∂} : Ω^{p,q}(M̂/U) → Ω^{p,q+1}(M̂/U) by

\begin{aligned}\overline{∂} & (ω_{i_{1},··· ,i_{p},j_{1},··· ,j_{q}}(z, w)dz^{i_{1}} ∧ ⋯ ∧ dz^{i_{p}} ∧ d\overline{z}^{j_{1}} ∧ ⋯ ∧ d\overline{z}^{j_{q}}) \\ & = ∑_{ℓ}(−1)^{p} \frac{∂ω_{i_{1},··· ,i_{p},j_{1},··· ,j_{q}}}{∂\overline{z}^{ℓ}} dz^{i_{1}} ∧ ⋯ ∧ dz^{i_{p}} ∧ d\overline{z}^{ℓ} ∧ d\overline{z}^{j_{1}} ∧ ⋯ ∧ d\overline{z}^{j_{q}}. \end{aligned}

We can define a wedge product ∧ between elements of Ω^{∗,∗}(M̂/U) in an obvious way. Thus (Ω^{0,∗}(M̂/U), \overline{∂}, ∧) is a differential graded algebra over O(U). We consider its germ as follows. Let V ⊂ U be open neighborhoods of 0. We put M̂(V) = π^{−1}(V). We consider pairs (ω, V) where ω ∈ Ω^{p,q}(M̂(V)/V). We say (ω, V) ∼ (ω′, V′) if ω = ω′ on M̂(V ∩ V′). The set of ∼ equivalence class of all such pairs is denoted by Ω^{p,q}(M̂/U)_{0}. It is obvious that Ω^{p,q}(M̂/U)_{0} is an O_{0} module. It is easy to see that

\frac{Ω^{p,q}(M̂/U)_{0}}{O_{0,+}Ω^{p,q}(M̂/U)_{0}} ≅ Ω^{p,q}(M).

We write the isomorphism by i. The following lemma is now obvious.

Lemma 1.7.1. ((Ω^{0,∗}(M̂/U)_{0}, \overline{∂}, ∧), i) is a deformation of (Ω^{0,∗}(M), \overline{∂}_{0}, ∧) over O_{0}.

Remark 1.7.1. Actually, to study deformation of complex structures, we need to study the deformation of the differential graded Lie algebra Γ(M; Λ^{0,∗} ⊗ T M) (defined Example 2.1.1), as is discussed by [61, 63, 71], instead of deformation of differential graded Lie algebra Ω^{p,q}(M).

Let (B, U) be a deformation of a holomorphic vector bundle E on M̂. In a similar way, we can define Ω^{p,q}(M̂/U; E) the set of all E valued p, q forms which is holomorphic with respect to the base direction. (M̂ = M × U.) We can also define its germ Ω^{p,q}(M̂/U; E)_{0}. We define \overline{∂}_{E+B} : Ω^{p,q}(M̂/U; E)_{0} → Ω^{p,q+1}(M̂/U; E)_{0} and module structure ∧ : Ω^{p,q}(M̂/U)_{0} ⊗ Ω^{p′,q′}(M̂/U; E)_{0} → Ω^{p+p′,q+q′}(M̂/U; E)_{0}. Thus we have

Lemma 1.7.2. ((Ω^{0,∗}(M̂/U; E)_{0}, \overline{∂}_{E+B}, ∧), i) we defined above is a deformation of (Ω^{0,∗}(M; E), \overline{∂}_{E}, ∧) over ((Ω^{0,∗}(M̂/U)_{0}, \overline{∂}, ∧), i).

The generalization to the case of families parametrized by a germ of analytic subspace as in §1.6 is straightforward. In the case when we do not move the complex structure of M, it is described as follows.

Let 𝔛 = (X, ℑ_{𝔛,0}), be a germ of analytic subspace (X ⊆ U). We put M̂ = M × X. Let B : U → Ω^{0,1}(M; End(E)) be as in Definition 1.6.6. We then have \overline{∂}_{E+B} : Ω^{p,q}(M̂/U; E)_{0} → Ω^{p,q+1}(M̂/U; E)_{0}. By Definition 1.6.6 we have (\overline{∂}_{E+B} ∘ \overline{∂}_{E+B})(ω) = ∑ f_{i}e_{i} where f_{i} ∈ ℑ_{𝔛,0}. We now put

Ω^{p,q}(M̂/𝔛; E)_{0} = \frac{Ω^{p,q}(M̂/U; E)_{0}}{ℑ_{𝔛,0}Ω^{p,q}(M̂/U; E)_{0}} .

Then \overline{∂}_{E+B} induces a homomorphism \overline{∂}_{E+B} : Ω^{p,q}(M̂/𝔛; E)_{0} → Ω^{p,q+1}(M̂/𝔛; E)_{0} such that \overline{∂}_{E+B} ∘ \overline{∂}_{E+B} = 0. We thus have

Lemma 1.7.3. ((Ω^{0,∗}(M̂/𝔛; E)_{0}, \overline{∂}_{E+B}, ∧), i) we defined above is a deformation of (Ω^{0,∗}(M; E), \overline{∂}_{E}, ∧) over ((Ω^{0,∗}(M) ⊗ O_{𝔛,0}, \overline{∂}, ∧), i).

Now we will study formal deformation. This means that we are going to study formal power series ring rather than convergent power series ring. We first review the formal power series ring and projective limit a bit. We consider the convergent power series ring O_{𝔛,0}. It is a local ring and its maximal ideal is O_{𝔛,0,+}. We consider the quotient ring O_{𝔛,0}/O_{𝔛,0,+}^{m}. It is finite dimensional as a vector space over ℂ. We put

Ô_{𝔛,0} = \underleftarrow{lim} O_{𝔛,0}/O_{𝔛,0,+}^{m}.  (1.27)

Here the right hand side is the projective limit. Let us recall its definition here for the convenience of the reader. We remark that there exists an obvious homomorphism π_{m,m′} : O_{𝔛,0}/O_{𝔛,0,+}^{m} → O_{𝔛,0}/O_{𝔛,0,+}^{m′} for m < m′.

Definition 1.7.4. Let S_{m} is a set for each m ∈ ℤ_{>0} and π_{m,m′} : S_{m} → S_{m′} be a map for m < m′ such that π_{m′,m′′} ∘ π_{m,m′} = π_{m,m′′}. We consider the direct product ∏ S_{m}. The projective limit \underleftarrow{lim} S_{m} is a subset of ∏ S_{m} consisting of (x_{1}, x_{2}, ⋯) such that π_{m,m′}(x_{m}) = x_{m′}.

In case when S_{m} are groups, rings, modules etc. and π_{m,m′} are homomorphisms, the projective limit \underleftarrow{lim} S_{m} is also a group, ring, module etc.

Remark 1.7.2. We defined projective limit only for family S_{m} parametrized by m ∈ ℤ_{≥0}. Projective limit can be defined for more general family. Remark 1.7.3. We define d : \underleftarrow{lim} S_{m} × \underleftarrow{lim} S_{m} → ℝ_{≥0} by

d((x_{1}, x_{2}, ⋯), (y_{1}, y_{2}, ⋯)) = exp (− inf {m | x_{m} ≠ y_{m}}) .

Then (\underleftarrow{lim} S_{m}, d) is a complete metric space.

Example 1.7.1. Let us consider the case when 𝔛 = U is an open neighborhood of 0 in ℂ^{N}. In this case, O_{U,0} is a convergent power series ring ℂ{z_{1}, ⋯, z_{N}} of N variables. Then O_{U,0}/O_{U,0,+}^{m} = ℂ[z_{1}, ⋯, z_{N}]/(z_{1}, ⋯, z_{N})^{m} is the set of all polynomials modulo the terms of order > m. Now let (P_{1}, ⋯, P_{m}, ⋯) ∈ ∏ O_{U,P_{0},0}/O_{U,P_{0},0}^{m} be an element of projective limit \underleftarrow{lim} O_{𝔛,0}/O_{𝔛,0,+}^{m}. It means that P_{m} coincides with P_{m′} up to order m. Therefore, (P_{1}, ⋯, P_{m}, ⋯) determines a formal power series of z_{i}. Namely we have Ô_{U,x_{0},0} ≅ ℂ[[z_{1}, ⋯, z_{N}]]. Here ℂ[[z_{1}, ⋯, z_{N}]] is a formal power series ring.

In general, if the ideal ℑ_{𝔛,0} is generated by f_{1}, ⋯, f_{m} ∈ O_{0} then
Ô_{𝔛,0} ≅ \frac{ℂ[[z_{1}, ⋯, z_{N}]]}{(f_{1}, ⋯, f_{m})}.  (1.28)

Here we regard f_{i} ∈ ℂ[[z_{1}, ⋯, z_{N}]] by taking its Taylor series at 0 and (f_{1}, ⋯, f_{m}) is an ideal generated by them.

We remark that the ring O_{𝔛,0}/O_{𝔛,+}^{m} in (1.27) is of finite dimension over ℂ (as a vector space). (In the case of a general local ring R over R, the ring R/R_{+}^{m} is an Artin algebra over R.) In other words, formal power series ring can be regarded as a projective limit of finite dimensional ℂ algebras (or of Artin R algebras). This is a reason why theory of formal deformation is based on “functor from Artin ring” [93], which we review here.

To make exposition elementary, we only consider the case when R is an algebraically closed field. (We usually take R = ℂ.) In this case, we do not need the notion of Artin R algebra and consider only an R algebra of finite dimension (as a vector space over R).

Definition 1.7.5. We define the category of finite dimensional local R algebra, abbreviated by {f. d. Alg. /R}, as follows. (1) Its object is a local R algebra R which is commutative with unit and is finite dimensional over R as a vector space. (2) The morphism R → R′ is an R algebra homomorphism.

Definition 1.7.6. A formal moduli functor is a covariant functor from {f. d. Alg. /R} to {Sets}, the category of sets.

For the reader who is not familiar with category theory, let us review what Definitions 1.7.5, 1.7.6, mean. (See [20] Exposé 11 for more detail on the relation of category theory to the theory of moduli.) Let 𝔉 be a formal moduli functor in the sense of Definition 1.7.6. It consists of two kinds of data.

One is 𝔉_{0} which associates a set 𝔉_{0}(R) to any local ring R which is commutative with unit and is of finite dimensional over R.

Let R, R′ be two such rings and let φ : R → R′ be an R algebra homomorphism. Then the second data 𝔉_{1} associates to φ a map 𝔉_{1}(φ) : 𝔉_{0}(R) → 𝔉_{0}(R′).

The condition for 𝔉_{0}, 𝔉_{1} to define a covariant functor is 𝔉_{1}(φ′ ∘ φ) = 𝔉_{1}(φ′) ∘ 𝔉_{1}(φ), where φ′ : R′ → R′′.

Our main example of formal moduli functor is one such that 𝔉_{0}(R) is the set of isomorphism classes of deformation of a given differential graded modules. To define it we first define :

Definition 1.7.7. Let (C_{R}, i), (C′_{R}, i′) be deformations of differential graded A module C over R. We say that (C_{R}, i) is isomorphic to (C′_{R}, i′) if there exists an isomorphism Φ : C_{R} → C′_{R} of differential graded A_{R} modules such that the induced isomorphism \overline{Φ} : C_{R}/R_{+}C_{R} → C′_{R}/R_{+}C′_{R} satisfies i′ ∘ \overline{Φ} = i.

We now define a functor 𝔇𝔢𝔣_{C} : {f. d. Alg. /R} → {Sets} for each differential graded A module C, where A is a differential graded ring over R. Let R be an object of {f. d. Alg. /R}.

Definition 1.7.8. 𝔇𝔢𝔣_{C,0}(R) is the set of all isomorphism classes of deformations of C over R. Let φ : R → R′ be a morphism of the category {f. d. Alg. /R}. Let C_{R} be a deformation of C over R. We put 𝔇𝔢𝔣_{C,1}(φ)(C_{R}) = C_{R} ⊗_{R} R′.

We remark that if C_{R} is isomorphic to C′_{R} in the sense of Definition 1.7.7 then C_{R} ⊗_{R} R′ is isomorphic to C′_{R} ⊗_{R} R′. Hence 𝔇𝔢𝔣_{C,1}(φ) : 𝔇𝔢𝔣_{C,0}(R) → 𝔇𝔢𝔣_{C,0}(R′) is well-defined. It is easy to see that 𝔇𝔢𝔣_{C} is a covariant functor : {f. d. Alg. /R} → {Sets}. Namely we can check 𝔇𝔢𝔣_{C,1}(φ′) ∘ 𝔇𝔢𝔣_{C,1}(φ) = 𝔇𝔢𝔣_{C,1}(φ′ ∘ φ).

To show a relation of moduli functor to moduli space, we need to discuss representability of the functor. In our case of formal deformation theory, we need a notion of pro-representability, which we define below.

Definition 1.7.9. Let ℜ be a local R algebra (commutative with unit). We say that ℜ is a pro {f. d. Alg. /R} object if the following holds. (1) Let us denote by ℜ_{+} the maximal ideal of ℜ. Then, for each m, the quotient ring ℜ/ℜ_{+}^{m} is an object of {f. d. Alg. /R}. (In other words, ℜ/ℜ_{+}^{m} is of finite dimension over R.) (2) ℜ ≅ \underleftarrow{lim} ℜ/ℜ_{+}^{m}. In other words, ℜ is complete with respect to the ℜ_{+} adic metric, (which is defined as in Remark 1.7.3).

Definition 1.7.10. Let ℜ be a pro {f. d. Alg. /R} object. We define a covariant functor 𝔉_{ℜ} : {f. d. Alg. /R} → {Sets} as follows. (1) For an object R of {f. d. Alg. /R}, 𝔉_{ℜ,0}(R) is the set of all R algebra homomorphisms : ℜ → R. (2) Let φ : R → R′ be a morphism in the category {f. d. Alg. /R}. Let ψ ∈ 𝔉_{ℜ,0}(R). Then (𝔉_{ℜ,1}(φ))(ψ) = φ ∘ ψ.

In case ℜ itself is an object of {f. d. Alg. /R} (namely ℜ is of finite dimension over R), then the functor 𝔉_{ℜ} defined in Definition 1.7.10 is the functor represented by ℜ in the usual sense of category theory.

Definition 1.7.11. A covariant functor {f. d. Alg. /R} → {Sets} is said to be pro representable if there exists a pro {f. d. Alg. /R} object ℜ such that 𝔉 is equivalent to 𝔉_{ℜ} defined in Definition 1.7.10

We recall that two functors 𝔉, 𝔉′ : {f. d. Alg. /R} → {Sets} are said to be equivalent to each other if the following holds : For each object R of {f. d. Alg. /R} there exists a bijection H_{R} : 𝔉_{0}(R) → 𝔉′_{0}(R) such that the following diagram commutes for any morphism φ : R → R′.

𝔉_{0}(R) ⟶^{H_{R}} 𝔉′_{0}(R)
𝔉_{1}(φ)↓  ↓^{𝔉′_{1}(φ)}
𝔉_{0}(R′) ⟶^{H_{R′}} 𝔉′_{0}(R′)

Diagram 3

We now define universal formal moduli space of deformation of differential graded module. Let A be a differential graded algebra over R and C be a differential graded A module.

Definition 1.7.12. A pro {f. d. Alg. /R} object ℜ is said to be a universal formal moduli space of the deformation of C if the functor 𝔉_{ℜ,0}(R) in Definition 1.7.10 is equivalent to the functor 𝔇𝔢𝔣_{C} in Definition 1.7.8.

Remark 1.7.4. It is more precise to say that Spec ℜ is a universal formal moduli space rather than to say ℜ is a universal moduli space. Here Spec ℜ is a formal scheme (see [45]). Since we do not introduce the notion of formal scheme, we say ℜ is a moduli space by abuse of notation.

Exercise 1.7.1. Prove that universal formal moduli space in the sense of Definition 1.7.12 is unique if it exists.

Now we consider our geometric situation of deformation of holomorphic vector bundles. Let 𝔛 = (X, ℑ_{𝔛,0}) be an analytic subspace and B : U → Ω^{0,1}(M; End(E)) be a deformation of E parametrized by X. We define Ô_{𝔛,0} by (1.27). Ô_{𝔛,0} is a pro {f. d. Alg. /ℂ} object.

Proposition 1.7.1. If the deformation B of E is universal, then Ô_{𝔛,0} is a universal moduli space of the deformation of (Ω^{∗}(M; E|_{M×{x_{0}}}), d, ∧).

On the contrary if there exists a universal moduli space ℜ of the deformation of (Ω^{∗}(M; E|_{M×{x_{0}}}), d, ∧), then there exists a universal family of holomorphic structures of ℜ on 𝔛 such that Ô_{𝔛,0} is isomorphic to ℜ.

Proof. (sketch) First we assume that our family is universal. Let ℜ be an object of {f. d. Alg. /ℂ}. We may write R = ℂ[[z_{1}, ⋯, z_{N}]]/(f_{1}, ⋯, f_{k}). f_{i} is a priori a formal power series. However using finite dimensionality of R we may take polynomials for f_{i}.

We consider a germ of analytic subspace 𝔜 = ({0}, (f_{1}, ⋯, f_{k})). It is easy to see that a morphism 𝔜 → 𝔛 corresponds one to one to the ℂ algebra homomorphism Ô_{𝔛,0} → R. It is an immediate consequence of Lemma 1.7.2 (which still holds in the case of deformation parametrized by a germ of analytic subspace) that the deformation of E parametrized by 𝔜 corresponds one to one to the deformation of (Ω^{∗}(M; E|_{M×{x_{0}}}), d, ∧) over R. We can then prove easily that Ô_{𝔛,0} is a universal moduli space of the deformation of (Ω^{∗}(M; E|_{M×{x_{0}}}), d, ∧).

The proof of converse is more involved since we need to see the relation between the deformation in the category of formal power series and of convergent power series. We do not try to do it here.

Homological algebra and Deformation theory

2.1 Homotopy theory of A_{∞} and L_{∞} algebras.

Now we are going to discuss less classical part of the story. We have so far studied the equation

\overline{∂}B + B ∘ B = 0.  (2.1)

The second term is of second order. We mentioned in §1.5 a possibility to consider the equation where there are terms of third or of higher order. To do so while keeping gauge invariance of the equation, we need to consider A_{∞} algebra, due to J.Stasheff [98], which we define in this section. In case we consider the deformation of complex manifolds rather than holomorphic vector bundle on it, we need to consider the equation

\overline{∂}B + \frac{1}{2}[B, B] = 0,  (2.2)

in place of (2.1). Here B ∈ Ω^{0,1}(M; T M) and [B, B] is a combination of wedge product in Λ^{0,1} factor and the bracket of vector field in T M factor. (See Example 2.1.1.) To generalize (2.2) so that it includes terms of third or of higher order we introduce the notion, L_{∞} algebra. ^{1}

To define A_{∞} and L_{∞} algebras we need to review coalgebra, coderivation etc.. Let C be a graded R module. (Here grading starts from 0.) We define its suspension C[1] by C[1]^{k} = C^{k+1}. Hereafter we denote deg x as the degree of element of x ∈ C and deg′ x is the degree of the same element regarded as an element of C[1]. Namely deg′ x = deg x − 1. We define its

1 We remark that A in A_{∞} algebra stands for associative and L in L_{∞} algebra stands for Lie.

Homotopy theory of A_{∞} and L_{∞} algebras. 39

Bar complex BC[1] by

B_{k}C[1] = C[1]^{k⊗}, BC[1] = ⨁_{k=0}^{∞} B_{k}C[1].

We remark B_{0}C[1] = R. We define the action of the group 𝔖_{k} of all permutations of k elements on B_{k}C[1] by

σ(x_{1} ⊗ ⋯ ⊗ x_{k}) = ±x_{σ(1)} ⊗ ⋯ ⊗ x_{σ(k)}.

where

± = (−1)^{∑_{i, j with i < j, σ(i) > σ(j)} deg′ x_{i} deg′ x_{j}}.  (2.3)

We define E_{k}C[1] to be the submodule consisting of fixed points of the 𝔖_{k} action on B_{k}C[1] and EC[1] = ⊕E_{k}C[1].

We define Δ : BC[1] → BC[1] ⊗ BC[1] by

Δ(x_{1} ⊗ ⋯ ⊗ x_{k}) = ∑_{i=0}^{k}(x_{1} ⊗ ⋯ ⊗ x_{i}) ⊗ (x_{i+1} ⊗ ⋯ ⊗ x_{k}).  (2.4)

Note the term of (2.4) in case of i = 0 is 1⊗(x_{1}⊗⋯⊗x_{k}) ∈ B_{0}C[1]⊗B_{k}C[1]. The restriction of Δ induces Δ : EC[1] → EC[1] ⊗ EC[1].

Definition 2.1.1. A graded coalgebra (D = ⊕D^{k}, Δ, ϵ) is a graded R module together with Δ : D → D ⊗ D, ϵ : D^{0} → R such that the following diagrams commute.

D ⊗_{R} D ⊗ D ⟵^{Δ⊗1} D ⊗ D
1⊗Δ↑  ^{Δ}↑
D ⊗ D ⟵^{Δ} D

Diagram 4

D ⟵^{ϵ⊗1} D ⊗ D ⟶^{1⊗ϵ} D
∥  ^{Δ}↑  ∥
D  D  D

Diagram 5

Coalgebra (D, Δ, ϵ) is said to be graded cocomutative if R ∘ Δ = Δ : D → D ⊗ D, where R(x ⊗ y) = (−1)^{deg x deg y}(y ⊗ x).

The following lemma is easy to check.

Lemma 2.1.1. (BC[1], Δ, ϵ) is a colagebra where ϵ is an obvious isomorphism B_{0}C[1] ≅ R. We define a degree of elements of BC[1] by deg′(x_{1} ⊗ ⋯ ⊗ x_{k}) = ∑ deg′ x_{i} = ∑ deg x_{i} − k. (EC[1], Δ, ϵ) is a graded cocomutative coalgebra. (We remark that we need to take deg′ and not deg in the definition of R.)

Definition 2.1.2. A graded homomorphism δ : D → D of degree 1 from a coalgebra D to itself is said to be a coderivation if the following diagram commutes.

D ⊗ D ⟵^{Δ} D
1 ˆ⊗δ+δ ˆ⊗1↑  ^{δ}↑
D ⊗ D ⟵^{Δ} D

Diagram 6 .

Here we define the graded tensor product A ˆ⊗B between two graded homomorphisms A,B by (A ˆ⊗B)(x ⊗ y) = (−1)^{deg B deg′ x}(A(x)) ⊗ (B(y)).

Lemma 2.1.2. For any sequence of homomorphisms f_{k} : B_{k}C[1] → C[1] of degree 1 for k = 1, 2, ⋯, there exists uniquely a coderivation δ : BC[1] → BC[1] whose Hom(B_{k}C[1], B_{1}C[1]) component is f_{k} and whose Hom(B_{0}C[1], B_{1}C[1]) component is zero. The same holds for EC[1] in place of BC[1].

Proof. Put

f̂_{k}(x_{1} ⊗ ⋯ ⊗ x_{n}) = ∑_{i=1}^{n−k+1}(−1)^{deg f_{k}(deg′ x_{1}+···+deg′ x_{i−1})}
x_{1} ⊗ ⋯ ⊗ x_{i−1} ⊗ f_{k}(x_{i} ⊗ ⋯ ⊗ x_{i+k−1})
⊗ x_{i+k} ⊗ ⋯ ⊗ x_{n},

(2.5)

and δ = ∑ f̂_{k}.

To simplify the formulae like (2.5) we introduce the following notation. Let D be a coalgebra. We define Δ^{k} : D → D^{k⊗} by

Δ^{k} = ⋯ (Δ ⊗ 1 ⊗ 1) ∘ (Δ ⊗ 1) ∘ Δ_{k − 1 times}.

Then, for an element x of D, we put

Δ^{k}(x) = ∑_{a} x_{a}^{(k;1)} ⊗ ⋯ ⊗ x_{a}^{(k;k)}.  (2.6)

In general, we use bold face letter such as x for elements of the bar complex BC[1] and roman letter such as x_{k} for elements of C[1].

Now Formula (2.5) can be written

f̂_{k}(x) = ∑_{a}(−1)^{deg f_{k} deg′ x_{a}^{(3;1)}}x_{a}^{(3;1)} ⊗ f_{k}(x_{a}^{(3;2)}) ⊗ x_{a}^{(3;1)}.

Now we are ready to define A_{∞} algebra and L_{∞} algebra.

Definition 2.1.3. A structure of A_{∞} algebra on C[1] is a series of R module homomorphisms 𝔪_{k} : B_{k}C[1] → C[1] (k = 1, 2, ⋯) of degree +1 such that the coderivation δ obtained by Lemma 2.1.2 satisfies δδ = 0. If we replace B by E, then it will be the definition of structure of L_{∞} algebra on C[1].

We can write the condition δδ = 0 more explicitly as follows.

\begin{matrix}∑_{i=1}^{n−k+1} & (−1)^{deg′ x_{1}+···+deg′ x_{i−1}} \\ 𝔪_{n−k+1}(x_{1} ⊗ ⋯ ⊗ x_{i−1} ⊗ 𝔪_{k}(x_{i} ⊗ ⋯ ⊗ x_{i+k−1}) \\ ⊗ x_{i+k} ⊗ ⋯ ⊗ x_{n}) = 0. \end{matrix}  (2.7)

In particular we have 𝔪_{1}𝔪_{1} = 0. Hence we can define 𝔪_{1} cohomology H(C, 𝔪_{1}).

Let (C, d, ·) be a graded differential algebra. We put

𝔪_{1}(x) = (−1)^{deg x}dx, 𝔪_{2}(x, y) = (−1)^{deg x(deg y+1)}x · y,

and 𝔪_{k} = 0 for k ≥ 3.

Lemma 2.1.3. 𝔪_{k} determines a structure of A_{∞} algebra on C.

Proof. It suffices to show that Hom(B_{k}C[1], C[1]) component of δδ is zero for k = 1, 2, 3. The case k = 1 is obvious. Let us check the case k = 3, and leave the case k = 2 to the reader. Let π_{1} : BC[1] → B_{1}C[1] = C[1] is the projection. Then we have :

π_{1}δδ(x ⊗ y ⊗ z) = (−1)^{deg′ x}𝔪_{2}(x, 𝔪_{2}(y, z)) + 𝔪_{2}(𝔪_{2}(x, y), z)

= (−1)^{deg x+1+deg y(deg z+1)+deg x(deg y+deg z+1)}x · (y · z)

+ (−1)^{deg x(deg y+1)+(deg x+deg y)(deg z+1)}(x · y) · z = 0.

The last equality follows from the associativity of ·. (Change of degree and sign is important so that the relation δδ = 0 becomes associativity relation.)

We next discuss the L_{∞} case.

Definition 2.1.4. A differential graded Lie algebra is a graded R module C together with operations [ , ] : C ⊗ C → C of degree 0 and d : C → C of degree 1 such that dd = 0 and (1) d[x, y] = [dx, y] + (−1)^{deg x}[x, dy]. (2) [x, y] = (−1)^{deg x deg y+1}[y, x]. (3) [[x, y], z]+(−1)^{(deg x+deg y) deg z}[[z, x], y]+ (−1)^{(deg y+deg z) deg x}[[y, z], x] = 0.

Example 2.1.1. Let M be a complex manifold and T_{ℂ}M be a complex tangent bundle. (Namely holomorphic vector bundle whose local frame is \frac{∂}{∂z^{i}} where z^{i}, i = 1, ⋯, n is a local complex coordinate.) Let C^{k} = Γ^{k}(M; T_{ℂ}M ⊗ Λ^{0,k}). We put d = \overline{∂} and

\begin{matrix}[f \frac{∂}{∂z^{i}} ⊗ d\overline{z}^{i_{1}} ∧ ⋯ ∧ d\overline{z}^{i_{k}}, g \frac{∂}{∂z^{j}} ⊗ d\overline{z}^{j_{1}} ∧ ⋯ d\overline{z}^{j_{ℓ}}] \\ = (f \frac{∂g}{∂z^{i}} \frac{∂}{∂z^{j}} − g \frac{∂f}{∂z^{j}} \frac{∂}{∂z^{i}} ) ⊗ d\overline{z}^{i_{1}} ∧ ⋯ d\overline{z}^{i_{k}} ⊗ d\overline{z}^{j_{1}} ∧ ⋯ d\overline{z}^{j_{ℓ}}. \end{matrix}

We obtain a differential graded Lie algebra.

Let (C, [ , ], d) be a differential graded Lie algebra. We put

𝔪_{1}(x) = (−1)^{deg x}dx, 𝔪_{2}(x, y) = (−1)^{deg x(deg y+1)}[x, y],

and 𝔪_{k} = 0 for k ≥ 3.

Lemma 2.1.4. 𝔪_{k} determines a structure of L_{∞} algebra on C.

Proof. Let us first remark that 𝔪_{2} is a well-defined operator on E_{2}C[1]. Namely we have

𝔪_{2}(x, y) = (−1)^{deg x(deg y+1)}[x, y] = (−1)^{deg x(deg y+1)+deg x deg y+1}[y, x]
= (−1)^{deg x(deg y+1)+deg x deg y+1+deg y(deg x+1)}𝔪_{2}(y, x)
= (−1)^{(deg x+1)(deg y+1)}𝔪_{2}(y, x).

It then suffices to show that Hom(E_{k}C[1], C[1]) component of δδ is zero for k = 1, 2, 3. The case k = 1 is obvious. Let us check the case k = 3, and leave the case k = 2 to the reader. Let π_{1} : EC[1] → E_{1}C[1] = C[1] is the projection.

We put x_{1} × ⋯ × x_{k} = ∑_{σ∈𝔖_{k}} ±x_{σ(1)} ⊗ ⋯ ⊗ x_{σ(k)}, were ± is as in Formula (2.3). Then we have :

\frac{1}{2}π_{1}δδ(x × y × z)

= 𝔪_{2}(𝔪_{2}(x, y), z) + (−1)^{deg′ z(deg′ x+deg′ y)}𝔪_{2}(𝔪_{2}(z, x), y)

+ (−1)^{deg′ x(deg′ y+deg′ z)}𝔪_{2}(𝔪_{2}(y, z), x)

= (−1)^{deg x(deg y+1)+(deg x+deg y)(deg z+1)}[[x, y], z]

+ (−1)^{deg z(deg x+1)+(deg y+deg z)(deg x+1)+(deg z+1)(deg x+deg y)}[[z, x], y]

+ (−1)^{deg y(deg z+1)+(deg y+deg z)(deg x+1)+(deg x+1)(deg y+deg z)}[[y, z], x] = 0.

(Note the calculation above is a bit problematic in case 2 is not invertible. We do not try to correct it since the case when R is a field of characteristic zero is our main concern).

Our next purpose is to define homotopy equivalence of A_{∞} and L_{∞} algebras. For this purpose we first define A_{∞} and L_{∞} homomorphisms.

Definition 2.1.5. Let (D, Δ, ϵ), (D′, Δ′, ϵ′) be a coalgebras. An R module homomorphism φ : D → D′ of degree 0 is said to be a coalgebra homomorphism if the following diagram commutes.

D′ ⊗ D′ ⟵^{Δ′} D′
φ⊗φ↑  ^{φ}↑
D ⊗ D ⟵^{Δ} D

Diagram 7

\begin{matrix}R & ⟵^{ϵ′} & D′ \\ ∥ & & ^{φ}↑ \\ R & ⟵^{ϵ} & D \end{matrix} Diagram 8

Lemma 2.1.5. Let φ_{k} : B_{k}C[1] → C′[1], k = 1, 2, ⋯ be a sequence of degree 0 R module homomorphisms. Then there exists a unique colagebra homomorphism φ̂ : BC[1] → BC′[1] such that its Hom(B_{k}C[1], B_{1}C′[1]) component is φ_{k}. The same statement holds if we replace B by E.

Proof. Let φ : BC[1] → C′[1] be a homomorphism whose Hom(B_{k}C[1], B_{1}C′[1]) component is φ_{k}. We then set (using notation (2.6))

φ̂(x) = ∑_{k} ∑_{a} φ(x_{a}^{k;1}) ⊗ ⋯ ⊗ φ(x_{a}^{k;k}).

It is easy to check that this φ̂ has a required property. The L_{∞} case is similar.

Definition 2.1.6. Let (C[1], 𝔪_{k}), (C′[1], 𝔪′_{k}) be A_{∞} algebras. A sequence of homomorphisms φ_{k} : B_{k}C[1] → C′[1] is said to be an A_{∞} homomorphism if the coalgebra homomorphism φ̂ : BC[1] → BC′[1] obtained by Lemma 2.1.5 satisfies φ̂ ∘ δ = δ ∘ φ̂. The definition of L_{∞} homomorphism is similar.

We can define a composition of A_{∞} and L_{∞} homomorphisms by { \atop φ ∘ ψ} = φ̂ ∘ ψ̂.

We next define a homotopy between A_{∞} and L_{∞} homomorphisms. The definition we give below is an analogy of a similar definition in the case of differential graded algebra given in [41] Chapter X. There is another way to define homotopy which is an analogy of one in [101] (we do not discuss later here).

Let (C[1], 𝔪_{k}) be an A_{∞} algebra. We define an A_{∞} algebra C[1] ⊗ R[t, dt] as follows.

Definition 2.1.7. An element of C[1]⊗R[t, dt] is written as P (t)+Q(t)dt, where P, Q ∈ C[t] are polynomials with coefficient on C. We put deg dt = 1.

The operator 𝔪_{k} is defined as follows. Let x_{i} = P_{i}(t) + Q_{i}(t)dt

𝔪_{1}(P (t) + Q(t)dt) = 𝔪_{1}(P (t)) − 𝔪_{1}(Q(t))dt − \frac{dP}{dt} dt  (2.8)
𝔪_{k}(x_{1}, ⋯, x_{k})
= 𝔪_{k}(P_{1}, ⋯, P_{k})
+ ∑_{i=1}^{k}(−1)^{deg′ P_{1}+··· deg′ P_{i−1}+1}𝔪_{k}(P_{1}, ⋯, Q_{i}, ⋯, P_{k})dt (2.9)

Here we extend 𝔪_{k} to B_{k}C[t] in an obvious way.

In case when (C[1], 𝔪_{k}) is an L_{∞} algebra, we define C[1] ⊗ R[t, dt] in the same way as an R module. The definition of operations 𝔪_{k} is also the same by using (2.8),(2.9).

We omit the proof of the A_{∞} and L_{∞} formulae. (See the final version of [33].) For t_{0} ∈ ℝ, we define an A_{∞} homomorphism Eval_{t=t_{0}} : C[1] ⊗ R[t, dt] → C[1] by

Eval_{t=t_{0}}(P (t) + Q(t)dt) = P (t_{0}).  (2.10)

More precisely, we define the B_{1}(C ⊗ R[t, dt])[1] → C[1] component by (2.10) and set all the other components to be 0.

Definition 2.1.8. Two A_{∞} homomorphisms φ, φ′ : C → C′ are said to be homotopic to each other, if there exists an A_{∞} homomorphism H : C → C′ ⊗ R[t, dt] such that Eval_{t=0} ∘H = φ, Eval_{t=1} ∘H = φ′. We define homotoy between L_{∞} homomorphisms in the same way.

In Theorems 2.1.1 and 2.1.2 below, we assume R contains ℚ.

Theorem 2.1.1. If φ is homotopic to φ′ and φ′ is homotopic to φ′′ then φ is homotopic to φ′′.

In case when C, C′ are differential graded algebras, Theorem 2.1.1 is proved in [41]. The general A_{∞} algebra case is similar and is proved in detail in [33]. We omit the proof in this article.

Definition 2.1.9. An A_{∞} homomorphism φ : C → C′ is said to be a homotopy equivalence if there exists an A_{∞} homomorphism ψ : C′ → C such that the compositions ψ ∘ φ, φ ∘ ψ are homotopic to identity. Two A_{∞} algebra are said to be homotopy equivalent if there exists a homotopy equivalence between them. Homotopy equivalence of the L_{∞} algebras is defined in the same way.

The following theorem is useful to show that an A_{∞} homomorphism to be a homotopy equivalence.

Theorem 2.1.2. If φ : C → C′ is an A_{∞} homomorphism which induces an isomorphism on 𝔪_{1} cohomology, then φ is a homotopy equivalence. The same holds for L_{∞} case.

Remark 2.1.1. Theorem 2.1.2 does not hold in the category of differential graded algebra. Namely if φ : C → C′ is a differential graded algebra homomorphism (of degree 0) which induces an isomorphism on cohomology. Then Theorem 2.1.2 implies that we can find a homotopy inverse of it which is an A_{∞} homomorphism. However it is in general not possible to find a homotopy inverse which is a differential graded algebra homomorphism.

Theorem 2.1.2 is proved in somewhat weaker version in the 2000 December version of [33]. The proof of the general case will be included in the final version of [33].

2.2 Maurer-Cartan equation and moduli functor.

We now generalize the Maurer-Cartan equation (1.5) to the case of A_{∞} and L_{∞} algebras. Namely we consider the equation

∑_{k} 𝔪_{k}(b, ⋯, b) = 0  A_{∞} case,  (2.11a)

∑_{k} \frac{1}{k!}𝔪_{k}(b, ⋯, b) = 0  L_{∞} case.  (2.11b)

where b ∈ C[1]^{0} = C^{1}. Note that (2.11a) coincides with (1.5) in the case of differential graded algebra, and (2.11b) coincides with (2.2) in the case of differential graded Lie algebra.

There is however one trouble to make sense of equations (2.11a), (2.11b). Namely the left hand side is an infinite sum in the case when infinitely many of 𝔪_{k} are nonzero. There are two ways to make sense of (2.11a),(2.11b). One is to define a topology on C and consider the case when the left hand side converges. (We may either consider non Archimedean valuation on our coefficient ring (R = ℂ[[T]] for example) or may consider the convergence in the classical sense (R = ℝ or ℂ). Both plays a role in the story of mirror symmetry.) The other possibility is to consider b which is nilpotent (namely product of several of them vanishes). This second point is related to the “functor from Artin ring” discussed in §1.7. Let us take this second point of view in this section. (The first point of view also appears later.) Let us again consider the case when R is an algebraically closed field of characteristic zero.

Let R be a finite dimensional local R algebra (commutative with unit). Let R_{+} be the maximal ideal of R. There exists N such that R_{+}^{N} = 0. Let C be an A_{∞} algebra. C_{R} = C ⊗_{R} R has a structure of A_{∞} algebra.

Let b ∈ C^{1} ⊗_{R} R_{+}. Obviously 𝔪_{k}(b, ⋯, b) = 0 if k > N. Hence equation (2.11) makes sense.

Definition 2.2.1. b is said to be a Maurer-Cartan element of C_{R} if it satisfies equation (2.11).

To simplify the notation we introduce the following notation.

e^{b} = ∑_{k=0}^{∞} b ⊗ ⋯ ⊗ b_{k times}  A_{∞} case,  (2.12a)

e^{b} = ∑_{k=0}^{∞} \frac{1}{k!}b ⊗ ⋯ ⊗ b_{k times}  L_{∞} case.  (2.12b)

We remark that 𝔪 : BC[1] → C[1] is a homomorphism which is 𝔪_{k} on B_{k}C[1]. (𝔪 : EC[1] → C[1] is similar.)

Then equation (2.11) can be written as 𝔪(e^{b}) = 0. Before going further let us explain the meaning of equation (2.11). Let us define a deformed boundary operator 𝔪_{1}^{b} by

𝔪_{1}^{b}(x) = 𝔪(e^{b}, x, e^{b})  A_{∞} case,  (2.13a)
𝔪_{1}^{b}(x) = 𝔪(e^{b}, x)  L_{∞} case.  (2.13b)

Lemma 2.2.1. 𝔪_{1}^{b}𝔪_{1}^{b} = 0 if and only if (2.11) is satisfied.

The proof is easy and is omitted. We can deform 𝔪_{k} also by

𝔪_{k}^{b}(x_{1}, ⋯, x_{k}) = 𝔪(e^{b}, x_{1}, e^{b}, x_{1}, ⋯, x_{k−1}, e^{b}, x_{k}, e^{b}), A_{∞} case,
𝔪_{k}^{b}(x_{1}, ⋯, x_{k}) = 𝔪(e^{b}, x_{1}, ⋯, x_{k})  L_{∞} case.

Then we obtain A_{∞} or L_{∞} algebra (C, 𝔪_{∗}^{b}). (See [33].)

We now define a functor M̃C(C) : {f. d. Alg. /R} → {Sets} as follows. M̃C(C)(R) is the set of all Maurer-Cartan elements of C_{R}. If ψ : R → R′ is a morphism in {f. d. Alg. /R} and if b is a Maurer-Cartan element of C_{R}, then (1 ⊗ ψ)(b) ∈ C_{R′} is a Maurer-Cartan elements of C_{R′}. Thus we obtain ψ_{∗} : M̃C(C)(R) → M̃C(C)(R′). We thus obtain a covariant functor M̃C(C) : {f. d. Alg. /R} → {Sets}.

However the set M̃C(C)(R) is usually too big. So we divide it by an appropriate gauge equivalence, which we define below.

Definition 2.2.2. Let b, b′ ∈ M̃C(C)(R). We say that b is gauge equivalent to b′ if there exists an element b̃ ∈ M̃C(C ⊗ R[t, dt])(R) such that Eval_{t=0} b̃ = b, Eval_{t=1} b̃ = b′. Here Eval_{t=t_{0}} : C_{R} ⊗ R[t, dt] → C_{R} is an A_{∞} homomorphism as in the last section.

In the theorem below, we assume R contains ℚ.

Theorem 2.2.1. Let b, b′, b′′ ∈ M̃C(C)(R). If b is gauge equivalent to b′ and b′ is gauge equivalent to b′′, then b is gauge equivalent to b′′.

The proof is similar to the proof of Theorem 2.1.1 and will be given in the final version of [33].

Let us rewrite the definition of gauge equivalence in a bit concrete way. Here we only discuss A_{∞} case. (L_{∞} case is similar.) Let b̃ is as in Definition 2.2.2. We put b̃ = b(t) + c(t)dt where b(t) ∈ C^{1} ⊗ R_{+}[t], c(t) ∈ C^{0} ⊗ R_{+}[t]. The equation (2.11a) for b̃ becomes

\frac{db(t)}{dt} + ∑_{k=1}^{N} ∑_{i=1}^{k} 𝔪_{k}(b(t)^{⊗i−1}, c(t), b(t)^{⊗k−i}) = 0,  (2.14)

∑_{k=1}^{N} 𝔪_{k}(b(t)^{⊗k}) = 0.  (2.15)

The condition Eval_{t=0} b̃ = b, Eval_{t=1} b̃ = b′ is a boundary condition b(0) = b, b(1) = b′. We remark that (2.14) and equation (2.11a) for b implies (2.15). In fact \frac{d}{dt} ∑_{k=1}^{N} 𝔪_{k}(b(t)^{⊗k}) is the left hand side of (2.14).

We remark that using (2.13a) equation (2.14) can be written as

\frac{db(t)}{dt} + 𝔪_{1}^{b(t)}(c(t)) = 0.  (2.16)

Let us consider the case when C is a differential graded algebra and R = ℂ or ℝ. Then (2.16) is

\frac{db(t)}{dt} + d(c(t)) − b(t) · c(t) + c(t) · b(t) = 0.  (2.17)

Let g(t) be the solution of the differential equation :

\frac{dg(t)}{dt} = g(t) · c(t),  g(0) = I  (2.18)

We remark that we use the fact that R is a ring with characteristic 0 to solve equation (2.18). We remark g(t) − 1 ∈ R_{+} for each t hence g(t) is invertible in the ring C_{R}^{0}.

Lemma 2.2.2. g(t)^{−1}𝔪_{1}^{b}(g(t)) = b − b(t).

Proof. We may assume b = 0 by replacing 𝔪_{1} by 𝔪_{1}^{b}. The lemma is obvious for t = 0. By (2.18), we have :

\begin{aligned}\frac{d}{dt}𝔪_{1}(g(t)) & = 𝔪_{1}(g(t) · c(t)) = d(g(t) · c(t)) \\ & = 𝔪_{1}(g(t)) · c(t) + g(t) · d(c(t)) = −g(t) · b(t) · c(t) + g(t) · d(c(t)) \end{aligned}

On the other hand, we have, by (2.18)

\begin{matrix}\frac{d}{dt}(g(t) · b(t)) = \frac{d}{dt}g(t) · b(t) + g(t) · \frac{d}{dt}b(t) \\ = g(t) · c(t) · b(t) − g(t) · d(c(t)) + g(t) · b(t) · c(t) − g(t) · c(t) · b(t). \end{matrix}

The lemma follows.

In case C = (Ω^{0,∗}(E), \overline{∂}_{E}, ∧) we studied in Chapter 1, we have g ∈ Γ(End(E)), 𝔪_{1}^{B} = \overline{∂}_{E−B}. Lemma 2.2.2 then implies \overline{∂}_{E+B} = g^{−1} ∘ \overline{∂}_{E} ∘ g. In other words g is an isomorphism from (E, \overline{∂}_{E+B}) to \overline{∂}_{E}. This justifies our terminology, gauge equivalence.

Definition 2.2.3. MC(C)(R) is the set of all gauge equivalence classes of elements of M̃C(C)(R).

If ψ : R → R′ is a morphism in {f. d. Alg. /R}, we can construct ψ_{∗} : MC(C)(R) → MC(C)(R′) in an obvious way. Hence MC(C) defines a functor : {f. d. Alg. /R} → {Sets}. We call it the Maurer-Cartan functor associated to A_{∞} or L_{∞} algebra C.

Our next goal is to show that Maurer-Cartan functor is homotopy invariant. Let φ_{k} : B_{k}C[1] → C′[1] be an A_{∞} or L_{∞} homomorphism. It induces φ̂ : BC[1] → BC′[1], or EC[1] → EC′[1].

Lemma 2.2.3. There exists φ_{∗} : C[1]^{0} → C′[1]^{0} such that φ̂(e^{b}) = e^{φ_{∗}(b)}

Proof. Put

φ_{∗}(b) = φ(e^{b}) = ∑_{k} φ_{k}(b, ⋯, b).

Here φ : BC[1] → C′[1] is a homomorphism which is φ_{k} on B_{k}C[1]. L_{∞} case is similar.

Using the fact that φ̂ is a chain map we have :

Lemma 2.2.4. If b ∈ M̃C(C)(R) then φ_{∗}(b) ∈ M̃C(C′)(R).

We also have the following :

Lemma 2.2.5. If b ∼ b′ and if φ_{k} is homotopic to φ′_{k} then φ_{∗}(b) ∼ φ′_{∗}(b′).

Proof. Let H : C → C′ ⊗ R[t, dt] be as in Definition 2.1.8. It induces H̃ : C ⊗ R[t, dt] → C′ ⊗ R[t, dt] as follows. We put H_{k}(v_{1}, ⋯, v_{k}) = H_{k}^{1}(v_{1}, ⋯, v_{k}) + H_{k}^{2}(v_{1}, ⋯, v_{k})dt, where H_{k}^{i} : B_{k}C[1] → C′[1] ⊗ R[t]. We extend H_{k}^{i} to B_{k}(C[1] ⊗ R[t]) → C′[1] ⊗ R[t] in an obvious way and denote it by the same symbol. Let x_{i} = P_{i}(t) + Q_{i}(t)dt. Then H̃_{k}(x_{1}, ⋯, x_{k}) = P (t) + Q(t)dt where

P (t) = H_{k}^{1}(P_{1}(t), ⋯, P_{k}(t)),
Q(t) = H_{k}^{2}(P_{1}(t), ⋯, P_{k}(t))

+ ∑_{i}(−1)^{deg′ P_{1}+··· deg′ P_{i−1}+1}H_{k}^{1}(P_{1}(t), ⋯, Q_{i}(t), ⋯, P_{k}(t)).

It is easy to check that H̃ is an A_{∞} homomorphism. Let b̃ ∈ M̃C(C ⊗ R[t, dt]) such that Eval_{t=0} b̃ = b, Eval_{t=1} b̃ = b. (Definition 2.2.2). Now H̃_{∗}(b̃) ∈ M̃C(C′ ⊗ R[t, dt]) and Eval_{t=0} H̃_{∗}(b̃) = φ_{∗}(b), Eval_{t=1} H̃_{∗}(b̃) = φ′_{∗}(b).

By Lemmata 2.2.4 and 2.2.5 we obtain a map φ_{∗}(R) : MC(C)(R) → MC(C′)(R). It is easy to see that the following diagram commutes for each morphism ψ : R → R′ in {f. d. Alg. /R}.

MC(C)(R) ⟶^{ψ_{∗}} MC(C)(R′)
φ_{∗}↓  ↓^{φ_{∗}}
MC(C′)(R) ⟶^{ψ_{∗}} MC(C′)(R′)

Diagram 9

Commutativity of Diagram 9 implies that φ_{∗} is a natural transformation of Maurer-Cartan functors : MC(C) → MC(C′). Moreover Lemma 2.2.5 implies that φ_{∗} : MC(C) → MC(C′) depends only on homotopy class of φ. The following theorem follows immediately.

Theorem 2.2.2. If C is homotopy equivalent to C′ then Maurer-Cartan functor MC(C) is equivalent to MC(C′).

Remark 2.2.1. Theorem 2.2.2 in the cases of differential graded algebra and of differential graded Lie algebra is due to Goldman-Milson [37, 38]. Its generalization to A_{∞} or L_{∞} algebra seems to have been a folklore and was quoted by several authors (without proof). (For example by Kontsevitch [69].) We give its rigorous proof here, assuming Theorems 2.1.1, 2.1.2, 2.2.1 which will be proved in the final version of [33].

Remark 2.2.2. We can state Theorem 2.2.2 in more functorial way as follows. Let {A_{∞} alg. /R}/ homotopy be the category whose object is the set of all homotopy equivalence classes of A_{∞} algebras over R and morphisms are homotopy class of A_{∞} homomorphisms ^{2}. Let

2 There exists a set theoretical trouble to say “set of all homotopy equivalence classes of A_{∞} algebras”. We can go around it by introducing universe in the same way as [13].

𝔉𝔲𝔫𝔨({f. d. Alg. /R}, {Sets}) be the category whose objects are the sets of all equivalence classes of covariant functors from {f. d. Alg. /R} to {Sets}. Then MC induces a functor : {A_{∞} alg. /R}/ homotopy → 𝔉𝔲𝔫𝔨({f. d. Alg. /R}, {Sets}).

2.3 Canonical model, Kuranishi map, and moduli space.

In this section, we apply Theorem 2.2.2 to construct a versal formal moduli space representing the Maurer-Cartan functor. Theorem 2.2.2 is useful to construct a moduli space because it enables us to replace a given A_{∞} algebra to another one which is homotopy equivalent to the original one but is easier to handle. A good representative of each homotopy class for our purpose is a canonical one, which we define below.

Definition 2.3.1. An A_{∞} (or L_{∞}) algebra (C, 𝔪_{∗}) is said to be canonical if 𝔪_{1} = 0.

Remark 2.3.1. In [69] Kontsevich called the same notion “minimal” L_{∞} algebra. In the case of differential graded algebra the name “minimal” was used by D. Sullivan in [101] for an important notion which is different from Definition 2.3.1. This is the reason we do not call it “minimal” but canonical. The name canonical may be justified by Proposition 2.3.1 below.

We say an A_{∞} (or L_{∞}) homomorphism φ : C → C′ is an isomorphism if there exists an A_{∞} (or L_{∞}) homomorphism φ′ : C′ → C such that the compositions φ′ ∘ φ and φ ∘ φ′ are equal to identity. Here the identity A_{∞} homomorphism id_{∗} is defined by id_{1} = id and id_{k} = 0 for k ≥ 2.

Proposition 2.3.1. A homotopy equivalence between canonical A_{∞} (or L_{∞}) algebras is an isomorphism.

Proof. The condition 𝔪_{1} = 0 implies that the 𝔪_{1} cohomology of C is isomorphic to C itself. Since homotopy equivalence φ_{∗} induces an isomorphism on 𝔪_{1} cohomology it follows that φ_{1}; C[1] → C[1] is an isomorphism. We can then easily prove that φ̂ : BC[1] → BC[1] is an isomorphisms. The converse of it is a cochain map and is a coalgebra map. Hence there exists ψ_{k} such that ψ̂ = φ̂^{−1}. ψ_{k} is the inverse of φ, as required.

Hereafter in this section we assume that R is a field of characteristic 0. We also assume that H(C, 𝔪_{1}) is finite dimensional.

Theorem 2.3.1. There exists a canonical A_{∞} algebra C_{can} homotopy equivalent to a given A_{∞} algebra C. The same holds for L_{∞} algebra.

Remark 2.3.2. Theorem 2.3.1 was first proved in [54]. See also [46, 78]. There might be some people who found it independently. (For example the author heard of a talk by A. Polishchuk discussing the same theorem in 1998 January at the Winter school held in Harvard University.) The proof below is similar to one by Kontsevich-Soibelman [70], and also to [33] (2000 December version) §A6.

Proof. The argument is similar to one we explained in §1.5. We prove the case of L_{∞} algebra to minimize the overlap with the argument of §1.5. Let C be an L_{∞} algebra. We put C_{can}^{k} = H^{k}(C, 𝔪_{1}). We first need an analogue of Theorem 1.4.1. Here we need to use the fact that our coefficient ring is a field. We have

Lemma 2.3.1. There exists a linear subspaces H^{k} ⊆ C^{k} projections Π_{H^{k}} : C^{k} → H^{k} and R linear maps G_{k} : C^{k} → C^{k−1} such that

G_{k+1} ∘ 𝔪_{1} + 𝔪_{1} ∘ G_{k} = 1 − Π_{H^{k}}.  (2.19)

Proof. We put Z_{k} = Ker 𝔪_{1} : C^{k} → C^{k+1}. Let H^{k} ⊆ Z_{k} be a linear subspace such that the restriction of the projection : Z_{k} → C_{can}^{k} = H^{k}(C, 𝔪_{1}) to H^{k} is an isomorphism. We put B^{k} = Im 𝔪_{1} : C^{k−1} → C^{k}. Then H^{k} ⊕ B^{k} = Z_{k}. We also choose I^{k} ⊆ C^{k} such that Z_{k} ⊕ I^{k} = C^{k}. It is easy to see that 𝔪_{1} induces an isomorphism I^{k−1} → B^{k}. Let G^{k} : B^{k} → I^{k−1} be an inverse of it. We extend G^{k} to C^{k} so that G^{k} = 0 on H^{k} and on I^{k}. It is easy to check (2.19).

We call G_{k} a propagator. We remark that in §1.4 we constructed a similar operator (See (1.13)).

Now we consider the set of trees which satisfies a bit milder condition than Condition 1.5.1. Namely we consider the following :

Condition 2.3.1. Γ satisfies (1),(3),(4),(5) of Condition 1.5.1 and (2)′ If v ∈ Vertex∫ (Γ) then ♯∂_{target}^{−1}(v) ≥ 2, ♯∂_{source}^{−1}(v) = 1.

We denote by Gr_{k} the set of all oriented graphs Γ satisfying Condition 2.3.1. (We do not take ribbon structure here, since we are studying L_{∞} algebra and not A_{∞} algebra.) We use the notations in §1.4. Remark 2.3.3. The relation of trees to A_{∞} or L_{∞} algebra had been known to algebraic topologists for a long time. See for example [11].

We first put φ_{1} = id and \overline{𝔪}_{1} = 0, and are going to define

\overline{𝔪}_{Γ} : E_{k}C_{can}[1] → C_{can}[1],  φ_{Γ} : E_{k}C_{can}[1] → C[1]  (2.20)

for Γ ∈ Gr_{k} inductively on k. We will then put : \overline{𝔪}_{k} = ∑_{γ∈Gr_{k}} \overline{𝔪}_{Γ}, φ_{k} = ∑_{γ∈Gr_{k}} φ_{Γ}.

Now let us assume that (2.20) is defined for Γ ∈ Gr_{ℓ}, ℓ < k. Let Γ ∈ Gr_{k}. Let v_{last} be its last vertex. Let e_{last} be the unique edge such that ∂_{target}(e_{last}) = v_{last}. We remove [0, 1]_{e_{last}} together with its two vertices from |Γ|. Then |Γ|\[0, 1]_{e_{last}} is a union ∪|Γ_{i}| of several elements Γ_{i} ∈ Gr_{k_{i}} with ∑_{i=1}^{ℓ} k_{i} = k. We remark that since we are using graph Γ which does not have a particular ribbon structure, there is no canonical way to order Γ_{1}, …, Γ_{ℓ}. So the construction below should be independent of the order.

Let x_{i} ∈ C_{can}[1]. Let 𝔖_{k} be the group of all permutation of k numbers {1, ⋯, k}. We put

y_{1,σ} = \overline{φ}_{Γ_{1}}(x_{σ(1)}, ⋯, x_{σ(k_{1})}), ⋯, y_{ℓ,σ} = \overline{φ}_{Γ_{ℓ}}(x_{σ(k−k_{ℓ}+1)}, ⋯, x_{σ(k)}),

and then

\begin{aligned}\overline{φ}_{Γ}(x_{1} ⊗ ⋯ ⊗ x_{k}) & = − ∑_{σ∈𝔖_{k}} ±\frac{1}{k_{1}! ⋯ k_{ℓ}!}G (𝔪_{ℓ}(y_{1,σ}, y_{2,σ}, ⋯, y_{ℓ,σ})) , \\ \overline{𝔪}_{Γ}(x_{1} ⊗ ⋯ ⊗ x_{k}) & = − ∑_{σ∈𝔖_{k}} ±\frac{1}{k_{1}! ⋯ k_{ℓ}!}Π_{H} (𝔪_{ℓ}(y_{1,σ}, y_{2,σ}, ⋯, y_{ℓ,σ})) , \end{aligned}

where ± is as in (2.3) and G is the homomorphism in Lemma 2.3.1. We now calculate

\begin{aligned}∑_{Γ∈Gr_{k}} & 𝔪_{1}(φ_{Γ}(x_{1} ⊗ ⋯ ⊗ x_{k})) \\ & = − ∑_{Γ∈Gr_{k}} ∑_{σ∈𝔖_{k}} ±𝔪_{1}(G (𝔪_{ℓ}(y_{1,σ}, y_{2,σ}, ⋯, y_{ℓ,σ}))) \\ & = ∑_{σ∈𝔖_{k}} ±𝔪_{ℓ}(y_{1,σ}, y_{2,σ}, ⋯, y_{ℓ,σ}) \\ & − ∑_{Γ∈Gr_{k}} ∑_{σ∈𝔖_{k}} ±G(𝔪_{1}(𝔪_{ℓ}(y_{1,σ}, y_{2,σ}, ⋯, y_{ℓ,σ}))). \end{aligned}  (2.21)

We are going to calculate the last line of (2.21) using L_{∞} relation of 𝔪. We have

\begin{matrix}− & 𝔪_{1}(𝔪_{ℓ}(y_{1,σ}, y_{2,σ}, ⋯, y_{ℓ,σ})) \\ = ∑_{m>1} ∑_{n_{1}+···+n_{m}=ℓ} ∑_{μ∈𝔖_{m}} ±𝔪_{m}(𝔪_{n_{1}}(y_{μ(1),σ}, ⋯), ⋯, 𝔪_{n_{m}}(⋯, y_{μ(ℓ),σ}))). \end{matrix} (2.22)

We let φ̂ : EC_{can}[1] → EC[1] be the coalgebra homomorphism induced by φ_{k} and \overline{δ} : EC_{can}[1] → EC_{can}[1] be the coderivation induced by \overline{𝔪}_{k}. The coderivation δ : EC[1] → EC[1] is induced by 𝔪_{∗}. We now prove the following lemma.

Lemma 2.3.2. We have δ ∘ φ̂ = φ̂ ∘ \overline{δ} and \overline{δ} ∘ \overline{δ} = 0.

Proof. We prove the equalities on 𝔊^{k}EC_{can}[1] = ⊕_{i≤k}E_{i}C_{can}[1] by induction on k. The case k = 1 is obvious. Let x ∈ E_{k}C_{can}[1]. By (2.21), (2.22) we have

𝔪_{1}(φ_{k}(x)) = − ∑_{ℓ>2} \overline{𝔪}_{ℓ}(φ̂(x)) + (G ∘ (𝔪 − 𝔪_{1}) ∘ δ ∘ φ̂)(x).  (2.23)

Here 𝔪 − 𝔪_{1} : EC[1] → C[1] is an operator which is zero on E_{1}C[1] and is 𝔪_{ℓ} on E_{ℓ}C[1], ℓ > 2. We want to apply induction hypothesis to calculate (δ ∘ φ̂)(x). We remark the following :

Sulemma 2.3.1. If Lemma 2.3.2 holds on 𝔊^{k−1}EC_{can}[1] then δ∘φ̂ = φ̂∘\overline{δ} as an equality of homomorphisms : 𝔊^{k}EC_{can}[1]/𝔊^{1}EC_{can}[1] → 𝔊^{k}EC[1]/𝔊^{1}EC[1].

The proof of sublemma is easy and is omitted.

Since 𝔪 − 𝔪_{1} is zero on 𝔊^{1}EC[1] it follows from Sublemma and induction hypothesis that

(G ∘ (𝔪 − 𝔪_{1}) ∘ δ ∘ φ̂)(x) = (G ∘ (𝔪 − 𝔪_{1}) ∘ φ̂ ∘ \overline{δ})(x).

It follows from definition that G ∘ (𝔪 − 𝔪_{1}) ∘ φ̂ = φ̂. We thus obtain δ ∘ φ̂ = φ̂ ∘ \overline{δ} on 𝔊^{k}EC_{can}[1]. Thus by induction δ ∘ φ̂ = φ̂ ∘ \overline{δ} holds.

The second formula \overline{δ} ∘ \overline{δ} = 0 follows from the first one as follows. We have

φ̂ ∘ \overline{δ} ∘ \overline{δ} = δ ∘ δ ∘ φ̂ = 0.  (2.24)

Since φ_{1} is an isomorphism and φ̂ preserves the filtration 𝔊 it follows that φ̂ is injective. Hence (2.24) implies \overline{δ} ∘ \overline{δ} = 0.

We thus constructed \overline{𝔪}_{k} : E_{k}C_{can}[1] → C_{can}[1], φ_{k} : E_{k}C_{can}[1] → C[1]. It is immediate from definition that φ_{1} induces an isomorphism on 𝔪_{1} cohomology. Therefore, by Theorem 2.1.2, φ_{k} is a homotopy equivalence of L_{∞} algebras. The proof of Theorem 2.3.1 is now complete.

We next use Theorem 2.3.1 (and Theorem 2.2.2) to construct versal family of deformations. We assume that the cohomology group H^{∗}(C, 𝔪_{1}) is finite dimensional. (We recall that we assume R to be a field of characteristic zero.) We replace C by a canonical one C_{can} using Theorem 2.3.1. Since the 𝔪_{1} cohomology of C_{can} is isomorphic to C_{can} itself, it follows that C_{can}^{k} is finite dimensional. Let e_{i}, i = 1, ⋯, b_{1} be a basis of C_{can}^{1}[1] and f_{i}, i = 1, ⋯, b_{2} be a basis of C_{can}^{2}[1]. We define elements P_{i} ∈ R[[X_{1}, ⋯, X_{b_{1}}]], i = 1, ⋯, b_{2} by

∑_{i=1}^{b_{2}} P_{i}(X_{1}, ⋯, X_{k})f_{i} = 𝔪 (exp(X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}})) .  (2.25)

Here exp(X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}}) is as in (2.13). It is easy to see that P_{i} is well-defined as a formal power series.

Definition 2.3.2. We define a pro {f. d. Alg. /R} object 𝔎_{C_{can}} by

𝔎_{C_{can}} ≅ \frac{R[[X_{1}, ⋯, X_{b_{1}}]]}{(P_{1}, ⋯, P_{b_{2}})}.

We call P_{1}, ⋯, P_{b_{2}} the formal Kuranishi map.

Lemma 2.3.3. If two A_{∞} (or L_{∞}) algebras C_{can} and C′_{can} are homotopy equivalent then 𝔎_{C_{can}} is isomorphic to 𝔎_{C′_{can}} as R algebras.

Proof. Let e′_{i} and f′_{j} be basis of C_{can}^{′0}[1], C_{can}^{′1}[1] respectively. We define F_{j}(X_{1}, ⋯, X_{b_{1}}), j = 1, ⋯, b_{1} by

\begin{aligned}F_{1}(X_{1}, ⋯, X_{b_{1}})e′_{1} & + ⋯ + F_{b_{1}}(X_{1}, ⋯, X_{b_{1}})e′_{b_{1}} \\ & = φ_{∗}(X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}}). \end{aligned}  (2.26)

here φ_{∗} is defined by e^{φ_{∗}x} = φ̂(e^{x}). Since φ_{1} : C_{can}^{1}[1] → C_{can}^{′1}[1] is a linear isomorphism we may choose f′_{j} so that f′_{j} = φ_{1}(f_{j}). It is easy to see that F^{∗} induces an R algebra isomorphism : R[[X′_{1}, ⋯, X′_{b_{1}}]]/(P′_{1}, ⋯, P′_{b_{2}}) → R[[X_{1}, ⋯, X_{b_{1}}]]/(P_{1}, ⋯, P_{b_{2}}).

Therefore by Theorem 2.3.1 we can define 𝔎_{C} = 𝔎_{C_{can}} for any A_{∞} (or L_{∞}) algebra C with finite dimensional 𝔪_{1} cohomology.

We put (C_{can})_{𝔎_{Ccan}} = C_{can}^{0}[1] ⊗_{R} 𝔎_{C_{can}} and define : 𝔟 = ∑ X_{i}e_{i} ∈ (C_{can}^{0})_{𝔎_{Ccan}} . Then we have 𝔪(e^{𝔟}) = 0 ∈ (C_{can}^{1})_{𝔎_{Ccan}} . Here we remark that 𝔪(e^{𝔟}) is an infinite sum ∑_{k} 𝔪_{k}(𝔟, ⋯, 𝔟) (in L_{∞} case we divide the terms by k!). But it converges in (C_{can}^{0})_{𝔎_{Ccan}} with respect to the topology induced by the non Archimedean valuation on the formal power series ring. (This means nothing but the infinite sum ∑_{k} 𝔪_{k}(𝔟, ⋯, 𝔟) makes sense as a formal power series.)

Thus we obtain an A_{∞} (or L_{∞}) algebra ((C_{can})_{𝔎_{Ccan}} , 𝔪^{𝔟}).^{3} The next lemma implies that this deformation is complete.

Lemma 2.3.4. Let R be a finite dimensional local R algebra and b ∈ M̃C(C_{can})(R). Then there exists an R algebra homomorphism ψ : 𝔎_{C_{can}} → R such that ψ(𝔟) = b.

Proof. We have a polynomials R_{1}(Y_{1}, ⋯, Y_{m}), ⋯, R_{N}(Y_{1}, ⋯, Y_{m}) of m variables such that R ≅ R[Y_{1}, ⋯, Y_{m}]/(R_{1}, ⋯, R_{N}) ≅ R[[Y_{1}, ⋯, Y_{m}]]/(R_{1}, ⋯, R_{N}). Let us write by 𝔞 the ideal generated by R_{1}, ⋯, R_{N}. We write

b ≡ F_{1}(Y_{1}, ⋯, Y_{m})e_{1} + ⋯ + F_{b_{1}}(Y_{1}, ⋯, Y_{m})e_{b_{1}} mod 𝔞.

3 In other words we have a deformation of C parametrized by a formal scheme Spec 𝔎_{C_{can}}.

Here e_{i} are basis of C_{can}^{1}. We define ψ̃ : R[[X_{1}, ⋯, X_{b_{1}}]] → R[[Y_{1}, ⋯, Y_{m}]] by ψ̃(X_{i}) = F_{i}(Y_{1}, ⋯, Y_{m}). Then ψ̃(X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}}) = b mod 𝔞. Therefore

ψ̃(δ(exp(X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}}))) = δ(e^{b}) = 0.

Hence, by definition, ψ̃ induces a homomorphism ψ : 𝔎_{C_{can}} → R.

We remark that the formal Kuranishi map P_{i} has no term of degree ≤ 1. It follows that the Zariski tangent space T_{0} Spec 𝔎_{C} can be identified with C_{can}^{1} = H^{1}(C; 𝔪_{1}). The Kodaira-Spencer map then is an identity. This fact together with Lemma 2.3.4 is a formal scheme analogue of Theorem 1.6.1.

We now prove Theorem 1.6.1. Let us take C^{∗} = Ω^{0,∗}(M; End(E)) and 𝔪_{k} is induced by \overline{∂}_{E} and ∘ by Lemma 2.1.3. Then by Theorem 2.3.1 we have \overline{𝔪}_{k} : B_{k}C_{can}[1] → C_{can}[1], φ_{k} : B_{k}C_{can}[1] → C[1], where C_{can}^{k} = Ext^{k}(E, E).

Proposition 2.3.2.

∥\overline{𝔪}_{k}(x_{1}, ⋯, x_{k})∥ ≤ C^{k}∥x_{1}∥ ⋯ ∥x_{k}∥,  (2.27)
∥φ_{k}(x_{1}, ⋯, x_{k})∥ ≤ C^{k}∥x_{1}∥ ⋯ ∥x_{k}∥,  (2.28)

where C is independent of k.

Here the norm in the right hand side of (2.28) is L_{ℓ}^{2} norm for any fixed ℓ. (The constant C in (2.28) may depend on ℓ.) The norm on C_{can} can be defined uniquely up to equivalence since C_{can} is finite dimensional.

We omit the proof of proposition since it is straight forward check using Properties (A),(B) we stated during the proof of Lemma 1.5.2 and the definition of \overline{𝔪}_{k} and φ_{k} given during the proof of Theorem 2.3.1. (In other words the proof is a analogue of the proof of Lemma 1.5.2.)

Proposition 2.3.2 implies that the formal Kuranishi map P : Ext^{1}(E, E) → Ext^{2}(E, E) actually converges in a neighborhood of the origin. We put 𝔛_{Kura} = P^{−1}(0) in the sense of Example 1.6.1. Then b̃ determines deformations parametrized by 𝔛_{Kura} as follows. We define

B_{Kura}(X_{1}, ⋯, X_{ℓ}) = φ (exp(b̃))

= ∑_{k} φ_{k}(X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}}, ⋯, X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}}). (2.29)

It is a formal power series of X_{i} with values in Ω^{0,1}(M, End(E)). By using (2.28) we find that there exists an open neighborhood U_{ℓ} of 0 in Ext^{1}(E, E) where (2.29) converges in L_{ℓ}^{2} sense for (X_{1}, ⋯, X_{b_{1}}) ∈ U. It is easy to see that B_{Kura} : U → Ω^{0,1}(M; End(E)) defines a deformation of E.^{4} We already showed that Kodaira-Spencer map is an isomorphism for this family. Let us prove the completeness. The proof is similar to the proof of Lemma 2.3.4, except we need to discuss convergence. Let ψ_{k} : B_{k}Ω^{0,∗}(M, End(E))[1] → Ext^{∗}(E, E) be a homotopy inverse of φ_{k} : B_{k}C_{can}[1] → C[1]. We recall that the existence of ψ_{k} follows from Theorem 2.1.2. We use (2.28), and check a proof of Theorem 2.1.2 carefully and obtain an estimate

∥ψ_{k}(x_{1}, ⋯, x_{k})∥ ≤ C^{k}∥x_{1}∥ ⋯ ∥x_{k}∥.  (2.30)

Now let B : U → Ω^{0,1}(M; End(E)) be a holomorphic map which defines a deformation of E parametrized by 𝔛. Here 𝔛 is a germ of analytic subspace. We consider ψ_{∗} ∘ B : U → C_{can}^{1} = Ext^{1}(E, E). (Here ψ_{∗} is defined by ψ̂(e^{x}) = e^{ψ_{∗}(x)}.) The map ψ_{∗} ∘ B is defined first as a formal power series. We then use (2.30) and the fact that B is a convergent power series, to show that ψ_{∗} ∘B converges in a small neighborhood of origin. By replacing U if necessary we may assume that it converges on U.

Now, in the same way as the proof of Lemma 2.3.4, we find that composition of ψ_{∗} ∘ B and Kuranishi map P : Ext^{1}(E, E) → Ext^{2}(E, E) vanishes. Hence we obtain a ring homomorphism Φ : O_{0,+}/(P_{1}, ⋯, P_{b_{2}}) → ℑ_{𝔛,0} by f ↦ f ∘ ψ_{∗} ∘ B. By definition (Definition 1.6.4), Φ is a morphism 𝔛 → 𝔛_{Kura}.

To complete the proof of Theorem 1.6.1, we need to show that the pull back of the deformation B_{Kura} : U → Ω^{0,1}(M; End(E)) by Φ is isomorphic to B. The pull back of B_{Kura} by Φ is ϕ_{∗} ∘ ψ_{∗} ∘ B : U → Ω^{0,1}(M; End(E)).

We use the fact that ϕ ∘ ψ is homotopic to identity to show that ϕ_{∗} ∘ ψ_{∗} ∘ B is gauge equivalent to B as follows. By definition of homotopy (Definition 2.1.8), there exists H : Ω^{0,∗}(M; End(E)) → Ω^{0,∗}(M; End(E))⊗ ℂ[t, dt] such that Eval_{t=0} ∘H = id, Eval_{t=1} ∘H = ϕ ∘ ψ. We can check the proof of Theorem 2.1.2 carefully again to find that an estimate similar to (2.26) holds for H. Then we have a map H_{∗} ∘B : U → (Ω^{0,∗}(M; End(E))⊗ ℂ[t, dt])^{1}, after shrinking U if necessary. We put H_{∗} ∘B = b(t, x)+c(t, x)dt, where x ∈ U, b(t, x) ∈ Ω^{0,1}(M; End(E)), c(t, x) ∈ Γ(M; End(E)). We use Lemma 2.2.2 here. Namely we solve equation (2.18) to obtain g(t)(x) ∈ Γ(M; End(E)). We use the estimate of H to show that g(t)(x) converges if x is in a neighborhood of 0. Now g(1) gives an isomorphism from the deformation B to ϕ_{∗} ∘ ψ_{∗} ∘ B. The proof of Theorem 1.6.1 is now complete.

We now turn to the proof of Theorem 1.3.3 and its formal analogue. We remark that the ring homomorphism π : 𝔎_{C} → R induces a homomorphism

π : H^{0}(C_{𝔎_{C}} , 𝔪b̃ ; 1) → H^{0}(C, 𝔪_{1}).  (2.31)

4 We can modify the family so that B converges in C^{∞} topology. See [53]

Theorem 2.3.2. If (2.31) is surjective, then the functor 𝔉_{𝔎_{C}} : {f. d. Alg. /R} → {Sets} is equivalent to the Maurer-Cartan functor MC(C) : {f. d. Alg. /R} → {Sets}.

Before proving Theorem 2.3.2, let us explain why it is a formal version of Theorem 1.3.3. We consider the case when our canonical A_{∞} (or L_{∞}) algebra satisfies (2.27). Then, as in the proof of Theorem 1.6.1, we find a germ of analytic subvarieties 𝔛_{Kura} in U ⊆ ℂ^{N} defined by Kuranishi map P_{i} ∈ O_{0,+} and B_{Kura} = b̃ : U → (C_{can}[1])^{0} defines a holomorphic family of A_{∞} (or L_{∞}) algebra parametrized by 𝔛_{Kura}. The operators 𝔪b̃ ; 1 define a holomorphic family of chain complexes

C_{can}^{0} →^{𝔪b̃ ; 1} C_{can}^{1} →^{𝔪b̃ ; 1} C_{can}^{2} → ⋯  (2.32)

We remark that H^{0}(C_{𝔎_{C}} , 𝔪b̃ ; 1) is the set of the germs of holomorphic maps s : U → C_{can}^{0} such that 𝔪b̃ ; 1(s) vanishes on 𝔛_{Kura}. Hence the surjectivity of (2.31) implies that we have a local frame s_{1}, ⋯, s_{b_{0}} of the kernel of (2.32) on 𝔛_{Kura} in a neighborhood of 0. We remark that the pointwise 0th cohomology of (2.32) is semi continuous. Hence the existence of the sections s_{i} implies that the pointwise 0th cohomology of (2.32) is of constant rank in a neighborhood of 0 in 𝔛_{Kura}. This is the assumption of Theorem 1.3.3. It is also easy to see that the assumption of Theorem 1.3.3 implies the surjectivity of (2.31).

Note that the discussion above shows that Theorem 2.3.2 implies Theorem 1.3.3 and its analogue stated in §1.6.

Proof. We now prove Theorem 2.3.2. We have already constructed a natural transformation 𝔉_{𝔎_{C}} → MC(C). Lemma 2.3.4 implies that this transformation is surjective. So it suffices to show that it is injective. In other words, it suffices to show the following :

Lemma 2.3.5. We assume (2.31) is surjective. If φ, φ′ : 𝔎_{C} → R are R algebra homomorphism, and if φ(b̃) ∼ φ′(b̃) in M̃C(C)(R), then φ = φ′.

Proof. We may assume C is canonical. Then H^{0}(C, 𝔪_{1}) = C^{0}. Let l_{1}, ⋯, l_{b_{0}} be a generator of C^{0}. By the surjectivity of (2.31) we have its lift l̃_{i} to C_{𝔎_{C}}^{0} such that 𝔪b̃ ; 1(l̃_{i}) = 0 in C_{𝔎_{C}}^{1} . It is easy to see that 𝔪b̃ ; 1 generate C_{𝔎_{C}}^{0} as a 𝔎_{C} module. Hence 𝔪b̃ ; 1 : C_{𝔎_{C}}^{0} → C_{𝔎_{C}}^{1} is zero.

Let e_{i} be a generator of C^{1} and 𝔞 ⊆ R[[X_{1}, ⋯, X_{b_{1}}]] be the ideal generated by formal Kuranishi maps. We recall b̃ ≡ ∑ X_{i}e_{i} mod 𝔞

Super space and odd vector field - an alternative formulation of L_{∞} algebra - 58

By assumption, we have b̂ ∈ M̃C(C)(R⊗R[t, dt]) such that Eval_{t=0}(b̂) = φ(b̃), Eval_{t=1}(b̂) = φ′(b̃). We put b̂ = x(t) + y(t)dt. We remark that x(t) ∈ (C ⊗ R[t])^{1} and 𝔪(e^{x(t)}) = 0. Let x(t) = ∑ X_{i}(t)e_{i}. Then φ̃(X_{i}) ↦ X_{i}(t) defines a ring homomorphism φ̃ : 𝔎_{C} → R[t]. Its composition with obvious homomorphism Eval_{t=0} : R[t] → R and with Eval_{t=1} : R[t] → R are equal to φ and φ′ respectively.

The condition that b̂ is a Maurer-Cartan element implies

\frac{d}{dt}x(t) = −𝔪_{1}^{b(t)}(y(t)).  (2.33)

Since b(t) = φ̃(b̃) and since 𝔪b̃ ; 1 : C_{𝔎_{C}}^{0} → C_{𝔎_{C}}^{1} is zero it follows that the right hand side of (2.33) is zero. Namely x(0) = x(t).

We remark that φ̃ (∑ X_{i}e_{i}) = x(t) and X_{i} generates 𝔎_{C}. Therefore \frac{d}{dt} φ̃(t) = 0. Hence φ = φ′. The proof of Lemma 2.3.5 and Theorem 2.3.2 is now complete.

Let us prove Theorem 1.3.1 here. Let us take a versal family constructed in the proof of Theorem 1.6.1 above and write it (𝔎_{C}, B_{Kura}). We already know that it is complete. Let (U, B) be another family such that the Kodaira-Spencer map is surjective. We may take a submanifold V ⊂ U such that the restriction of Kodaira-Spencer map to T_{0}V is an isomorphism. Since (𝔎_{C}, B_{Kura}) is complete, we have a morphism (𝔉, Φ) : (V, B) → (𝔎_{C}, B_{Kura}). Using Lemma 1.3.2 and the fact the Kodaira-Spencer map of both deformations are isomorphisms, we find that d_{0}𝔉 : T_{0}V → T_{0}𝔛 is an isomorphism. Then, using the fact that V is a manifold, we can apply implicit function theorem to prove that 𝔉 is an isomorphism in a neighborhood of zero. (We remark that we do not need to assume that 𝔛 is a manifold in a neighborhood of zero to apply implicit function theorem here.) Hence the (V, B) is isomorphic to (𝔎_{C}, B_{Kura}) and is complete. Therefore, (U, B) is complete.

2.4 Super space and odd vector field - an alternative formulation of L_{∞} algebra -

According to [2, 65], we can rewrite the contents of §2.2, 2.3 using the terminology of formal geometry. In the case of L_{∞} algebra we need to use supper formal manifold and in the case of A_{∞} algebra we need a kind of “noncommutative geometry”. [21] uses the formalism of [2, 65]. It is more useful in L_{∞} case than A_{∞} case, since the case of super manifold is closer to usual geometry than noncommutative geometry. So we discuss only L_{∞} case here. Our argument is very brief here.

We start with explaining formal super manifold (only) in the case we concern with. Let us consider V = ⊕V^{k} a graded vector space. Let V_{ev}, V_{od} be the sum of their even or odd degree parts, respectively. We regard V_{ev} as the “Bosonic” part and V_{od} as the “Fermionic” part. This mean nothing but we regard ring of functions on it as ∏_{k,ℓ} S_{k}V_{ev}^{∗} ⊗ Λ^{ℓ}V_{od}^{∗}, where S_{k} denotes the kth symmetric power. We remark that ∏_{k,ℓ} S_{k}V_{ev}^{∗} ⊗ Λ^{ℓ}V_{od}^{∗} is a dual to EV. (Note EV is a direct sum hence only a finite sum is allowed. Its dual is a direct product.)

Definition 2.4.1. The ring of function on formal supper manifold C[1] is the dual vector space EC[1]^{∗} of EC[1].

Let e_{i} be a basis of C[1]^{ev} = C^{od} and f_{i} be a basis of C[1]^{od} = C^{ev}. An element of C is written as a finite sum : ∑ x^{i}e_{i} + ∑ y^{j}f_{j}. Then x^{i} and y^{j} be basis of the dual vector space C^{∗}. Hence element of EC[1]^{∗} can be written uniquely as

h = ∑_{k} ∑_{j_{1}<···<j_{k}} h_{j_{1},··· ,j_{k}}(x^{1}, x^{2}, ⋯)y^{j_{1}} ∧ ⋯ ∧ y^{j_{k}}  (2.34)

where h_{j_{1},··· ,j_{k}} ∈ R[[x^{1}, x^{2}, ⋯]]. The ring structure is determined by y^{i} ∧ y^{j} = −y^{j} ∧ y^{i} and the ring structure on R[[x^{1}, x^{2}, ⋯]].

Definition 2.4.2. A formal vector field on the formal super manifold C[1] is an expression

𝔙 = ∑ V_{i}^{x} \frac{∂}{∂x^{i}} + ∑ V_{i}^{y} \frac{∂}{∂y^{i}}

where V_{i} ∈ EC[1]^{∗}. For h as in (2.34) and 𝔙 as above, we put :

\begin{aligned}𝔙(h) = & ∑_{k,ℓ} ∑_{j_{1}<···<j_{k}} V_{ℓ}^{x} \frac{∂h_{j_{1},··· ,j_{k}}}{∂x^{i}} y^{j_{1}} ∧ ⋯ ∧ y^{j_{k}} \\ & + ∑_{k} ∑_{j_{1}<···<j_{k}} ∑_{i=1}^{k}(−1)^{i−1}V_{j_{i}}^{y}h_{j_{1},··· ,j_{k}}y^{j_{1}} ∧ ⋯ ∧ ŷ^{j_{i}} ∧ ⋯ ∧ y^{j_{k}}. \end{aligned} (2.35)

(2.35) is characterized by \frac{∂}{∂x^{i}} x^{j} = δ_{ij}, \frac{∂}{∂y^{i}} x^{j} = 0, \frac{∂}{∂y^{i}} y^{j} = δ_{ij}, \frac{∂}{∂x^{i}} y^{j} = 0, and 𝔙(hh′) = 𝔙(h)h′ + (−1)^{deg 𝔙 deg h}h𝔙(h′). Here we define degree by : deg \frac{∂}{∂x^{i}} = deg x^{i} = 0, − deg \frac{∂}{∂y^{i}} = deg y^{i} = 1.

It is easy to see the following :

Lemma 2.4.1. If deg 𝔙 and deg 𝔚 are even then there exists an super vector field [𝔙, 𝔚] such that [𝔙, 𝔚](h) = (𝔙𝔚 − 𝔚𝔙)(h). If deg Θ and deg Ξ are odd then there exists a super vector field {Θ, Ξ} such that {Θ, Ξ}(h) = (ΘΞ + ΞΘ)(h).

It is obvious that [𝔙, 𝔙] = 0 for super vector field 𝔙 of even degree. However {Θ, Θ} ≠ 0 in general for super vector field Θ of odd degree. Actually we have :

Lemma 2.4.2. We assume 2 is invertible on R. Then, supper vector field Θ of degree 1 satisfying {Θ, Θ} = 0 corresponds one to one to the L_{∞} structure on C[1].

Proof. Let Θ defines a derivation Θ : EC[1]^{∗} → EC[1]^{∗}. Its dual defines a coderivation δ : EC[1] → EC[1]. δδ = 0 is equivalent to {Θ, Θ} = 2Θ^{2} = 0. The lemma follows.

Let (C, 𝔪), (C′, 𝔪′) be L_{∞} algebras and Θ, Θ′ be odd vector fields corresponding to them by Lemma 2.4.2. Let φ_{k} be an L_{∞} homomorphism. It induces a coalgebra homomorphism φ̂ : EC[1] → EC′[1]. Hence its dual is an algebra homomorphism φ̂^{∗} : EC′[1]^{∗} → EC[1]^{∗}. That is a morphism of formal super manifold C[1] → C′[1]. Since φ̂ ∘ δ = δ ∘ φ̂ it follows that φ̂^{∗} ∘ Θ′ = Θ ∘ φ̂^{∗}. Thus, L_{∞} homomorphism will become a morphism of formal super manifold preserving odd vector fields on it. One may continue and translate various other operations of L_{∞} algebras to the language of super manifolds. We do not try to do it here.

Application to Mirror symmetry

3.1 Novikov ring and filtered A_{∞}, L_{∞} algebras.

In this chapter, we explain relations of the discussion of deformation theory in Chapter 1,2 to Mirror symmetry.

We explain a construction of [33] which associates an “A_{∞} algebra” to a Lagrangian submanifold of a symplectic manifold (satisfying some conditions we will explain later). (We review various notions of symplectic geometry we need at the beginning of §3.2.) Actually what we will associate to a Lagrangian submanifold is a slight modification of A_{∞} algebra, which we call filtered A_{∞} algebra. To define it we first need to define universal Novikov ring. We will discuss universal Novikov ring more in §3.5. Novikov introduced a kind of formal power series ring in [81] to study Morse theory of closed one form. It was applied by [51] etc. to infinite dimensional situation of Floer homology, (which may be regarded as a Morse theory of closed one form on loop space). Namely Floer homology which we will discuss in §3.3, is defined as a module over Novikov ring. In order to use the same ring independent of the symplectic manifold (and of its Lagrangian submanifold) we use a ring which we call universal Novikov ring. We define it below.

Let R be a commutative ring. We consider the formal sum x = ∑_{i} a_{i}T^{λ_{i}} satisfying the following conditions.

Condition 3.1.1. (1) a_{i} ∈ R, (2) λ_{i} ∈ ℝ, (3) λ_{i} < λ_{i+1}, (4) lim_{i→∞} = ∞.

Definition 3.1.1. The set of all formal sums x = ∑_{i} a_{i}T^{λ_{i}} satisfying Conditions 3.1.1 is called the universal Novikov ring and is written as Λ_{R,nov}. Λ_{R,nov} becomes a ring (R algebra) by an obvious definition of sum and multiplication. We replace (2) by λ_{i} ≥ 0. We then obtain a subring Λ_{R,nov,0}. We replace (2) by λ_{i} > 0. We then obtain an ideal Λ_{R,nov,+} of Λ_{R,nov}. (We omit R in case no confusion can occur.)

Novikov ring and filtered A_{∞}, L_{∞} algebras. 62

In case R is a field, Λ_{R,nov,0} is a local ring with maximal ring Λ_{R,nov,+}. We define a filtration 𝔉 on Λ_{nov} by

𝔉^{λ}Λ_{nov} = {x | x is as in (3.1) satisfying Condition 3.1.1 and λ_{i} ≥ λ.}

𝔉^{λ}Λ_{nov} is a filtration. Namely 𝔉^{λ}Λ_{nov} is a subabelian group (with respect to +) and 𝔉^{λ}Λ_{nov} · 𝔉^{λ′}Λ_{nov} ⊆ 𝔉^{λ+λ′}Λ_{nov}. We remark that 𝔉 induces a filtration on Λ_{nov,0} and Λ_{nov,+} Remark 3.1.1. We remark that Λ_{nov,0} is not a Noether ring, since the ascending sequence of ideals 𝔉^{1/i}Λ_{nov} does not stop. This fact makes harder to study algebra or module over it.

A filtered Λ_{nov,0} module is a Λ_{nov,0} module C together with filtration 𝔉^{λ}C such that 𝔉^{λ}Λ_{nov} · 𝔉^{λ′}C ⊆ 𝔉^{λ+λ′}C. We say that a Λ_{nov} module homomorphism φ : C → C′ is a filtered Λ_{nov,0} module homomorphism if φ(𝔉^{λ}C) ⊂ 𝔉^{λ}C′.

Filtration defines a metric on Λ_{nov} and a module on it by d(x, y) = exp (− inf{λ | x − y ∈ 𝔉^{λ}C}) . Λ_{nov}, Λ_{nov,0} and Λ_{nov,+} are complete with respect to this metric. From now on we assume all filtered Λ_{nov,0} modules are complete. We also assume that all filtered A_{∞} (or L_{∞}) algebras are completion of free Λ_{nov,0} module.

If C, C′ are filtered Λ_{nov,0} module, we define a filtration on its tensor product C ⊗_{Λ_{nov,0}} C′ by

𝔉^{λ}(C ⊗_{Λ_{nov,0}} C′) = ⋃_{μ} 𝔉^{μ}C ⊗_{Λ_{nov,0}} 𝔉^{λ−μ}C′.

The tensor product C ⊗_{Λ_{nov,0}} C′ is not complete with respect to the metric induced by this filtration. We denote by C ˆ⊗_{Λ_{nov,0}}C′ the completion.

Graded filtered Λ_{nov,0} module is defined in an obvious way. Let C be a graded filtered Λ_{nov,0} module. We consider

B_{k}C[1] = C[1] ˆ⊗_{Λ_{nov,0}} ⋯ ˆ⊗_{Λ_{nov,0}}C[1]_{k times}

Let B̂C[1] be the completion of the direct sum ⊕_{k}B_{k}C[1]. E_{k}C[1] and ÊC[1] are defined by taking its submodule which is invariant under the action of 𝔖_{k}.

They are formal coalgebra. Here formal coalgebra is defined replacing ⊗ by ˆ⊗ in the definition of coalgebra. We can define coderivation and cohomomorphism for formal coalgebra in the same way. (We assume them to be filtered.) The following analogy of Lemmata 2.1.2 2.1.5 holds.

Lemma 3.1.1. Let f_{k} : B_{k}C[1] → C[1], k = 0, 1, ⋯, be a sequence of filtered homomorphisms of degree 1. Then there exists a unique coderivation δ : B̂C[1] → B̂C[1] whose restriction to B_{k}C[1] is f_{k}.

Let φ_{k} : B_{k}C[1] → C′[1], k = 0, 1, ⋯, be a sequence of filtered homomorphisms of degree 0. We assume φ_{0}(Λ_{nov,0}) ⊆ 𝔉^{λ_{0}}B_{1}C′[1] for some positive λ_{0}. Then there exists a coalgebra homomorphism φ̂ : B̂C[1] → B̂C[1] whose Hom(B_{C}k[1], B_{1}C′[1]) component is φ_{k}.

The same statement holds when we replace B by E.

We remark that we include f_{0} and φ_{0} here but not in Lemmata 2.1.2 2.1.5.

Proof. We prove A_{∞} case only. We put

f̂_{0}(x_{1} ⊗ ⋯ ⊗ x_{k})
= φ_{0}(1) ⊗ x_{1} ⊗ ⋯ ⊗ x_{k} + (−1)^{deg′ x_{1}}x_{1} ⊗ φ_{0}(1) ⊗ x_{2} ⊗ ⋯ ⊗ x_{k}
+ ⋯ + (−1)^{deg′ x_{1}+···+deg′ x_{k}}x_{1} ⊗ ⋯ ⊗ x_{k} ⊗ φ_{0}(1).

We define f̂_{k}, k ≥ 1 in the same way as the proof of Lemma 2.1.2. Then we can prove that δ = f̂_{0} + ⋯ + f̂_{k} + ⋯ converges by using the fact f_{i} preserves filtration.

We next define φ̂. We put

e^{φ(1)} = ∑_{k} φ(1) ⊗ ⋯ ⊗ φ(1)_{k times}.

By assumption it converges in B̂C′[1]. Now we put

φ̂(x) = ∑_{k} ∑_{a} e^{φ(1)} ⊗ \overline{φ}(x_{a}^{k;1}) ⊗ e^{φ(1)} ⊗ ⋯ ⊗ e^{φ(1)} ⊗ \overline{φ}(x_{a}^{k;k}) ⊗ e^{φ(1)}.

Here \overline{φ} is φ_{k} on B_{k}C[1] k ≠ 0 and is zero on B_{0}C[1]. It is easy to check that φ̂ converges and is a coalgebra homomorphism.

Remark 3.1.2. B̂C[1] has another filtration different from 𝔉^{λ}B̂C[1]. Namely we put 𝔊^{k}B̂C[1] = ⊕_{i≤k}B_{i}C[1]. We call the filtration 𝔉 the energy filtration and 𝔊 the number filtration. In case f_{0} ≠ 0 or φ_{0} ≠ 0, f̂ or φ̂ does not preserve the number filtration. They preserve energy filtration. We also remark that B̂C[1] is not complete with respect to the number filtration.

Definition 3.1.2. A structure of filtered A_{∞} algebra on filtered graded Λ_{nov,0} module C is a series of filtered homomorphisms 𝔪_{k} : B_{k}C[1] → C[1], k = 0, 1, ⋯ of degree 1 such that δδ = 0 where δ = ∑ 𝔪̂_{k} : B̂_{k}C[1] → C[1] is obtained by Lemma 3.1.1. We also assume that 𝔪_{0}(Λ_{nov,0}) ⊆ 𝔉^{λ_{0}}B_{1}C[1].

A sequence of homomorphisms φ_{k} : B_{k}C[1] → C′[1] is a filtered A_{∞} homomorphism between filtered A_{∞} algebras if φ_{0}(Λ_{nov,0}) ⊆ 𝔉^{λ_{0}}B_{1}C′[1] and the homomorphism φ̂ obtained by Lemma 3.1.1 satisfies δφ̂ = φ̂δ.

L_{∞} can be defined in a similar way.

We will explain how to modify the argument of the last chapter to our filtered situation later after we introduce our main example.

3.2 Review of a part of global symplectic geometry.

In this section, we review several points on global symplectic geometry, especially those related to pseudoholomorphic curve, which we need for our main construction. [52, 75, 76] are standard references of them.

A symplectic manifold is a pair (M, ω) where ω is a closed 2 form on M such that it is nondegenerate as an anti symmetric two form on T_{p}M for each p ∈ M. M is automatically even dimensional. Let 2n be its dimension. Then ω^{n} is nowhere vanishing 2n form on M and hence determines an orientation. A Lagrangian submanifold of a symplectic manifold (M, ω) is an n dimensional closed submanifold L such that ω|_{L} = 0. We remark that if the dimension of submanifold of M is strictly larger than n then the restriction of ω to L cannot vanish.

A typical example of symplectic manifold is a Kähler manifold. Especially if M is a projective variety that is a complex submanifold of ℂP^{n} then it is a symplectic manifold, whose symplectic structure is obtained as a pull back of the Fubini-Study form ω on ℂP^{n} which is defined by π^{∗}ω = −4\sqrt{−1}∂\overline{∂} log(|z^{0}|^{2} + ⋯ + |z^{n}|^{2}) where π : ℂ^{n+1} → ℂP^{n} is the projection.

Another important example of symplectic manifold is a cotangent bundle T^{∗}N of any smooth manifold N. The symplectic form on T^{∗}N is given by dθ. Here θ is a one form on T^{∗}N such that θ(X) = u(π_{∗}(X)) where X ∈ T_{u}T^{∗}N and π : T^{∗}N → N is the natural projection. If x^{1}, ⋯, x^{n} is a coordinate of N then element of T^{∗}N is written as p_{1}dx^{1} + ⋯ + p_{n}dx^{n}. Hence p_{i}, x^{j}, i, j = 1, ⋯, n is a coordinate of T^{∗}N. Using this coordinate our symplectic form ω on T^{∗}N is ω = dp_{1} ∧ dx^{1} + ⋯ + dp_{n} ∧ dx^{n}.

An example of Lagrangian submanifold of a projective variety is a submanifold consisting of real valued points. Namely let M ⊆ ℂP^{n} be a complex submanifold which is preserved by complex conjugataction τ. We assume L = {x ∈ M|τ(x) = x} is a submanifold of dimension n = dim M/2. Then we can show L is a Lagrangian submanifold.

There are also various examples of Lagrangian submanifolds in T^{∗}N. One is a conormal bundle T_{K}^{∗}N of a submanifold K of N. Here

T_{K}^{∗}N = {(x, u) ∈ T^{∗}M|x ∈ N, u|_{T_{x}N} = 0}.

In fact, if K is defined by equations x^{k+1} = ⋯ = x^{n} = 0 then T_{K}^{∗}N is defined by equations x^{k+1} = ⋯ = x^{n} = 0, p_{1} = ⋯ = p_{k} = 0. Hence ω zero on T_{K}^{∗}N.

Another example is a graph of a closed one form u which is defined as follows. For a one form u on N, we put G_{u} = {(x, u(x))|x ∈ N}. Let us define a diffeomorphism i : N → G_{u} by i(x) = (x, u(x)) . We can show that the pull back of θ by i is u itself. It follows that i^{∗}ω = du. Therefore G_{u} is a Lagrangian submanifold if and only if u is closed.

Symplectic geometry has a long history. There are many interesting results and their applications. However, for a long time, it seems that there had been only few results in symplectic geometry which is really of global nature. For example, the following question was open for a long time. Does there exists a pair of symplectic manifolds (M, ω) and (M, ω′) on a same manifold M such that [ω] = [ω′] ∈ H^{2}(M; ℝ) but there is no diffeomorphism φ : M → M with φ^{∗}ω′ = ω. The reason why such results were not known seems to me that there was basically no general technique which can be applied to study global symplectic geometry. Arnold in 1960’s formulates a series of conjectures which is related to global problem of symplectic geometry. (See for example [3].) Roughly speaking, those questions ask whether there exists “symplectic topology”. Around the beginning of 1980’ several works appeared which shows that such “symplectic topology” does exist and is extremely rich. Among those results, Gromov’s one in [42] is quite remarkable. In [42], Gromov introduced a new technique, pseudoholomorphic curve, to study global structure of symplectic manifold. Let us briefly review it here. Let (M, ω) be a symplectic manifold. An almost complex structure J : T M → T M is a tensor such that JJ = −1.

Definition 3.2.1. J is said to be compatible with ω if (1) ω(JX, JY) = ω(X, Y), (2) ω(X, JX) > 0 and (3) ω(X, JX) = 0 implies X = 0.

It is proved in [42] (see [75, 76]) that compatible almost complex structure always exists and the set of compatible almost complex structures is contractible ^{1}.

The idea by Gromov is to apply techniques of complex geometry to almost complex manifold (M, J) to get information of a symplectic manifold (M, ω).

There are basically two kinds of methods in complex geometry. One is to use holomorphic functions or holomorphic maps defined on M, and the other is to use holomorphic maps to M. We remark here, in case (M, J) is almost complex (namely in case J is not integrable), there are not so many holomorphic functions on M. Hence the first kinds of methods are hard to apply in our case of (M, J).

The important observation by Gromov is that, even in case when (M, J) is not integrable, there are many holomorphic maps to (M, J) from Riemann surface (complex one dimensional manifold). The basic reason of it is that any almost complex structure on real two dimensional manifold is automatically integrable.

The method then initiated by Gromov is to study moduli space of holomorphic maps from Riemann surface to (M, J) to get information of

1 It follows that the Chern classes of (the tangent bundle of) symplectic manifold is well-defined.

(M, ω). Gromov called holomorphic map from Riemann surface to an almost complex manifold, a pseudoholomorphic curve. The contractibility of the set of almost complex structures guarantees that any invariant obtained by using compatible almost complex structure is an invariant of symplectic manifold if it is independent of continuous change of compatible almost complex structures.

Using an existence of symplectic structure compatible to J, Gromov proved various compactness results of the moduli space of pseudoholomorphic maps, hence its fundamental cycle in principle defines such an invariant. Ruan [89] made this construction (which somehow was implicit in [42]) more explicit.

This invariant in turn was found to be an invariant of topological sigma model with target space M, which Witten [107] introduced in an informal way. The invariant obtained in this way is now called Gromov-Witten invariant. See [89, 90, 76, 34] more about it.

Our main concern here is its relative version. Namely we consider a Lagrangian submanifold L of M and study a map φ : D^{2} → M such that :

Condition 3.2.1. (1) φ is pseudoholomorphic. Namely J ∘ dφ = dφ ∘ j_{D^{2}}. Here j_{D^{2}} is the standard complex structure of D^{2}. (2) φ(∂D^{2}) ⊆ L.

Gromov [42] already studied moduli space of such φ to get information of Lagrangian submanifolds in ℂ^{n}. (For example he proved that there does not exist simply connected compact Lagrangian submanifold in ℂ^{n}.) Floer [22] used a similar idea to study problems of intersection of Lagrangian submanifolds especially to study the following problem due to Arnold [3].

Problem 3.2.1. Let L ⊂ M be a Lagrangian submanifold, and let φ : M → M be a Hamiltonian diffeomorphism (which we will define below). We assume L is transversal to φ(L). Then, under “some” condition, we have an estimate

♯L ∩ φ(L) ≥ ∑ rank H_{k}(L; ℤ_{2}).  (3.1)

Let us define Hamiltonian diffeomorphism. Let (M, ω) be a symplectic manifold and f be a function on it. There exists a vector field X_{f} such that ω(X_{f}, V) = df(V) holds for any vector V. X_{f} is called the Hamiltonian vector field. We now consider f : M × [0, 1] → ℝ. Let f_{t}(x) = f(x, t). It induces a one parameter family of vector fields X_{f_{t}}. We define a family of diffeomorphisms φ_{t} : M → M by

\frac{d}{dt}φ_{t}(x) = X_{f_{t}}(φ_{t}(x)), φ_{0}(x) = x.  (3.2)

Definition 3.2.2. φ : M → M is called a Hamiltonian diffeomorphism if there exists f_{t}(x) = f(x, t) such that φ_{1} = φ, and φ_{t} is defined by (3.2).

One can prove easily that Hamiltonian diffeomorphism is a symplectic diffeomorphism, namely φ^{∗}ω = ω.

Floer [22] proved (3.1) in case π_{2}(M, L) = 0. He used new homology theory which is now called Floer homology for this purpose. Let us briefly explain it here. Let L_{1}, L_{2} be two Lagrangian submanifolds such that π_{2}(M, L_{i}) = 0. We assume that L_{1} is transversal to L_{2}. We consider ℤ_{2} vector space CF (L_{1}, L_{2}) whose basis is identified to the intersection points p ∈ L_{1} ∩ L_{2}. Namely we put

CF (L_{1}, L_{2}) = ⨁_{p∈L_{1}∩L_{2}} ℤ_{2}[p].  (3.3)

Floer defined a degree of each [p] and defined a boundary operator on it as follows. Let p, q ∈ L_{1} ∩ L_{2}. We consider maps φ : D^{2} → M satisfying the following conditions. We put ∂D_{+}^{2} = {z ∈ ∂D| Im z > 0}, ∂D_{−}^{2} = {z ∈ ∂D| Im z < 0}.

Condition 3.2.2. (1) φ is pseudoholomorphic. Namely J ∘ dφ = dφ ∘ j_{D^{2}}. Here j_{D^{2}} is the standard complex structure of D^{2}. (2) φ(∂_{+}D^{2}) ⊂ L_{1}, φ(∂_{−}D^{2}) ⊂ L_{2}. (3) φ(−1) = p, φ(1) = q.

Let M̃(p, q; L_{1}, L_{2}) be the moduli space of all such maps φ. The group Aut(D^{2}; {±1}) of biholomorphic maps D^{2} → D^{2} preserving ±1 acts on M̃(p, q; L_{1}, L_{2}). This group is isomorphic to ℝ. We denote by M(p, q; L_{1}, L_{2}) the quotient space. Floer proved the following :

Theorem 3.2.1. Under the assumption π_{2}(M, L_{i}) = 0, there exists μ : L_{1} ∩ L_{2} → ℤ such that the following holds after taking a “generic perturbation^{2}” of the pseudoholomorphic curve equation J ∘ dφ = dφ ∘ j_{D^{2}}.

(1) M(p, q; L_{1}, L_{2}) is a smooth manifold of dimension μ(q) − μ(p) − 1.

(2) If μ(q) − μ(p) − 1 = 0, then M(p, q; L_{1}, L_{2}) consists of finitely many points.

(3) If μ(q) − μ(p) − 1 = 1, then M(p, q; L_{1}, L_{2}) can be compatcified to CM(p, q; L_{1}, L_{2}) which is a 1 dimensional manifold with boundary.

(4) In the situation of (3) the boundary of CM(p, q; L_{1}, L_{2}) is identified with

⋃_{r∈L_{1}∩L_{2},μ(r)=μ(q)+1} M(p, r; L_{1}, L_{2}) × M(r, q; L_{1}, L_{2}).  (3.4)

(Note that (3.4) is of finite order by (2).) We will not discuss the proof of this theorem which is now classical. μ is called the Maslov-Viterbo index.

2 We do not explain precise meaning of it in the article.

(4) can be explained by the following Figure.

Figure 3

Now the definition of Floer homology is as follows. We put deg[p] = μ(p) hence CF (L_{1}, L_{2}) is a graded ℤ_{2} vector space. We next define

⟨δ[p], [q]⟩ ≡ ♯M(p, q; L_{1}, L_{2}) mod 2

in case μ(q) − μ(p) − 1 = 0 and put

δ[p] = ∑⟨δ[p], [q]⟩[q].

δ is an operator of degree 1. We show

Corollary 3.2.1. δ ∘ δ = 0.

Proof. Let us calculate the coefficient of [q] in δδ([p]). We may write it as

∑_{r}⟨δ[p], [r]⟩⟨δ[r], [q]⟩.

It suffices to consider the case μ(q) − μ(p) − 1 = 1. Namely we can apply (4) of Theorem 3.2.1. We then have

∑_{r}⟨δ[p], [r]⟩⟨δ[r], [q]⟩

= ⋃_{r∈L_{1}∩L_{2},μ(r)=μ(q)+1} ♯(M(p, r; L_{1}, L_{2}) × M(r, q; L_{1}, L_{2}))

= ♯∂M(p, q; L_{1}, L_{2}) ≡ 0 mod 2,

since the order of the boundary of one dimensional compact manifold is even.

We thus define a Floer cohomology by

HF (L_{1}, L_{2}) = H(CF (L_{1}, L_{2}), δ).

Floer proved the following two properties of it.

Theorem 3.2.2. We assume π_{2}(M, L_{i}) = 0. If φ_{i} are Hamiltonian diffeomorphisms then HF (L_{1}, L_{2}) ≅ HF (φ_{1}L_{1}, φ_{2}L_{2}).

Theorem 3.2.3. We assume π_{2}(M, L) = 0. Then HF (L, L) ≅ H(L; ℤ_{2}).

It is easy from definition to see that

rank HF (L_{1}, L_{2}) ≥ ♯L_{1} ∩ L_{2}.  (3.5)

(3.1) follows from Theorems 3.2.2,3.2.2 and (3.2).

We remark that our discussion so far assumed π_{2}(M, L) = 0. After Floer, Oh [82] relaxed the condition π_{2}(M, L) = 0. His assumption is that Lagrangian submanifold is monotone and its minimal Maslov number is ≥ 3. We do not explain this condition.

Actually there is an example where (3.1) does not hold in general.

Example 3.2.1. Let us consider S^{2}. Any one dimensional submanifold of it is a Lagrangian submanifold. Let L be a circle which is in a small neighborhood of north pole. We can find easily a Hamiltonian diffeomorphism φ such that φ(L) ∩ L = ∅. On the other hand, H_{∗}(L) = H_{∗}(S^{1}) ≠ 0.

Therefore, something wrong should happen if we try to generalize the story of Floer to more general Lagrangian submanifold. For example, in the case of Example 3.2.1 it turns out that Floer homology is not defined for such a Lagrangian submanifold.

Thus finding a good condition for Floer homology to be defined is important for its application to global symplectic geometry. Later it was found that the same problem is also closely related to mirror symmetry. This is the main point of our story. So before stating the results (and conjectures) precisely, we first explain the rough story.

The author together with Oh, Ohta, Ono^{3} developed an obstruction theory for the well-definedness of the Floer homology of Lagrangian submanifold [33]. (Our projects starts around 1997 just after necessary analytic machinery is completed in [34].) We found that there are a series of obstructions which takes value in the cohomology of L such that if they all vanish then the Floer homology is well-defined.

However, Floer homology thus defined actually depends on various choices involved. Namely there exists a moduli space associated to a Lagrangian submanifold and Floer homology is well-defined as a family parametrized by this moduli space. The condition that the obstruction vanish is equivalent to the condition that this moduli space is nonempty. The algebraic machinery we need to establish it is one we developed in the last chapter (its filtered version precisely). Namely the moduli space parametrizing Floer homology is a moduli space representing appropriate Maurer-Cartan functor.

On the other hand, the author proposed to generalize the story of Lagrangian intersection Floer homology to the case when there are three or more Lagrangian submanifolds involved ^{4}. He then found that there appears an A_{∞} structure [23], in early 1990’s. Then Kontsevich proposed a conjecture that the A_{∞} structure on Floer homology of Lagrangian submanifold should be a “mirror” of a similar A_{∞} structure on sheaf cohomol-

3 Kontsevich gave us an important suggestion to start this research. 4 The author was inspired by the idea by Donaldson and Segal when he start this project.

From Lagrangian submanifold to A_{∞} algebra. 70

ogy of complex manifold (that is the A_{∞} algebra related to the complex Ω^{0,∗}(End(E)) introduced in Chapters 1 and 2). Namely Kontsevich proposed a homological mirror symmetry conjecture in [66, 69]. It roughly states that there are pairs of symplectic manifold M and complex manifold M^{∧} such that Lagrangian submanifold of M corresponds to a coherent sheaf on M^{∧} and Floer homology of Lagrangian submanifolds in M corresponds to the sheaf cohomology on M^{∧}. Moreover A_{∞} structure on Floer homology on M corresponds to Yoneda and Massey-Yoneda product on sheaf cohomology on M^{∧}.

We remark that homological mirror symmetry conjecture was found before “second string theory revolution”. Later Brane theory becomes important. Homological Mirror symmetry conjecture can then be regarded naturally as a correspondence of Branes and as a part of various dualities. After that and after Strominger-Yau-Zaslow’s important proposal [100] to construct mirror manifold by using dual torus fibration, several people began to be interested in homological mirror symmetry. Among them, Polishchuk-Zaslow [86] proved a part of it in the case of elliptic curve. (Kontsevich [66] discussed the case of elliptic curve earlier.) Just after that the author generalized it partially to the case of complex torus of higher dimension [27]. The main tool used in [27] is a result of [32] which calculates the A_{∞} structure of Floer homology in the case of cotangent bundle. Some more papers on mirror symmetry of Abelian variety appeared after that. There are other important works by P. Seidel and his coauthors (such as [96, 95, 94, 57]) which are also related to homological mirror symmetry conjecture.

While studying the case of complex tori, the author found various interesting and delicate phenomena happen in the study of Floer homology of Lagrangian submanifolds, its product (A_{∞}) structure, and also its family version ([31]). The study of Floer homology thus tied more with homological algebra and with deformation theory, which we explained in Chapters 1 and 2. Homological mirror symmetry conjecture now becomes more precise after being involved much in homological algebra and deformation theory. For example, we conjecture the coincidence of the two moduli spaces, one for deformation of Floer homology of Lagrangian submanifolds and the other for deformation of coherent sheaves or of vector bundles.

3.3 From Lagrangian submanifold to A_{∞} algebra.

Now, after explaining rough story and history, let us discuss the construction of filtered A_{∞} algebra associated to Lagrangian submanifolds. Here we consider the case we have only one Lagrangian submanifold L rather than a pair of Lagrangian submanifolds as in Floer’s case. (The case there are two or more Langrangian submanifolds are mentioned at the end of §3.4 and discussed in detail in [33, 30].)

The condition π_{2}(M, L) = 0 we assumed before implies that there exist no maps φ : (D^{2}, ∂D^{2}) → (M, L) satisfying Condition 3.2.1. In fact, since φ is zero homotopic it follows that ∫_{D^{2}} φ^{∗}ω = 0. We can easily show that if φ is pseudoholomorphic and is non constant then ∫_{D^{2}} φ^{∗}ω > 0. This is the basic reason why Theorem 3.2.2 holds. In other words, presence of pseudoholomorphic disk φ : (D^{2}, ∂D^{2}) → (M, L) deforms usual homology group H(L) to Floer homology group HF (L). (See §3.4).

Let us discuss moduli space of pseudoholomorphic disks satisfying Condition 3.2.1. Let β ∈ π_{2}(M, L). We use the following moduli space.

Definition 3.3.1. The moduli space M_{k+1}(L; β) is the set of all ∼ equivalence classes of pairs (φ, z⃗) where :

(1) φ : (D^{2}, ∂D^{2}) → (M, L) satisfies Condition 3.2.1. (2) The homotopy class of φ is β. (3) z⃗ = (z_{0}, ⋯, z_{k}) where z_{i} ∈ ∂D^{2}. We assume that z_{0}, ⋯, z_{k} respects the cyclic order of ∂D^{2}.

We say (φ, z⃗) ∼ (φ′, z′) if there exists a biholomorphic automorphism u : D^{2} → D^{2} such that φ′ = φ ∘ u, z_{i} = u(z′_{i}).

The basic task we need to carry out in order to use the moduli space M_{k+1}(L; β) to various problems is :

(A) (Compactifiation) Find an appropriate compactification CM_{k+1}(L; β) of it.

(B) (Transversality) Find an appropriate perturbation of the pseudoholomorphic curve equation J ∘ dφ = dφ ∘ j_{D^{2}}, so that the moduli space CM_{k+1}(L; β) will become a “smooth manifold” after perturbation.

(C) (Index theory) Calculate the dimension of CM_{k+1}(L; β).

(D) (Orientation) Find a condition on M, L under which CM_{k+1}(L; β) is oriented.

These are package of results one needs to establish topological field theory by nonlinear partial differential equation and moduli space of its solutions. (Donaldson [18] first used such package of results to establish invariants of 4 manifolds (Donaldon invariants). Gromov applied Donaldson’s idea to the moduli space of pseudoholomorphic curves (from closed Riemann surface).)

For point (A), we can easily modify Kontsevich’s ([67]) notion of stable map so that it can be applied to the case of Riemann surface with boundary (disk). See [33]. For point (B), there is now a general theory developed in [34] which can be applied to various situations in a uniform way. Hence basically there is nothing new to work out but we can just apply [34]^{5}. The key notion we use to carry out (B) is space with Kuranishi structure. We will explain it informally later. Actually it is a smooth analogue of the notion of complex analytic space discussed in §1.6.

Now the package (A),(B),(C),(D) in our situation can be stated as follows.

Theorem 3.3.1. ([33]) There exists μ : π_{2}(M; L) → ℤ (the Maslov index) with the following properties.

(1) CM_{k+1}(L; β) is a compact space with Kuranishi structure (with corners), of dimension μ(β) + n + k − 1.

(2) CM_{k+1}(L; β) is oriented if L is relatively spin, in the sense defined later. Relative spin structure determines an orientation of CM_{k+1}(L; β).

(3) The boundary of CM_{k+1}(L; β) is described by fiber products of various CM_{k′+1}(L; β′) where k′ ≤ k and ω ∩ β′ ≤ ω ∩ β.

Explanation of the notion used in the statement of Theorem 3.3.1 will follow (after a few remarks). The statement (3) is a bit vague since we do not mention which fiber product appears. We do not explain it since the main focus of this article is on algebraic formalism and we want to minimize the explanation of geometric analysis parts of the story. Remark 3.3.1. We explain two more properties of the Maslov index μ in Theorem 3.3.1. Let π_{2}(M) → π_{2}(M, L) → π_{1}(L) be a part of homotopy exact sequence of the pair M, L. Then the composition π_{2}(M) → π_{2}(M, L) →^{μ} ℤ coincides with 2c^{1}(M). Here c^{1}(M) ∈ H^{2}(M; ℤ) determines

π_{2}(M) → ℤ. Also π_{2}(M) → π_{2}(L) →^{w^{1}} ℤ_{2} coincides with μ modulo 2. Here w^{1} : π_{1}(L) → {0, 1} is the first Stiefel Whitney class of L. (Namely w^{1}(γ) = 0 if orientation of T L is preserved along the loop γ. In particular μ is even valued if L is oriented.

Let us now explain notions used in Theorem 3.3.1. We will explain two notions, relative spin structure and a space with Kuranishi structure with corners. To define relative spin structure, let us recall some well-known facts on vector bundle on manifolds. First, for any vector bundle E, there exists a characteristic class w^{2}(E) ∈ H^{2}(L; ℤ_{2}), Stiefel-Whitney class such that the structure group of E is reduced to Spinor group if and only if w^{2}(L) = 0. Second, a real vector bundle E on 3 manifold is trivial if it is oriented and spin. (Namely w_{1}(E) = w_{2}(E) = 0.)

Definition 3.3.2. Let L be a submanifold of M. Then L is said to be relatively spin if L is oriented and if there exists a cohomology class st ∈ H^{2}(M; ℤ_{2}) such that w^{2}(T L) = i^{∗}st.

5 There is one point to clarify about transversality of CM_{k+1}(L; β) which is not included in the general theory. That is the problem of transversality at diagonal ⊂ L^{k+1}. This is point (2) mentioned in the discussion after Theorem 3.3.2, and is handled in [33].

We take an oriented vector bundle E on M such that w^{2}(E) = st. Then the relative spin structure of L is a trivialization of T L ⊕ E on the two skeleton of L.

In the case when L is spin, we may take st = 0. Hence relative spin structure corresponds one to one to the spin structure of L. We refer [33] for more detail on orientation of our moduli space.

We next briefly explain Kuranishi structure with corners. We consider an open neighborhood U ⊂ ℝ^{n_{1}} × ℝ_{≥0}^{n_{2}} of 0. Let F = (f^{1}, ⋯, f^{m}) : U → ℝ^{m} be a smooth map. We consider the set F^{−1}(0). As in the case of analytic subset and analytic subspace (which we discussed in §1.6), the subset F^{−1}(0) of U does not contain enough information in case df^{1}, ⋯, df^{m} are not linearly independent. For example, in a simplest situation, we need to count the order of such a space (that is a moduli space of pseudoholomorphic curves). Let us suppose that n_{1} = 1, n_{2} = 0, m = 1 and f(x) = x^{2}. (Here x ∈ ℝ.) Set theoretically, f^{−1}(0) consists of one point. But if we perturb the equation x^{2} = 0 a bit and consider x^{2} = ϵ, then the number of the solutions is zero (if we count it with sign). Hence we need to “remember” additional information coming from the equation F (x) = 0 itself. So we need something more than a subset of U. In the case when F is a complex analytic function, this is exactly the idea of analytic space explained in §1.6. In the case when F is a polynomial, this is the idea of scheme. What we need here is its C^{∞} analogue. (Working in C^{∞} category is inevitable in studying moduli space of pseudoholomorphic curves, (especially with Lagrangian boundary condition), since our problem is strictly real one and it is impossible to assume complex analyticity of the equation.) In case we work in C^{∞} category, considering the ring of germs at 0 and dividing it by ideals generated by f^{i} does not seem to work. This is because the ring of smooth functions does not have some nice properties which are enjoyed by the ring of holomorphic functions. So in place of considering rings (or ringed spaces), we regard a pair (U, F) itself as a chart of our “space”. We then define appropriate notion of coordinate change (or equivalently define a way to glue charts). We thus obtain a space with Kuranishi structure, roughly.

More precisely, we need to include one more point into the story. Namely, in general, our moduli space is not given as F^{−1}(0) but as F^{−1}(0)/Γ locally. Let us explain the notation. Γ is a finite group. We assume that there is a linear actions of Γ on ℝ^{n_{1}} and on ℝ^{m}. We assume that this action preserves U. Moreover we assume that F is Γ equivariant. Hence the zero points set F^{−1}(0) has a Γ action and we may consider the quotient space F^{−1}(0)/Γ. By Kuranishi structure in Theorem 3.3.1, we mean an object gluing such triples (U, F, Γ) in an appropriate sense. (Hence those objects are smooth analogue of Deligne-Mumford stack^{6}, see [15].) We do not try

6 In some other situation like gauge theory, we need to consider the case when Γ is of

to define what we mean by gluing (U, F, Γ). See [34].

We said that our Kuranishi structure is one with corners, since U is an open subset of ℝ^{n_{1}} × ℝ_{≥0}^{n_{2}}. We can define a boundary of a space with Kuranishi structure with corners.

Once we obtain oriented Kuranishi structure with corners, we can define its fundamental chain. It is a chain and is not a cycle in general since we are studying analogy of manifold with boundary or corners. We remark that, already in the situation of Theorem 3.2.1 (3), the moduli space CM(p, q; L_{1}, L_{2}) was a one dimensional manifold with boundary. Studying its boundary is the main part of the proof of the basic equality δδ = 0. The same situation will occur in our more general setting. This is the main difference between the case of pseudoholomorphic curve from closed Riemann surface (where everything can be discussed in the level of homology) and our case of pseudoholomorphic curve from disk (where we need to work in the chain level). This causes various technical troubles which are treated in [33].

Now we finished a brief explanation of the statement of Theorem 3.3.1 and apply it to construct an A_{∞} algebra.

Let L be a Lagrangian submanifold of M. We assume that L is relatively spin and fix relative spin structure. Then orientation of the moduli spaces CM_{k+1}(L; β) is induced. We next assume that the Maslov index μ : π_{2}(M, L) → ℤ is zero. Then, the dimension of our moduli space CM_{k+1}(L; β) is n + k − 1 and is independent of β.

Remark 3.3.2. We remark that this in particular implies that c^{1}(M) is zero on π_{2}(M) by Remark 3.3.1. Calabi-Yau manifold has this property. We also remark that if L is a special Lagrangian submanifold (see [100]) of Calabi-Yau manifold then Maslov index μ : π_{2}(M, L) → ℤ is zero. (See for example [33] for its proof.) The case of special Lagrangian submanifold in Calabi-Yau manifold is believed to be the most important in Mirror symmetry.

Definition 3.3.3. The evaluation map ev = (ev_{0}, ⋯, ev_{k}) : CM_{k+1}(L; β) → L^{k+1} is defined by ev[φ, z⃗] = (φ(z_{0}), ⋯, φ(z_{k})) where z⃗ = (z_{0}, ⋯, z_{k}).

As we mentioned already Theorem 3.3.1 and general theory of Kuranishi structure developed in [34] imply that there exists a fundamental chain [CM_{k+1}(L; β)] ∈ S_{n+k−1}(CM_{k+1}(L; β)). Here we may regard it as a singular chain. We use it to define an operator 𝔪_{k}.

We first take a countably generated complex of singular chains of L over ℚ. (The choice of this subcomplex is based on delicate technical argument which we do not mention in this article.) We write it S(L). We consider the tensor product C^{∗}(L) = S_{n−∗}(L) ˆ⊗_{ℚ}Λ_{ℚ,nov,0}. Here ˆ⊗_{ℚ} is a

positive dimension. It will then be an analogy of Artin stack.

completion with respect to the metric induced by the filtration on Λ_{ℚ,nov,0}. We are going to define a structure of filtered A_{∞} algebra on it.

Now we define 𝔪_{k} as follows. Let us take an element P_{i} of C^{d_{i}}(L). It is a chain of degree n − d_{i}. We now take the fiber product

CM_{k+1}(L; β) ×_{(ev_{1},··· ,ev_{k})} (P_{1} × ⋯ × P_{k}).

It is a ℚ chain of dimension n + k − 2 − ∑ d_{i}. We use the evaluation map ev_{0} to regard it as a chain in L. We thus obtain

ev_{∗} (CM_{k+1}(L; β) ×_{(ev_{1},··· ,ev_{k})} (P_{1} × ⋯ × P_{k})) ∈ S_{n+k−2−∑ d_{i}}(L). (3.6)

We regard (3.6) as an element of C^{∑ d_{i}+2−k}(L) and write it 𝔪_{k,β}(P_{1}, ⋯, P_{k}). In case β = 0 we need to define 𝔪_{k,β} in a bit different way. Roughly speaking we “put”

𝔪_{1,0}(P) = ∂P,  𝔪_{2,0}(P_{1}, P_{2}) = P_{1} ∩ P_{2}.  (3.7)

However (3.7) itself is not correct as we will mention later.

Definition 3.3.4. We put :

𝔪_{k}(P_{1}, ⋯, P_{k}) = ∑_{β} T^{[ω]∩β}𝔪_{k,β}(P_{1}, ⋯, P_{k}),  (3.8)

and extends it to a Λ_{ℚ,nov,0} module homomorphism.

Then the main theorem of [33] is :

Theorem 3.3.2. 𝔪_{k} defines a structure of filtered A_{∞} algebra on C^{∗}(L).

Before mentioning various serious and delicate points in the rigorous argument to justify Definition 3.3.4 and proving Theorem 3.3.2, let us explain parts of the arguments which are easier to explain.

(A) We first need to show that (3.8) converges in C^{∗}(L). This is nontrivial since there are infinitely many terms involved. However we can prove it by using Gromov compactness. Namely Gromov compactness implies that, for each E, there is only a finitely many β such that ∫_{β} ω < E and that CM_{k+1}(L; β) is nonempty. This means that modulo 𝔉^{E}CM_{k+1}(L; β) there is only finitely many terms in (3.8), where 𝔉^{E}CM_{k+1}(L; β) is the energy filtration. This implies the convergence of (3.8).

(B) We next check that the degree is correct. The degree of P_{i} is d_{i} − 1 after shifted. On the other hand the degree of the right hand side of (3.8) is ∑ d_{i} − k + 1 after shifted. Hence the degree of 𝔪_{k} (after shifted) is 1 as required.

(C) Next we check the condition that 𝔪_{0} ≡ 0 modulo Λ_{ℚ,nov,0}. This is immediate from 𝔪_{0,β} = 0 if β = 0.

(D) The proof of the fact that 𝔪_{k} satisfies the A_{∞} relation is based on Theorem 3.3.1 (3) and is roughly as follows. We study 𝔪_{1,0} ∘ 𝔪_{k,β}. Since 𝔪_{1,0} is the usual boundary operator by (3.7), it follows that this composition is obtained by using the boundary of the moduli space CM_{k+1}(L; β). Theorem 3.3.1 (3) asserts that the boundary of CM_{k+1}(L; β) is described as a fiber product of various similar moduli spaces. Taking fiber product of the moduli spaces corresponds to taking a composition of the operators obtained by it. Hence by looking which kinds of fiber product appears in the compactification, we find A_{∞} relation. Roughly the boundary of the moduli space CM_{k+1}(L; β) is described by the following figure.

Figure 4

Now we mention more delicate parts of argument to justify Definition 3.3.4 and proving Theorem 3.3.2. We discuss them only briefly since those points are not our main concern in this article ^{7}.

(1) We need to specify orientation to define the right hand side of (3.6) as a ℚ chain. Basically relative spin structure gives a way to define orientation of CM_{k+1}(L; β). Moreover P_{i} (which is Poincaré dual to a cochain over ℚ) is cooriented. So we obtain a coorientaion of the fiber product. However it is rather a delicate problem to handle the orientation of the fiber product and check that A_{∞} formula is correct with sign. Actually we need more than 60 pages in [33] for this purpose. It is preferable to find a simpler argument.

(2) The formula (3.7) itself actually can not be justified. The reason is that P is not transversal to P so we can not put 𝔪_{2,0}(P, P) = P ∩ P. This is the problem of transversality at diagonal. The way to overcome is as follows. We perturb diagonal and define operations 𝔪_{k,0} inductively on k so that we can define it in the chain level on a countably generated subcompex S(L) of singular chain complex and it is an A_{∞} algebra homotopy equivalent to the De-Rham complex. In other words, we need to use A_{∞} algebra for 𝔪_{k,0}, which corresponds to the rational homotopy theory ^{8}.

7 However the author emphasizes that those ‘technical detail’ (which the author together with Oh, Ohta, Ono spend almost 5 years to work out) is the main part of the story and asserting results without working out that kinds of detail is extremely dangerous. 8 It is realized by several specialists of surgery theory that transversality at diagonal is one of the most essential point in differential topology.

(3) We need to take a complex S(L) carefully so that it is countable (otherwise we can not use Bair’s category theorem to achieve transverality) and the right hand side of (3.6) is again contained in the same complex.

After this brief explanation on the proof of Theorem 3.3.2, we continue our story. Our next task is to state that the A_{∞} algebra in Theorem 3.3.2 is independent of various choices involved. Precisely it is invariant up to homotopy equivalence. Let us define homotopy equivalence of filtered A_{∞} algebra. It is similar to the case of usual A_{∞} algebra described in Chapter 2. There is a few points to modify, which we explain below.

Let C be a filtered A_{∞} algebra. We first define a filtered A_{∞} algebra C[1] ˆ⊗_{Λ_{R,nov}}Λ_{R,nov}[t, dt]. The definition is almost the same as Definition 2.1.7. One important difference is that we take completion of the tensor product here. Namely the element of C[1] ˆ⊗_{Λ_{R,nov}}Λ_{R,nov}[t, dt] is written as P (t) + Q(t)dt where P (t), Q(t) are infinite sum P (t) = ∑ t^{i}P_{i}, Q(t) = ∑ t^{i}Q_{i} where P_{i}, Q_{i} ∈ 𝔉^{λ_{i}}C with λ_{i} → ∞. The operation 𝔪_{k} on C[1] ˆ⊗_{Λ_{R,nov}}Λ_{R,nov}[t, dt] is defined in the same way as Definition 2.1.7.

We write C[1][t, dt] in place of C[1] ˆ⊗_{Λ_{R,nov}}Λ_{R,nov}[t, dt] hereafter. We remark that we can define a filtered A_{∞} homomorphism Eval_{t+t_{0}} :

C[1][t, dt] → C[1]. In the same way as (2.10) as follows. Let P (t)+Q(t)dt ∈ Λ_{R,nov}[t, dt]. We put :

Eval_{t+t_{0}}(P (t) + Q(t)dt) = ∑ t_{0}^{i}P_{i}  (3.9)

(3.9) is an infinite sum but it converges in C[1].

Now we can define homotoy between filtered A_{∞} (or L_{∞}) algebras in the same way as Definition 2.1.8. Theorem 2.1.1 holds in the case of filtered A_{∞} (or L_{∞}) algebra. The definition of homotopy equivalence is also the same as Definition 2.1.9.

Now we have :

Theorem 3.3.3. The A_{∞} algebra is independent of the various choices involved (for example to compatible almost complex structures) up to homotopy equivalence. If φ is a symplectic diffeomorphism then (C^{∗}(L), 𝔪_{∗}) is homotopy equivalent to (C^{∗}(φ(L)), 𝔪_{∗}).

We omit the proof which is in [33]^{9}.

9 The definition of homotoy equivalence we gave in the December 2000 version of [33] looks different from one we gave above. We proved Theorem 3.3.3 in [33] using the definition there. We will rewrite the proof of it and prove the homotopy equivalence in the sense defined above in the final version of [33]. The two definitions are actually equivalent to each other.

Maurer-Cartan equation for filtered A_{∞} algebra. 78

3.4 Maurer-Cartan equation for filtered A_{∞} algebra.

In this section, we explain the way to modify the argument of §2.3 to filtered A_{∞} algebra, especially to the filtered A_{∞} algebra (C^{∗}(L), 𝔪_{∗}) in §3.3. (We only discuss the case of filtered A_{∞} algebra C to save notation. All the argument are parallel in the case of filtered L_{∞} algebra.) The Maurer-Cartan equation is

δ(e^{b}) = 0, where b ∈ ∪_{λ>0}𝔉^{λ}C^{1}.  (3.10)

Since e^{b} = ∑_{k=0}^{∞} b ⊗ ⋯ ⊗ b_{k times} converges as an element of B̂C[1], left hand

side of (3.10) makes sense. This point is different from §2.3.

In the same way as §2.3, solution of (3.10) defines a filtered A_{∞} algebra (C, 𝔪^{b}) by

𝔪_{k}^{b}(x_{1}, ⋯, x_{k}) = 𝔪(e^{b}, x_{1}, e^{b}, ⋯, e^{b}, x_{k}, e^{b}).

We remark that (C, 𝔪^{b}) is a filtered A_{∞} algebra if b ∈ ∪_{λ>0}𝔉^{λ}C^{1}, without assuming δ(e^{b}) = 0. The Maurer-Cartan equation δ(e^{b}) = 0 is equivalent to 𝔪_{0}^{b} = 0. Namely it is equivalent to the condition that (C, 𝔪^{b}) is an A_{∞} algebra.

Let us elaborate the assumption b ∈ ∪_{λ>0}𝔉^{λ}C^{1}. Let us reduce the coefficient ring of our filtered A_{∞} algebra C to R = Λ_{nov,0}/Λ_{nov,+} and obtain \overline{C} = C ⊗_{Λ_{nov,0}} R. Then \overline{C} together with induced operations \overline{𝔪}_{k} is an A_{∞} algebra over R. (Note \overline{𝔪}_{0} = 0 by our assumption that 𝔪_{0} ≡ 0 mod Λ_{nov,+}.)

Hence (C, 𝔪) is a “deformation” over Λ_{nov,0} in the sense similar to Definition 1.7.2. (However since 𝔪_{0} ≠ 0 it is not strictly so.) The condition b ∈ ∪_{λ>0}𝔉^{λ}C^{1} implies (C, 𝔪^{b}) ⊗_{Λ_{nov,0}} R ≅ (\overline{C}, \overline{𝔪}). Namely (C, 𝔪^{b}) is a deformation of (\overline{C}, \overline{𝔪}). In case when b satisfies Maurer-Cartan equation, 𝔪_{0}^{b} = 0, it is a deformation of (\overline{C}, \overline{𝔪}) in the sense of Definition 1.7.2 strictly.

Thus when we study the set of solutions of (3.10), we are studying moduli space of deformations, that is deformations of deformations. We will explain in the next section that studying it is natural in mirror symmetry.

We expand equation (3.10) and obtain

𝔪_{0}(1) + 𝔪_{1}(b) + 𝔪_{2}(b, b) + ⋯ = 0.  (3.11)

Another important difference between (3.11) and (2.11) is that (3.11) is inhomogeneous. (Namely there is a term 𝔪_{0}(1).) As a consequence b = 0 is not a solution of (3.11). Actually there are cases where (3.11) has no solutions.

Definition 3.4.1. We say that filtered A_{∞} algebra C is unobstructed if (3.11) has solution. We say that b is a bounding chain (or Maurer-Cartan element) of C if (3.10) is satisfied.

We now define gauge equivalence of solutions of (3.10) as follows.

Definition 3.4.2. Let b, b′ are bounding chains of C. Then b is said to be gauge equivalent to b′ (and is written as b ∼ b′), if there exists b̃ a bounding chain of C[t, dt] such that Eval_{t=0}(b̃) = b, Eval_{t=1}(b̃) = b′.

We can prove that b ∼ b′, b′ ∼ b′′ imply b ∼ b′′ in a similar way to the proof of Theorem 2.2.1. (See the final version of [33].) Lemma 2.2.5 can be generalized to our situation in the same way.

Definition 3.4.3. We denote by M̃C(C) the sets of all bounding chains of filtered A_{∞} algebra C. The sets of gauge equivalence class of bounding chains is denoted by MC(C).

The following Theorem 3.4.1 follows from what we already explained. Let consider a category {filtered A_{∞} alg./R} whose object is a filtered A_{∞} algebra over R and whose morphism is a filtered A_{∞} homomorphism. We consider its quotient category^{10} {filtered A_{∞} alg./R}/homotopy whose object is a homotopy equivalence class of filtered A_{∞} algebra over R and whose morphism is a homotopy class of A_{∞} homomorphism.

Theorem 3.4.1. C ↦ MC(C) induces a functor : {filtered A_{∞} alg./R}/homotopy → {Sets}.

Theorem 3.4.1 implies that the set MC(C) is invariant of homotopy type of C. However ‘invariant as sets’ does not mean so much. This is the reason we state Theorem 3.4.1 by using quotient category as above. A better way to state homotopy invariance of MC(C) is given later in this section (Proposition 3.4.2).

To study MC(C) we use canonical model as in §2.3.

Definition 3.4.4. A filtered A_{∞} algebra (C, 𝔪) is said to be canonical if 𝔪_{0} = 0 and 𝔪_{1} ≡ 0 mod Λ_{nov,+}.

To generalize Theorem 2.3.1 we assume a kind of finiteness condition for our A_{∞} algebra C.

Definition 3.4.5. A filtered A_{∞} algebra C is said to be weakly finite if it is homotopy equivalent to C′ which is finitely generated as Λ_{nov,0} module.

We need one more assumption. We recall that we assumed that C is a completion of free Λ_{nov,0} module (as a Λ_{nov,0} module). So, as Λ_{nov,0} module, we have C ≅ \overline{C} ⊗ Λ_{nov,0}. Hence, an R module homomorphism \overline{f}_{k} : B_{k}\overline{C}[1] → \overline{C}[1] induces a filtered Λ_{nov,0} module homomorphism f_{k} : B_{k}C[1] → C[1].

10 See for example [47, 55, 35] for its definition

Definition 3.4.6. A filtered A_{∞} algebra C is said to be strongly gapped if there exists λ_{i} and \overline{𝔪}_{k,i} : B_{k}\overline{C}[1] → \overline{C}[1] such that

𝔪_{k} = ∑_{i} T^{λ_{i}}𝔪_{k,i},  lim_{i→∞} λ_{i} = ∞.  (3.12)

Our main example (C(L), 𝔪_{k}) is strongly gapped by definition. We proved in [33] §A4, that it is weakly finite.

Theorem 3.4.2. For any weakly finite, strongly gapped, and unobstructed filtered A_{∞} algebra, there exists a canonical filtered A_{∞} algebra homotopy equivalence to it.

We remark that if we require 𝔪_{1} = 0 in the definition of canonical filtered A_{∞} algebra, then we cannot prove Theorem 3.4.2. This is because we need an analogue of Lemma 2.3.1, which does not hold over Λ_{nov,0} but only over a field. ^{11}.

Proof. (Sketch) We consider 𝔪_{1,0} = \overline{𝔪}_{0}⊗1 as in (3.12) (here we put λ_{0} = 0). We remark that 𝔪_{1,0} ∘ 𝔪_{1,0} = 0 follows from 𝔪_{0} ≡ 0 mod Λ_{nov,+}. We use the assumption that R is a field to obtain a decomposition Ker 𝔪_{1,0} = Im 𝔪_{1,0} ⊕ H(C; 𝔪_{1,0}). We put C_{can}^{∗} = H^{∗}(C; 𝔪_{1,0}). We then obtain a propagator G_{k} such that G_{k+1} ∘ 𝔪_{1,0} + 𝔪_{1,0} ∘ G_{k} = 1 − Π_{C_{can}^{k}}.

The construction of the structure 𝔪_{k}^{can} of filtered A_{∞} algebra on C_{can} and of filtered A_{∞} homomorphism φ_{k} : B_{k}C_{can} → C then goes in a similar way to the proof of Theorem 2.3.1.

The are two differences however. One is that we use graphs such that interior vertex may have 1 or 2 edges. (In Condition 2.3.1, we assumed interior vertices have at least 3 edges.) Second, we assign λ_{i} to each vertex.

Each interior vertex of our graph then corresponds to a term 𝔪_{k,λ_{i}} in (3.12) with (k, λ_{i}) ≠ (1, 0). The rest of the construction of 𝔪_{k}^{can}, φ_{k} is similar to the proof of Theorem 2.3.1 and is omitted. (See the final version of [33].)

To complete the proof, we need the following two results. Let φ : C → C′ be a filtered A_{∞} homomorphism of weakly finite strongly gapped A_{∞} algebra. We assume C is unobstructed and let b ∈ C is a bounding chain.

Theorem 3.4.3. If φ_{1} : H(C, 𝔪_{1}^{b}) → H(C′, 𝔪_{1}^{φ_{∗}b}) is an isomorphism, then φ is a homotopy equivalence.

Proposition 3.4.1. If \overline{φ}_{1} : H(\overline{C}, \overline{𝔪}_{1}^{b}) → H(\overline{C}′, \overline{𝔪}_{1}^{\overline{φ}_{∗}b}) is an isomorphism, then φ_{1} : H(C, 𝔪_{1}^{b}) → H(C′, 𝔪_{1}^{φ_{∗}b}) is an isomorphism.

11 On the other hand, if we use the field Λ_{nov} as a coefficient ring, we can prove a lemma similar to Lemma 2.3.1. However the propagator G obtained over Λ_{nov} coefficient does not preserve energy filtration. As a consequence, if we try to define operators \overline{𝔪}_{k} using G, it does not converge.

Theorem 3.4.3 is an analogue of Theorem 2.2.1 and can be proved in the same way. Proposition 3.4.1 will follow from a spectral sequence which we explain later (Theorem 3.4.5). It is easy to see that φ_{k} : B_{k}C_{can} → C satisfies the assumption of Proposition 3.4.1. Hence it is a homotopy equivalence.

We can show that homotopy equivalence between canonical A_{∞} algebras are isomorphism in the same way as Proposition 2.3.1

We now use Theorem 3.4.2 to define formal Kuranishi map as follows. Let C be a canonical A_{∞} algebra. Let e_{i} and f_{i} are basis of C^{1} and C^{2} respectively. We b_{i} = rank_{R} \overline{C}^{i}. We take formal parameters X_{1}, ⋯, X_{b_{1}}, X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ} and put

Λ_{nov,0}[[X_{1}, ⋯, X_{b_{1}}]] = Λ_{nov,0} ˆ⊗_{R}R[[X_{1}, ⋯, X_{b_{1}}]]
Λ_{nov,0}⟨X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}⟩ = Λ_{nov,0} ˆ⊗_{R}R[X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}]

We remark that Λ_{nov,0}⟨X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}⟩ consists of elements

∑ a_{i_{1},··· ,i_{b_{1}}} X_{1}^{i_{1}} ⋯ X_{b_{1}}^{i_{b_{1}}}

such that a_{i_{1},··· ,i_{b_{1}}} ∈ 𝔉^{λ_{i_{1},··· ,i_{b_{1}}}} Λ_{nov,0} with

lim_{min{i_{1},··· ,i_{b_{1}} }→∞} λ_{i_{1},··· ,i_{b_{1}}} = ∞.

Λ_{nov,0}⟨X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}⟩ is called strictly convergent ring in Rigid analytic geometry. See [12].

We define P_{j} ∈ Λ_{nov,0}[[X_{1}, ⋯, X_{b_{1}}]], P_{j}^{ϵ} ∈ Λ_{nov,0}⟨X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}⟩ by

∑ P_{j}(X_{1}, ⋯, X_{b_{1}})f_{j} = 𝔪 (exp(X_{1}e_{1} + ⋯ + X_{b_{1}}e_{b_{1}}))
∑ P_{j}^{ϵ}(X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ})f_{j}, = 𝔪 (exp(T^{ϵ}X_{1}^{ϵ}e_{1} + ⋯ + T^{ϵ}X_{b_{1}}^{ϵ}e_{b_{1}})) .

Definition 3.4.7. We put 𝔎_{C} = Λ_{nov,0}[[X_{1}, ⋯, X_{b_{1}}]]/(P_{1}, ⋯, P_{b_{2}}). We also put : 𝔎_{C}^{ϵ} = Λ_{nov,0}⟨X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}⟩/(P_{1}^{ϵ}, ⋯, P_{b_{2}}^{ϵ}). For ϵ < δ we define a homomorphism π_{ϵ,δ} : Λ_{nov,0}⟨X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}⟩ →

Λ_{nov,0}⟨X_{1}^{δ}, ⋯, X_{b_{1}}^{δ}⟩ by π_{ϵ,δ}(X_{i}^{ϵ}) = T^{δ−ϵ}X_{i}^{δ}. It induces π_{ϵ,δ} : 𝔎_{C}^{ϵ} → 𝔎_{C}^{δ}. We then define 𝔎_{C}^{+} = \underleftarrow{lim} 𝔎_{C}^{ϵ}.

We can prove easily the following proposition.

Proposition 3.4.2. The isomorphism classes (as Λ_{nov,0} algebras) of 𝔎_{C}, 𝔎_{C}^{ϵ}, 𝔎_{C}^{+} are independent of the homotopy equivalence of C.

We define b̃ = ∑ X_{i}e_{i}, b̃^{ϵ} = ∑ T^{ϵ}X_{i}^{ϵ}e_{i}. b̃^{ϵ} defines b̃^{+} ∈ 𝔎_{C}^{+}. They satisfy Maurer-Cartan equation (3.10). Therefore (C_{𝔎_{C}} , 𝔪b̃), (C_{𝔎_{C}^{ϵ}} , 𝔪b̃^{ϵ}), (C_{𝔎_{C}^{+}}, 𝔪b̃^{+}) are A_{∞} algebras. (Here we put C_{𝔎_{C}} = C ˆ⊗_{Λ_{nov,0}}𝔎_{C} etc.)

We remark that 𝔎_{C} is a complete local ring whose maximal ideal is generated by T^{λ} (for all λ > 0), and X_{i}. Hence it parametrizes a deformation of \overline{C} which is an A_{∞} algebra over R.

On the other hand, 𝔎_{C}^{ϵ} is not a local ring. Namely its spectrum has many (closed) points. This is equivalent to the fact that we can define f_{a_{1},··· ,a_{b_{1}}} : Λ_{nov,0}⟨X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ}⟩ → Λ_{nov,0} by f_{a_{1},··· ,a_{b_{1}}} (P (X_{1}^{ϵ}, ⋯, X_{b_{1}}^{ϵ})) = P (a_{1}, ⋯, a_{b_{1}}). (In other words Spec(𝔎_{C}^{ϵ}) is infinitesimally small in T direction but is of positive size in X_{i} direction.) Hence 𝔎_{C}^{+} is not a local ring either. We consider the ideal Λ_{nov,+}𝔎_{C}^{+} of it and put \overline{𝔎}_{C}^{+} = 𝔎_{C}^{+}/(Λ_{nov,+}𝔎_{C}^{+}). We then consider a deformation C\overline{𝔎}_{C}^{+} = (C_{𝔎_{C}^{+}}, 𝔪b̃^{+}) ⊗_{𝔎_{C}^{+}} \overline{𝔎}_{C}^{+}. C\overline{𝔎}_{C}^{+} is a de-

formation of \overline{C} and is the restriction of the family (C_{𝔎_{C}^{+}}, 𝔪b̃^{+}) to its “sub space” defined by T = 0. We find easily see that C\overline{𝔎}_{C}^{+} is a trivial defor-

mation. Namely it is isomorphic (\overline{C}, \overline{𝔪}) ⊗_{R} \overline{𝔎}_{C}^{+}. Thus, (C_{𝔎_{C}^{+}}, 𝔪b̃^{+}) is a deformation of C whose restriction to T = 0 is trivial.

(C_{𝔎_{C}} , 𝔪b̃), (C_{𝔎_{C}^{+}}, 𝔪b̃^{+}) have the following completeness properties (Lemma 3.4.1) similar to Lemma 2.3.4. To state the lemma, we need some notations.

Let {filtered complete Artin local/Λ_{nov,0}} be the category of filtered complete Artin local Λ_{nov,0} algebra. {filtered complete/Λ_{nov,0}} is the category of filtered complete Λ_{nov,0} algebra. (They can be defined in the same way as Definition 1.7.5.)

To each a filtered A_{∞} algebra C (which may not be canonical), we define functors MC(C) : {filtered complete Artin local/Λ_{nov,0}} → {Sets}, MC^{+}(C) : {filtered complete/Λ_{nov,0}} → {Sets} as follows.

If ℜ be a filtered complete Artin local Λ_{nov,0} algebra, then MC(C)(ℜ) is the set of all gauge equivalence classes of b^{+} ∈ C_{ℜ}^{+} such that b ≡ 0 mod ℜ_{+} and that b satisfies the Maurer-Cartan equation. (Here ℜ_{+} is a maximal ideal of ℜ.) If R is a filtered complete Λ_{nov,0} algebra, then MC^{+}(C)(R) is a set of all gauge equivalence classes of b such that b ≡ 0 mod Λ_{nov,+}R and b satisfies the Maurer-Cartan equation.

Lemma 3.4.1. If b ∈ MC(C)(ℜ), then there exists a Λ_{nov,0} algebra homomorphism φ : 𝔎_{C} → ℜ such that φ(b̃) = b. If b ∈ MC^{+}(C)(R), then there exists a Λ_{nov,0} algebra homomorphism ψ : 𝔎_{C}^{+} → R such that ψ(b̃^{+}) = b.

The proof of Lemma 3.4.1 is similar to the proof of Lemma 2.3.4. We next discuss universality. We need a similar condition to one in Theorem 2.3.2. We consider the homomorphisms :

π : H^{0}(C_{𝔎_{C}} , 𝔪b̃ ; 1) → H^{0}(\overline{C}, \overline{𝔪}_{1}),  (3.13a)
π : H^{0}(C_{𝔎_{C}^{+}}, 𝔪b̃^{+} ; 1) → H^{0}(\overline{C}, \overline{𝔪}_{1}).  (3.13b)

Lemma 3.4.2. If (3.13) is surjective then the homomorphisms φ and ψ in Lemma 3.4.1 is unique.

The proof is the same as the proof of Theorem 2.3.2.

These two lemmata immediately imply the following Theorem 3.4.4. We need some more notation to state it. We define another functors 𝔉_{𝔎_{C}} : {filtered complete Artin local/Λ_{nov,0}} → {Sets} so that 𝔉_{𝔎_{C}} (ℜ) is the set of all Λ_{nov,0} algebra homomorphisms 𝔎_{C} → ℜ. We define 𝔉_{𝔎_{C}^{+}} : {filtered complete/Λ_{nov,0}} → {Sets} so that 𝔉_{𝔎_{C}^{+}}(R) is the set of all Λ_{nov,0} algebra homomorphisms ψ : 𝔎_{C}^{+} → R.

Theorem 3.4.4. If (3.13) is surjective then the functor MC(C) is equivalent to 𝔉_{𝔎_{C}} and MC^{+}(C) is equivalent to 𝔉_{𝔎_{C}^{+}}.

Let us consider the case of the filtered A_{∞} algebra C(L) of Lagrangian submanifold defined in the last section. We assume that L is connected and C(L) is unobstructed. The A_{∞} algebra \overline{C}(L) = C(L)/(Λ_{nov,+} · C(L)) is one by rational homotopy type of L. In particular H^{0}(\overline{C}(L); \overline{𝔪}_{1}) = R since L is connected. Its generator is the fundamental cycle [L]. We proved in [33] that [L] is gives nonzero element in H^{0}(C_{𝔎_{C}} , 𝔪b̃ ; 1), H^{0}(C_{𝔎_{C}^{+}}, 𝔪b̃ ; 1).^{12} Therefore (3.13) is surjective in this case. Hence 𝔉_{𝔎_{C}}, 𝔉_{𝔎_{C}^{+}} are universal moduli spaces of appropriate Maurer-Cartan functors. This fact may be related to the stability of the mirror object in complex side. (Compare [104].)

We recall that 𝔎_{C}^{+} represents the moduli functor of deformations of C such that its restriction to T = 0 is trivial. This moduli functor is not infinitesimal one, since it is a functor from {filtered complete/Λ_{nov,0}} whose object is not necessary Artin or local. So it makes sense to say a point of it. (On the other hand, 𝔎_{C} represents a moduli functor of infinitesimal deformation of C.)

Theorem 3.4.4 implies MC(C) = Hom_{Λ_{nov,0}}(𝔎_{C}^{+}, Λ_{nov,0}) if (3.13) is surjective. (Here the right hand side is the set of all Λ_{nov,0} algebra homomorphisms which is continuous with respect to the Λ_{nov,+} adic topology.) In other words, MC(C) is the set of R valued points^{13} of 𝔎_{C}^{+}.

We next explain a spectral sequence which describe a relation of (C, 𝔪_{1}^{b}) to (\overline{C}, \overline{𝔪}_{1}). In the case of filtered A_{∞} algebra C(L) of Lagrangian submanifold, it gives a relation between cohomology of L and Floer cohomology H(C, 𝔪_{1}^{b}). In the case when π_{2}(M, L) = 0, Theorem 3.2.3 asserts that Floer cohomology is isomorphic to the usual cohomology. The reason for it was there was no holomorphic disk when π_{2}(M, L) = 0. If this assumption is not satisfied, then Floer cohomology may not be equal to the usual

12 We proved in [33] that it is a (homotopy) unit of our A_{∞} algebra C(L). In particular, it gives a nontrivial element of the cohomology. 13 See for example [48] for its definition.

cohomology of Lagrangian submanifold. The spectral sequence we discuss below describe the procedure how they are deformed.

Remark 3.4.1. A translation of this phenomenon to physics language might be, “instanton effect changes the dimension of the moduli space of the vacuum since it change the mass of some particle from zero to positive number”.

Let b be a bounding chain of C. In general, if there exists a filtration on chain complex, we obtain a spectral sequence. (See any text book of homological algebra). Filtered A_{∞} algebra C has a filtration (energy filtration) and hence (C, 𝔪_{1}^{b}) is a filtered complex. However the filtration is parametrized by real number and not by integer. So we fix sufficiently small λ_{0} > 0 and use the filtration F^{k}C = 𝔉^{kλ_{0}}C.

The other trouble is that the ring Λ_{nov,0} is not Noetherian. It causes a serious trouble when proving convergence of spectral sequence. This problem is overcome in [33] §A4. We then obtain the following theorem.

Theorem 3.4.5. We assume that C is weakly finite and strongly gapped. Let b be a bounding chain of it. Then, there exists a spectral sequence E_{r}^{p,q} with the following properties.

(1) E_{2}^{p,q} ≅ H(\overline{C}, \overline{𝔪}_{1}) ⊗_{R} F^{q}Λ_{nov,0}/F^{q+1}Λ_{nov,0}.

(2) There exists a filtration F^{q}H^{p}(C, 𝔪_{1}^{b}) on H^{p}(C, 𝔪_{1}^{b}) and r_{0} such that E_{r_{0}}^{p,q} ≅ E_{r_{0}+1}^{p,q} ≅ ⋯ ≅ E_{∞}^{p,q} ≅ F^{q}H^{p}(C, 𝔪_{1}^{b})/F^{q+1}H^{p}(C, 𝔪_{1}^{b}).

Theorem 3.4.5 was proved by Y.G.Oh [83] in the case of monotone Lagrangian submanifolds with minimal Maslov number ≥ 3. See [33] §A4 for the proof of Theorem 3.4.5. We remark Proposition 3.4.1 follows from Theorem 3.4.5.

Before going to the next section, we explain very briefly the case when there are more than one Lagrangian submanifolds. See [33] on the case when there are two Lagrangian submanifolds and [30] for the case when there are 3 or more Lagrangian submanifolds.

Let L_{1}, L_{2} be two Lagrangian submanifolds. We assume that their Maslov index are zero. We also assume that they are relatively spin, namely we assume that there exists st ∈ H^{2}(M; ℤ_{2}) which reduces to the second Stiefel-Whitney class of L_{i}. We assume that we can take the same st for both of the Lagrangian sumanifolds. We then obtain filtered A_{∞} algebras (C(L_{i}), 𝔪_{i}). The Lagrangian intersection Floer homology then is a filtered A_{∞} bimodule CF (L_{1}, L_{2}). Namely it is a left (C(L_{1}), 𝔪_{i}) and right (C(L_{2}), 𝔪_{i}) module. We do not define A_{∞} bimodule here. (See [33] for its definition.)

In case there are three of more Lagrangian submanifolds L_{i}, then we can define a product operations.

We can use the ring 𝔎_{C(L_{i})}^{+} defined in Definition 3.4.7 to rewrite Lagrangian intersection Floer cohomology and their product structures as follows.

Theorem 3.4.6. Let L_{i} be a countable set of mutually transversal Lagrangian submanifolds. We assume that their Maslov index are all zero. We also assume that there exists st ∈ H^{2}(M; ℤ_{2}) which restricts to w^{2}(L_{i}) for any i. We fix relative spin structure for each L_{i}.

Then for each i, j, there finitely generated exists 𝔎_{C(L_{i})}^{+} 𝔎_{C(L_{j})}^{+} differential graded bimodule (D(L_{i}, L_{j}), 𝔪_{1}) and operations

\begin{matrix}𝔪_{k} : D(L_{i_{1}}, L_{i_{2}}) & ˆ⊗_{𝔎_{C}(L_{2})}D(L_{i_{2}}, L_{i_{3}}) ˆ⊗_{𝔎_{C(L_{3})}^{+}} ⋯ \\ ⋯ & ˆ⊗_{𝔎_{C(L_{k−1})}^{+}}D(L_{i_{k−1}}, L_{i_{k}}) → D(L_{i_{1}}, L_{i_{k}}), \end{matrix}

which satisfies A_{∞} formula.

See [33] ([31]) for the proof of Theorem 3.4.6. As 𝔎_{C}(L_{i}) ˆ⊗_{Λ_{nov,0}}𝔎_{C}(L_{j}) module, D(L_{i}, L_{j}) is :

D(L_{i}, L_{j}) ≅ ⨁_{p∈L_{1}∩L_{2}} 𝔎_{C}(L_{i}) ˆ⊗_{Λ_{nov,0}}Λ_{nov,0}[p] ˆ⊗_{Λ_{nov,0}}𝔎_{C}(L_{j}).

We remark that Spec(𝔎_{C(L_{i})}^{+}) is a moduli space parametrizing a deformation of A_{∞} algebra C(L_{i}). Bimodule over 𝔎_{C(L_{i})}^{+}, 𝔎_{C(L_{j})}^{+} is regarded as a coherent sheaf over the product Spec(𝔎_{C(L_{i})}^{+}) ×_{Spec Λ_{nov,0}} Spec(𝔎_{C(L_{j})}^{+}). Hence its cohomology sheaf (which is an object of derived category of coherent sheaves on Spec(𝔎_{C(L_{i})}^{+}) ×_{Spec Λ_{nov,0}} Spec(𝔎_{C(L_{j})}^{+})) is a family of Floer homologies. (This is only a local family. To study global family we need more. See [31].) We will discuss the mirror object of one constructed in Theorem 3.4.5. Remark 3.4.2. The bimodule D(L_{i}, L_{j}) and operations in Theorem 3.4.5 is invariant of various choices involved, for example the choice of compatible almost complex structure J and of various perturbations. However it is not independent of Hamiltonian diffeomphism. Namely D(L_{1}, L_{2}) ≠ D(φ_{1}(L_{1}), φ_{2}(L_{2})) in general. However Floer homology coincides if we change the coefficient ring to Λ_{nov}. See [33] for the proof. Remark 3.4.3. We remark that we assumed that Maslov index π_{2}(M, L) → ℤ vanishes in this section. The reason we need it is that otherwise the operator 𝔪_{k} does not preserve degree, because the dimension of the moduli space M_{k+1}(L; β) depends on the cohomology class β. This assumption is not used anywhere else. So by considering ℤ_{2} graded Floer homology we may remove this assumption without difficulty. For the application to symplectic geometry, to study the general case is necessary. However the homomorphism (3.13) then may not be surjective. Including the case Maslov index is nonzero leads us the notion of extended moduli space. (See [87].) In the side of complex geometry, we may consider the ℤ_{2} graded chain complex of coherent sheaves. Including such objects also leads us extended moduli space.

Remark 3.4.4. Usually in Mirror symmetry we include a flat U(1) bundle on L. Actually such a parameter is already included in our story. Namely our parameter T may be regarded a complex number. Its real part which is related to H^{1}(L; ℝ) by the spectral sequence in Theorem 3.4.5. H^{1}(L; ℝ) parametrize a deformation of our Lagrangian submanifold L. Then the imaginary part H^{1}(L; \sqrt{−1}ℝ) corresponds to the deformation of the trivial bundle on L to a flat (nontrivial) U(1) bunlde.

3.5 Homological mirror symmetry.

We now are going back to the complex geometry side of the story and explain more what is expected to be a mirror of the construction of §§3.2,3.3, and 3.4. Actually the author is not an expert of complex geometry part of the story. There are many deep mathematics involved, some of which the author does not know enough. So he is afraid that there might be some error in this section. He dares to write them here since it seems almost impossible to find a person who has enough knowledge to all of many aspects of the mirror symmetry. For example the person who have enough backgroud of both symplectic part and complex part of the story is rare.

We remark that the author is much influenced by Kontsevich-Soibelman [70] while writing this section.

We first introduce some more Novikov rings. Let ♢ be a sub semigroup of ℝ (namely ♢ ⊆ ℝ such that a, b ∈ ♢ implies a+b ∈ ♢). We put Λ_{R,nov}^{♢} =

{∑ a_{i}T^{λ_{i}} | λ_{i} ∈ ♢} . For example Λ_{R,nov}^{ℝ_{≥0}} = Λ_{R,nov,0}, Λ_{R,nov}^{ℝ_{>0}} = Λ_{R,nov,+}.

On the other hand, Λ_{R,nov}^{ℤ_{≥0}} is the formal power series ring R[[T]] and Λ_{R,nov}^{ℤ} is the Laurant polynomial ring R[[T]][T^{−1}].

The example which appeared in symplectic geometry of Lagrangian submanifold is the case when ♢ is the semigroup generated by the symplectic areas (that is the integration of symplectic form) of pseudoholomorphic disks. In the case studied by Novikov himself, that is the case of Morse theory of closed one form θ, the semigroup ♢ is the set of all θ ∩ ℓ for ℓ ∈ π_{1}(M).

In mirror symmetry, we consider the case Λ_{R,nov}^{ℚ_{≥0}}, and its maximal ideal

Λ_{R,nov,+}^{ℚ_{≥0}} = Λ_{R,nov}^{ℚ_{>0}} and Λ_{R,nov}^{ℚ}. One may also take : Λ_{R,nov}^{ℚ,0} = ⋃_{m} Λ_{R,nov}^{ℤ[1/m]},

Λ_{R,nov}^{ℚ_{≥0},0} = Λ_{R,nov}^{ℚ,0} ∩ Λ_{R,nov}^{ℚ_{≥0}}, Λ_{R,nov,+}^{ℚ_{≥0},0} = Λ_{R,nov}^{ℚ,0} ∩ Λ_{R,nov}^{ℚ_{>0}}.

Exercise 3.5.1. Prove that Λ_{R,nov}^{ℚ,0} is the algebraic closure of R[[T]][T^{−1}] if R is a algebraically closed field.

Elements of Λ_{ℂ,nov}^{ℚ_{≥0},0} is a “formal Puiseux series” ∑_{k=0}^{∞} a_{k}T^{k/n}. The ring Λ_{R,nov}^{ℚ} is a completion of Λ_{R,nov}^{ℚ,0}.

The geometric meaning of these rings are as follows. The ring ℂ[[T]] is a ring of functions of one variables. Since we are considering formal power series we may say it is a “ring of holomorphic functions on D^{2}(0), disk of radius zero”. Elements of ℂ[[T]][T^{−1}] is meromorphic function that is a function which is defined outside the origin but have only a pole at 0. Hence ℂ[[T]][T^{−1}] may be regarded as a “ring of holomorphic functions on D^{2}(0)\{0}”.

Considering Λ_{R,nov}^{ℤ[1/n]} corresponds to taking an n fold Galois cover D_{n}^{2}(0) → D^{2}(0). Hence elements of its sum Λ_{R,nov}^{ℚ,0} may be regarded as a inductive limit \underrightarrow{lim} O(D_{n}^{2}(0)\\{0\}). In algebraic geometry, it is impossible to consider universal cover in the usual sense, so one considers system of finite covers and take its limit. In that sense Λ_{R,nov}^{ℚ,0} is a ring of functions of “algebraic universal cover” of D^{2}(0)\{0}.

The universal Novikov ring Λ_{nov,0} we used in §3.3, 3.4 is of more transcendental nature and may be regarded as a ring of holomorphic functions of the “usual universal cover” of D^{2}(0)\{0}.

Now let us explain how they appears in the complex geometry side of the mirror symmetry. Let us start with a symplectic manifold (M, ω). Mirror symmetry predicts that there exists a complex manifold (M^{∨}, J) which is a mirror to (M, ω) under some assumption. (It is not conjectured that any symplectic manifold has a mirror, however.)

To be more precise, we have to modify it a bit. First in mirror symmetry, one usually include a closed 2 form B on M, which is called a B field. The sum Ω = ω + \sqrt{−1}B is called a complexified symplectic structure and (M, Ω) is expected to correspond to a complex manifold (M^{∨}, J) = (M, Ω)^{∨}. We need to include B since the moduli space of complex structure is complex analytic object so its mirror (moduli space of symplectic manifold) should be modified so that it will be a complex object.

Furthermore, when we have a symplectic manifold (M, ω) we actually have a family of them that is a family (M, zω) where z is a complex number such that Re z > 0. (Then (Re z)ω becomes the symplectic form and (Im z)ω becomes a B field.) Here I would like to state two points which seems to be widely accepted among researchers of mirror symmetry.

(⋆) The mirror (M, ω)^{∨} exists only if ω ∈ H^{2}(M; ℚ). Namely ω is rational homology class. (⋆⋆) If ω ∈ H^{2}(M; ℤ) and k ∈ ℤ then (M, zω)^{∨} = (M, (z + k\sqrt{−1})ω)^{∨}.

The author does not try to explain the reason that why they are believed. Let us assume them. We then may assume ω ∈ H^{2}(M; ℤ) since we can replace ω by nω (n ∈ ℤ) if necessary. We put q = e^{−2πz}.

Remark 3.5.1. We remark that, in Definition 3.3.4, T appeared as T^{ω∩β}. Hence if we put T = e^{−1} then then T^{zω∩β} will be q^{ω∩β} the same form as in Definition 3.3.4. Thus by redefining T = q, the formal parameter T in the §3.3,3.4 may be regarded as a coordinate of the disk parametrizing the mirror family. (See [31] for more detail about this point.)

(⋆⋆) above implies that we have a family of complex manifolds (M^{∨}, J_{q}) = (M, zω)^{∨} parametrized by q ∈ D^{2}(1)\{0}. Another conjecture (see [44, 70]) predicts that such a family is a maximal degenerate family of Calabi-Yau manifolds. Here we recall the definition of maximal degenerate family briefly. See [73, 70] for more detail.

Let π : M̂ → D^{2}\{0} be a family of complex manifold parametrized by a unit disk D^{2} minus origin. We are interested in the case when the fibers are smooth. We assume that is extends to a flat family π^{+} : \overline{M̂} → D^{2} over D^{2} but assume the fiber of the origin (π^{+})^{−1}(0) is singular.

The story here is related to the theory of variation of Hodge structures. (See [39].) We have a fiber bundle H^{k}(M) → D^{2}\{0} whose fiber at q ∈ D^{2}\{0} is the cohomology group of (π^{+})^{−1}(q) = (M, J_{q}). The bundle H^{k}(M) is a flat bundle and the flat connection is the famous Gauss-Manin connection. Let us denote its monodromy on H^{n}(M) by ρ : H^{n}(M) → H^{n}(M). (Here n = dim_{ℂ} M.) (Since π_{1}(D^{2}\{0}) = ℤ we only need to consider the generator.) It is known that the eigenvalue of ρ are all roots of unity. Namely ρ^{N} − 1 is nilpotent for some N. It is also known that (ρ^{N} − 1)^{n+1} = 0.

Definition 3.5.1. The family π : M̂ → D^{2}\{0} is said to be a maximal degenerate family if (ρ^{N} − 1)^{n} ≠ 0, (ρ^{N} − 1)^{n+1} = 0

In [44, 70], interesting conjecture is proposed about the behavior of Calabi-Yau metric on (M, J_{q}) when q → 0 for a maximal degenerate family M̂ → D^{2}\{0}, by using Gromov-Hausdorff convergence (see [43]) of Riemannian manifolds.

We remark that maximal degenerate family is the opposite extreme to the case when monodromy is given by a Dehn twist along symplectic sphere, which is studied in detail by [96, 57, 95, 94].

Now let us return to our situation of π : M̂ → D^{2}\{0}. We obtain a differential graded algebra Ω^{0,∗}(M̂/D^{2}\{0}) over O(D^{2}\{0}), in the same way as we explained at the beginning of §1.7. The ring O(D^{2}\{0}) is different from the ring of meromorphic functions on D^{2}. Namely O(D^{2}\{0}) contains a function which has an essential singularity at 0. However using an extension of our family to 0, one can find a differential graded algebra Ĉ over O(D^{2}) so that Ĉ ⊗_{O(D^{2})} O(D^{2}\{0}) is homotopy equivalent to Ω^{0,∗}(M̂/D^{2}\{0}). Moreover Ĉ ⊗_{O(D^{2})} ℂ[[T]][T^{−1}] is independent to the extension but depends only on the family over D^{2}\{0}.

However, the differential graded algebra Ĉ over ℂ[[T]] depends on the choice of the extension of our family at 0. The choice of such extension is called a choice of model sometimes. It seems that no canonical choice of such model is know in the general situation, and seems to be related to a deep problem in algebraic geomegry. A natural choice of a model is known in some case. For example in the case of elliptic curve [59, 80].

The discussion on the Novikov ring at beginning of this section suggests that it is important to consider the family parametrized not only by ℂ[[T]] but also by Λ_{ℂ,nov}^{ℚ_{≥0},0} or Λ_{ℂ,nov}^{ℚ_{≥0}}. To obtain such a family, we first take an n fold cover of the base D^{2}\{0}, pullback the family, and, consider the “limit” when n → ∞. It seems that what Kontsevich-Soibelman [70] suggested about the relation of Mirror symmetry and Rigid analytic geometry is somehow related to this point. Let me mention just one example to show this relation.

Example 3.5.1. Let us consider a (real) 2 torus T^{2}. Its symplectic form ω is unique up to constant. Its mirror (T^{2}, zω)^{∨} is ℂ/(ℤ⊕\sqrt{−1}zℤ). This gives a standard family of elliptic curve parametrized by q = e^{−2πz} ∈ D^{2}\{0}.

The monodromy matrix is (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}). In this case there is a canonical choice

of the model. Namely we put type I singular fiber (in the classification of Kodaira [59]) over origin. Now we replace q by q^{1/n}. Then the monodromy

matrix will become (\begin{matrix}1 & n \\ 0 & 1 \end{matrix}). Hence the the singular fiber will become of

type I_{n}.

Now, what happens when n → ∞? We will have a dense set of singular points which consists of all rational points of S^{1} and its completion (that is S^{1}) may be regarded a limit. This seems to be the picture of Rigid analytic geometry ([12, 9]).

This S^{1} in turn will be the Gromov-Hausdorff limit of the Riemannian manifold ℂ/(ℤ ⊕ \sqrt{−1}zℤ) equipped with Calabi-Yau metric (which is nothing but the flat metric in this case) with diameter normalized to 1.

This seems to be the simplest case of the picture proposed by Kontsevich-Soibelman [70].

Now let us try to formulate homological mirror symmetry conjecture. Suppose we have a Lagrangian submanifold L of a Calabi-Yau manifold (M, ω) which is relatively spin and Maslov index π_{2}(M, L) → ℤ is zero. We define :

Definition 3.5.2. L is said to be rational if ∫_{D^{1}} φ^{∗}ω ∈ ℚ for any [φ] ∈ π_{2}(M, L).

Remark 3.5.2. The definition of rationality here is tentative one. For example let us consider the case of symplectic 2-torus (T^{2}, ω). (2 is the real dimension.) In T^{2} = ℝ^{2}/(ℤ ⊕ ℤ), we consider Lagrangian submanifolds ℝ/ℤ×{a}, where a ∈ ℝ/ℤ. I would rather like to call it rational only when a ∈ ℚ/ℤ. But in the sense of Definition 3.5.2 it is always rational. Such a trouble might disappear in the case when M is simply connected.

To a rational Lagrangian submanifold L, we can associate an A_{∞} algebra (C(L), 𝔪) over Λ_{nov,0}^{ℚ,0}. Actually, using the fact that π_{2}(M, L) is finitely generated, we can define it over Λ_{nov,0}^{ℤ[1/m]} for some m ∈ ℤ_{>0}.

We suppose that we have a mirror family (M, zω)^{∨} parametrized by q = e^{−z}.^{14} By Remark 3.5.1 we may identify q with T in §3.3, 3.4. We take an m hold cover of our mirror family and get a family π_{m} : M̂_{m} → D^{2}\{0}. Mirror object is expected to be a family of holomorphic vector bundle over this family. But we need a bit more general object than vector bundle, namely an object of the derived category of coherent sheaves. In our situation, it may be regarded as a complex

Ê_{1} ⟶^{δ_{1}} Ê_{2} ⟶^{δ_{2}} Ê_{3} ⟶^{δ_{3}} ⋯ ⟶^{δ_{N−1}} Ê_{N}.  (3.14)

here Ê_{i} is a family of holomorphic vector bundles over π_{m} : M̂_{m} → D^{2}\{0} and δ_{i} are holomorphic section of Hom(Ê_{i}, Ê_{i+1}) over M̂_{m}. We write such an object 𝔼.

One can define a differential graded algebra describing a deformation of this objects. For example we can proceed as follows ^{15}. We put

Ω^{k}(𝔼, 𝔼) = ⨁_{i,j:i≤j≤i+k} Ω^{0,k+i−j}(M̂_{m}/O(D^{2}\{0}); Hom(E_{i}, E_{j})). (3.15)

For φ ∈ Ω^{k}(𝔼, 𝔼) we denote its Ω^{0,p−ℓ}(M̂_{m}/O(D^{2}\{0}); Hom(E_{i}, E_{i+ℓ})) component by φ_{i,ℓ}. We then put

(dφ)_{i,j} = ±\overline{∂}_{E_{j}} ∘ φ_{i,j} ± φ_{i,j} ∘ \overline{∂}_{E_{i}} ± δ_{j} ∘ φ_{i,j−1} ± φ_{i+1,j} ∘ δ_{i}

We also put

(φ ∘ ϕ)_{i,j} = ∑_{ℓ} ±φ_{ℓ,j} ∘ ϕ_{i,ℓ}

(See [27] Chapter 4 for sign.) We can check (Ω^{∗}(𝔼, 𝔼), d, ∘) is a differential graded algebra and hence an A_{∞} algebra over O(D^{2}\{0}) (where T^{1/m} is the coordinate). If we can extend 𝔼 to a family over D^{2} then we have a differential graded algebra over O(D^{2}). We formalize it as in §1.7 and

14 Construction of mirror family is now studied extensively by several people. In this paper the author do not try to discuss it. 15 Compare [27] Chapter 4. There we developed a similar but more general construction in the case of twisted complex.

obtain a differential graded algebra over Λ_{nov,0}^{ℤ[1/m]} or Λ_{nov}^{ℤ[1/m]}. We remark that a differential graded algebra over Λ_{nov,0}^{ℤ[1/m]} depends on choice of the extension (model).

Now a part of homological mirror symmetry conjecture is stated as follows.

Conjecture 3.5.1. Let L be a rational Lagrangian submanifold which is relatively spin and its Maslov index is zero. We assume that (C(L), 𝔪) is unobstructed.

Then there exists an object 𝔼 as in (3.15) together with its extension to 0 ∈ D_{m}^{2}, such that (C(L), 𝔪) is homotopy equivalent to (Ω^{∗}(𝔼, 𝔼), d, ∘) as an A_{∞} algebra over Λ_{nov,0}^{ℤ[1/m]}.

We proceed to the case corresponding to Theorem 3.4.5 as follows. Let us consider the case when we have two objects 𝔼 and 𝔽 of derived category of coherent sheaves on M̂_{m} → D^{2}\{0}. Then we can define Ω^{∗}(𝔼, 𝔽) in a way similar to (3.15). It is a differential graded bimodule over (Ω^{∗}(𝔼, 𝔼), d, ∘) and (Ω^{∗}(𝔽, 𝔽), d, ∘). It induces a 𝔎_{Ω^{∗}(𝔼,𝔼),}, 𝔎_{Ω^{∗}(𝔽,𝔽)} differential graded bimodule D^{∗}(𝔼, 𝔽).

Conjecture 3.5.2. If Lagrangian submanifold L_{i} corresponds to 𝔼_{i} by Conjecture 3.5.1 then the differential graded bimodule D^{∗}(𝔼_{i}, 𝔼_{j}) is chain homotopy equivalent to the differential graded bimodule D^{∗}(L_{i}, L_{j}).

We remark that Conjecture 3.5.1 implies 𝔎_{C(L_{i})} ≅ 𝔎_{Ω(𝔼_{i},𝔼_{i})}. And hence D^{∗}(𝔼_{i}, 𝔼_{j}), D^{∗}(L_{i}, L_{j}) are both 𝔎_{C(L_{i})} 𝔎_{C(L_{j})} differential graded bimodule.

We can continue and can state the coincidence of product structures. In the complex side it is Massey-Yoneda product induced by obvious composition operators ∘ : Ω^{k}(𝔼_{1}, 𝔼_{2}) ⊗_{Λ_{nov,0}^{ℤ[1/m]}} Ω^{ℓ}(𝔼_{2}, 𝔼_{3}) → Ω^{k+ℓ}(𝔼_{1}, 𝔼_{3}). The most natural way to do so is to use the notion of filtered A_{∞} category defined in [30]. We add two remarks.

Remark 3.5.3. In Conjecture 3.5.1 we took Λ_{nov,0}^{ℤ[1/m]} with large m as a coefficient ring. However the number m depends on 𝔼. So to have a better statement it is natural to use Λ_{nov,0}^{ℚ,0} or Λ_{nov}^{ℚ,0} in place of Λ_{nov,0}^{ℤ[1/m]} by taking limit. To go to the limit however, we have to clarify the following point. Let us denote the mth branched cover of D^{2} by D_{m}^{2}. (D_{m}^{2} is actually D^{2} but its coordinate is T^{1/m}.) Let M̂_{m} → D_{m}^{2}\{0}. Let M̂_{m}^{+} → D_{m}^{2} be a model of M̂_{m}. (that is an extension of it to the origin.) Let us consider M̂_{mm′} → D_{mm′}^{2}\{0}. We want to find M̂_{mm′}^{+} → D_{mm′}^{2} together with M̂_{mm′}^{+} → M̂_{m}^{+}. The naive choice, that is the fiber product D_{mm′}^{2} ×_{D_{m}^{2}} M̂_{m}^{+}, does not seem to be a good choice. For example this is not correct choice in case of Example 3.5.1. In the case of Abelian variety, [80] seems to give an appropriate choice. The author does not know whether the choice of such systems M̂_{m}^{+} → D_{m}^{2} together with maps M̂_{mm′}^{+} → M̂_{m}^{+} are know in the general case of, say, Calabi-Yau manifolds.

We suppose that such a choice is given. Then we can say of the coherent sheaves of such a projective system. In other words, we may consider a derived category of coherent sheaf over M̂ equipped with a kind of Etal topology. Then, A_{∞} category we obtain is defined over Λ_{nov}^{ℚ,0}. The coherent sheaves on Berkovich spectra [9] (as discussed in [70]) might be related to such objects. The author does not have enough knowledge to discuss them at the time of writing this article.

We finally explain a mirror to Example 1.6.4. Let us first consider T^{2} = ℝ^{2}/ℤ^{2}. Let x, y be coordinates of ℝ^{2}. We take a Lagrangian submanifold L_{k} of T^{2} defined by y = −kx, where k ∈ ℤ_{≥0}. We use mirror symmetry of elliptic curve by [86] then L_{0} will becomes to the trivial bundle on mirror 𝕋 (elliptic curve) and L_{k} becomes a complex line bundle L.

We remark in this case Floer’s condition π_{2}(T^{2}, L_{k}) = 0 is satisfied. Hence Floer homology is defined as in §3.3. We find that the intersections of L_{0} and L_{k} consists of k points. We can calculate its Maslov-Viterbo index and find that it is 1. Hence HF^{0}(L_{0}, L_{k}) = 0, HF^{1}(L_{0}, L_{k}) = ℂ^{k}. We thus in a situation mirror to Example 1.6.4.

Now we take two Lagrangian submanifolds L_{(1)} = L_{k} × L_{0} and L_{(2)} = L_{0} × L_{k} on direct product T^{2} × T^{2}. We take symplectic form ω on it such that (T^{2} × {0}) ∩ ω = ({0} × ω) ∩ ω. Then the mirror family is a product 𝕋_{q} × 𝕋_{q}. When q = e^{−z} converges to 0 the two factors will degenerates as in Example 3.5.1. It is easy to see that this family is maximally degenerate family.

Now it is easy to see that L_{(1)}, L_{(2)} are mirror of pr_{1}^{∗} L and pr_{2}^{∗} L, respectively. The mirror of E = pr_{1}^{∗} L⊕pr_{2}^{∗} L should be the union L_{(1)}∪L_{(2)}. It is immersed however. So the construction of §3.3,3.4 does not apply directly. However we can modify it as follows. Take C(L_{1}) ⊕ C(L_{2}) and add two generators [p^{12}], [p^{21}] to each intersection points p ∈ L_{1}∩L_{2}. There are k^{2} intersection points p ∈ L_{1} ∩ L_{2} which we write p_{ij} i, j = 1, ⋯, k. Now

C(L_{1}) ⊕ C(L_{2}) ⊕ ⨁_{ij}(Λ_{nov,0}[p_{ij}^{12}] ⊕ Λ_{nov,0}[p_{ij}^{21}])

is our complex. (See [1].) The boundary operator 𝔪_{1} is nontrivial on C(L_{i}) and hence 𝔪_{1} cohomology is

H^{∗}(L_{1}) ⊕ H^{∗}(L_{2}) ⊕ ⨁_{ij}(Λ_{nov,0}[p_{ij}^{12}] ⊕ Λ_{nov,0}[p_{ij}^{21}]).  (3.16)

We can define a structure of A_{∞} algebra on it. To calculate formal Kuranishi map we need to calculate the product structure. Since there is no holomorphic disk which bounds L_{i}, it follows that the operator 𝔪_{2} is equal to the usual cup product and 𝔪_{3} and higher are zero, on the first two components. We remark also that there is no pseudoholomorphic disk bounding union of two Lagrangian submanifolds L_{1}, L_{2}, other than trivial one. The trivial disk contributes

𝔪_{2}(p_{ij}^{12}, p_{ij}^{21}) = [p_{ij;1}], 𝔪_{2}(p_{ij}^{21}, p_{ij}^{12}) = −[p_{ij;2}]  (3.17)

Where p_{ij;1} is a singular 0 chain p_{ij} ∈ L_{(1)} and p_{ij;2} is a singular 0 chain p_{ij} ∈ L_{(2)}. All other products are zero. We remark that H^{2}(L_{i}) = H_{0}(L_{i}). Hence the degree 2 part of (3.16) is ℂ^{2}. (3.17) show that 𝔪_{2}(p_{ij}^{12}, p_{ij}^{21}) 𝔪_{2}(p_{ij}^{21}, p_{ij}^{12}) will be the first and second factor of H^{2}(L_{i}) ⊕ H^{2}(L_{j}) = ℂ^{2}, respectively. We thus find :

∑_{ij} x_{ij}[p_{ij;1}] + y_{ij}[p_{ij;2}] ↦ (∑_{ij} x_{ij}y_{ij}, − ∑_{ij} x_{ij}y_{ij})

is the (nonzero part of) Kuranishi map. This map coincides with one in Example 1.6.4.

In this example, the operator is independent of q = e^{−z}. We can find an example that the operator actually depends on q (and is a theta function of it) in the case M^{∨} = 𝕋^{3}. See Chaper 4 of [27] and Chaper 7 of (2000 December version of) [33]. More example seems to be found in Physics literature.

References

[1] M. Akaho. An Intersection theory of Lagrangian Immersion. PhD thesis, Kyoto Univ., 1997.

[2] M. Alexandrov, A. Schwarz, M. Kontsevich, and O. Zaboronsky. The geometry of the master equation and topological quantum field theory. Intern. J. Modern Phys. A, 12:1405–1429, 1997.

[3] V. Arnol′d. Sur une propriété topologique des applications globalement canoniques de la mécanique classique. C. R. Acad. Sci. Paris, 261:3719– 3722, 1965.

[4] M. Artin. Deformation of Singularities. Tata Institute, 1976.

[5] S. Axerlod and I. Singer. Chern-Simons perturbatoin theory I. In S. Catto and A. Rocha, editors, Proc. XXth International Conference on Differential Theoretic method in Theoretical Physics, pages 3 – 45. World Scientifique, 1991.

[6] S. Axerlod and I. Singer. Chern-Simons perturbatoin theory II. J. of Diff. Geom., 39:173–213, 1991.

Homological mirror symmetry. 94

[7] S. Barannikov and M. Kontsevich. Frobenius manifolds and formality of Lie algebras of polyvector fields. Internat. Math. Res. Notices, 4:201–215, [8] K. Behrend. and B. Fantechi. The intrinsic normal cone. Invent. Math.,

128(1):45–88, 1997.

[9] V. Berkovich. Spectral theory and analytic geometry over non-Archimedean fields. American Mathematical Society, Providence, RI, 1990.

[10] M. Bershadsky, M. Ceccoti, S. Ooguri, and C. Vafa. Kodaira-Spencer theory of gravity and exact results for quantum string amplitude. Commun. Math. Phys., 165:311–427, 1994.

[11] J.M. Boardman and R.M. Vogt. Homotopy Invariant Algebraic Structures on Topological Spaces. Springer, 1973. Lecture Notes in Math. 347.

[12] S. Bosch, U. Güntzer, and R. Remmert. Non-Archimedean Analysis. Springer, 1984. Grundlehren der mathematischen Wissenschaften.

[13] N. Bourbaki. Universe. In Théorie des Topos et Cohomologie Etale des Schémas, pages 185 – 217. Springer, 1970. Lecture Notes in Math. 269.

[14] I. Ciocan-Fontanine and M. Kapranov. Derived Quot schemes. Ann. Sci. École Norm. Sup. (4), 34(3):403–440, 2001.

[15] P. Deligne. and D. Mumford. The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math., 36:75–109, 1969.

[16] S. Donaldson. Connection, cohomology, and intersectoion forms of 4manifolds. J. of Diff. Geom., 24:275–341, 1986.

[17] S. Donaldson. Irrationality and the h-cobordism conjecture. J. Differential Geom., 26(1):141–168, 1987.

[18] S. Donaldson. Polynomial invariants for smooth four-manifolds. Topology, 29(3):257–315, 1990.

[19] A. Douady. Le problém de modules pour les sou-espaces analytiques compact d’un espace analytique donné. Ann. Inst. Fourier, 16:1–95, 1966.

[20] A. Douady, A.Grothendieck, and C.Houzel. Familles d’Espace Complexes et Fondements de La Geométrié Analytique. Secrétariat mathématique, 1960/61. Séminaire Henri Cartan 13.

[21] Y. Eliashberg, A. Givental, and H. Hofer. Introduction to symplectic field theory. Geom. Funct. Anal., Special Volume, Part II:560–673, 2000. GAFA 2000 (Tel Aviv, 1999).

[22] A. Floer. Morse theory for Lagrangian intersectoins. J. Diff. Geom., 28:513 – 547, 1988.

[23] K. Fukaya. Morse homotopy, A^{∞}-categories, and Floer homologies. In H. J. Kim, editor, Proc. of the 1993 Garc Workshop on Geometry and Topology, volume 18 of Lecture Notes series, pages 1 – 102. Seoul Nat. Univ., 1993. http://www.kusm.kyoto-u.ac.jp/ ∼fukaya/ fukaya.html.

[24] K. Fukaya. Morse homotopy and Chern-Simons perturbation theory. Comm. Math. Phys., 181(1):37–90, 1996.

[25] K. Fukaya. Floer homology for 3-manifolds with boundary I. preprint, never to appear, http://www.kusm.kyoto-u.ac.jp/∼fukaya fukaya.html, 1997.

[26] K. Fukaya. Anti-self-dual equation on 4-manifolds with degenerate metric. Geom. Funct. Anal., 8(3):466–528, 1998.

[27] K. Fukaya. Mirror symmetry of abelian variety and multi theta functions. to appear in J. Alg. Geom., 1998.

[28] K. Fukaya. Floer homology and mirror symmetry I. preprint, http://www.kusm.kyoto-u.ac.jp/∼fukaya fukaya.html, 1999.

[29] K. Fukaya. Symplectic Geometry. Iwanami Shoten Publisher, Tokyo Japan, 1999. In Japanese.

[30] K. Fukaya. Floer homology and mirror symmetry II. to appear in Advanced Studies in Pure Math. Minimal Surfaces, Geometric Analysis and Symplectic Geometry, 2001.

[31] K. Fukaya. Floer homology for families - report of a project in progress -. to appear in Contemporary Math., 2001.

[32] K. Fukaya and Y.G. Oh. Zero-loop open strings in the cotangent bundle and Morse homotopy. Asian J. Math., 1:99 – 180, 1997.

[33] K. Fukaya, Y.G. Oh, H.Ohta, and K.Ono. Langrangian intersection Floer theory -anomaly and obstruction-. preprint, http://www.kusm.kyotou.ac.jp/∼fukaya/ fukaya.html, 2000.

[34] K. Fukaya and K. Ono. Arnold conjecture and Gromov-Witten invariants. Topology, 38:933 – 1048, 1999.

[35] S. Gelfand and Y. Manin. Methods of homological algebra. Springer-Verlag, Berlin, 1996. Translated from the 1988 Russian original.

[36] D. Gilberg and N. Trudinger. Elliptic Partial Differential Equations of Second Order, volume 224. Springer, 1970.

[37] W. Goldman and J.Millson. The deformation theory of representation of fundamental groups in compact Kähler manifolds. Publ. Math. I.H.E.S., 67:43–96, 1988.

[38] W. Goldman and J.Millson. The homotopy invariance of Kuranishi space. Illinois J. Math., 34:337–367, 1990.

[39] P. Griffiths, editor. Topics in transcendental algebraic geometry, Princeton, NJ, 1984. Princeton University Press.

[40] P. Griffiths and J. Harris. Principles of algebraic geometry. Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics.

[41] P. Griffiths and J. Morgan. Rational homotopy theory and differential form, volume 16 of Progress in Math. Birkhäuser, 1981.

[42] M. Gromov. Pseudoholomorphic curves in symplectic manifolds. Invent. Math., 82(2):307–347, 1985.

[43] M. Gromov. Metric structures for Riemannian and non-Riemannian spaces. Birkhäuser Boston Inc., Boston, MA, 1999. Based on the 1981 French original [MR 85e:53051], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.

[44] M. Gross and P. Wilson. Large Complex structure limits of K3 surfaces. preprint, math.DG/0008018l, 2000.

[45] A. Grothendieck and C.Houzel. Fondements de la Géometrie Algébraique. Secrétariat mathématique, 1962.

[46] V. Gugenheim and J. Stasheff. On perturbations and A_{∞}-structures. Bull. Soc. Math. Belg. Sér. A, 38:237–246 (1987), 1986.

[47] R. Hartshorne. Residues and Duality, volume 20 of Lecture notes in Mathematics. Springer-Verlag, 1966.

[48] R. Hartshorne. Algebraic geometry. Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52.

[49] V. Hinich and V. Schechtman. Deformation theory and Lie algebra homology. I. Algebra Colloq., 4(2):213–240, 1997.

[50] V. Hinich and V. Schechtman. Deformation theory and Lie algebra homology. II. Algebra Colloq., 4(3):291–316, 1997.

[51] H. Hofer. and D. Salamon. Floer homology and Novikov rings. In The Floer memorial volume, pages 483–524. Birkhäuser, Basel, 1995.

[52] H. Hofer and E. Zehnder. Symplectic invariants and Hamiltonian dynamics. Birkhäuser Verlag, Basel, 1994.

[53] Y. Itagaki. Convergence of the normalized solution of Maurer-Cartan equation in the Baranikov-Kontsevich construction, 1999. Master Thesis, Kyoto Univ.

[54] T. V. Kadeishvili. The algebraic structure in the homology of an A(∞)- algebra. Soobshch. Akad. Nauk Gruzin. SSR, 108(2):249–252 (1983), 1982.

[55] M. Kashiwara and P.Shapira. Sheaves on manifolds, volume 292 of Grund. der Math. Springer-Verlag, 1990.

[56] B. Keller. Introduction to A-infinity algebras and modules. math.RA/9910179.

[57] M. Khovanov and P. Seidel. Quivers, Floer cohomology, and braid group actions. preprint, MathQA/0006056.

[58] S. Kobayashi. Differential geometry of complex vector bundles. Princeton University Press, Princeton, NJ, 1987.

[59] K. Kodaira. On compact analytic surfaces II. Ann. Math. 99, 77:523 – 626, 1963.

[60] K. Kodaira, L. Nirenberg, and D.C. Spencer. On the existence of deformations of complex analytic structures. Ann. of Math., 68:450–459, 1958.

[61] K. Kodaira and D.C. Spencer. On deformations of complex analytic structures I, II. Ann. of Math., 67:328–466, 1958.

[62] K. Kodaira and D.C. Spencer. A theorem of completeness for complex analytic structures. Acta Math., 100:281–294, 1958.

[63] K. Kodaira and D.C. Spencer. On deformations of complex analytic structures III. Ann. of Math., 71:43–76, 1960.

[64] M. Kontsevich. A_{∞} algebras in mirror symmetry. MPI Arbeitstagung talk, 1993.

[65] M. Kontsevich. Formal (non)commutative symplectic geometry. In The Gelfand Mathematical Seminars, 1990–1992. Birkhäuser, 1993.

[66] M. Kontsevich. Feynman diagram and low dimensional topology. In Proceeding of the First European Congress of Mathematics, pages 97–122. Birkhäuser, 1994.

[67] M. Kontsevich. Enumeration of rational curves via torus actions. In The moduli space of curves (Texel Island, 1994), pages 335–368. Birkhäuser Boston, Boston, MA, 1995.

[68] M. Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, Zürich, volume I, pages 120 – 139. Birkhäuser, 1995.

[69] M. Kontsevich. Deformation quantization of poisson manifolds i. preprint, q-alg/9709040, 1997.

[70] M. Kontsevich and Y. Soibelman. Homological mirror symmetry and torus fibrations. In Symplectic Geometry and Mirror Symmetry. World Sci. Press, 2001.

[71] M. Kuranishi. On the loccally complete families of complex analytic structures. Ann. of Math., 75:536–577, 1962.

[72] J. Li and G. Tian. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc., 11(1):119–174, 1998.

[73] B. Lian, A. Todorov, and S.T. Yau. Maximal unipotent monodromy for complete intersection CY manifolds. preprint, math.AG/0008061, 2000.

[74] Y. Manin. Frobenius manifolds, quantum cohomology, and moduli spaces. American Mathematical Society, Providence, RI, 1999.

[75] D. McDuff and D. Salamon. J-holomorphic curves and quantum cohomology, volume 6 of University Lecture Seires. Amer. Math. Soc., 1994.

[76] D. McDuff and D. Salamon. Introduction to Symplectic topology. Oxford Science PUbl., 1995.

[77] S. Merkulov. Frobenius_{∞} invariants of homotopy Gerstenhaber algebras. I. Duke Math. J., 105(3):411–461, 2000.

[78] S.A. Merkulov. Strongly homotopy algebras of a Kähler manifold. Internat. Math. Res. Notices, 3:153–164, 1999.

[79] D. Mumford. Geometric invariant theory. Springer, 1965.

[80] A. Néron. Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Inst. Hautes Études Sci. Publ.Math. No., 21:128, 1964.

[81] S. Novikov. Multivalued functions and functional - an analogue of the Morse theory. Sov. Math. Dokl., 24:222 – 225, 1981.

[82] Y.G. Oh. Floer cohomology of Lagrangian intersections and pseudoholomorphic disks I,II. Comm. Pure and Appl. Math., 46:949 – 994, 995 – 1012, 1993.

[83] Y.G. Oh. Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings. Internat. Math. Res. Notices, 7:305–346, 1996.

[84] H. Ohta. Obstraction to and deformation of Lagrangian intersection Floer cohomology. In Symplectic Geometry and Mirror Symmetry. World Sci. Press, 2001.

[85] V.P. Paramodov. Deformation of complex spaces. Russian Math. Surveys, 31-3:129–197, 1976.

[86] A. Polishchuk and E. Zaslow. Categorical mirror symmetry: the elliptic curve. Adv. Theor. Math. Phys., 2:443 – 470, 1998.

[87] Z. Ran. Thickening Calabi-Yau moduli spaces. In Mirror symmetry, II, pages 393–400. Amer. Math. Soc., Providence, RI, 1997.

[88] D.B. Ray and I.M. Singer. R-torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7:145–210, 1971.

[89] Y. Ruan. Topological sigma model and Donaldson-type invariants in Gromov theory. Duke Math. J., 83(2):461–500, 1996.

[90] Y. Ruan and G. Tian. A mathematical theory of quantum cohomology. J. Differential Geom., 42(2):259–367, 1995.

[91] K. Saito. Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci., 19(3):1231–1264, 1983.

[92] M. Schlessinger and J. Stasheff. The Lie algebra structure of tangent cohomology and deformation theory. J. Pure Appl. Algebra, 38(2-3):313–322, 1985.

[93] V.P. Schlessinger. Functor from Artin rings. Trans. Amer. Math. Soc., 130:208–222, 1968.

[94] P. Seidel. More about vanishing cycles and mutations. In Symplectic Geometry and Mirror Symmetry. World Sci. Press, 2001.

[95] P. Seidel. Vanishing cycles and mutations. In Proc. of the 3rd European Congress. Birkhauser, 2002.

[96] P. Seidel and R. Thomas. Braid group actions on derived categories of coherent sheaves. Duke Math. J., 108(1):37–108, 2001.

[97] V. Smirnov. Simplicial and operad methods in algebraic topology. American Mathematical Society, Providence, RI, 2001. Translated from the Russian manuscript by G. L. Rybnikov.

[98] J. Stasheff. Homotopy associativity of H-spaces. I, II. Trans. Amer. Math. Soc. 108;275-292, 1963 ; ibid., 108:293–312, 1963.

[99] J. Stasheff. Deformation theory and the Batalin-Vilkovisky master equation. In Deformation theory and symplectic geometry (Ascona, 1996), pages 271–284. Kluwer Acad. Publ., Dordrecht, 1997.

[100] A. Strominger, S.T. Yau, and E. Zaslow. Mirror symmetry is T -duality. Nuclear Phys. B, 479(1-2):243–259, 1996.

[101] D. Sullivan. Infinitesimal calculations in topology. Publ. I.H.E.S., 47:269– 331, 1978.

[102] C. Taubes. Self-dual connections on 4-manifolds with indefinite intersection matrix. J. of Diff. Geom., 19:517–560, 1984.

[103] R. Thomas. An obstructed bundle on a Calabi-Yau 3-fold. Adv. Theor. Math. Phys., 3(3):567–576, 1999.

[104] R. Thomas. Moment map, monodromy and mirror manifolds. In Symplectic Geometry and Mirror Symmetry. World Sci. Press, 2001.

[105] G.N. Tyurina. Locally semiuniversal flat deformations of isolated singuralities of complex spaces. Izv. Akad. Nauk. SSSR Ser. Math., 33:1026–1058, 1969.

[106] F. Warner. Foundations of Differential Manifolds and Lie Groups, volume 94. Springer, 1971.

[107] E. Witten. Topological sigma models. Comm. Math. Phys., 118(3):411–449,