Non-Commutative Periods and Mirror Symmetry in Higher Dimensions

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Non-Commutative Periods and Mirror Symmetry in Higher Dimensions

Serguei Barannikov

To cite this version:

Serguei Barannikov. Non-Commutative Periods and Mirror Symmetry in Higher Dimen- sions. Communications in Mathematical Physics, Springer Verlag, 2002, 228 (2), pp.281-325.

�10.1007/s002200200656�. �hal-00486486v2�

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arXiv:math/9903124v1 [math.AG] 22 Mar 1999

Generalized periods and mirror symmetry in dimensions n > 3

^∗

Sergey Barannikov^†

Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, 91440, France

Abstract

The predictions of the Mirror Symmetry are extended in dimensions n > 3 and are proven for projective complete intersections Calabi-Yau varieties. Precisely, we prove that the total collection of rational Gromov- Witten invariants of such variety can be expressed in terms of certain invariants of a new generalization of variation of Hodge structures attached to the dual variety.

To formulate the general principles of Mirror Symmetry in arbitrary dimension it is necessary to introduce the “extended moduli space of complex structures”M. We show that the moduli spaceMis the base of generalized variation of Hodge structures. An analogM → ⊕kH^k(Xⁿ,C)[n−

k] of the classical period map is described and is shown to be a local isomorphism. The invariants of the generalized variations of Hodge structures are introduced. It is proven that their generating function satisfies the system of WDVV-equations exactly as in the case of Gromov-Witten invariants.

The basic technical tool utilized is the Deformation theory.

Contents

1 Introduction 2

2 Basics of deformation theory 5

2.1 Moduli spaces via differential graded Lie algebras . . . . 5 2.2 Equivalence of Deformation functors . . . . 7 2.3 Formal manifolds and odd vector fields . . . . 8 3 Extended moduli spaces of complex structures 9 3.1 The sheaf of graded Lie algebras . . . . 9 3.2 The (formal) moduli space associated with ΛT . . . . 10 3.3 Moduli space ofA∞−deformations of D^bCoh(X). . . . 12

∗Based on the author’s Ph.D. thesis accepted December 1998

†Alfred P. Sloan Fellow

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4 Generalized periods. 14

4.1 Classical periods. . . . 14

4.2 The generalized periods mapM → ⊕kH^k(X,C)[n−k] . . . . 15

5 Invariants of generalized variations of Hodge structures 21 5.1 Nonlinear limits of families of affine spaces . . . . 22

5.2 Tangent sheafTM. . . . 23

5.3 Family of moduli spaces . . . . 23

5.4 Affine structure onM~,~6= 0 . . . . 25

5.5 Flags and flat connections overA¹_C . . . . 28

5.6 The bundleMd^W →CP¹ and the map Φ^W. . . . 29

5.7 Canonical connection on M →f A¹_Cand symmetry vector field. . . 30

5.8 Invariants of the generalized VHS. . . . 32

5.9 Flat metric onM. . . . 34

5.10 Flat identity . . . . 38

5.11 Dependence on the base pointπ∈ M^classical . . . . 39

5.12 Frobenius manifolds . . . . 40

5.13 Family of Frobenius manifold structures onM . . . . 41

6 Mirror Symmetry in dimensionsn >3 41 6.1 Projective hypersurfaces . . . . 42

6.2 Complete intersections . . . . 42

6.3 Weight filtration . . . . 43

6.4 q-expansion . . . . 44 6.5 Invariants of the generalized VHS and Gromov-Witten invariants 45

1 Introduction

This work is devoted to the description of the collection of all rational Gromov- Witten invariants of Calabi-Yau varieties in arbitrary dimension via the invariants of a certain new generalization of variations of Hodge structures attached to the mirror dual varieties.

The first discovery in this direction was the striking prediction made by Candelas, de la Ossa, Green and Parkes [COGP] for the numbers of rational curves on quintic threefold inP⁴in terms of the periods of some “dual” family of Calabi-Yau threefolds.

These numbers of rational curves on quintic are the simplest examples of rational Gromov-Witten invariants. According to the theory of Gromov-Witten invariants (see [KM]) the collection of all rational Gromov-Witten invariants of a projective algebraic manifoldY is encoded in the generating function

(Potential) F(t), t ∈ H^∗(Y,C) (1.1) considered as series over the semigroup ringQ[B] whereB is the semigroup of effective one-dimensional algebraic cycles modulo numerical equivalences. The total space of the cohomology groupsH^∗(Y,C) is considered here as a complex

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supermanifold. It is convenient to choose a graded basis{∆a}inH^∗(Y,C). Let us denote by {t^a} the dual set of linear coordinates and choose some generic representatives {Γa} of the homology classes dual to {∆a} . Intuitively, the Taylor coefficientsN(a1, . . . , an;β) in the series expansion

F(t) = X

n;a1,...,an;β

1

n!N(a1, . . . , an;β)q^βt^a¹. . . t^aⁿ (1.2) count the numbers of algebraic maps f : C → Y where C is a rational curve withnmarked points{x1, . . . , xn} such thatf(xi)∈Γai andf∗([C]) =β ∈B.

However a lot of work is needed in order to give the precise definition for these numbers (see [BM]).

The conjectures of [COGP] were partially extended to higher dimensions in [BvS] (see also [GMP]) where the formulas describing hypothetically a subset of the Gromov-Witten invariants corresponding to the restriction of the third derivative of the potential to the subspace of the second cohomology group

∂³F(t)_t∈H₂_(Y,C) (1.3)

were proposed.

The next important achievement was made by A. Givental ([G]) who has established the conjectures from [COGP] and [BvS] which allow to express

∂³F(t)|H²(Y,C) in terms of the classical periods associated with the mirror dual family.

We construct a generalization of the classical periods map in order to find the expression for the whole generating functionF(t). In other words the aim of our work is to identify the total collection of rational Gromov-Witten invariants for the Calabi-Yau varieties of dimensionn >3 with certain invariants coming from a generalization of the theory of variations of Hodge structure on the dual varieties. We prove the coincidence of the two types of invariants for projective complete intersections Calabi-Yau varieties and its duals.

In section 3 the “extended moduli space of complex structures”Mis introduced. LetX be a complex manifold with trivial canonical sheaf. Let us denote by JX the corresponding complex structure on the underlying C^∞−manifold XC^∞. Recall that the tangent space to the classical moduli space of complex structures at a smooth point is identified canonically with H¹(X,TX). Ac- cording to Kodaira-Spencer theory a complex structure onXC^∞ close toJX is described by an elementγ∈Ω^0,1(X,TX) satisfying Maurer-Cartan equation

∂γ¯ +1

2[γ, γ] = 0 (1.4)

The correspondence: complex structureJ →γJ can be described as follows. A complex structure on X may be defined as a decomposition TR⊗C=T ⊕T of the complexified tangent space into the sum of complex conjugate subspaces which constitute formally integrable distributions (Newlander-Nirenberg theorem). A deformation of such decomposition corresponds to a graph of a linear

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mapT →T, i.e. an elementγ∈Ω^0,1(X,TX). The equation (1.4) is the condition of the formal integrability ofT. The elementsγ, γ^′describing the equivalent complex structures are related via the action of the group corresponding to the Lie algebra Ω^0,0(X,TX). The extended moduli space of complex structuresM is described similarly by the elements¹

γ ∈ ⊕p,qΩ^0,q(X,Λ^pTX)[p−q−1] (1.5) satisfying the eq. (1.4). The technic of the deformation theory which allows one to associate the moduli spaceM with the differential graded Lie algebra g= ⊕q,pΩ^0,q(X,Λ^pTX)[p−q−1] is recalled in §2. A trick from the rational homotopy theory (see [DGMS]) allows one to prove that the moduli spaceMis smooth with the tangent space at the base point [X] canonically isomorphic to

⊕p,qH^q(X,Λ^pTX)[p−q] (1.6) We demonstrate in§3.3 using the formality theorem from [K1] that the supermoduli spaceMparametrizes theA∞−deformations ofD^bCoh(X).

The section§4 is devoted to the descrpition of the generalized period map.

The condition c1(TX) = 0 implies that there exist nonvanishing holomorphic n−form ΩXt,n= dimCX for everyt∈ M^classical. If one fixes a hyperplaneL⊂ Hⁿ(X,C) transversal to the last component of the Hodge filtrationF^≥nat the base point [X0] then it allows one to define the classical period mapM^classical→ Hⁿ(X,C). It sends a point (t) of the moduli space of complex structures to the cohomology class of the holomorphicn−form Ω^L_X_t normalized so that [Ω^L_X_t]− [ΩX0]∈L. The theorem 1 and the proposition 4.2.4 describe a generalization of this map for the extended moduli spaceM. HereW is an increasing filtration on the total sum of cohomology groupsH^∗(X,C) complementary to the Hodge filtration. It turns out that our generalized period map

Π^W :M → ⊕kH^k(X,C)[n−k], n= dimCX (1.7) is locally an isomorphism.

The map Π^W arises from a structure which may be understood as certain generalization of the variation of Hodge structures (genaralized VHS) having the moduli spaceMas the base. This is explained in sections 5. We introduce also in this section the invariants of such generalized VHS. One of the important properties of these invariants is the fact that their generating function satisfies the system of WDVV-equations exactly as in the case of Gromov-Witten invariants.

In the section 6 we prove that the rational Gromov-Witten invariants of projective complete intersections Calabi-Yau manifolds coincide with the invariants of the generalized VHS introduced in section 5 which correspond to their mirror pairs. One reformulation of this result is the equality

C_αβ^γ (τ) =X

i

((∂Π)⁻¹)^γ_i∂α∂βΠⁱ

1For a graded objectAwe denote byA[k] the tensor product ofAwith the trivial object concentrated in degree (−k)

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where∂_αβγ³ F =P

δgγδC_αβ^δ (τ) is the third derivative of the generating function for rational Gromov-Witten invariants and Πⁱ(τ) is the vector of generalized periods depending on the point of the extended moduli space M associated with the mirror dual variety.

The importance of the problem of constructing a moduli space with the properties similar to M for understanding the Mirror Symmetry phenomena was anticipated in [W1]. It was conjectured in [K2] that such a moduli space is related with hypothetical moduli space ofA∞−deformations ofD^bCoh(X). The definition of the moduli spaceMappeared for the first time in [BK]. The relations of our results with homological mirror symmetry conjecture are discussed in§3.3 and at the end of§5.4.

Acknowledgements. I am thankful to my dissertation adviser Maxim Kont- sevich for support and constant interes in my work. I would like also to acknowl- edge the stimulating atmosphere of the Institute des Hautes Etudes Scientifiques in Bures-sur-Yvette where this work was conducted. My research is supported by the Alfred P. Sloan Fellowship.

All the moduli spaces which we consider are Z−graded manifolds, in other words they are supermanifolds with additionalZ−grading on the structure sheaf compatible with theZ2-grading. To simplify notations we replace degt^a bya in superscripts.

2 Basics of deformation theory

We recall here the basics of the Deformation theory which is the main technical tool used throughout the text. The material presented here is well-known to the specialists (see for example [SchSt],[GM],[K1]). The Deformation theory was developed in the work of a number of mathematicians (P.Deligne, V.Drinfeld, B.Feigin, W.Goldman and J.Millson, A.Grothendieck, M.Kontsevich, M.Schle- ssinger, J.Stasheff. . . ). Unfortunately many of their results remained unpub- lished for a long time.

We assume that we work over a fieldk of characteristic zero.

2.1 Moduli spaces via differential graded Lie algebras

The principal strategy of Deformation theory may be described as follows. Given some mathematical structure²Aone can associate toAthe differential graded Lie algebra³ Der^∗(A) defined canonically up to quasi-isomorphisms. Recall that the differential graded Lie algebra is a graded vector space equipped with

2for exampleAcan be an associative algebra, complex manifold, vector bundle etc.

3sometimes it is more convenient to work with more general notion ofL∞−algebra

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differential and graded skew-symmetric bracket satisfying a list of axioms g=⊕kg^k, d:g^k→g^k+1, d²= 0, [·,·] :g^k⊗g^l→g^k+l

d[γ1, γ2] = [dγ1, γ2] + (−1)^γ¹[γ1, dγ2], [γ2, γ1] =−(−1)^γ¹^γ²[γ1, γ2] (2.1) [γ1[γ2, γ3]] + (−1)^γ³^(γ¹^+γ²⁾[γ3[γ1, γ2]] + (−1)^γ¹^(γ²^+γ³⁾[γ2[γ3, γ1]] = 0 The correspondenceA→Der^∗(A) may be viewed as a kind of “derived functor”(or rather “derived correspondence”) with respect to the standard correspondence A →Der(A) which associates to A its Lie algebra of infinitesimal automorphisms. Then the equivalence classes of deformations of the structure A are described in terms ofDer^∗(A). In the standard approach of the deformation theory one considers inductevely the deformations up to the given order 1,2, . . . , N, . . .. In other words, the algebras of functions on the standard pa- rameter spaces of deformations are the Artin algebras with residue field k (in our context they will beZ−graded generally). Recall that such an algebraAis isomorphic to a direct sumk⊕m wherekis a copy of the base field andmis a finite-dimensional commutative nilpotent algebra (Z−graded in general). Even more concretly, any Artin algebra with the residue fieldk is isomorphic to an algebra of the formk[ti]i∈S/I, whereI is an idealI⊃t^Nk[ti] andS is a finite set of (graded) generators. Then the deformationsAe_/Aof the structureAover an Artin algebraAare described by solutions to Maurer-Cartan equation

dγ+1

2[γ, γ] = 0, γ∈(Der^∗(A)⊗m)¹ (2.2) The equivalent deformations (Ae/A)1≃(Ae/A)2correspond to the solutions from the same orbit of the group associated with the nilpotent Lie algebra (Der^∗(A)⊗

m)⁰. This Lie algebra acts on the space (g⊗m)¹ by

α∈(g⊗m)⁰→ γ˙ =dα+ [α, γ] (2.3) It is convinient to introduce functor Defg associated with a differential graded Lie algebrag

Defg(A) ={dγ+1

2[γ, γ] = 0|γ∈(g⊗m)¹}/Γ⁰(A) (2.4) which acts from the category of Artin algebras with residue fieldkto the category of sets. We will denote Def_g⁰the corresponding functor in the more widely known case when only Artin algebras concentrated in degree 0 are involved.

Sometimes we denote the more general functor onZ−graded Artin algebras by Def_g^Z. The description above of the deformations ofAin terms ofDer^∗(A) may be rephrased now by saying that the functor which associates to Athe set of equivalence classes of deformationsAe/Ais isomorphic to DefDer^∗(A).

Furthermore, in the cases when the actual moduli space of deformations of Aexists, the functor Defgis equivalent to the functor Homcontinuous( ˆO,·) where Oˆ is the pro-Artin⁴ algebra which is equal to the completion of the algebra of functions on the actual moduli space of deformations ofA.

4=projective limit of Artin algebras

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Given a differential graded Lie algebragsuch that the functor Defgis equivalent to the functor represented by some pro-Artin algebraOg one can define the formal moduli spaceMg associated to gby proclaimingOg to be “the algebra of functions onMg”.

The basic tool to deal with differential graded Lie algebras and formal moduli spaces associated to them is provided by the theorem on quasi-isomorphisms which is described in the next subsection.

2.2 Equivalence of Deformation functors

We need to recall first the following homotopy generalization of the notion of the morphism between two differential graded Lie algebras.

A sequence of linear maps

F1:g1→g2

F2: Λ²(g1)→g2[−1] (2.5) F3: Λ³(g1)→g2[−2]

· · ·

defines anL∞−morphism of differentialZ−graded Lie algebrasg1 andg2 if dFn(γ1∧. . .∧γn)−X

i

±Fn(γ1∧. . .∧dγi∧. . .∧γn) =

=1 2

X

k,l≥1, k+l=n

1 k!l!

X

σ∈Sn

±[Fk(γσ(1)∧. . .∧γσ(k)), Fl(γσ(k+1)∧. . .∧γσ(k+l))] +

+X

i<j

±Fn−1([γi, γj]∧γ1∧. . .∧γn) (2.6) In particular, the first map is a morphism of complexes which respects the Lie brackets up to homotopy defined by the second map, which itself respects the Lie brackets up to higher homotopies and so on.

An L∞−map F = {Fn}, F : g1 → g2 defines a natural transformations of the functorsF∗: Defg₁ →Defg₂. Ifγ∈(g1⊗m)¹is a solution to Maurer-Cartan equation then

F∗(γ) :=

X∞ n=1

1

n!Fn(γ∧. . .∧γ) (2.7) is a solution to Maurer-Cartan equation in (g2⊗m)¹

Recall that an L∞−morphism F = {Fn}, F : g1 → g2 is called quasi- isomorphism if its linear part F1 induces an isomorphism of cohomology of complexes (g1, d1) and (g2, d2).

Basic Theorem of Deformation Theory.If F = {Fn} : g1 → g2 is an L∞−morphism from g1 to g2 which is a quasi-isomorphism then the natural transformation of deformation functorsF∗:Def_g₁ →Def_g₂ is an isomorphism.

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2.3 Formal manifolds and odd vector fields

Here we recall the geometric picture of the theory presented above. In particular we give an interpretation of the notions ofL∞−morphism and of the moduli spaceMg described by differential graded Lie algebrag.

The set of mapsFn: Λⁿg1→g2[1−n] can be thought of as the set of Taylor coefficients of a formal mapF :g1[1]→g2[1] preserving the zero. Namely, the algebra of formal power series on a (super) vector spaceg[1] can be identified with the dual to the free cocommutative coalgebra Symm(g[1]) cogenerated by g[1]. Then all geometric objects associated with theZ−graded (formal) manifold g[1] may be described in terms of this coalgebra. In particular, a map of formal manifoldsg1[1] →g2[1] corresponds to a coalgebra morphism Symm(g1[1])→ Symm(g2[1]). A map of free algebras is defined uniquely by its restriction to the set of generators. Similarly, a map of coalgebras Symm(g1[1])→Symm(g2[1]) is defined uniquely by its components Fn : Sⁿ(g1[1]) → g2[1], n ≥0. Recall that in the category ofZ−graded vector spacesSⁿ(V[1]) = Λⁿ(V)[n]. The set of maps (2.5) defines naturally a (formal power series) map ofZ−graded manifolds g1[1] →g2[1] preserving the origins, since the corresponding constant term is zero: F0 = 0. The condition (2.6) can be translated into geometric terms as follows.

The structure of the differential graded Lie algebra on a graded vector space gis interpreted as a degree one vector fieldQg, Q²_g= 0 on the vector spaceg[1]

preserving the origin. Namely, the structure maps

d:g→g[1], [·,·] :S²(g[1])→g[2] (2.8) are the components of uniquely defined degree one derivationQgof the coalgebra Symm(g). It corresponds to the vector field

Q(γ) =dγ+1

2[γ, γ] (2.9)

The relations (2.1) which are satisfied by the structure maps (2.8) are exactly equivalent to the condition⁵ Q²_g= 0.

The equations (2.6) are then reformulated as the single condition

F∗(Qg1) =Qg2 (2.10)

This picture implies that the moduli space described by a differential graded Lie algebragcan be thought of as a “nonlinear cohomology” of the operatorQg. The precise meaning of this may be described as follows. One has a subscheme

”KerQ” :={γ∈g[1]|Q(γ) = 0} (2.11) of zeroes of the vector fieldQg. The vector fields of the form [Qg, α] whereαis an arbitrary constant vector field ong[1] of degree−1 span a distribution “ImQ”.

5The differential graded Lie algebras correspond to vector fields whose Taylor expansion contains only linear or quadratic terms. An arbitrary (formal) vector fieldQ, Q²= 0, Q(0) = 0 of degree one ong[1] is equivalent by definition to the structure ofL∞−algebra ong

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This distribution is tangent to the subscheme “KerQ” since [Q,[Q, α]] = 0. The moduli spaceMg corresponds in geometric terms to the “quotent” of the sub- manifold “KerQ” defined by the zeroes of the vector fieldQby the distribution

“ImQ”. The functor Defg describing the moduli space Mg is then identified with the natural functor describing the “quotent” “(KerQ/ImQ)”.

3 Extended moduli spaces of complex structures

We recall here following [BK] the definition of the extended moduli space of complex structuresM. We also recall the arguments showing that it is a smooth moduli space with the tangent space

T[X]=⊕p, qH^q(X,Λ^pTX)[p−q]

3.1 The sheaf of graded Lie algebras

LetX,dimCX =nbe a smooth projective algebraic manifold such thatc1(TX)∈ Pic(X) is zero. We assume thatXis defined over the complex numbers although most of our constructions are valid forX defined over an arbitrary algebraically closed fieldkof characteristic zero.

Consider the coherent sheaf ofZ−graded Lie algebras

g=⊕kg^k[k], g^k:= Λ^1−kTX (3.1) endowed with the Schouten-Nijenhuys bracket. This bracket is uniquely defined by the following conditions:

(1) Forv1, v2 ∈ TX the commutator [v1, v2] is the standard bracket on vector fields.

(2) For v ∈ TX, f ∈ OX the commutator [v, f] ∈ OX is the Lie derivative Lievf.

(3) The wedge product and the Lie bracket define the structure of an odd Poisson algebra(=Gerstenhaber algebra) ong[−1], in other words

[v1, v2∧v3] = [v1, v2]∧v3+ (−1)^(v¹^+1)v²v2∧[v1, v3] (3.2) The technic of Deformation theory associates to the sheafga (formal) moduli space.

To describe this moduli space forX/C6 let us take the Dolbeault resolution ofgand consider the differential graded Lie algebra

g=⊕kg^k[k], g^k :=⊕k=q−p+1Ω^0,q(X,Λ^pTX) (3.3)

6In the case of the manifold over arbitrary field of characteristic zero one should work with

“simplicial Lie algebra” of ˇCech cochains.

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endowed with the differential ¯∂and the extension of Schouten-Nijenhuys bracket by the cup-product of differential forms. Informally, the moduli spaceMasso- ciated withgmay be understood as the moduli space of solutions to Maurer- Cartan equation (1.4) ingoverZ−graded bases modulo gauge equivalences.

We show in §3.3 that the moduli space associated togby the deformation theory parametrizes theA∞−deformations ofD^bCoh(X). On the other hand, because of the natural embeddingT ⊂ΛT the deformations controlled by the sheaf ΛT generalize the deformations of complex structures onX.

3.2 The (formal) moduli space associated with ΛT

To introduce the moduli space M we use the technic of deformation theory explained in§2. The moduli space is described by the deformation functor Defg

associated tog(see§2.1). The algebra of functions on the moduli spaceMis by definition the algebra representing the functor Defg. First we need to introduce the odd “Laplacian” operator acting on the space g. The behaviour of this operator with respect to various algebraic structure on the graded vector space gis described by the Batalin-Vilkovisky formalism (see for example [Schw]).

Odd Laplacian

It follows from the conditionc1(TX) = 0 that there exists an everywhere nonvanishing holomorphicn−form Ω∈Γ(X,ΛⁿT_X^∗). It is defined canonically up to a multiplication by a constant. Let us fix a choice of Ω. It induces isomorphism of complexes (Ω^0,∗(X,Λ^pTX),∂)¯ ≃ (Ω^0,∗(X,Ω^n−p),∂);¯ γ 7→ γ ⊢ Ω. One can define then the differential ∆ ongby the formula

(∆γ)⊢Ω =∂(γ⊢Ω) (3.4)

The Lie bracket ongsatisfies the following identity (Tian-Todorov lemma) [γ1, γ2] = (−1)^degγ¹⁺¹(∆(γ1∧γ2)−(∆γ1)∧γ2−(−1)^degγ¹⁺¹γ1∧∆γ2) (3.5) where degγ=q−p+ 1 for γ∈Ω^0,q(X,Λ^pTX). In particular ∆ is a derivation of the differential graded Lie algebra structure.

Diagram of quasi-isomorphisms Denote byHthe graded vector space

H=⊕kH^k, dimH^k = X

q−p=k

dimH^q(X,Λ^pTX) (3.6) Denote by C[[t_H]] the graded algebra of formal power series on H. We recall here the proof from [BK] of the nonobstructedness of the Z−graded moduli space associated to g. Analogous arguments will be applied later for similar differential graded Lie algebras. It is convinient to fix some choice of a set{t^a} of linear coordinates onH.

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Proposition 3.2.1. The deformation functor associated to gis isomorphic to the functor represented by the algebraC[[t_H]]. Equivalently, there exists a versal solution⁷ to the Maurer-Cartan equation

∂γ(t) +¯ 1

2[γ(t), γ(t)] = 0 (3.7)

in formal power series with values ing γ(t) =X

a

γat^a+ 1 2!

X

a1,a2

γa1a2t^a¹t^a²+. . .∈(g⊗C[[tb _H]])¹ (3.8)

Remark 3.2.2. The solution of the form (3.8) is versal iff the cohomology classes[γa]form a basis of cohomology of the complex (g,∂).¯

Proof. The idea is to use the well-known trick from rational homotopy theory of K¨ahler manifolds (see [DGMS]) and the theorem from §2.2. Notice it follows from the equation (3.5) that in the following diagram all arrows are the morphisms of differential graded Lie algebras

(g,∂)¯ ←(Ker∆,∂)¯ →(Ker∆/Im∆, d:= 0) (3.9) The∂∂−lemma (see [GH]) implies that both arrows are in fact quasi-isomorphisms.¯

Corollary 3.1. One can associate togthe smooth (formal)⁸moduli space M, ObM ≃ C[[t_H]]. The tangent space to M at the base point [X] is canonically isomorphic to theZ−graded vector space⊕q,pH^q(X,Λ^pTX)[p−q].

Remark 3.2.3. The differential Lie algebragand the associated moduli space were first introduced in [BK] where g was considered with a slightly different grading. The k−th graded component of the similar differential graded Lie algebrag^˜^∗ defined in [BK] is g^˜^k =⊕q+p−1=kΩ^0,q(X,Λ^pTX). The corresponding moduli spaces considered asZ₂−graded (formal) manifolds are canonically isomorphic since the two gradings agree modZ₂.

Remark 3.2.4. The classical moduli space of complex structuresM^classical is naturally a subspace of M. The classical deformations are parametrized by t∈ M such thatγ(t)is equivalent to a solution with values in Ω^0,1(X, TX).

7 This means that any other solution over arbitrary Artin algebra (or projective limit of Artin algebras) is equivalent to a solution obtained from this solution via a base change. It is also a ”minimal” solution having these properties. It is common to use the adjective “versal”

in such situations.

8For our purposes it is sufficient to work on the level of formal manifolds. In fact the standard technic of Kuranishi spaces may be used to show that all our moduli spaces exist on the level of analytic manifolds, i.e. all the power series representing the versal solutions may be chosen to be convergent

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3.3 Moduli space of A_∞−deformations of D^bCoh(X).

Here we sketch demonstration of the fact that the supermoduli spaceMparamet- rizes theA∞−deformations ofD^b(Coh(X). The notion ofA∞−algebra was introduced by J.Stasheff. TheA∞−categories in the context of mirror symmetry were considered for the first time in [K2]. The main result which is used in this subsection is the formality theorem from [K1].

Recall (see [St]) that the structure of the A∞−algebra on a graded vector space A is defined by odd degree one derivation M of the free coassociative coalgebra generated byA[1]

C(A[1]) :=A[1]⊕(A[1])^⊗2⊕. . .

such that [M, M] = 0. Degree one derivations of the free coalgebra are in one- to-one correspondence with collections of their componentsmk:A^⊗k →A[2−k]

for allk∈N. The condition [M, M] = 0 is translated into the infinite number of identities:

X

k+l=n+1 k−1X

i=0

±mk(x1, . . . , xi, ml(xi+1, . . . , xi+l), xi+l+1, . . . , xn) = 0 (3.10) The first identity means that m1 is a differential on the graded vector space A. The next identity says thatm1 is a derivation with respect to the product defined bym2. The third identity is the associativity of the productm2 up to the homotopy defined bym3. The next identity is the compatibility ofm3with the productm2up to higher homotopies and so on.

There exists a homotopy generalization of the notion of module over the associative algebra.

Definition 3.3.1. The structure of module over an A∞−algebra A on graded vector space E is a degree one connection M^E on the free comodule generated byE

C(A[1], E) :=E⊕(E⊗A[1])⊕. . .⊕(E⊗(A[1])^⊗k)⊕. . . (3.11) such that[M^E, M^E] = 0.

Recall that a connection on a moduleMover differential coalgebra (C, d) is a linear operator∇:M→Mcompatible with the coalgebra module structure:

∆^M◦ ∇ = (∇ ⊗ Id±Id ⊗ d)◦∆^M. Again such connections acting on free modules are in one-to-one correspondence with collections of linear mapsm^E_i : A^⊗i⊗E→E[1−i] satisfying an infinite number of identities similar to (3.10).

For example, ifAis in fact a differential graded associative algebra andm^E_i = 0 fori≥2, thenE is a differential graded module overA.

Recall that anA∞−categoryX is a collection of objectsObX together with graded vector spaces of morphisms Hom^∗(E1, E2) for any pair E1, E2 ∈ C and

”higher compositions”

mk(E0, . . . , Ek) : Hom^∗(E0, E1)⊗. . .⊗Hom^∗(Ek−1, Ek)→Hom^∗(E1, Ek)[2−k]

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defined for anyk≥0 andE0, . . . , Ek ∈Ob(X). The mapsmk(E0, . . . Ek) should satisfy an infinite number of “associativity up to homotopy” constraints similar to (3.10) which can be formulated by saying that⊕i<jHom^∗(Ei, Ej) should form an A∞−algebra. In particular m1(E) is a differential on Hom^∗(E, E). One assumes usually also that the identity morphism 1E ∈ Hom⁰(E, E) is singled out and satisfiesm2(1E, f) =f, mk(f1, . . . ,1E, . . . , fk−1) = 0 for anyk6= 2.

One can show that for anA∞−algebraAthe collection of allA∞−modules forms an A∞−category which we will denote by M od∞(A). For example the space Hom^∗(E1, E2) is defined as the space of all comodule morphisms C(A[1], E1)→ C(A[1], E2). This is the same as arbitrary collection of linear maps fk :A^⊗k⊗E1→E2[−k] fork≥0. The differential acting on Hom^∗(E1, E2) is given bym1(φ) =M^E²◦φ±φ◦M^E¹.

The deformations of theA∞−categoryM od∞(A) are described by the functor Def^Z associated with the differential graded Lie algebra which consists of derivations of free coassociative coalgebra generated by A[1] equipped with counit

Der^∗(C1(A[1])), C1(A[1]) =k·1⊕A[1]⊕. . .⊕(A[1])^⊗n⊕. . . (3.12) The differential is the commutator withM ∈Der¹(C(A[1]))⊂Der¹(C1(A[1])).

This is the differential graded Lie algebra of Hochschild cochainsC^∗(A, A)[1].

Let us suppose that we are given a solution to Maurer-Cartan equation Γ(t)∈ (C(A, A)[1]b⊗MR)¹ where R = k⊕MR is a (pro)-Artin algebra assumed to be Z−graded. Then the objects of the deformed category (M od^∞)/R can be described as pairs

(E,MfÊ), E∈Ob(M od∞), MfÊ=MÊ+ ΓÊ, (3.13) ΓÊ∈(Connec^∗_Γ(C(A[1], E))b⊗MR)¹,[MÊ+ ΓÊ, MÊ+ ΓÊ] = 0

Here MfÊ =MÊ+ ΓÊ is a connection depending on the parameters t∈R on the free comodule C(A[1], E) which lifts the derivation M + Γ acting on the coalgebraC1(A[1]).

One of the ways to describe theA∞−category structure onD^bCoh(X) is to identify it with the category which is obtained from the differential graded category of finitely generated projective modules over the algebraAX= (Ω^0,∗(X),∂)¯ by taking the same collection of objects and by taking H⁰ of the complex Hom^∗(E₁^∗,E₂^∗) as the linear space of morphisms between two objects E1,E2. Then one can use the homotopy technic (see for example [M,P]) to construct the A∞−category structure on D^bCoh(X). The A∞−categoryD^bCoh(X) is A∞−equivalent to the category of finitely generated projective modules over AX.

Let g^˜^∗ denotes our differential graded Lie algebra of Dolbeault forms with coefficients in polyvector fields which is considered with slightly different grading (g^˜^∗)^k = ⊕q+p−1=kΩ^0,q(X,Λ^pT). Since the two gradings agree mod2 the corresponding supermoduli spaces are the same. It is possible to prove using the formality theorem from [K1] that the differential graded Lie algebrag^˜^∗ is