Hodge theoretic aspects of mirror symmetry
aa r X i v : . [ m a t h . AG ] M a y Hodge theoretic aspectsof mirror symmetry
L.Katzarkov M.Kontsevich T.Pantev
Abstract
We discuss the Hodge theory of algebraic non-commutative spaces and analyzehow this theory interacts with the Calabi-Yau condition and with mirror symmetry.We develop an abstract theory of non-commutative Hodge structures, investigate ex-istence and variations, and propose explicit construction and classification techniques.We study the main examples of non-commutative Hodge structures coming from asymplectic or a complex geometry possibly twisted by a potential. We study the inter-actions of the new Hodge theoretic invariants with mirror symmetry and derive non-commutative analogues of some of the more interesting consequences of Hodge theorysuch as unobstructedness and the construction of canonical coordinates on moduli.
Contents
Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2
Meromorphic connections on the complex line . . . . . . . . . 72.1.3
Stokes data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.4
The definition of a nc -Hodge structure . . . . . . . . . . . . . . 102.1.5 Variations of nc -Hodge structures . . . . . . . . . . . . . . . . . 122.1.6 Relation to other definitions . . . . . . . . . . . . . . . . . . . . . 132.1.7
Relation to usual Hodge theory . . . . . . . . . . . . . . . . . . 142.1.8 nc -Hodge structures of exponential type . . . . . . . . . . . . . 161.2 Hodge structures in nc geometry . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Categorical nc -geometry . . . . . . . . . . . . . . . . . . . . . . . 242.2.2 The main conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.3
Cyclic homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.4
The degeneration conjecture . . . . . . . . . . . . . . . . . . . . 282.2.5
The meromorphic connection in the u -direction . . . . . . . . 292.2.6 The Q -structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2.7 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3
Gluing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.1 nc -De Rham data . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.2 nc -Betti data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4 Structure results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.1
A quiver description of nc -Betti data . . . . . . . . . . . . . . . 452.4.2 Gluing of nc -Hodge structures . . . . . . . . . . . . . . . . . . . 452.5 Deformations of nc -spaces and gluing . . . . . . . . . . . . . . . . . . . 492.5.1 The cohomological Hochschild complex . . . . . . . . . . . . . . 492.5.2
Corrections by constants . . . . . . . . . . . . . . . . . . . . . . . 522.5.3
Singular deformations . . . . . . . . . . . . . . . . . . . . . . . . . 53 A -model Hodge structures . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 B -model Hodge structures. . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Mirror symmetry examples . . . . . . . . . . . . . . . . . . . . . . . . . 75
Canonical coordinates for Calabi-Yau variations . . . . . . . . . . . . 774.1.1
Variations over supermanifolds . . . . . . . . . . . . . . . . . . . 774.1.2
Calabi-Yau variations . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.3
Decorated Calabi-Yau variations . . . . . . . . . . . . . . . . . . 794.1.4
Generalized decorations . . . . . . . . . . . . . . . . . . . . . . . 824.1.5
Formal variations of Calabi-Yau type . . . . . . . . . . . . . . . 834.2
Algebraic framework: dg Batalin-Vilkovisky algebras . . . . . . . . . 834.2.1
Preliminaries on L ∞ algebras . . . . . . . . . . . . . . . . . . . . 844.2.2 DG Batalin-Vilkovisky algebras . . . . . . . . . . . . . . . . . . 862.2.3
Geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . 894.2.4
Relation to Calabi-Yau variations of nc -Hodge structures . . 904.3 B -model framework: manifolds with anticanonical sections . . . . . 914.3.1 The classical Tian-Todorov theorem. . . . . . . . . . . . . . . . 914.3.2
Moduli of Calabi-Yau manifolds. . . . . . . . . . . . . . . . . . . 924.3.3
Generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.3.4
Mixed Hodge theory in a nutshell. . . . . . . . . . . . . . . . . . 994.3.5
The moduli stack of Fano varieties. . . . . . . . . . . . . . . . . 1004.3.6
Algebras for the Landau-Ginzburg model. . . . . . . . . . . . . 1014.4
Categorical framework: spherical functors . . . . . . . . . . . . . . . . 1024.4.1
Calabi-Yau nc -spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 1024.4.2 Spherical functors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.5 A -model framework: symplectic Landau-Ginzburg models . . . . . . 1074.5.1 Symplectic geometry with potentials. . . . . . . . . . . . . . . . 1074.5.2
Categories of branes . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.5.3
Mirror symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
This paper is a first in a series aiming to develop a general procedure associating a 2-dimensional cohomological field theory in the sense [KM94] (CohFT in short) to a certainstructure in derived algebraic geometry. More precisely, for any Calabi-Yau A ∞ -categorysatisfying appropriate finiteness conditions (smoothness and compactness), and such that anoncommutative analog of the Hodge ⇒ de Rham spectral sequence collapses, we associatean infinite-dimensional family of CohFTs. The additional parameters needed to specify theCohFT are of a purely cohomological nature. Conjecturally, our procedure applied to theFukaya category should give (higher genus) Gromov-Witten invariants of the underlyingsymplectic manifold.This program was first outlined by the second author in several talks given in 2003-2004,and some aspects of it were later developed in depth by K.Costello [Cos07b, Cos05, Cos07c,Cos07a]. The whole picture turns out to be very intricate, and in the process of writingwe realized that we have to address a large variety of problems. In this installment we donot discuss the general plan of our approach but rather focus on those features of A ∞ or dgcategories that can be captured by Hodge theoretic constructions. We propose a formalism3hat starts with Homological Mirror Symmetry and extrapolates a geometric picture for therequisite categories that makes them amenable to study via old and new Hodge theory. Ourhope is that this geometric treatment will provide new invariants and will expand the scopeof possible applications in symplectic geometry and algebraic geometry.Mirror symmetry was introduced in physics as a special duality between two N = 2 superconformal field theories. Traditionally a N = 2 super conformal field theory is constructed asa quantization of a non-linear σ -model with target a compact Calabi-Yau manifold equippedwith a Ricci flat K¨ahler metric and a closed 2-form - the so called B -field. Two Calabi-Yaumanifolds X and Y form a mirror pair X | Y if the associated N = 2 super conformal fieldtheories are mirror dual to each other [CK99].Homological Mirror Symmetry was introduced in 1994 by the second author for thecase of Calabi-Yau manifolds but today the realm of its applicability is much broader. Inparticular many of our considerations in the present work are governed by an analogue ofHomological Mirror Symmetry for geometries with potentials. We study the effect of suchmirror symmetry on the associated categories of D -branes and especially on the associatednon-commutative Hodge structures on homological invariants, i.e. on the Hochschild andcyclic homology and cohomology of such categories. We study mirror pairs consisting of acompact manifold on one side, and of a Landau-Ginzburg model with a proper potential ona non-compact manifold having c = 0 on the other. We formulate the mirror symmetryconjecture on the Hodge theoretic level in both directions. That is, we compare the non-commutative Hodge structures associated with a compact complex manifold and a mirrorholomorphic Landau-Ginzburg model, and also the non-commutative Hodge structures as-sociated with a compact complex manifold with a chosen smooth anticanonical divisor andwith the mirror symplectic Landau-Ginzburg model. This picture is clearly non-symmetricand has to be generalized. In order to completely understand the Hodge theoretic aspect ofmirror symmetry, one will have to allow for non-trivial potentials on both sides of the dual-ity and include the novel toric dualities between formal Landau-Ginzburg models of Clarke[Cla08] and the new smooth variations of non-commutative Hodge structures of Calabi-Yautype that we attach to anticanonical Q -divisors in section 4.3. We plan to return to such ageneralization in a future work.Due to its foundational nature the paper comes out somewhat long winded and technicalfor which we apologize. It is organized in three major parts:The first part introduces and develops the abstract theory of non-commutative ( nc )Hodgestructures. This theory is a variant of the formalism of semi-infinite Hodge structures that4as introduced by Barannikov [Bar01, Bar02a, Bar02b]. We discuss the general theory of nc -Hodge structures in the abstract and analyze the various ways in which the Betti, deRham and Hodge filtration data can be specified. In particular we compare nc and ordinaryHodge theory and explain how nc -Hodge theory fits within the setup of categorical non-commutative geometry. We also pay special attention to the nc -aspect of Hodge theory andits interaction with the classification of irregular connections on the line via topological data.One of the most useful technical results in this part is the gluing Theorem 2.35 which allowsus to assemble nc -Hodge structures out of some simple geometric ingredients. This theoremis used later in the paper for constructing nc -Hodge structures attached to geometries witha potential.The second part explains how symplectic and complex geometry give rise to nc -Hodgestructures and how these structures can be viewed as interesting invariants of Gromov-Wittentheory, projective geometry and the theory of algebraic cycles. In particular we analyze theBetti part of the nc -Hodge theory of a projective space (viewed as a symplectic manifold) anduse this analysis to propose a general conjecture for the integral structure on the cohomologyof the Fukaya category of a general compact symplectic manifold. The formula for the integralstructure uses only genus zero Gromov-Witten invariants and characteristic classes of thetangent bundle. Our conjecture is in complete agreement with the recent work of Iritani[Iri07] who made a similar proposal based on mirror symmetry for toric Fano orbifolds. Wealso discuss in detail the origin of the Stokes data for holomorphic geometries with potentialsand investigate the possible categorical incarnations of this data.In the third part we study nc -Hodge structures and their variations under the Calabi-Yaucondition. We extend and generalize the standard treatment of the deformation theory ofCalabi-Yau spaces in order to get a theory which works equally well in the nc -context and tobe able to properly define the canonical coordinates in Homological Mirror Symmetry. Weapproach the deformation-obstruction problem both algebraically and by Hodge theoreticmeans and we obtain unobstructedness results, generalized pre Frobenius structures andsome interesting geometric properties of period domains for nc -Hodge structure. We alsostudy global and infinitesimal deformations and describe different constructions of Betti andde Rham nc -Hodge data for ordinary geometry, relative geometry, geometry with potentialsand abstract nc -geometry. Acknowledgments:
Throughout the preparation of this work we have benefited from dis-5ussions with many people who generously shared their thoughts and insights with us. Spe-cial thanks are due to A D. Auroux, M. Abouzaid, .Bondal, R.Donagi, V.Golyshev, M.Gross,A.Losev, D.Orlov, C.Simpson, Y.Soibelman, Y.Tschinkel, A.Todorov, and B.To¨en for experthelp, encouragement and advice. We would also like to thank the University of Miami forproviding the productive research environment in which most of this work was done. Duringvarious stages of this work we have enjoyed the hospitality of several outstanding researchinstitutions. We thank the IAS, the IHES, the Centre Interfacultaire Bernoulli at the EPFL,and the ESI for the excellent working conditions they have provided. The first and thirdauthor would especially like to thank the organizers of the conference “From tQFT to tt ∗ and integrability” at the University of Augsburg, for giving them an opportunity to speakand for the invitation to contribute to the proceedings volume of the conference.During the preparation of this work Ludmil Katzarkov was partially supported by theFocused Research Grant DMS-0652633 and a research grant DMS-0600800 from the NationalScience Foundation, and a FWF grant P20778. Tony Pantev was partially supported by NSFFocused Research Grant DMS-0139799, NSF Research Training Group Grant DMS-0636606,and NSF grant DMS-0700446. In this section we will discuss the notion of a pure non-commutative ( nc ) Hodge structure.The nc -Hodge structures are analogues of the classical notion of a pure Hodge structure on acomplex vector space. Both the nc -Hodge structures discussed presently and Simpson’s non-abelian Hodge structures [Sim97a] generalize classical Hodge theory. In Simpson’s theory,one allows for non-linearity in the substrate of the Hodge structure: the non-abelian Hodgestructures of [Sim97a] are given by imposing Hodge and weight filtrations on non-lineartopological invariants of a K¨ahler space, e.g. on cohomology with non-abelian coefficients,or on the homotopy type. In contrast the nc -Hodge structures discussed in this paper consistof a novel filtration-type data (the twistor structure of [Sim97b, Her03, Sab05b]) which arestill specified on a vector space, e.g. on the periodic cyclic homology of an algebra.Similarly to ordinary Hodge theory nc -Hodge structures arise naturally on the de Rhamcohomology of non-commutative spaces of categorical origin.6 .1 Hodge structures We will give several different descriptions of a nc -Hodge structure in terms of local data. Webegin with the notion of a rational and unpolarized nc -Hodge structures, ignoring for thetime being the existence of polarizations and integral lattices. The nc -Hodge structures will be described in terms of geometric data on the puncturedcomplex line, so we fix once and for all a coordinate u on C and the compactification C ⊂ P .We will write C [[ u ]] for the algebra of formal power series in u , and C (( u )) for the field offormal Laurent series in u . Similarly, we will write C { u } for the algebra of power series in u having positive radius of convergence, and C { u } [ u − ] for the field of meromorphic Laurentseries in u with a pole at most at u = 0. We will need some standard notions and facts related to meromorphic connections onRiemann surfaces. We briefly recall those next. More details can be found in e.g. [Sab02,chapter II].Let M be a finite dimensional vector space over C { u } [ u − ], and let ∇ be a meromorphicconnection on M . Explicitly ∇ is given by a C -linear map ∇ ddu : M → M which satisfiesthe Leibniz rule for multiplication by elements in C { u } [ u − ]. A holomorphic extensionof M is a free finitely generated C { u } -submodule H ⊂ M , such that H ⊗ C { u } C { u } [ u − ] = M . Traditionally (see e.g. [Sab02, section 0.8]) a holomorphic extension is called a lattice . Wewill avoid this classical terminology since it may lead to confusion with the integral latticestructures that we need.As usual the data ( M , ∇ ) or ( H , ∇ ) should be viewed as local models for geometric dataon a Riemann surface: ( M , ∇ ) is the local model of a germ of a meromorphic bundle withconnection on a Riemann surface, and ( H , ∇ ) is the local model of a holomorphic bundlewith meromorphic connection on a Riemann surface.7uppose ( M , ∇ ) is a meromorphic bundle with connection over C { u } [ u − ] and let H ⊂ M be a holomorphic extension. We say that H is logarithmic with respect to ∇ if ∇ ( H ) ⊂ H duu . We say that ( M , ∇ ) has at most a regular singularity at H ⊂ M which is logarithmic with respect to ∇ . Remark 2.1
The Riemann-Hilbert correspondence implies (see e.g. [Sab02, II.3.7]) thatthe functor of taking the germ at 0 ∈ P : finite rank algebraic vector bundleswith connections on A − { } witha regular singularity at ∞ G / / finite dimensional C { u } [ u − ]-vector spaces with meromor-phic connections is an equivalence of categories. For future reference we choose once and for all an inverse B of G . We will call B the algebraization functor or the Birkhoff extension functor .Suppose H is a holomorphic bundle over C { u } equipped with a meromorphic connection ∇ . Let M = H ⊗ C { u } C { u } [ u − ] and let ( M, ∇ ) = B ( M , ∇ ) be the corresponding Birkhoffextension. The algebraic bundle M on A − { } admits a natural extension to a holomorphicbundle H on A : on a small punctured disc centered at 0 ∈ A , the bundle M is analyticallyisomorphic to M , and so H gives us an extension to the full disc. In particular G and B can be promoted to a pair of inverse equivalences finite rank algebraic vector bundleson A equipped with a meromor-phic connection with poles at mostat 0 and ∞ , and a regular singular-ity at ∞ G / / finite rank free C { u } -modulesequipped with a meromorphicconnection B o o which we will denote again by G and B . Let ( H , ∇ ) be a holomorphic bundle with meromorphic connection over C { u } . We willneed the Deligne-Malgrange description of the associated meromorphic connection ( M , ∇ )via a filtered sheaf on the circle. We briefly recall this description next. More details can8e found in [Mal83] and [BV85]. Let ( M, ∇ ) := B (( M , ∇ )) be the Birkhoff extension of( M , ∇ ) to P . Consider the circle S := C × / R × + . The sheaf of local ∇ -horizontal sectionsof M an on C × is a locally constant sheaf on C × , which by contractability of R × + induces alocally constant sheaf S of C -vector spaces on S .The sheaf S is equipped with a natural local filtration by subsheaves { S ≤ ω } ω ∈ Del , where (i)
Del is the complex local system on S for which for every open U ⊂ S the space ofsections Del ( U ) is defined to be the space of all holomorphic one forms ω on the sectorSec( U ) := (cid:8) re iϕ (cid:12)(cid:12) r > , ϕ ∈ U (cid:9) which are of the form ω = X a ∈ Q a< − c a u a du, where at most finitely many c a = 0 and the branches u a are chosen arbitrarily.Note that the germs of sections of Del are naturally ordered: if ω ′ , ω ′′ ∈ Del ( U ), ϕ ∈ U ,and if ω ′ − ω ′′ = c a u a + (cid:18) higher or-der terms (cid:19) , then one sets ω ′ < ϕ ω ′′ ⇔ Re (cid:18) c a e iϕ ( a +1) a + 1 (cid:19) < . (ii) For every ϕ ∈ S and every ω ∈ Del ϕ the stalk( S ≤ ω ) ϕ ⊂ S ϕ is defined to be the subspace( S ≤ ω ) ϕ := s ∈ S ϕ = Γ (cid:0) R × + e iϕ , M an (cid:1) ∇ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − R ω s has moderate growthin the direction ϕ , i.e. (cid:13)(cid:13)(cid:13) e − R ω s (cid:13)(cid:13)(cid:13) | R × + e iϕ = O (cid:0) r − N (cid:1) , when r → N ≫ Here k•k is the Hermitian norm of a section of M computed in some (any) meromorphictrivialization of M an near u = 0. 9 efinition 2.2 The filtration we just defined is the
Deligne-Malgrange-Stokes filtra-tion , and the
Del -filtered sheaf S is called the Stokes structure associated to ( M , ∇ ) . Remark 2.3
The Deligne-Malgrange-Stokes filtration satisfies the following property. Firstof all, there exists a covariantly local system of finite sets
Del ( M , ∇ ) ⊂ Del canonicallyassociated with ( M , ∇ ) such that the filtration by all of Del is determined by a filtration byall
Del ( M , ∇ ) ( U ) and all consecutive factors are non-trivial at all points of S except finitelymany (called the directions of the Stokes rays). Outside the Stokes rays the set Del M , ∇ ( φ )is totally ordered. It is easy to see that when we cross a Stokes ray then the order changesby flipping the order on several disjoint intervals (e.g. { , , , , , } → { , , , , , } ).Moreover, on the subquotients corresponding to these intervals, two filtrations coming fromthe left and from the right of the anti Stokes ray are opposed to each other. This impliesthat the graded pieces with respect to the Deligne-Malgrange-Stokes filtration are locallyconstant sytems of vector spaces on the total space of stalks of the sheaf Del ( M , ∇ ) (which isa disjoint union of finite coverings of S ). Remark 2.4
A fundamental theorem of Deligne and Malgrange [Mal83, Theorem 4.2],[BV85, Theorem 4.7.3] asserts that the functor ( M , ∇ ) (cid:0) S , { S ≤ ω } ω ∈ Del (cid:1) is an equiva-lence between the category of meromorphic connections over C { u } [ u − ] and the category of Del -filtered local systems on S satisfying the conditions described in Remark 2.3. We willuse this equivalence to define the Betti part of a nc -Hodge structure. nc -Hodge structure After these preliminaries we are now ready to define nc -Hodge structures. Definition 2.5 A rational pure nc -Hodge structure consists of the data ( H, E B , iso ) ,where • H is a Z / -graded algebraic vector bundle on A . E B is a local system of finite dimensional Z / -graded Q -vector spaces on A − { } . • iso is an analytic isomorphism of holomorphic vector bundles on A − { } : iso : E B ⊗ O A −{ } ∼ = −→ H | A −{ } . Note:
The isomorphism iso induces a natural flat holomorphic connection ∇ on H | A −{ } .These data have to satisfy the following axioms: (nc-filtration axiom) ∇ is a meromorphic connection on H with a pole of order ≤ at u = 0 and a regular singularity at ∞ . More precisely, there exist: • a holomorphic frame of H near u = 0 in which ∇ = d + X k ≥− A k u k ! du with A k ∈ Mat r × r ( C ) , r = rank H . • a meromorphic frame of H near u = ∞ in which ∇ = d + X k ≥− B k u − k ! d ( u − ) and B k ∈ Mat r × r ( C ) . ( Q -structure axiom) The Q -structure E B on ( H, ∇ ) is compatible with Stokes data. Moreprecisely, let (cid:0) S , { S ≤ ω } ω ∈ Del (cid:1) be the Stokes structure corresponding to the germ ( H , ∇ ) := G ( H, ∇ ) , and let S B ⊂ S be the Q -structure on S induced from E B viathe isomorphism iso . We require that the Deligne-Malgrange-Stokes filtration on S isdefined over Q , i.e. ( S ≤ ω ∩ S B ) ⊗ Q C = S ≤ ω for all local sections ω ∈ Del . (opposedness axiom) The Q -structure S B induces a real structure on S and hence a com-plex conjugation τ : S → S . Let b H be the holomorphic bundle on P obtained as the luing of H alg |{| u |≤ } and (cid:16) γ ∗ H alg (cid:17) |{| u |≥ } via τ , where where γ : P → P is the realstructure on P which fixes the unit circle, i.e. γ ( u ) := 1 / ¯ u . Then we require that b H be holomorphically trivial, i.e. b H ∼ = O ⊕ r P .A morphism f : ( H , E B, , iso ) → ( H , E B, , iso ) of nc -Hodge structures is a pair f = ( f, f B ) , where f : H → H , is an algebraic map of vector bundles which intertwinesthe connections, and f B : E B, → E B, is a map of Q -local systems, such that f ◦ iso = iso ◦ ( f B ⊗ id O ) . We will write ( Q - nc HS ) for the category of pure nc -Hodge structures. Remark 2.6
The meromorphic connection ( M, ∇ ) where M = H ⊗ C [ u ] C [ u, u − ] can bethought of as the de Rham data of the nc -Hodge structure, the local system S B of rationalvector spaces over S endowed with the rational Stokes filtration (see Q -structure axiom)can be thought of as the Betti data, and the holomorphic extension H of M can be thoughtof as the analogue of the Hodge filtration. nc -Hodge structures One can also define variations of nc -Hodge structures: Definition 2.7
Let S be a complex manifold. A variation of pure nc -Hodge structures over S is a triple ( H, E B , iso ) , where • H is a holomorphic Z / -graded vector bundle on A × S which is algebraic in the A -direction. • E B is a local system of Z / -graded Q -vector spaces on ( A − { } ) × S . • iso is an analytic isomorphism of holomorphic vector bundles iso : E B ⊗ O ( A −{ } ) × S ∼ = → H | ( A −{ } ) × S . Let ∇ be the induced meromorphic connection on H . The data ( H, E B , iso ) should satisfy: nc-filtration axiom) The connection ∇ has a regular singularity along {∞}× S and Poincar´erank ≤ along { } × S , i.e. u · ∇ ∂∂u : H → H is a holomorphic differential operator on H of order ≤ . (Griffiths transversality axiom) For every locally defined vector field ξ ∈ T S we have that u · ∇ ξ : H → H, is a holomorphic differential operator on H of order ≤ . ( Q -structure axiom) The Stokes structure on the local system S on S × S is well defined,i.e. the steps in the Deligne-Malgrange-Stokes filtration on S are sheaves on S × S .Furthermore the Q -structure E B is compatible with the Stokes data as in Definition 2.5. (opposedness axiom) The relative version of the gluing construction for nc -Hodge struc-tures gives a globally defined complex vector bundle b H on P × S , which is holomor-phically trivial in the P direction. Moreover, with respect to the extension b H theconnection ∇ is meromorphic with Poincar´e rank one along ( { } × S ) ∪ ( {∞} × S ) . Various special cases and partial versions of our notion of a nc -Hodge structure have beenstudied before in slightly different but related setups. We list a few of the relevant notionsand references without going into detailed comparisons: • A version of ( Z -graded) nc -Hodge structures appears in the fundamental work ofK.Saito (see [Sai83, Sai98b, Sai98a] and references therein) on the Hodge theoreticinvariants of quasi-homogeneous hypersurface singularities under the name weightsystem . • A version of the notion of a variation of (complex) nc -Hodge structure appears in thework of Cecotti-Vafa in Conformal Field Theory [CV91, CV93a, CV93b, BCOV94]under the name tt ∗ -geometry . 13 Various versions of the notion of a (complex,polarized) nc -Hodge structure appear inalgebraic geometry and non-abelian Hodge theory in the works of Simpson [Sim97a,Sim97b] and T.Mochizuki [Moc06a, Moc06b, Moc07a, Moc07b] under the names of (tame or wild) harmonic bundle or pure twistor structure , and in the workof Sabbah [Sab05b] under the name integrable pure twistor D -module . • The analytic germ of a (complex) variation of nc -Hodge structures appears in mirrorsymmetry in the work of Barannikov [Bar01, Bar02a, Bar02b] and Barannikov and thesecond author [BK98] under the name semi-infinite Hodge structure . The integralstructures on semi-infinite Hodge structure were recently introduced and studied in thework of Iritani [Iri07]. • A version of the notion of a (real) nc -Hodge structure appears in singularity the-ory in the work of Hertling [Her03, Her06] and Hertling-Sevenchek [HS07] under thename TER structure . Hertling and Sevenchek also consider polarized and mixed nc -Hodge structures. Those appear under the names TERP structure and mixedTERP structure respectively. In particular in [HS07] Hertling and Sevenchek studythe degenerations of of TERP structures and prove a version of Schmid’s nilpotentorbit theorem which gives rise to the notion of a limiting mixed TERP structure. De-generations of variants of nc -Hodge structures, as well as limiting mixed nc -Hodgestructures appear also in the works of Sabbah [Sab05a] and S.Szabo [Sza07]. Recall (see e.g. [Del71]) that a pure rational Hodge structure of weight w is a triple( V, F • V, V Q ) where: • V is a complex vector space, • V Q ⊂ V is a Q -subspace such that V = V Q ⊗ Q C , and • F • V is a Hodge filtration of weight w on V , i.e F • V is a decreasing finite exhaustivefiltration by complex subspaces which satisfies F p V ⊕ F w +1 − p V = V , where the complexconjugation on V is the one given by the real structure V R = V Q ⊗ R ⊂ V .14 pure Hodge structure is a direct sum of pure Hodge structures of various weights,and a morphism of pure Hodge structures is a linear map of complex vector spaces, whichmaps the rational structures into each other and is strictly compatible with the filtrations.We will write ( Q -HS) for the category of pure rational Hodge structures. It is well known[Del71] that ( Q -HS) is an abelian Q -linear tensor category. For every w ∈ Z we have a ⊗ -invertible object in ( Q -HS) of pure weight 2 w : the Tate Hodge structure Q ( w ) givenby Q ( w ) := ( C , F • , Q ), where F i = C for i ≤ w and F i = { } for i > w .It turns out that pure Hodge structures can be viewed as nc -Hodge structures. Thisis achieved through a version of the Rees module construction (see e.g. [Sim97a]) whichconverts a filtered vector space into a bundle over the affine line A . Specifically, given apure Hodge structure ( V, F • V, V Q ) of weight w we consider the rank one meromorphic bundlewith connection T w := (cid:18) C { u } [ u − ] , d − w · duu (cid:19) and we set • H := H w mod 2 := P i u − i F i V { u } viewed as a C { u } -submodule in C { u } [ u − ] ⊗ C V . Clearly, this submodule is preserved by the operator ∇ u ddu for the connection ∇ := (cid:0) d − w · duu (cid:1) ⊗ id V , i.e. ( H , ∇ ) is a logarithmic holomorphic extension of themeromorphic bundle with connection T w ⊗ C V . Note:
Consider the algebraization ( H, ∇ ) = B ( H , ∇ ) of ( H , ∇ ). The fiber H := H/ ( u − H of H at 1 ∈ A is canonically identified with V . By definition the connection ∇ on H has monodromy ( − w id V and so preserves any rational subspace in V . • E B := E w mod 2 B - the Q -local system on A − { } defined as the subsheaf E B ⊂ H consisting of sections whose value at 1 is in V Q ⊂ V = H/ ( u − H . In other words E B is the locally constant sheaf on A − { } with fiber V Q and monodromy ( − w id V Q . • iso is the isomorphism of complex local systems, corresponding to the embedding E B ⊂ H . Remark 2.8
On every simply connected open (in the analytic topology) subset U ⊂ A − { } the bundle with connection T w has a horizontal section u w/ . In partic-ular on such opens we have H | U = P i u w/ u − i F i [ u ].15he data ( H, E B , iso ) satisfy tautologically the ( Q -structure axiom) and the (opposednessaxiom) from Definition 2.5. Indeed, the ( Q -structure axiom) is satisfied since by definition ∇ has a regular singularity at 0 and so S ≤ ω = S or 0 for all ω . The (opposedness axiom) is satisfied as it is equivalent in the case of regular singularities to the oposedness propertyin the definition of the usual Hodge structures.Thus, the assignment ( V, F • V, V Q ) → ( H, E B , iso ) gives a functor n : ( Q -HS) → ( Q - nc HS )which by definition factors through the orbit category (see e.g. [Kel05] for the definition ofan orbit category) π : ( Q -HS) → ( Q -HS) / ( • ⊗ Q (1)) , i.e we have N = n ◦ π for a functor N : ( Q -HS) / ( • ⊗ Q (1)) → ( Q - nc HS ) . The proof of the following statement is an immediate consequence from the definition.
Lemma 2.9
The functor N is fully faithful and its essential image consists of all nc -Hodgestructures that have regular singularities and monodromy = id on H and = − id on H . Remark 2.10
It is straightforward to check that the functor N can also be defined infamilies and embeds the category of variations of Hodge structures (modulo the Tate twist)into the category of variations of nc -Hodge structures. nc -Hodge structures of exponential type As we saw in section 2.1.7 the usual Hodge structures give rise to special nc -Hodge struc-tures with regular singularities. The nc -Hodge structures with regular singularities are alsoimportant because they can serve as building blocks of general nc -Hodge structures. Let16 H, E B , iso ) be a nc -Hodge structure, let ( H , ∇ ) = G (( H, ∇ )) be the germ of ( H, ∇ ) atzero, and assume that A − = 0, i.e. ∇ has a second order pole. According to Turrittin-Levelt formal decomposition theorem (see e.g. [Mal79], [BV85], [Sab02, II.5.7 and II.5.9])we can find a finite base change p N : C → C , p N ( t ) := t N = u , so that p ∗ N ( H , ∇ )[ t − ] isformally isomorphic to a direct sum of regular singular connections on meromorphic bundlesmultiplied by exponents of Laurent polynomials. More precisely we can find polynomial tails g i ( t ) ∈ C [ t − ], C { t } [ t − ]-vector spaces R i and meromorphic connections( ∇ i ) ddt : R i → R i , each with at most regular singularity at 0, so that we have an isomorphism of formal mero-morphic connections over C (( t )):Ψ : p ∗ N ( H , ∇ ) O C { t } [ t − ] C (( t )) ∼ = −→ m M i =1 E g i O C { t } [ t − ] ( R i , ∇ i ) O C { t } [ t − ] C (( t )) . Here E f denotes the rank one holomorphic bundle with meromorphic connection( C { t } , d − df ), and ( R i , ∇ i ) denote meromorphic bundles with connections having regularsingularities. Remark 2.11
The bundle E f has a non-vanishing horizontal section, namely e f . In partic-ular the multivalued flat sections of E g i ⊗ ( R i , ∇ i ) are given by multiplying multivalued flatsections of ( R i , ∇ i ) by e g i .In the examples coming from Mirror Symmetry that we are interested in, the base change p N is not needed for the decomposition to work. In this case we can take g i ( u ) = c i /u where c , . . . , c m ∈ C denote the distinct eigenvalues of A − . Because of this we introduce thefollowing definition (see also [HS07, Definition 8.1]): Definition 2.12
We say that a nc -Hodge structure ( H, E B , iso ) is of exponential type ifthere exists a formal isomorphism Ψ : ( H ⊗ C { u } C [[ u ]] , ∇ ) ∼ = → m M i =1 (cid:16) E c i /u ⊗ ( R i , ∇ i ) (cid:17) ⊗ C { u } C [[ u ]] where ( R i , ∇ i ) are meromorphic bundles with connections with regular singularities and c , . . . , c m ∈ C denote the distinct eigenvalues of A − . emark 2.13 • There are various sufficient conditions that will guarantee that a given nc -Hodge structure is decomposable without base change. For instance, this will be the caseif A − has distinct eigenvalues, or if A − = 0. More generally, it suffices to require that wecan find holomorphic functions ℓ i ( u ) ∈ C { u } so that ℓ i (0) = c i for i = 1 , . . . , m and thecharacteristic polynomial of u A ( u ) is det ( c · id − u A ( u )) = Q mi =1 ( c − ℓ i ) ν i . • Not every irregular connection with a pole of order two is of exponential type. Indeed therank two connection ∇ = d − u − u − u − ! has a horizontal section e − u − u e − u − ! , and so one needs a quadratic base change for the formal decomposition to work for thisconnection. • If a nc -Hodge structure ( H, E B , iso ) is of exponential type, then one can check (see [HS07,Lemma 8.2]) that for each i = 1 , . . . , m we can find a unique holomorphic extension H c i ⊂ R i in which the connection has a second order pole and so that Ψ induces a formal isomorphismof holomorphic bundles with meromorphic connectionsΨ : ( H , ∇ ) ⊗ C [[ u ]] ∼ = −→ m M i =1 E c i /u O C { u } ( H c i , ∇ i ) ⊗ C [[ u ]] , over C [[ u ]].The nc -Hodge structures with regular singularities or the nc -Hodge structures of exponentialtype comprise full subcategories( Q - nc HS ) reg ⊂ ( Q - nc HS ) exp ⊂ ( Q - nc HS )in ( Q - nc HS ). In fact, in the exponential type case one can state the nc -Hodge structureaxioms in an easier way. The simplification comes from the fact that in this case the Deligne-Malgrange-Stokes filtration is given by subsheaves S ≤ λ of S that are labeled by λ ∈ R andconsisting of solutions decaying faster than O (cid:16) exp (cid:16) λ + o (1) r (cid:17)(cid:17) , r = | u | . Indeed, tracing18hrough the definition one sees that in the exponential case for a ray defined by ϕ thejumps of the steps of the Deligne-Malgrange-Stokes filtration occur exactly at the numbersRe( c i e − iϕ ). Furthermore, the associated graded pieces for the filtration are local systemson the circle and in fact coincide with the regular pieces ( R i , ∇ i ) that appear in the formaldecomposition of the connection. Hence one arrives at the following Definition 2.14 A rational pure nc -Hodge structure of exponential type consistsof the data ( H, E B , iso ) , where • H is a Z / -graded algebraic vector bundle on A . • E B is a local system of finite dimensional Z / -graded Q -vector spaces on A − { } . • iso is an analytic isomorphism of holomorphic vector bundles on A − { } : iso : E B ⊗ O A −{ } ∼ = −→ H | A −{ } . These data have to satisfy the following axioms: (nc-filtration axiom) exp
The connection ∇ induced from iso is a meromorphic connectionof exponential type on H with a pole of order ≤ at u = 0 and a regular singularity at ∞ . More precisely, there exist: • a holomorphic frame of H near u = 0 in which ∇ = d + X k ≥− A k u k ! du with A k ∈ Mat r × r ( C ) , r = rank C { u } H . • a holomorphic frame of H near u = ∞ in which ∇ = d + X k ≥− B k u − k ! d ( u − ) and B k ∈ Mat r × r ( C ) . a formal isomorphism over C (( u )) : ( H [ u − ] , ∇ ) ∼ = → m M i =1 E c i /u ⊗ ( R i , ∇ i ) where ( R i , ∇ i ) are meromorphic bundles with connections with regular singulari-ties and c , . . . , c m ∈ C denote the distinct eigenvalues of A − . ( Q -structure axiom) exp The Q -structure E B on ( H, ∇ ) is compatible with Stokes data in thefollowing sense. The filtration { S ≤ λ } λ ∈ R of S by the subsheaves S ≤ λ , whose stalk at ϕ ∈ S is given by ( S ≤ λ ) ϕ := s ∈ S ϕ = Γ (cid:0) R × + e iϕ , H (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s is a ∇ -horizontal section of H over the ray R × + e iϕ , for which (cid:13)(cid:13) s (cid:0) re iϕ (cid:1)(cid:13)(cid:13) = O (cid:18) exp (cid:18) λ + o (1) r (cid:19)(cid:19) . when r → . is defined over Q , i.e. ( S ≤ λ ∩ S B ) ⊗ Q C = S ≤ λ for all λ ∈ R . (opposedness axiom) exp = (opposedness axiom) Remark 2.15
It is instructive to understand more explicitly the behavior of the Deligne-Malgrange-Stokes filtration for nc -Hodge structures (or more generally irregular connections)of exponential type. As before we denote by S the complex local system on the circle S corresponding to a nc -Hodge structure for which A − has distinct eigenvalues c , . . . c m .By definition, for every ϕ , the steps in the Deligne-Malgrange-Stokes filtration ( S ≤ λ ) ϕ jump exactly when λ crosses one of the numbers Re( c k e − iϕ ). More invariantly, the as-signment ϕ ∈ S
7→ {
Re( c e − iϕ ) , . . . , Re( c k e − iϕ ) } ⊂ R is a sheaf Λ of finite sets of realnumbers (possibly with repetitions) on S . For a general value of ϕ , the real numbers { Re( c e − iϕ ) , . . . , Re( c k e − iϕ ) } are all distinct but for finitely many special values of ϕ someof Re( c e − iϕ ) , . . . , Re( c k e − iϕ ) will coalesce. More precisely we have the Stokes rays R > · i ( c b − c a ) and the associated set SD ⊂ [0 , π ) of Stokes directions: i.e. ϕ ∈ SD , if and20nly if there is some pair a = b s.t. c a − c b = re i ( π + ϕ ) for some r >
0. Clearly for everyopen arc U ⊂ S , which does not intersect SD the restriction Λ | U is a local system of finitesets of cardinality m . Moreover the values ϕ ∈ SD are precisely the ones for which some ofRe( c e − iϕ ) , . . . , Re( c k e − iϕ ) become equal to each other.Now recall that for any given ϕ ∈ S , the subspaces ( S ≤ λ ) ϕ ⊂ S ϕ do not change if wemove λ ∈ R continuously without passing through some element of Λ ϕ . In other words, wecan label the steps of the Deligne-Malgrange-Stokes filtration by local sections of Λ, andso that at each ϕ ∈ S the steps are ordered according to the order on Λ ϕ induced fromthe embedding Λ ϕ ⊂ R . The finite set SD ⊂ S of Stokes directions breaks the circle intodisjoint arcs. Over each such arc U we have that Λ | U is a local system of finite sets ofreal numbers with m linearly ordered flat sections and the steps Deligne-Malgrange-Stokesfiltration of S | U are labeled naturally by these sections. If we move from U to an adjacent arc U ′ by passing across a Stokes direction φ ∈ SD , then some of the elements in the labellingset get identified at φ and get reordered when we cross over to U ′ (see Figure 1). λ λ λ λ λ ′ λ ′ λ ′ λ ′ U U ′ S R Stokesdirection φ Figure 1: The system of labels for the Deligne-Malgrange-Stokes filtration.In fact, if λ < . . . < λ m are the ordered flat sections of Λ | U , and λ ′ < . . . < λ ′ m are theordered flat sections of Λ | U ′ , then the transition from the λ ’s to the λ ′ ’s is always such thatcertain groups of consecutive λ ’s are totally reordered into groups of consecutive λ ′ ’s. For21nstance in Figure 1 the passage from { λ , λ , λ , λ } to { λ ′ , λ ′ , λ ′ , λ ′ } across the Stokespoint φ ∈ SD has the effect of relabelling: λ λ ′ , λ λ ′ , λ λ ′ , and λ λ ′ .This behavior of the labelling set and the behavior of the associated filtration can be sys-tematized in the following: Definition 2.16
Let S be a finite dimensional local system of Z / -graded complex vectorspaces over S . Let c , . . . , c m be distinct complex numbers, let Λ be the sheaf of finite sets ofreal numbers on S given by ϕ
7→ {
Re( c e − iϕ ) , . . . , Re( c e − iϕ ) } , and let SD ⊂ S be the associated set of Stokes directions.An abstract Deligne-Malgrange-Stokes filtration of S of exponential type andexponents ( c , . . . , c m ) is a filtration by subsheaves S ≤ λ such that: • S ≤ λ is labeled by local continuous sections λ of Λ and is locally constant on any arcwhich does not intersect SD . • Suppose ϕ ∈ SD , and let U, U ′ ⊂ S − SD be the two arcs adjacent at ϕ . Let λ < · · · < λ m and λ ′ < · · · < λ ′ m be the ordered flat sections of Λ | U and Λ | U ′ respectively.Trivialize S on U ∪ U ′ ∪ { ϕ } by identifying the flat sections with the elements of thefiber S ϕ and let ⊂ F ≤ λ ⊂ . . . ⊂ F ≤ λ m ⊂ S ϕ , and ⊂ F ′≤ λ ′ ⊂ . . . ⊂ F ′≤ λ ′ m ⊂ S ϕ be the filtrations corresponding to this trivialization and the filtrations S ≤ λ on U and U ′ respectively, i.e. F ≤ λ i := lim ψ ∈ Uψ → ϕ ( S ≤ λ i ) ψ and F ′≤ λ ′ i := lim ψ ∈ U ′ ψ → ϕ (cid:0) S ≤ λ ′ i (cid:1) ψ Let ≤ i < j ≤ i < j ≤ · · · ≤ i k < j k ≤ m be the sequence of integers such that λ a = λ ′ a for a [ i , j ] ∪ [ i , j ] ∪ · · · ∪ [ i k , j k ] , and for each interval [ i s , j s ] we have that λ ′ j s = λ i s , λ j s − = λ i s +1 , . . . λ ′ i s = λ j s . Then we require that: – for each a [ i , j ] ∪ [ i , j ] ∪ · · · ∪ [ i k , j k ] we have F ≤ λ a = F ′ λ ′ a ; – for each s = 1 , . . . , k , F ≤ λ js = F ≤ λ ′ js and the filtrations F ≤ λ is /F ≤ λ is − ⊂ F ≤ λ is +1 /F ≤ λ is − ⊂ · · · ⊂ F ≤ λ js /F ≤ λ is − F ′≤ λ ′ is /F ′≤ λ ′ is − ⊂ F ′≤ λ ′ is +1 /F ′≤ λ ′ is − ⊂ · · · ⊂ F ′≤ λ ′ js /F ′≤ λ ′ is − re ( j s − i s ) -opposed. Remark 2.17
The above discussion generalizes immediately from connections of exponen-tial type to arbitrary meromorphic connections (see remark 2.3). One gets a collection ofcurves drawn on the boundary of the cylinder which can be interpreted as a projection to0-jets of a Legendrian link in the contact manifold of 1-jets of functions on S .The categories of nc -Hodge structures, of nc -Hodge structures of exponential type, or of nc -Hodge structures with regular singularities all behave similarly to ordinary Hodge structures.For instance one can introduce the notion of polarization on nc -Hodge structures, whichspecializes to the usual notion in the case of ordinary Hodge structures. (This will not beneeded for our discussion so we will not spell it out here. The interested reader may wish toconsult [Her06, HS07, Kon08] for the details of the definition.) In fact we have the following Lemma 2.18
The categories ( Q - nc HS ) reg ⊂ ( Q - nc HS ) exp ⊂ ( Q - nc HS ) are Q -linear abeliancategories. The respective categories of polarizable nc -Hodge structures are semi-simple. Proof:
The statement is a manifestation of Simpson’s
Meta-Theorem from [Sim97b]. Theopposedness axiom implies that the respective categories are abelian and the existence ofpolarizations implies the semi-simplicity. The proofs follow verbatim the argument in usualHodge theory or the argument in [Sim97b]. Alternatively one can use the comparison state-ment [HS07, Lemma 3.9] identifying the nc -Hodge structures with pure twistor structuresand then invoke [Sim97b, Lemma 1.3 and Lemma 3.1]. (cid:3) The bundles with connections ( H c i , ∇ i ) can be thought of as the regular singular constituentsof the nc -Hodge structure ( H, E B , iso ). The ( H c i , ∇ i )’s are invariants of the nc -Hodge struc-ture but of course they do not give a complete set of invariants (see the third point in 2.13).As usual we need additional Stokes data (see e.g. [Sab02]) in order to reconstruct the pair( H , ∇ ) from its regular constituents. To understand how the rest of nc -Hodge structurearises from the constituents we need to understand how the rational structure E B interactswith the Stokes data. This process is very similar to the interaction between Betti, de Rhamand Dolbeault cohomology in ordinary Hodge theory and we will describe it in detail insection 2.3. 23he nc -Hodge structures one finds in geometric examples are very often regular (e.g. in thecase of ordinary Hodge structures) or at worst have exponential type. It is also expectedthat the nc -Hodge structures arising in mirror symmetry will always be of exponential typebut at the moment this is only supported by experimental evidence.We will discuss in detail some of this evidence in the subsequent sections. Before we getto the examples however, it will be instructive to comment on the reason for introducing the nc -Hodge structures at the first place. The geometric significance of these structures stemsfrom the fact that they appear naturally on the cohomology of non-commutative spaces ofcategorical nature. nc geometry The version of non-commutative geometry that is most relevant to nc -Hodge structures isthe one in which a proxy for the notion of a non-commutative space ( nc -space) is a category,thought of as the (unbounded) derived category of quasi-coherent sheaves on that space. nc -geometry The basic notion here is:
Definition 2.19 A graded complex nc -space (respectively a complex nc -space ) is a C -linear differential graded (respectively Z / -graded) category C which is homotopy completeand cocomplete. Notation:
We will often write C X for the category to signify that it describes the sheaftheory of some nc -space X , even when we do not have a geometric construction of X .The categorical point of view to non-commutative geometry goes back to the works of Bondal[Bon93], Bondal-Orlov [BO01, BO02] with many non-trivial examples computed in the laterworks of Orlov [Orl04, Orl05b, Orl05a], Caldero-Keller [CK05, CK06], Aroux, Orlov, and thefirst author, [AKO04, AKO06], Kuznetsov [Kuz05b, Kuz05a, Kuz06], etc. More recently thisapproach to nc -geometry became the central part of a long term research program initiated24y the second author and was studied systematically in the works of the second author andSoibelman [KS06b, Kon08], To¨en [To¨e07a], and To¨en-Vaquie [TV05]. Remark 2.20 (i)
We do not spell out here the notions of homotopy completeness andcocompleteness in dg categories since on one hand they are quite technical and on the otherhand will not be used later in the paper. It is worth mentioning though that some effortis required to define these notions. In the original approach of the second author describedin his 2005 IAS lectures and in his 2007 course at the University of Miami the homotopycompleteness and cocompletness in C was defined by a universal property for homotopycoherent diagrams of objects in the dg category labeled by simplicial sets. Alternatively[To¨e07c] one may use the model category ( C op − mod ) of C op -dg modules, whose equivalencesare the quasi-isomorphisms, and whose fibrations are the epimorphisms. In these terms onesays that C is homotopy complete if the full subcategory of ( C op − mod ) consisting of quasi-representable objects is preserved by all small homotopy limits (defined via the given modelstructure). Similarly we say that C is homotopy cocomplete if C op is homotopy complete. (ii) Note that in the above definition the category C is automatically triangulated as followsalready from the existence of finite homotopy limits, and Karoubi closed by the standardmapping telescope construction [BN93]. Example 2.21
The two main types of nc -spaces are the following: usual schemes: Usual complex schemes can be viewed as (graded) nc -spaces. Given ascheme X over C , the corresponding category C X is the derived category D ( Qcoh ( X ) w )of quasi-coherent sheaves on X taken with an appropriate dg enhancement (see [BK91]).In particular, the closed point pt = Spec( C ) corresponds to the category C pt of com-plexes of C -vector spaces. modules over an algebra: If A is a differential graded (or Z / C , then we get a nc -space ncSpec ( A ) such that C ncSpec ( A ) = ( A − mod )is the category of dg modules over A which admit an exhaustive increasing filtrationwhose associated graded are sums of shifts of A .25o illustrate how the above notion of a nc -space fits with the nc Hodge structures we willconcentrate on the case of nc -affine spaces, i.e. nc -spaces equivalent to ncSpec ( A ) for somedifferential Z / A over C . Note that because of derived Morita equivalencesan affine nc -space X does not determine an algebra A uniquely, i.e. different algebras cangive rise to the same nc -space. Remark 2.22
The condition is not as restrictive as it appears at a first glance. In factalmost all nc -spaces that one encounters in practice are affine. For instance usual quasi-compact quasi-separated schemes of finite type over C are affine when viewed as nc -spaces.This follows from a deep theorem of Bondal and van den Bergh [BvdB03] which asserts thatfor such a scheme X the category C X = D ( Qcoh ( X )) has a compact generator E . That is,we can find an object E ∈ C X so thatHom( E , • ) : C X → C pt commutes with homotopy colimits and has a zero kernel. In particular the dg algebracomputing the category C X is given in terms of the generator E , i.e. C X ∼ = ( Hom ( E , E ) op − mod ) . Suppose now that X = ncSpec ( A ). Recall that an object E ∈ C X = ( A − mod ) is perfect ifHom( E , • ) preserves small homotopy colimits. We will write Perf X for the full subcategoryof perfect objects in C X . We now have the following definition (see e.g. [KS06b, Kon08] or[TV05]): Definition 2.23
A complex differential Z / -graded algebra is called smooth: if A ∈ Perf ncSpec ( A ⊗ A op ) ; compact: if dim C H • ( A, d A ) < + ∞ or equivalently if A ∈ Perf pt . Note:
One can check (see e.g. [KS06b] or [TV05]) that the properties of X being smoothad compact do not depend on the choice of the algebra A which computes C X . Also, for a26sual scheme X of finite type over C , smoothness and compactness in the scheme-theoreticsense are equivalent to smoothness and compactness in the nc -sense. The analogy with commutative geometry suggests that one should look for pure nc -Hodgestructures on the cohomology of smooth an proper nc -spaces. More precisely we have thefollowing basic conjecture Conjecture 2.24
Let X be a smooth and compact nc -space over C . Then the periodiccyclic homology HP • ( C X ) of C X carries a natural functorial pure Q - nc -Hodge structure withregular singularities.Furthermore if the Z / -grading on X can be refined to a Z -grading, then the nc -Hodgestructure on HP • ( C X ) is an ordinary pure Hodge structure, i.e. belongs to the essentialimage of the functor N . There are some natural candidates for the various ingredients of the conjectural nc -Hodgestructure on HP • ( C X ). Assuming that X ∼ = ncSpec ( A ) is nc -affine, we can compute HP • ( C X ) in terms of A . Namely HP • ( C X ) = HP • ( A ) = HP • (cid:0) C red • ( A, A )(( u )) , ∂ + u · B (cid:1) , where • u is an even formal variable (of degree 2 in the Z -graded case); • C red − k +1 ( A, A )(( u )) := A ⊗ ( A/ C · A ) ⊗ k ⊗ C (( u )), for all k ≥ • ∂ = b + δ , where b ( a ⊗ · · · ⊗ a n ) := n − X i =0 ( − deg( a ⊗···⊗ a i ) a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n + ( − deg( a ⊗···⊗ a n )(deg( a n )+1)+1 a n a ⊗ · · · ⊗ a n − ,
27s the Hochschild differential, and δ ( a ⊗ · · · ⊗ a n ) := n X i =0 ( − deg( a ⊗···⊗ a i − ) a ⊗ · · · ⊗ d A a i ⊗ · · · ⊗ a n is the differential induced from d A via the Leibniz rule; • B ( a ⊗ · · · ⊗ a n ) := n X i =0 ( − (deg( a ⊗···⊗ a i ) − a i +1 ⊗···⊗ a n ) − A ⊗ a i +1 ⊗ · · · ⊗ a n ⊗ a ⊗ · · · ⊗ a i , is Connes’ cyclic differential. nc -Hodgestructure Note that by construction HP • ( C X ) is a module over C (( u )). We can also look at thenegative cyclic homology HC −• ( C X ) of C X . By definition HC −• ( C X ) is the cohomology ofthe complex (cid:0) C red • ( A, A )[[ u ]] , ∂ + u · B (cid:1) , and so is a module over C [[ u ]]. The specialization HC −• ( C X ) /uHC −• ( C X )of this module at u = 0 maps to the cohomology of the complex( C red • ( A, A ) , ∂ )of reduced Hochschild chains for A which by definition is the Hochschild homology HH • ( A )of A . The Hochschild-to-cyclic spectral sequence implies that(2.2.1) dim C (( u )) HP • ( A ) ≤ dim C HH • ( A )If X is a smooth and compact nc -space, the Hochschild chain complex of C X is the derivedtensor product over A ⊗ A op of a perfect complex with finite dimensional cohomology with28tself. In particular HH • ( C X ) := HH • ( A ) is a finite dimensional C -vector space, and soby (2.2.1) we have that HP • ( C X ) is finite dimensional over C (( u )). Thus the C [[ u ]]-module HC −• ( C X ) is finitely generated and so corresponds to the formal germ at u = 0 of an algebraic Z / A C . The fiber of this sheaf at u = 0 is HH • ( C X ) and the genericfiber is HP • ( C X ). In [KS06b, Kon08] the second author proposed the so called degenerationconjecture asserting that for a smooth and compact nc -space X = ncSpec ( A ) we havean equality of dimensions in (2.2.1). In other words the degeneration conjecture assertthat for a smooth and compact nc -space the C [[ u ]]-module HC −• ( C X ) is free of finite rankand thus corresponds to an algebraic vector bundle on the one dimensional formal disc D := Spf( C [[ u ]]). Remark 2.25
There is a lot of evidence supporting the validity of this conjecture. The workof Weibel [Wei96] shows that if X is a usual quasi-compact and quasi-separated complexscheme the Hochschild and periodic cyclic homology of X viewed as a nc -space can be iden-tified with the algebraic de Rham and Dolbeault cohomology of X respectively. Combinedwith the degeneration of the Hodge-to-de-Rham spectral sequence in the smooth proper casethis shows that the degeneration conjecture holds true for usual schemes. Also recently in avery exciting sequence of papers [Kal07a, Kal06] Kaledin proved the degeneration conjecturefor graded nc -spaces X = ncSpec ( A ) for which A is concentrated in non-negative degrees.The case of graded nc -spaces X = ncSpec ( A ) for which A is concentrated in non-positivedegrees was also settled by Shklyarov [Shk08]. The general graded case and the Z / u -direction The next observation is that the C { u } [ u − ]-module HP • ( C X ) comes equipped with a naturalmeromorphic connection. Indeed, recall that by the work of Getzler [Get93] there is a versionof the Gauss-Manin connection which exists on the periodic cyclic homology of any flat familyof differential graded algebras (see also [Tsy07, Kal07b]). An analogous statement holds inthe Z / A x over the formal disc Spf C [[ x ]] with a formal parameter x ,29s an operator ∇ GM u ∂∂x : H • ( C red ( A x , A x )[[ u, x ]] , ∂ A x + u · B A x ) → H • ( C red ( A x , A x )[[ u, x ]] , ∂ A x + u · B A x )satisfying the Leibniz rule with respect to the multiplications by u and x (compare this withthe (Griffiths transversality axiom) in Definition 2.7 from Section 2.1.5).Suppose now A is a differential Z / m A , differential d A , anda strict unit 1 A . Then we can form a flat family A → A − { } of differential Z / A − { } . The fiber A t of A over a point t ∈ A C − { } is the d( Z / Z / A and m A t := t · m A ,d A t := t · d A , A t := t − · A . Looking at the scaling properties of ∂ and B we see that the identity morphism on the levelof cochains induces a natural isomorphism(2.2.2) H • (cid:0) C red • ( A t , A t )[[ u ]] , ∂ A t + u · B A t (cid:1) ∼ = / / H • ( C red • ( A, A )[[ u ]] , ∂ + ut − · B ) . This isomorphism does not come from a quasi-isomorphism of complexes, as the identitymap is not a morphims of complexes: the differentials do not coincide but differ by thefactor t . If A is smooth and compact, then the negative cyclic homology of the family ofalgebras A t gives rise to an algebraic vector bundle g HC − on the product ( A − { } ) × D .Here D := Spf C [[ u ]] denotes the one dimensional formal disc. We will write ( t, u ) for thecoordinates on ( A −{ } ) × D . We will be interested in fact only in the formal neigborhood ofpoint t = 1 where we can choose as a local coordinate x := log( t ). The Getzler-Gauss-Maninconnection then can be viewed as a relative holomorphic connection ∇ GM on g HC − whichdifferentiates only along A C − { } . On the other hand the formal completion of the group C × at 1 acts on ( A C − { } ) × D by ( t, u ) ( µt, µ u ) for µ ∈ C × . The isomorphism (2.2.2) givesrise to a C × -equivariant structure on the vector bundle g HC − and the infinitesimal actionof d/dµ associated with this equivariant structure gives a holomorphic differential operator Λ ∈ Diff ≤ ( g HC − , g HC − ) with symbol equal to (cid:18) t ∂∂t + 2 u ∂∂u (cid:19) · id g HC − . ∇ u ∂∂u := u · Λ − ∇ GM ut ∂∂t is a first order differential operator on g HC − with symbol u ∂∂u · id g HC − and so after restricting g HC − to { } × D this operator gives a meromorphic connection ∇ onthe C [[ u ]]-module HC −• ( C X ) with at most a second order pole at u = 0. Note also that ifthe algebra A is Z -graded, then the family A t t is easily seen to be trivial and the connection ∇ has the first order pole at u = 0 with monodromy equal to ( − parity . Q -structure The categorical origin of the rational (or integral) structure of the conjectural nc -Hodgestructure is more mysterious. Conceptually the correct rational structure should come fromthe Betti cohomology or, say, the topological K-theory of the nc -space. There are two naturalapproaches to constructing the rational structure E B ⊂ HP • ( C X ): (a) The soft algebra approach ([Kon08]). Let again X = ncSpec ( A ) be an affine nc -space, and assume X is compact. By analogy with the classical geometric case one expectsthat there should exist a nuclear Frech´et d( Z / A C ∞ so that • The K-theory of A C ∞ satisfies Bott periodicity, i.e. K i ( A C ∞ ) = K i +2 ( A C ∞ ) for all i ≥ • There is a homomorphism ϕ : A → A C ∞ of d( Z / ϕ ∗ : HP • ( A ) → HP • ( A C ∞ ) is an isomorphism, and the image of the Chern character map ch : K • ( A C ∞ ) → HP • ( A C ∞ )is an integral lattice, and hence gives a rational structure E B ⊂ HP • ( A ).31 ote: If X is a smooth and compact complex variety and if E ∈
Perf ( X ) is a vector bundlegenerating C X , then one may take A := A , • ( X, E ∨ ⊗ E ) , ¯ ∂ ) A C ∞ := A , ( X, E ∨ ⊗ E ) . Note that the algebra A C ∞ is Morita equivalent to C ∞ ( X ). (b) The semi-topological K-theory approach (Bondal, To¨en, [To¨e07b]). Assumeagain that X = ncSpec ( A ) is a smooth and compact graded nc -affine nc -space. Considerthe moduli stack M X of all objects in Perf X . This is an ∞ -stack which by a theorem ofTo¨en and Vaquie [TV05] is locally geometric and locally of finite presentation. Moreover forany a, b ∈ N the substack M [ a,b ] X ⊂ M X consisting of objects of amplitude in the interval[ a, b ] is a geometric b − a + 1-stack. The functor sending a complex scheme to the underlyingtopological space in the analytic topology gives rise by a left Kan extension to a topologicalrealization functor | • | : Ho ( Stacks / C ) → Ho(
Top )from the homotopy category of stacks to the homotopy category of complex spaces. FollowingFriedlander-Walker [FW05] we define the semi-topological K-group of the nc -space X to be K st ( X ) := π ( | M X | ) . The group structure here is induced by the direct sum ⊕ of A -modules: the monoid( π ( | M X | ) , ⊕ ) is actually a group. To see this note that for any A -module E we havethat [ E ⊕ E [1]] = 0 in π ( | M X | ). Indeed we have distinguished triangles E / / / / E [1] / / E [1] E / / E ⊕ E [1] / / E [1] / / E [1]the first of which corresponds to id ∈ Ext ( E [1] , E ) = Hom( E, E ), and the second corre-sponds to 0 ∈ Ext ( E [1] , E ) = Hom( E, E ). Since Ext ( E [1] , E ) = Hom( E, E ) is a vectorspace, it follows that id deforms to 0 continuously and so the second terms in the abovetriangles represent the same point in π ( | M X | ).More generally ⊕ makes | M X | into an H -space K st ( X ) which is the degree zero part ofa natural spectrum. Using this one can define K sti ( X ) for all i ≥ C X is triangulated it is a module over the category Perf pt of com-plexes of C -vector spaces with finite dimensional total cohomology. In particular K st • ( X ) isa graded module over K st • (pt). It can be checked that K st (pt) = BU = K top (pt) , and so K st • ( X ) is a graded Z [ u ]-module (deg u = 2).Now we can define K top • ( X ) := K st • ( X )[ u − ] = K st • ( X ) ⊗ Z [ u ] Z [ u, u − ] . Again one expects that there is a Chern character map ch : K top • ( X ) → HP • ( C X )whose image gives a rational structure E B on HP • ( C X ). Note: If X is a smooth and compact complex variety, then the Friedlander-Walker com-parison theorem [FW01] implies that K top ( D ( QCoh ( X ))) ∼ = K top ( X top ), where X top is thetopological space underlying X . Even though we have some good candidates for the ingredients H , ∇ , E B of the conjectural nc -Hodge structure associated with a nc -space, there are several important problems thatneed to be addressed before one can prove Conjecture 2.24: • show that the connection ∇ has regular singularities (this is automatically true in the Z -graded case); • show that ∇ preserves the rational structure; • show that the opposedness axiom hold.One can show that for a smooth compact nc-space the coefficient A − in the u -connection isa nilpotent operator, which gives an evidence in favor of the regular singularity question.33n fact Conjecture 2.24 and the above questions are special cases of a general conjecture whichpredicts the existence of a general nc -Hodge structure on the periodic cyclic homology of acurved d( Z / In this section we discuss how general nc -Hodge structures of exponential type can be gluedtogether out of nc -Hodge structures with regular singularities and additional gluing data. nc -De Rham data The de Rham part of a nc -Hodge structure of exponential type can be prescribed in threeequivalent ways: nc dR(i) A pair ( M , ∇ ), where M is a finite dimensional vector space over C { u } [ u − ] and ∇ is a meromorphic connection. These data should satisfy the following Property nc dR(i): There exist: • a frame e = ( e , . . . , e r ) of M over C { u } [ u − ] in which ∇ = d + X k ≥− A k u k ! du with A k ∈ Mat r × r ( C ), r = rank C { u } [ u − ] M . In other words, there is a holomorphicextension H = C { u } e ⊕ . . . ⊕ C { u } e r in which ∇ has at most a second order pole. • a formal isomorphism over C (( u )):( M , ∇ ) ⊗ C { u } [ u − ] C (( u )) ∼ = → m M i =1 E c i /u ⊗ ( R i , ∇ i )34here ( R i , ∇ i ) are meromorphic bundles with connections with regular singularitiesand c , . . . , c m ∈ C denote the distinct eigenvalues of A − . nc dR(ii) A pair ( M, ∇ ), where M is an algebraic vector bundle on A − { } and ∇ is aconnection on M . These data should satisfy the following Property nc dR(ii): M can be extended to an algebraic vector bundle f M on P , and • with respect to this extension and appropriate local trivializations at zero and infinitywe must have ∇ = d + X k ≥− A k u k ! du near 0 , ∇ = d + X k ≥− B k u − k ! d ( u − ) near ∞ . In other words ∇ : f M → f M ⊗ O P Ω P (2 · { } + {∞} ). • There is a formal isomorphism over C (( u )):( M, ∇ ) ⊗ C [ u,u − ] C (( u )) ∼ = → m M i =1 E c i /u ⊗ ( R i , ∇ i )where ( R i , ∇ i ) are meromorphic bundles with connections with regular singularitiesand c , . . . , c m ∈ C denote the distinct eigenvalues of A − . nc dR(iii) An algebraic holonomic D -module M on A . The D -module M should alsosatisfy the following Property nc dR(iii): M has regular singularities and H • DR ( A , M ) = 0.The nc -de Rham data of types nc dR(i) , nc dR(ii) , and nc dR(iii) form obvious full sub-categories in the categories of meromorphic connections over C { u } [ u − ], algebraic vectorbundles with connections on A − { } , and coherent algebraic D -modules on A respectively.We have the following 35 emma 2.26 The categories of nc -de Rham data of types nc dR(i) , nc dR(ii) , and nc dR(iii) are all equivalent. Proof.
In essence we have already discussed the equivalence nc dR(i) ⇔ nc dR(ii) inRemark 2.1. Explicitly ( M , ∇ ) = G (( M, ∇ )) = ( M ⊗ C [ u,u − ] C { u } [ u − ] , ∇ ).We define a functor F : (data (iii) ) → (data (ii) ) as follows. Let M be a regular holo-nomic algebraic D -module on A with trivial de Rham cohomology. Denote the coordinateon A by v . The vanishing of de Rham cohomology means that the action ddv : M → M is an invertible operator. Consider the algebraic Fourier transform Φ M which is the samevector space as M endowed with action of the Weyl algebra defined by˜ v := ddvdd ˜ v := − v where ˜ v is the coordinate on the dual line. By our assumptions Φ M is a holonomic D -module on which ˜ v acts invertibly. Hence Φ M is the direct image of a holonomic D -module Φ M ′ on A − { } under the embedding (cid:0) A − { } (cid:1) ֒ → A = Spec( C [˜ v ])Finally, making the change of coordinates u = 1 / ˜ v we obtain a D -module M on A − { } with coordinate u .We claim that F ( M ) := M obtained in this way satisfies the property nc dR(ii) , and thatby this construction one obtains all such modules. It follows from the well-known propertiesof the Fourier transform that Φ M has no singularities in A − { } and the its singularityat ˜ v = 0 is regular. Hence M is a vector bundle on A − { } endowed with connection withregular singularity at ∞ . It only remains to to use the well-known fact (see e.g. [Mal91,Chapters IX-XI] or [Kat90, Theorem 2.10.16]) that the exponential type property for M isequivalent to the property of M to have only regular singularities. (cid:3) Remark 2.27
The characterization of the exponential type property in terms of the Fouriertransform can be stated more precisely (see [Mal91, Chapters IX-XI] or [Kat90, Theo-rem 2.10.16]): For an algebraic holonomic D -module M on the complex affine line, thefollowing two conditions are equivalent: 36) M has regular singularities;2) the Fourier transform Φ M of M has no singularities outside 0, its singularity at 0 isregular, and its singularity at infinity is of exponential type.Explicitly Φ M being of exponential type at infinity means that if x is a coordinate on A centered at 0, then after passing to the formal completion ( Φ M ) ⊗ C [ x ] C (( x − )) the resultingmodule will be isomorphic to a finite sum m M i =1 E c i x ⊗ ( R i , ∇ i )where ( R i , ∇ i ) are D -modules with a regular singularity at infinity. Remark 2.28
Note that the de Rham data nc dR(i) is analytic in nature, whereas nc dR(ii) and nc dR(iii) are algebraic. In fact from the proof it is clear that nc dR(ii) and nc dR(iii) and their equivalence still make sense if we replace C with any field of characteristic zero. nc -Betti data The (rational) Betti part of a nc -Hodge structure of exponential type can be prescribed infour ways: nc B(i)
A (middle perversity) perverse sheaf G • of Q -vector spaces on the Riemann surface C (taken with the analytic topology) satisfying the following Property nc B(i): R Γ( C , G • ) = 0. nc B(ii)
A constructible sheaf F of Q -vector spaces on the Riemann surface C (taken withthe analytic topology) satisfying the following Property nc B(ii): R Γ( C , F ) = 0. 37 c B(iii)
A finite collection of distinct points S = { c , . . . , c n } ⊂ C , and • a collection U , U , . . . , U n of finite dimensional non-zero Q -vector spaces, • a collection of linear maps T ij : U j → U i , for all i, j = 1 , . . . , n ,satisfying the following Property nc B(iii): T ii ∈ GL ( U i ). nc B(iv)
A local system S of Q -vector spaces on S equipped with a filtration { S ≤ λ } λ ∈ R bysubsheaves of Q -vector spaces, satisfying the following Property nc B(iv):
The filtration { S ≤ λ ⊗ C } λ ∈ R of S ⊗ C is a Deligne-Malgrange-Stokesfiltration of exponential type. In other words, there exist complex numbers c , . . . , c n ∈ C so that: • For every ϕ ∈ S , the filtration { ( S ≤ λ ⊗ C ) ϕ } λ ∈ R of the stalk ( S ⊗ C ) ϕ jumpsexactly at the real numbers { Re ( c k e − iϕ ) } nk =1 . • The associated graded sheaves of S ⊗ C with respect to { S ≤ λ ⊗ C } λ ∈ R are localsystems on S .Again there are natural equivalences of the different types of Betti data (for nc B(iii) theequivalence depends on certain choices of paths as one can see from the proof of Theorem 2.29and the statement of Lemma 2.30.). Consider the full subcategories ( nc B(i) ) and ( nc B(ii) )of nc -Betti data of types nc B(i) and nc B(ii) in the category of perverse sheaves of Q -vectorspaces on C and in the category of constructible sheaves of Q -vector spaces on C respectively.We have the following Theorem 2.29
The categories of nc -Betti data of types nc B(i) and nc B(ii) are naturallyequivalent. More precisely, the natural functors H − : D b constr ( C , Q ) → Constr ( C , Q ) and [1] : Constr ( C , Q ) → D b constr ( C , Q ) induce mutually inverse equivalences of the full subcategories ( nc B(i) ) ⊂ D b constr ( C , Q ) and ( nc B(ii) ) ⊂ Constr ( C , Q ) . roof. First we look at the data nc B(i) more closely. Suppose X is a complex analyticspace underlying a complex quasi-projective variety. Recall (see e.g. [BBD82, KS94, Dim04])that a bounded complex G • of sheaves of C -vector spaces on X is called a (middle perversity)perverse sheaf if it has constructible cohomology sheaves H k ( G • ) and if • for all k ∈ Z , we have dim R { x ∈ X | H − k ( i ∗ x G • ) = 0 } ≤ k , • for all k ∈ Z , we have dim R { x ∈ X | H k ( i ! x G • ) = 0 } ≤ k .Here i x : x ֒ → X denotes the inclusion of the point x in X .For future reference we will write D b constr ( X, Q ) for the derived category of complexes of Q -vector spaces on X with constructible cohomology, Perv ( X, Q ) ⊂ D b constr ( X, Q ) for the fullsubcategory of middle perversity perverse sheaves, and Constr ( X, Q ) ⊂ D b constr ( X, Q ) for thefull subcategory of constructible sheaves.From the definition it is clear that if G • is a perverse sheaf on C , then G • has at mosttwo non-trivial cohomology sheaves H − ( G • ) and H ( G • ). Moreover the support of H ( G • )has dimension ≤
0. Now the cohomology R Γ • ( G • ) = H • ( C , G • ) can be computed via thehypercohomology spectral sequence E pq = H p ( C , H q ( G • )) ⇒ H p + q ( C , G • ) . Since G • has only two cohomology sheaves, the E sheet of this spectral sequence is0 H ( C , H ( G • )) , , ZZZZZZZZZZZZZZZZZZZZZZZZZZZ H ( C , H ( G • )) , , YYYYYYYYYYYYYYYYYYYYYYYYYYY H ( C , H ( G • )) · · ·− H ( C , H − ( G • )) H ( C , H − ( G • )) H ( C , H − ( G • )) · · · H p ( C , H q ( G • )) =0 for all q and all p >
1. Furthermore since H ( G • ) has at most zero dimensional supportwe have H ( C , H ( G • )) = 0. In particular the spectral sequence degenerates at E and theonly potentially non-trivial cohomology groups of G • are H − ( C , G • ) = H ( C , H − ( G • )) , and H ( C , G • ) = H ( C , H − ( G • )) ⊕ H ( C , H ( G • )) . R Γ( G • ) = 0 we get that H ( C , H ( G • )) = 0, i.e. that H ( G • ) = 0. In other words G • = F [1] for some constructible sheaf F with R Γ( F ) = 0.To finish the proof of the theorem we need to show that for every constructible sheaf F with R Γ( F ) = 0, the object F [1] will be perverse (for the middle perversity). For this wewill have to look more closely at constructible sheaves on the complex line.Suppose F is a constructible sheaf of Q vector spaces on C . Then there is a finite set S = { c , . . . , c n } of points in C so that C − S is the maximal open set on which F restrictsto a local system. Let F := F | C − S denote this local system. Let C − S j ֒ → C i ← ֓ S be thenatural inclusions and let ϕ : F → j ∗ j ∗ F = j ∗ F be the adjunction homomorphism.Before we can describe F and F via the quiver-like data of type nc B(iii) we will needto make some rigidifying choices. First we fix a base point c ∈ C − S . For i = 1 , . . . , n we choose a collection of a small disjoint discs D i ⊂ C , each D i centered at c i . For eachdisc we fix a point o i ∈ ∂ D i and denote by l i the loop starting and ending at o i and tracing ∂ D i once in the counterclockwise direction. We fix an ordered system of non-intersectingpaths { a i } ni =1 ⊂ C − ( ∪ ni =1 D i ) which connect the base point c with the each of the o i as inFigure 2. c i c o i D i a i Figure 2: A system of paths for S ⊂ C .Let mon l i : F o i → F o i be the monodromy operator associated with the local system F andthe loop l i . The stalk ( j ∗ F ) c i of the constructible sheaf j ∗ F at c i can be identified naturallywith the subspace F mon l i o i of invariants for the local monodromy. Taking stalks at each c i ∈ S
40e get Q -vector spaces F c i and the adjunction map ϕ : F → j ∗ F induces linear maps ϕ c i : F c i → F mon c i o i ⊂ F o i . Note that, by descent, specifying the constructible sheaf F is equivalent to specifying thecollection of points S ⊂ C , the local system F on C − S , the collection of vector spaces { F c i } ni =1 and the collection of linear maps { ϕ c i } ni =1 . In particular, the compactly supportedpullback of F [1] via the inclusion i c i : { c i } ֒ → C can be computed in terms of these linearalgebraic data and is given explicitly by the complex i ! c i ( F [1]) = [ F c i ϕ c i / / F o i − mon l i / / F o i − . By definition F [1] is a perverse sheaf iff for all c i ∈ S the complex of vector spaces i ! c i ( F [1])has no cohomology in strictly negative degrees, i.e. iff ϕ c i is injective for all i = 1 , . . . , n .Next we rewrite the condition R Γ( C , F ) = 0 in terms of the descent data( F , { F c i } , { ϕ c i } ). To simplify notation let U := F c , V i = F c i for i = 1 , . . . , n . Let T i : U → U be the monodromy operator for the local system F and the c -based loop γ i obtained by first tracing the path a i from c to o i , then tracing the loop l i , and then tracingback a i in the opposite direction. Similarly we have linear maps ψ i : V i → U T i ⊂ U obtainedby conjugating ϕ c i : V i → F o i with the operator of parallel transport in F along the path a i .The descent data for F with respect to the open cover C = ( C − S ) ∪ ( ∪ ni =1 D i ) are nowcompletely encoded in the linear algebraic data ( U, { V i } ni =1 , { T i } ni =1 , { ψ i } ni =1 ). Cover C bythe two opens C − S and ∪ ni =1 D i . The intersection of these two opens is the disjoint union ofpunctured discs ` ni =1 ( D i − c i ). The Mayer-Vietoris sequence for F and this cover identifies R Γ( C , F ) with the complex: L ni =1 V i ⊕ ni =1 ψ i / / U ⊕ n ⊕ ni =1 (1 − T i ) / / U ⊕ n L L U id ⊕ nU iiiiiiiiiiiiiiiiiii ⊕ ni =1 (1 − T i ) / / U ⊕ n − id ⊕ nU jjjjjjjjjjjjjjjj Q -vector spaces: R Γ( C , F ) ∼ = L ni =1 V i ⊕ ni =1 ψ i , , ZZZZZZZZZZZZZZZ L U ⊕ n U id ⊕ nU dddddddddddddddddd ψ i : V i → U are injective for all i = 1 , . . . , n , and(b) the map U → ⊕ ni =1 U/V i is an isomorphism.Thus the acyclicity of R Γ( C , F ) implies the perversity of F [1]. The theorem is proven. (cid:3) The conditions (a) and (b) from the proof of Theorem 2.29 suggest a better way of recordingthe linear algebraic content of F . Namely, if we set U i := U/V i , then we can use (b) toidentify U with ⊕ ni =1 U i , V i with ⊕ j = i U j and the map ψ i : V i ֒ → U with the natural inclusion ⊕ j = i U j ⊂ ⊕ ni =1 U i . The only thing left is the data of the monodromy operators T i ∈ GL ( U ), i = 1 , . . . , n . However for each i we have embedding V i (cid:31) (cid:127) ψ i / / Ker h U (1 − T i ) −→ U i and so under the decomposition U = ⊕ ni =1 U i the automorphism T i has a block form T i = · · · T i · · ·
00 1 · · · T i · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · T ii · · ·
00 0 · · · T i +1 ,i · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · T ni · · · where T i | U i = P nj =1 T ji , and T ji : U i → U j . The linear maps T ji are unconstrained except forthe obvious condition that for all i the map T i should be invertible, which is equivalent to T ii : U i → U i being invertible for all i = 1 , . . . , n . Also since S was chosen to be such that C − S is the maximal open on which F is a local system, it follows that U i = { } for all i = 1 , . . . , n .In other words we have proven the following Lemma 2.30
Fix the set of points S = { c , . . . , c n } and choose the discs { D i } i = n and thesystem of paths { a i } ni =1 . The functor assigning to a constructible sheaf F with singularitiesat S the data ( { U i } ni =1 , { T ij } ) establishes an equivalence between the groupoid of all data oftype nc B(ii) with singularities exactly at S and all data of type nc B(iii) with the given S . nc - de Rham and Betti data is provided as usual by the Riemann-Hilbert correspondence. This is tautological but we record it for future reference: Lemma 2.31
The de Rham functor: M → cone M ⊗ C [ u ] O an A ∂∂u / / M ⊗ C [ u ] O an A ! establishes an equivalence between the categories ( nc dR(iii) ) and ( nc B(i) ) ⊗ C . Finally, note that Theorem 2.29, together with Lemma 2.31, and Deligne’s classification[BV85, Theorem 4.7.3] of germs of irregular connections give immediately:
Lemma 2.32
The data data ( nc B(ii) ) and ( nc B(iv) ) are equivalent. Proof.
Let F be a constructible sheaf of Q -vector spaces on C . Define a local system S of Q -vector spaces on S as the restriction of F to the circle “at infinity”, i.e. define the stalkof S at ϕ ∈ S to be S ϕ := lim r → + ∞ F re iϕ . Next, for any λ ∈ R and any ϕ ∈ S consider the half-plane H ϕ,λ := ( λ + { u ∈ C | Re( u ) ≥ } ) · e iϕ , as shown on Figure 3.Now suppose that R Γ( C , F ) = 0. By the long exact sequence for the cohomology of thepair H ϕ,λ ⊂ C we get that H i ( C , H ϕ,λ ; F ) = 0 unless i = 1. The Deligne-Malgrange-Stokesfiltration on S is then given explicitly by S ϕ, ≤ λ := H ( C , H ϕ,λ ; F ) ⊂ S ϕ . (cid:3) For the purposes of nc -Hodge theory all these statements can be summarized in the following43 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) λe iϕ λ ≥ H ϕ,λ or (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) λe iϕ λ ≤ H ϕ,λ Figure 3: The half-plane H ϕ,λ . Theorem 2.33
There is natural equivalence of categories triples ( H, E B , iso ) satisfyingthe (nc-filtration axiom) exp and the ( Q -structureaxiom) exp ↔ quadruples (( H, ∇ ) , F B , f ), where • H is an algebraic Z / C and ∇ is a meromor-phic connection on H satisfying the (nc-filtration axiom) exp ; • F B ∈ Constr ( C , Q ), satisfying R Γ( C , F B ) = 0; • f is an isomorphism f : F B ⊗ C → DR (cid:0) Φ (cid:2) ι ∗ (cid:0) ( H, ∇ ) | A −{ } (cid:1)(cid:3)(cid:1) in D b constr ( C , C ) Here as before DR is the de Rham complex functor from the derived category of regular holonomic D -modules to the derived category of constructible sheaves, ι is the inclusion map ι : A − { } ֒ → A given by ι ( v ) = v − , and Φ ( • ) is the Fourier-Laplace transform for D -modules on A . Proof.
Follows immediately from previous equivalences. (cid:3) .4 Structure results In this section we collect a few results clarifying the structure properties of the nc -Hodgestructures of exponential type. nc -Betti data Since the gluing data nc B(iii) are of essentially combinatorial nature, it is natural to lookfor a quiver interpretation of this data. To that end consider the algebra(2.4.1) A n := * p ,. . . , p n T , T − ,. . . , T − nn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p + p + . . . + p n = 1 p i p j = p j p i for i = j , p i = p i T − ii p i T p i = p i T p i T − ii = p i + This is the path algebra of the complete quiver having n ordered vertices, n − n arrowsconnecting all pairs of distinct vertices, and 2 n -loops - two at each vertex, with the onlyrelations being that the two loops at every given vertex are inverses to each other.Note that our description of the gluing data nc B(iii) now immediately gives the following
Lemma 2.34
For a given set of points S = { c , . . . , c n } ⊂ C , the category of gluing data nc B(iii) with singularities at S is equivalent to the category of finite dimensional represen-tations of A n . In particular since the braid group B n on n -strands acts naturally on the data nc B(iii) we get a homomorphism B n → Aut( A n ) from the braid group to the group of algebraautomorphisms of A n . nc -Hodge structures It is natural to expect that the usual classification of connections with second order poles interms of formal regular type and Stokes multipliers can be promoted to a similar classificationof nc -Hodge structures. The search for such a classification leads naturally to the followingtheorem: Theorem 2.35
Let { ( H, E B , iso ) } be a nc -Hodge structure of exponential type. Then speci-fying { ( H, E B , iso ) } is equivalent to specifying the following data: regular type) A finite set S = { c , . . . , c n } ⊂ C and a collection { (( R i , ∇ i ) , E B,i , iso i ) } ni =1 of nc -Hodge structures with regular singularities. (gluing data) A base point c ∈ C − S , a collection of discs { D i } ni =1 and paths { a i } ni =1 ,chosen as in the proof of Theorem 2.29, and for every i = j , i, j ∈ { , . . . , n } a mapof rational vector spaces T ij : ( E B,j ) c −→ ( E B,i ) c Proof.
It will be convenient to introduce formal counterparts to the de Rham parts of the nc -Hodge structures appearing in the statement of the theorem. We consider the following: formal(a) A pair ( M for , ∇ for ), where M for is a finite dimensional vector space over C (( u ))and ∇ for is a meromorphic connection on M for of exponential type. formal(b) A finite set of points S = { c , . . . , c n } ⊂ C and a collection { ( R for i , ∇ for i ) } ni =1 where each R for i is a non-zero finite dimensional vector space over C (( u )) and each ∇ for i is ameromorphic connection on R for i with a regular singularity. formal(c) A finite collection of points S = { c , . . . , c n } ⊂ C , and • a collection U , U , . . . , U n of finite dimensional non-zero Q -vector spaces, • a collection of linear maps T ii ∈ GL ( U i ), for all i = 1 , . . . , n ,By Remark 2.13 the natural functor from the category of data formal(b) to the categoryof data formal(a) , which is given by( formal(b) ) / / ( formal(a) ) (cid:0) S ; { ( R for i , ∇ for i ) } ni =1 (cid:1) / / L ni =1 E c i /u ⊗ ( R for i , ∇ for i ) =: ( M for , ∇ for )is an equivalence of categories.Also we have the following Lemma 2.36
The categories of data formal(b) and formal(c) are naturally equivalent. roof. Indeed, consider the category C of all data consisting of a finite set of points S = { c , . . . , c n } ⊂ C and a collection { ( R i , ∇ i ) } ni =1 where each R i is a non-zero finitedimensional vector space over C { u } [ u − ] and each ∇ i is a meromorphic connection on R i with a regular singularity and non-trivial monodromy. Then we have natural functors( formal(b) ) C ( • ) ⊗ C (( u )) rrrrrrrrrrr mon % % LLLLLLLLLLL ( formal(c) )where ( • ) ⊗ C (( u )) is the passage to a formal completion and mon is given by assigning toeach ( R i , ∇ i ) the pair ( U i , T i ), where U i is the fiber of the Birkhoff extension B ( R i , ∇ i ) of( R i , ∇ i ) at 1 ∈ A , and T i is the monodromy of B ( R i , ∇ i ) around the unit circle traced inthe positive direction.This proves the lemma since mon is an equivalence by the Riemann-Hilbert correspon-dence and ( • ) ⊗ C (( u )) is an equivalence by the formal decomposition theorem [Sab02, II.5.7]). (cid:3) Note that these equivalences are compatible with the corresponding equivalence of an-alytic de Rham data and Betti data. More precisely we have a commutative diagram offunctors(2.4.2) ( nc dR(i) ) / / O O (cid:15) (cid:15) ( formal(a) ) O O (cid:15) (cid:15) ( nc B(iii) ) / / ( formal(c) )Here the right vertical equivalence is the composition of the equivalences ( formal(a) ) ∼ =( formal(b) ) ∼ = ( formal(c) ) that we just discussed. The left vertical equivalence is thecomposition of the equivalence ( nc dR(i)) ∼ = ( nc dR(iii)) given in Lemma 2.26, the equivalence( nc dR(iii)) ∼ = ( nc B(i)) from Lemma 2.31, the equivalence ( nc B(i)) ∼ = ( nc B(ii)) given inTheorem 2.29, and the equivalence ( nc B(ii)) ∼ = ( nc B(iii)) from Lemma 2.30.Horizontally we have the forgetful functors( nc dR(i) ) / / ( formal(a) )( M , ∇ ) (cid:31) / / ( M , ∇ ) ⊗ C { u } [ u − ] C (( u )) , nc B(iii) ) / / ( formal(c) ) (cid:0) S ; { U i } ni =1 , { T ij } ni,j =1 (cid:1) (cid:31) / / ( S ; { U i } ni =1 , { T ii } ni =1 ) . Next we need the following
Lemma 2.37
Suppose that ( M , ∇ ) is some de Rham data of type nc dR(i) and let ( M for , ∇ for ) = ( M , ∇ ) ⊗ C { u } [ u − ] C (( u )) be the corresponding formal data. Then: (a) the map (cid:18) C { u } -submodules H ⊂ M , onwhich ∇ has a pole of order ≤ (cid:19) ( • ) ⊗ C [[ u ]] / / C [[ u ]] -submodules H for ⊂ M for ,on which ∇ for has a pole of order ≤ , is bijective. (b) If Ψ : ( M for , ∇ for ) → L ni =1 E c i /u ⊗ ( R for i , ∇ for i ) is a formal isomorphism, then the map C [[ u ]] -submodules H for ⊂ M for ,on which ∇ for has a pole of order ≤ C [[ u ]] -submodules H for i ⊂ R for i ,for all i = 1 , . . . , n , on which ∇ for i has a pole of order ≤ , Ψ o o is bijective. Proof. (a)
Pick some frame e of M over C { u } [ u − ] and let H := C { u }· e ⊂ M be the sub-module of all sections in M that are holomorphic in this frame. Now any C { u } -submodule H ⊂ M , on which ∇ has a pole of order ≤ C { u } -submodule of M which iscommensurable with H , i.e. we will have u N H ⊂ H ⊂ u − N H for N ≫
1. However theformal completion functor ( • ) ⊗ C { u } C [[ u ]] establishes an isomorphism between the Grass-manian GL r ( C { u } [ u − ]) /GL r ( C { u } ) and the affine Grassmanian GL r ( C (( u )) /GL r ( C [[ u ]]).But this map preserves the condition that a submodule H is invariant under ∇ u d/du whichproves (a) . 48 b) As already mentioned in Remark 2.13 this is proven in [HS07, Lemma 8.2]. Alternativelywe can reason as in the proof of part (a) . Let H be a C [[ u ]]-submodule in M for whichis commensurable with H , for and preserved by ∇ u ddu . The operator ∇ u ddu acts on theinfinite-dimensional topological complex vector space M for with finitely many infinite Jordanblocks with eigenvalues { c , . . . , c n } . The corresponding generalized eigenspaces are exactlymodules E c i /u R for i . Hence H for = ⊕ i (cid:16) H for ∩ E c i /u R for i (cid:17) Therefore we obtain extensions R for i with second order poles and regular singularity.Combining the previous lemma with the equivalences in diagram (2.4.2) and the descriptionof nc -Hodge structures from Section 2.1.8 gives the theorem. (cid:3) nc -spaces and gluing In this section we will briefly examine how the gluing construction for nc -Hodge structuresvaries with parameters. In particular, we will look at deformations of nc -spaces and theway the gluing data for the nc -Hodge structures on the cohomology of these spaces interactswith the appearance of a curvature in the d( Z /
2g algebra computing the sheaf theory of thespace.
Suppose X = ncSpec A is a nc -affine nc -space. Recall that the cohomological Hochschildcomplex is defined as C • ( A, A ) := Y n ≥ Hom C − Vect (cid:0) ( Π A ) ⊗ n , A (cid:1) , Its shift Π C • ( A, A ) is a Lie superalgebra with respect to the Gerstenhaber bracket [Ger64],and can be interpreted as the Lie algebra of continuous derivations of the free topological alge-bra Q n ≥ (( Π A ) ⊗ n ) ∨ . The multiplication m A and differential d A of A combine into a cochain γ A := m A + d A ∈ C • ( A, A ) satisfying [ γ A , γ A ] = 0.The formal deformation theory of X is controlled by a d( Z / Π C • ( A, A ) endowed with the differential [ γ A , • ]. It is convenient to consider also the reducedHochschild complex C • red ( A, A ) := Y n ≥ Hom C − Vect (cid:0) ( Π ( A/ C · A )) ⊗ n , A (cid:1) , C • ( A, A ). The reduced complex is (after the parity change)a dg Lie subalgebra in Π C • ( A, A ). Moreover it is quasi-isomorphic to Π C • ( A, A ). Hence,for deformation theory purposes one can replace Π C • ( A, A ) by Π C • red ( A, A ).Let γ = P i ≥ γ i t i ∈ tC evenred ( A, A )[[ t ]] be a formal path consisting of solutions of theMaurer-Cartan equation, i.e. d γ + 12 [ γ , γ ] = 0 ( ⇔ [ γ + γ A , γ + γ A ] = 0 ) . Such a solution defines so called formal deformation of the d( Z / A as a weak(or curved) A ∞ -algebra (see e.g. [LH03] for the definition and [Sch03] for a more detailedanalysis). We can use the cochain γ + γ A ∈ C even ( A, A )[[ t ]] to twist the notion of an A -module. We will write A γ for the (weak) A ∞ -algebra over C [[ t ]] corresponding to A and γ + γ A and ( A γ − mod ) for the C [[ t ]]-linear dg category of all modules over A γ . By definition( A γ − mod ) is the category of dg modules over a bar-type resolution of A γ [KS06b]. As analgebra the relevant bar dg algebra is the completed tensor product(2.5.1) Y n ≥ (cid:0) ( Π A ) ⊗ n (cid:1) ∨ b ⊗ C [[ t ]]where the algebra structure comes from the usual algebra structure on C [[ t ]] and the tensoralgebra structure on Q n ≥ (( Π A ) ⊗ n ) ∨ . Thus for every γ ∈ tC evenred ( A, A )[[ t ]] which solves theMauer-Cartan equation we get a differential γ + γ A on the graded algebra (2.5.1). The bardg algebra of A γ is now defined as the dg algebra B γ := Y n ≥ (cid:0) ( Π A ) ⊗ n (cid:1) ∨ b ⊗ C [[ t ]] , γ + γ A ! . The dg category ( A γ − mod ) is by definition the category of dg modules over B γ which aretopologically free as modules of the underlying algebra, i.e. after forgetting the differential,and also satisfying the condition of unitality at t = 0.As before this category can be viewed as the category C X γ := ( A γ − mod ) of quasi-coherentsheaves on a nc -affine nc -space X γ → D defined over the formal disc D = Spf( C [[ t ]]). Moregenerally we will get a nc -space X over the formal scheme of solutions to the Maurer-Cartanequation and X γ → D is the restriction of X to the formal path γ + γ A sitting inside thatformal scheme.Similarly we can use γ to twist the notion of a Hochschild cohomology class for A . Namelywe can consider the Hochschild cohomology of the A ∞ -algebra A γ . It is given explicitly as50he cohomology HH • γ ( A ) := H • ( C • ( A, A )[[ t ]] , [ γ + γ A , • ]) , and is a commutative algebra with respect to the cup product. Note also that the algebra HH • γ ( A ) comes equipped with a unit [1 A ] and a distinguished even element [ γ + γ A ], i.e. astructure similar to the one discussed in Section 2.2.5. Remark 2.38 • If γ has no component of degree zero, i.e. if γ ∈ tC evenred , + ( A, A )[[ t ]] , where C • red , + ( A, A ) = Y n ≥ Hom C − Vect (cid:0) ( Π ( A/ C · A )) ⊗ n , A (cid:1) , then A γ is an honest (strong) A ∞ -algebra, and the category ( A γ − mod ) will typically havemany interesting objects. Furthermore, in this case smoothness and compactness are stableunder deformations. That is, if A is smooth (respectively compact) over C , then A γ issmooth (respectively compact) over C [[ t ]]. • If the n = 0 component of γ is non-trivial, i.e. if the corresponding A ∞ structure hasa non-trivial m , then the category ( A γ − mod ) may have no non-zero objects. The basicexample of this is when A = C and γ = t · A .If the original algebra A has the degeneration property, then it is easy to see that the Hodge-to-de Rham spectral sequence will degenerate for the periodic cyclic homology of A γ . Inother words the formal nc -space X will give rise to a variation of nc -Hodge structures overthe formal scheme of solutions of the Maurer-Cartan equation for A . When we have a non-trivial n = 0 component in γ this may lead to a paradoxical situation in which we have afamily of nc -spaces over D which has no sheaves over the generic point but has non-trivialde Rham cohomology (i.e. periodic cyclic homology) generically. This suggests the followingimportant Question 2.39
What is the geometrical meaning of HH • γ ( A ), HH • ( A γ ), HH −• ( A γ ), and HP • ( A γ ), when γ has non-trivial n = 0 component and the objects of ( A γ − mod ) dissapearover D × ? 51 emark 2.40 Note that if γ solves the Maurer-Cartan equation, then for any c ∈ t C [[ t ]],the cochain γ + c · A will also solve the Maurer-Cartan equation . So we have a naturalmechanism for modifying formal paths of solutions of the Maurer-Cartan equation. We willexploit this mechanism in the next section. The unpleasant phenomenon of having nc -spaces with no sheaves and non-trivial cohomologyat the generic point is related to the gluing description for nc -Hodge structures. The ideais that the A -modules that dissapear at the generic point of D may reappear again if wemodify the weak A ∞ algebra A γ appropriately. The periodic cyclic homologies of the differentadmissible modifications of A γ then correspond to the regular pieces in the gluing descriptionof the nc -de Rham data given by HP • ( A γ ). More precisely we have the following Conjecture 2.41
Suppose that A is a smooth and compact d ( Z / g algebra. Let γ ∈ tC evenred ( A, A )[[ t ]] be a formal even path of solutions of the Maurer-Cartan equation for A .Then the periodic cyclic homology HP • ( A γ ) carries a canonical functorial structure of avariation of Q - nc -Hodge structures of exponential type over D = Spf( C [[ t ]]) . Furthermorethere exists a positive integer N and a finite collection of pairwise distinct Puiseux series c i = X j ≥ c i,j t jN , c i,j ∈ C such that: • The series c i are the distinct eigenvalues of the operator of mutiplication by the class [ γ + γ A ] in the supercommutative algebra HH • γ ( A ) b ⊗ C [[ t ]] C (( t )) . • For each i the category ( A γ + c i · A − mod ) is a non-trivial C (cid:2)(cid:2) t /N (cid:3)(cid:3) -linear d ( Z / g cate-gory which are smooth and compact over C (cid:2)(cid:2) t /N (cid:3)(cid:3) and is computed by a d ( Z / g alge-bra B i defined over C (cid:2)(cid:2) t /N (cid:3)(cid:3) and quasi-isomorphic to the (weak) A ∞ -algebra A γ + c i · A . • The Hochschild homologies HH • ( B i ) are flat C (cid:2)(cid:2) t /N (cid:3)(cid:3) -modules and we have X i rk C [[ t /N ]]( HH • ( B i )) = rk C [[ t /N ]] HH • ( A γ ) = dim C HH • ( A ) . In fact this is the main reason for all the hassle with the unit and the reduced complex in this section. The variation of nc -Hodge structures HP • ( A γ ) viewed as a variation over C (cid:2)(cid:2) t /N (cid:3)(cid:3) has as regular constituents the variations of nc -Hodge structures on HP • ( B i ) whoseexistence is predicted by Conjecture 2.24. In particular Conjecture 2.41 says that the categorical and Hodge theoretic content of thealgebra A γ consists of the following data: (categories) A finite collection of smooth and compact C (cid:2)(cid:2) t /N (cid:3)(cid:3) -linear d( Z / B i − mod ). (gluing) A finite collection of distinct Piuseux series c i ∈ C (cid:2)(cid:2) t /N (cid:3)(cid:3) , and formal nc -gluingdata which glues the variations of regular nc -Hodge structures on HP • ( B i ) into avariation of nc -Hodge structure of exponential type over C (cid:2)(cid:2) t /N (cid:3)(cid:3) .In the above discussion we have tacitly replaces the analytic setting from Section 2.3 by aformal setting. One can check that both the de Rham and Betti data make sense here, e.g.one can speak about homotopy classes of non-intersecting paths to points c i thinking about t as a small real positive parameter. Remark 2.42
This situation is analogous to a well known setup in singularity theory.Namely, if we have a germ of an isolated hypersurface singularity given by an equation f = 0, and if we have a deformation of f which has several critical values, then the Milnornumber of the original singularity is equal to the sum of the Milnor numbers of the simplercritical points of the deformed function. In fact, as we will see in section 3.2 the singularitysetup is a rigorous manifestation of the above conjectural picture. Suppose next that A is compact but not smooth (or smooth but non-compact) d( Z / γ ∈ tC evenred ( A, A )[[ t ]] be a formal path of solutions of the Maurer-Cartan equation. We expect that the usual definition of smoothness and compactness canbe modified to give a notion of smoothness together with compactness of A γ at the genericpoint, i.e. over C (( t )), even when the objects in ( A γ − mod ) dissapear over C (( t )).53n the case when A γ is smooth and compact over C (( t )), i.e. when the deformation givenby γ is a smoothing deformation, we also expect Conjecture 2.41 to hold at the genericpoint. More precisely, we expect to have Puiseux series c i as above for which the associatedcategories ( A γ + c i A − mod ) are non-trivial and smooth and compact over C (cid:0)(cid:0) t /N (cid:1)(cid:1) . Wealso expect that the periodic cyclic homology HP • ( A γ ) is equipped with a variation of nc -Hodge structures of exponential type over C (( t )) so that the periodic cyclic homologies ofthe categories ( A γ + c i A − mod ) are the regular pieces of this variation after we base changeto C (cid:0)(cid:0) t /N (cid:1)(cid:1) . Finally, the Puiseux series { c i } should be the eigenvalues of the operator ofmultiplication by [ γ + γ A ] ∈ HH • ( A γ ) b ⊗ C [[ t ]] C (( t )). In this section we discuss examples of nc -Hodge structures arising from smooth and compactCalabi-Yau geometries and we study how these structures are affected by mirror symmetry.Specifically we look at a generalization of Homological Mirror Symmetry which relates cate-gories of boundary topological field theories (or D -branes) associated with the following twotypes of geometric backgrounds: A -model backgrounds: Pairs (
X, ω ), where X is a compact C ∞ -manifold, and ω is asymplectic form on X satisfying a convergence property (see below). B -model backgrounds: Pairs w : Y → disc ⊂ C , where Y is a complex manifold withtrivial canonical class, and w is a proper holomorphic map.We will explain how each such background (both in the A and the B model) gives rise tothe geometric and Hodge theoretic data described in Section 2.5.2. Namely we get: • A finite collection { Z A/Bi } of smooth compact nc -spaces. In fact { Z A/Bi } will be(see Section 4.4.1 for the definition) odd/even Calabi-Yau nc -spaces of dimension (cid:0) dim R X mod 2 (cid:1)(cid:14) (dim C Y mod 2). • Complex numbers c A/Bi and Betti gluing data n T A/Bij o for the regular nc -Hodge struc-tures on the periodic cyclic homology of Z A/Bi .In particular the data (cid:0) HC −• (cid:0) Z Ai (cid:1) , (cid:8) c Ai (cid:9) , (cid:8) T Aij (cid:9)(cid:1) and (cid:0) HC −• (cid:0) Z Bi (cid:1) , (cid:8) c Bi (cid:9) , (cid:8) T Bij (cid:9)(cid:1) each glueinto a nc -Hodge structure of exponential type. The generalized Homological MirrorSymmetry Conjecture now asserts that if two A / B -model backgrounds ( X, ω )/( Y, w ) are54irror to each other, then the associated nc -geometry and nc -Hodge structure packages areisomorphic: (cid:0) Z Ai , (cid:8) c Ai (cid:9) , (cid:8) T Aij (cid:9)(cid:1) ∼ = (cid:0) Z Bi , (cid:8) c Bi (cid:9) , (cid:8) T Bij (cid:9)(cid:1) . A -model Hodge structures: symplectic manifolds Suppose (
X, ω ) is a compact symplectic manifold of dimension dim R X = 2 d . In the casewhen X is a Calabi-Yau variety (in particular c ( X ) = 0) one has a family of superconformalfield theories attached to X in the large volume limit (i.e. after the rescaling ω → ω/ ~ where0 < ~ ≪ A -twist gives a topological quantum field theory (see [HKK + Z -gradedFukaya category associated to ( X, ω/ ~ ). One the other side, Gromov-Witten invariantscan be defined for an arbitrary compact symplectic manifold, not necessarily the one with c ( X ) = 0. Our goal in this section is to describe what is an analog of the Fukaya categoryfor general ( X, ω ).Namely, it is expected that for (
X, ω ) of large volume the Fukaya category of (
X, ω ) is aweak Z / A ∞ -category which will satisfy the generalized smoothness and compact-ness properties conjectured in Section 2.5.3. Briefly this should work as follows. FollowingFukaya-Oh-Ohta-Ono [FOOO07] consider a finite collection L = { L i } of transversal ori-ented spin Lagrangian submanifolds in X and form a “degenerate” version Fuk L of Fukaya’scategory which only involves the L i . More precisely we take Ob ( Fuk L ) = { L i } , and defineHom Fuk L ( L i , L j ) = C L i ∩ L j , i = j,A • ( L i , C ) , i = j. Here C L i ∩ L j is taken with the ordinary algebra structure but is put in degree equal to theMaslov grading mod 2, and A • ( L i , C ) is the dg algebra of C ∞ differential forms on L i .We consider a 1-parameter family of symplectic manifolds(3.1.1) (cid:16) X, ω ~ (cid:17) , ~ ∈ R > , ~ → . It will be convenient to introduce a new parameter q := exp( − / ~ ) (note that q → ~ → C q the usual Novikov ring: C q := ( ∞ X i =0 a i q E i (cid:12)(cid:12)(cid:12)(cid:12) formal series where a i ∈ C and E i ∈ R with lim i →∞ E i = + ∞ . )
55n the case [ ω ] ∈ H ( X, Z ) one can replace the Novikov ring C q by more familiar algebra C (( q )) of Laurent series. The three-point genus zero Gromov-Witten invariants of the sym-plectic family (3.1.1) give rise (see e.g. [KM94],[LT98],[Sie99], [CK99],[FO01]) to a C q -valued(small) quantum deformation of the cup product on H • ( X, C ): ∗ q : H • ( X, C ) ⊗ → H • ( X, C ) ⊗ C q Conjecturally the series for the quantum product is absolutely convergent for sufficientlysmall q .What is constructed in [FOOO07] is a solution γ of the Maurer-Cartan equation in thecohomological Hochschild complex of Fuk L with coefficients in the series in C q with strictlypositive exponents (equal to the areas of non-trivial pseudo-holomorphic discs). The meaningof the quantum product is the cup-product in the Hochschild cohomology of the deformedweak category.The d Z /
2g category
Fuk L over C q is compact but not smooth. If the collection L ischosen to be big enough, i.e. if it generates the full Fukaya category, then Fuk L is the largevolume limit of Fuk ( X, ω ), i.e. the limit in which all disc instantons for ω are supressed.Now the formalism of Section 2.5.3 should associate with Fuk L = ( A − mod ) and γ a finite collection { c i } of formal series in positive powers of q and a collection { Fuk i } ofnon-trivial smooth and compact modifications of the Fukaya category whose Hochschildhomologies are the regular singularity constitutents of the Hochschild homology of the q -family of Fukaya categories near the large volume limit. In this geometric context, weexpect that the { c i } are the eigenvalues of the quantum multiplication operator c ( T X ) ∗ q ( • )acting on H • ( X, C ) ⊗ C [[ u ]]. Some evidence for this comes from the observation that when c ( T X ) vanishes in H ( X, Z ), then the Fukaya category is Z -graded thus is a fixed point ofthe renormalization group. There is also a more explicit direct argument identifying theclass c ( T X ) with the infnitesimal generator of the renormalization group, but we will notdiscuss it here.The formalism of Section 2.5.3 now predicts that the periodic cyclic homology of theFukaya category, which additively should be the same as the de Rham cohomology of X ,should carry a natural nc -Hodge structure satisfying the degeneration conjecture from Sec-tion 2.2.4. This expectation is supported by ample evidence coming from mirror symmetryfor Calabi-Yau complete intersections. Here we present further evidence by describing anatural nc -Hodge structure on the de Rham cohomology of a symplectic manifold and byshowing that as ω approaches the large volume limit this structure fits in a natural variation56f nc -Hodge structures.Using the quantum product ∗ q we will attach to ( X, ω ) a variation (( H , ∇ ) , E B , iso ) of nc -Hodge structures over a small disc { q ∈ C | | q | < r } in the q -plane. First we describe the nc -Hodge filtration ( H , ∇ ) and its variation in the q -direction: • H := H • ( X, C ) ⊗ C { u, q } and H := M k = d mod 2 H k ( X, C ) ! ⊗ C { u, q } H := M k = d +1 mod 2 H k ( X, C ) ! ⊗ C { u, q }• ∇ is a meromorphic connection on H with poles along the coordinate axes u = 0 and q = 0, given by ∇ ∂∂u := ∂∂u + u − ( κ X ∗ q • ) + u − Gr , ∇ ∂∂q := ∂∂q − q − u − ([ ω ] ∗ q • ) , where: κ X ∈ H ( X, Z ) denotes the first Chern class of the cotangent bundle of X computedw.r.t. any ω -compatible almost complex structure, and Gr : H → H is the grading operator defined to be Gr | H k ( X, C ) := k − d id H k ( X, C ) .The data ( H , ∇ ) define a q -variation of (the de Rham part of) nc -Hodge structures. Definingthe Q -structure is much more delicate. To gain some insight into the shape of the rationallocal system E B one can look at the monodromy in the q direction of the algebraic bundlewith connection ( H, ∇ ) | ( A −{ } ) ×{ q ∈ C | | q | Let X = CP n − and let ω be the Fubini-Studi form. Let ( H, ∇ ) be theholomorphic bundle with meromorphic connection on ( A − { } ) × { q ∈ C | | q | < R } defined bove. Let ψ ∈ H be a holomorphic section which is covariantly constant with respect to ∇ .Then(a) For every u = 0 , ψ = 0 the limit (in a sector of the q -plane) ψ cl ( u ) = lim q → (cid:18) exp (cid:18) − log( q ) u ([ ω ] ∧ ( • )) (cid:19)(cid:19) ψ exists. Furthermore, ψ cl satisfies the differential equation (cid:18) ddu + u − κ X ∧ + u − Gr (cid:19) ψ cl = 0 . (b) The vector ψ const ( u ) := exp(log( u ) Gr ) exp (cid:18) log( u ) u κ X ∧ ( • ) (cid:19) ψ cl ∈ H • ( X, C ) is independent of u .(c) Define the rational structure E B ⊂ H ∇ as the subsheaf of all covariantly constantsections ψ for which the vector ψ const ∈ H • ( X, C ) belongs to the image of the map H • ( X, Q ) d / / H • ( X, C ) b Γ( X ) ∧ ( • ) / / H • ( X, C ) , where d ∈ GL ( H • ( X, C )) is the operator of multiplication by (2 πi ) k/ on H k ( X, C ) ,and b Γ( X ) is a new characteristic class of X defined as b Γ( X ) := exp C ch ( T X ) + X n ≥ ζ ( n ) n ch n ( T X ) ! , where C = lim n →∞ (cid:18) · · · + 1 n − ln( n ) (cid:19) is Euler’s constant, and ζ ( s ) is Riemann’s zeta function.Then the inclusion E B ⊂ H ∇ is compatible with Stokes data, i.e. the rational structure E B satisfies ( Q -structure axiom) exp . The calculation presented below was known already to B.Dubrovin [Dub98, Section 4.2.1],where he also obtained a Taylor expansions of a power of a Gamma function in quantumcohomology, although he did not identify it with a characteristic class.58 roof of Proposition 3.1. In the standard basis { , h, h , . . . , h n − } of H • ( P n − , C ) theconnection ∇ on H is given by ∇ ∂∂u = ∂∂u + u − nqn 0. . . . . . n + u − − n 0. . . . . .0 n − ∇ ∂∂q = ∂∂q − q − u − q , If ψ = P ni =1 ψ i h i − is a local section of H , a straightforward check shows that the conditionon ψ to be ∇ -horizontal is solved by the following ansatz: ψ n = u − n Z Γ u,q exp( F ) n − Y i =1 dz i z i ψ n − = (cid:18) uq ∂∂q (cid:19) ψ n ψ n − = (cid:18) uq ∂∂q (cid:19) ψ n · · · ψ = (cid:18) uq ∂∂q (cid:19) n − ψ n . Here F is the function on ( C × ) n − with coordinates z , . . . , z n − depending on parameters u, q = 0 and given by F ( z , z , . . . , z n − ; u, q ) := u − (cid:18) z + z + . . . z n − + qz z . . . z n − (cid:19) . The integral is taken over some fixed ( n − u,q in ( C × ) n − (depending on the parameters u , q ) which is going to infinity in directionswhere Re( F ) → −∞ . 59ore generally, the domain of integration Γ u,q used for defining ψ n can be taken to be a( n − C × ) n − . The rapid decay homology cycleson smooth complex algebraic varieties are the natural domains of integration for periods ofcohomology classes of irregular connections. The rapid decay homology was introducedand studied by Hien [Hie07, Hie08], following previous works of Sabbah [Sab00] and Bloch-Esnault [BE04]. In particular by a recent work of Mochizuki [Moc08a, Moc08b] and Hien[Hie08]it follows that (after a birational base change) taking periods induces a perfect pairingbetween the de Rham cohomology of an irregular connection and the rapid decay homology.This powerful general theory is not really needed in our case where the manifold is the affinealgebraic torus ( C × ) n − but it does provide a useful perspective.Explicitly the non-compact cycles that we will use to generate horizontal sections of( H, ∇ ) will be the ( n − X , Z ) constructed as fol-lows. Start with a smooth projective compactification X of ( C × ) n − with a normal crossingboundary divisor D which is adapted to F in the sense that if u and q are nonzero, thedivisors of zeroes and poles of F in X do not intersect with each other, and locally at pointsof D the function F can be written as a product of an invertible holomorphic function and amonomial in the local coordinates. Let X be the real oriented blow-up of X along the divisor D . Now consider the real boundary ∂ X of X , i.e. the union of all the boundary divisorsof the real oriented blow-up. The boundary ∂ X contains a natural open real semi-algebraicsubset Z ⊂ ∂ X consisting of all points b ∈ ∂ X , such that |F ( z ; u, q ) | → ∞ when z → b , andfor points z ∈ t ( C × ) n − near b the argument of F ( z ; u, q ) lies strictly in the left half-plane of C . Note that the real blow-up X has the same homotopy type as X − D = ( C × ) ( n − and sorelative cycles on ( X , Z ) can be thought of as non-compact cycles on ( C × ) ( n − . Moreoversince Z is defined by our condition on the argument of F , it follows that relative cycles withboundaries in Z give rise to well defined integrals of exp( F ) Q z − i dz i .Next observe the integrals over relative cycles with integral coefficients, i.e. elementsin H n − ( X , Z ; Z ) give rise to a covariantly constant integral lattice in the bundle ( H, ∇ ).Furthermore the Deligne-Malgrange-Stokes filtration is integral with respect to this lattice.Indeed if we fix a real number λ , then whenever Re ( F ) < λ · | u | − , it follows that | exp( F ) | < exp( λ · | u | − ) when u → 0. Hence the steps of the Deligne-Malgrange-Stokes filtration of( H, ∇ ) are easy to describe in this language: they correspond to periods of exp( F ) Q z − i dz i on relative cycles on ( X , Z ) whose boundary is contained in half-planes of the form Re( F ) < const. The periods over cycles with integral coefficients and the same boundary propertythen give a full integral lattice in each such step.60ow to finish the proof of the proposition we have just to calculate the limiting lattice(which is independent of u and q ) consisting of vectors ψ const ∈ H • ( X, C ) defined in termsof ψ by the formula in part (b) of the statement of the proposition.For a general ∇ -horizontal local section ψ = P ni =1 ψ i h i − in a sector at 0 in the q -plane(for given u = 0) one has an asymptotic expansion of ψ at q → ψ n = n − X i =0 a i ( u )(log q ) i + O ( q (log q ) n ) + . . . , Then we have that the “classical limit” (at q → ψ cl ( u ) = ( n − u n − a n − ( u )( n − u n − a n − ( u )...0! u a ( u ) . Now we restrict to the case where all variables are real, u < q > { ( z , . . . , z n ) ∈ C n | z i > ∀ i } .Function ψ n = ψ n ( u, q ) decays exponentially fast at q → + ∞ for a given u < 0, henceone can extract its asymptotic expansion at q → + ∞ Z ψ n q s dqq = ∞ X i =0 a i ( u ) i !( − i s i +1 + O (1) , s → . + ∞ Z ψ n q s dqq = u − n + ∞ Z · · · + ∞ Z | {z } n times dqq n − Y i =1 dz i z i exp (cid:18) u − (cid:18) z + z + . . . z n − + qz z . . . z n − (cid:19)(cid:19) q s = u − n + ∞ Z · · · + ∞ Z | {z } n − n − Y i =1 dz i z i exp u − n − X i z i ! · Z + ∞ exp (cid:18) quz z . . . z n − (cid:19) q s dqq | {z } || Γ( s )( − uz z ...z n − ) s = u − n ( − u ) s Γ( s ) Z + ∞ · · · Z + ∞ n − Y i =1 (cid:18) dz i z i z si exp z i u (cid:19) = u − n ( − u ) s Γ( s ) (( − u ) s Γ( s )) n − = u − n ( − u ) ns Γ( s ) n . The conclusion is that the chosen branch ψ cl ( u ) is completely defined by the expansion u − n ( − u ) ns Γ( s ) n = ψ cl ,n ( u )( − u ) s + ψ cl ,n − ( u )( − u ) s + · · · + ψ cl , ( u )( − u ) n − s n + O (1) , s → q = 0)(2 π √− i u i i , for i = 0 , . . . , n − ψ cl satisfies the differential equation (cid:18) ddu + u − κ X ∧ + u − Gr (cid:19) ψ cl = 0 . q → 0) of the equation ∇ ∂∂u ( ψ ) = 0One can check that the operator ddu + u − κ X ∧ + u − Gr can be written asexp (cid:18) − log( u ) u κ X ∧ ( • ) (cid:19) exp( − log( u ) Gr ) ◦ ddu ◦ exp(log( u ) Gr ) exp (cid:18) log( u ) u κ X ∧ ( • ) (cid:19) This follows from the commutation relation[ κ X ∧ ( • ) , Gr ] = − κ X ∧ ( • )Finally, in the above formulas one can replace log( u ) by log( − u ) (and also u − n by( − u ) − n with principal values at the domain u < 0. Having this modification in mind, weconclude that the vector ψ const = ψ const ( u ) := exp(log( − u ) Gr ) exp (cid:18) log( − u ) u κ X ∧ ( • ) (cid:19) ψ cl ∈ H • ( X, C )is independent of u , and in particular it coincides with ψ cl ( − u = − ψ const ( u ) and ψ cl ( u ) are identity matrices. Therefore the vector ψ const isgiven by Taylor coefficients ψ const , s + · · · + ψ const ,n s n − = s n Γ( s ) n + O ( s n ) = Γ(1 + s ) n + O ( s n )We see that ψ const ∈ H • ( X, C ) (after rescaling by operator d from the Proposition) with thevalue of the multiplicative characteristic class associated with the series Γ(1+ s ) = 1+ O ( s ) ∈ C [[ s ]] and the tangent bundle T X , because [ T X ] = n [ O (1)] − [ O ] for X = CP n , and by theclassical expansion log(Γ(1 + s )) = C s + X k ≥ ζ ( k ) k s k The action of the monodromy corresponds (up to torsion) to the multiplication by κ X ∈ H • ( X, Z ). (cid:3) The previous proposition suggests the following general definition: Definition 3.2 The rational structure on ( H, ∇ ) is the local subsystem E B ⊂ H | A −{ } ofmultivalued ∇ -horizontal sections whose values at belong to the image of H • ( X, Q ) d / / H • ( X, C ) b Γ( T X ) ∧ ( • ) / / H • ( X, C ) , here d ∈ GL ( H • ( X, C )) is the operator of multiplication by (2 πi ) k/ on H k ( X, C ) , and b Γ( T X ) is a new characteristic class of X defined as b Γ( T X ) := d Y i =1 Γ(1 + λ i ) , where Γ( s ) is the classical gamma function and λ i are the Chern roots of T X computed inany ω -admissible almost complex structure. Remark 3.3 Apart from the calculation in Proposition 3.1 there are a few other (loose)motivations for this definition: • The class b Γ appears in the context of deformation quantization in the work of the secondauthor [Kon99, Section 4.6]. • The number χ ( X ) ζ (3) appears in the mirror formula for the quintic threefold. • Golyshev’s description [Gol01, Gol07] of the nc -motives associated with the Landau-Ginzburg mirror of a toric Fano involves similar hypergeometric series. • The same class b Γ was derived and a definition similar to Definition 3.2 was proposed in therecent work of Iritani [Iri07] for the case of toric orbifolds by tracing out the mirror imageof rational structure of the mirror Landau-Ginzburg model. Conjecture 3.4 The triple ( H, E B , iso ) associated above with a symplectic manifold ( X, ω ) is a variation of nc -Hodge structures of exponential type. Remark 3.5 (i) In general it is not clear if the ( Q -structure axiom) exp holds in thiscase. It does hold trivially in the graded case, i.e. when X is a Calabi-Yau. (ii) At the moment the “exponential type” part of the conjecture is not supported byany evidence beyond the graded case in which the nc -Hodge structure is regular. It ispossible that for non-K¨ahler symplectic manifolds the nc -Hodge structure on the de Rhamcohomology is not of exponential type. 64 .2 B -model Hodge structures: holomorphic Landau-Ginzburgmodels Suppose we have an algebraic map w : Y → C , where Y is a smooth quasi-projectivemanifold and w has a compact critical locus crit( w ) ⊂ Y . Let S = { c , . . . , c m } ⊂ C denotethe critical values of w .A pair ( Y, w ) like that is called a holomorphic Landau-Ginzburg model and oftenarises (see e.g. [HV00, HKK + Y, w ) giverise to a natural nc -space nc ( Y, w ). The category C nc ( Y, w ) can be described in two equivalentways (in fact these descriptions are valid even if the critical locus of w is not compact). Firstnote that it is enough to define Perf C nc ( Y, w ) since that the category C nc ( Y, w ) can be thought ofas the homotopy colimit completion of Perf C nc ( Y, w ) . For the latter we have two models: Perf C nc ( Y, w ) as a category of matrix factorizations: This model was proposed originallyby the second author as a mathematical description of the category of D -branes andwas subsequently studied extensively in the physics and mathematics literature, see[KL03, KL04] and [Orl04, Orl05b, Orl05a].A matrix factorization on ( Y, w ) is a pair ( E = E ⊕ E , d E ∈ End( E ) opp ), where E is a Z / Y , and d E is an odd endomorphism satisfying d E = w · id E .In the case when Y is affine the Z / E, d E ) , ( F, d F ) of homo-morphisms between two matrix factorizations is defined as Hom(( E, d E ) , ( F, d F ) :=(Hom( E, F ) , d ) where for a ϕ : E → F we have dϕ := ϕ ◦ d E − d F ◦ ϕ . For general Y the same definition works if we replace Hom( E, F ) by some acyclic model, e.g. ifwe use the Dolbeault resolution. The resulting category MF ( Y, w ) of matrix factoriza-tions is a C -linear d( Z / Perf C nc ( Y, w ) to be the derived category D b ( MF ( Y, w )) of the category of matrix factorizations.To construct D b ( MF ( Y, w )) one notes that in addition to being a d( Z / MF ( Y, w ) can also be viewed as a curved d( Z / w (see e.g. [PP05] for the definition) or as a Z / A ∞ -category, i.e. an A ∞ category with an m -operation given by w (see e.g. [Sch03, LH03] for the definition).65n particular we can form the associated homotopy category (in the A ∞ -sense) whichby definition will be the derived category of matrix factorizations.Alternatively, one can use the following two step construction proposed by Orlov. Firstwe pass to the homotopy category of MF ( Y, w ), i.e. we consider the category whoseobjects are matrix factorizations and whose morphisms are given by the quotient ofHom(( E, d E ) , ( F, d F )) by homotopy equivalences. Next (following the standard wis-dom) we need to quotient Ho( MF ( Y, w )) by the subcategory of acyclic factorizations.Since the matrix factorizations are not complexes, they do not have cohomology andso we can not define acyclicity in the usual way. But there is another point of view onacyclicity. If we have a short exact sequence of usual complexes, then the total complexof this diagram will be an acyclic complex. So we define acyclic matrix factorizationsas the total matrix factorization of an exact sequence of factorizations. With thisdefinition we get a thick subcategory in the homotopy category Ho( MF ( Y, w )) matrixfactorizations and then we can pass to the Serre quotient of Ho( MF ( Y, w )) by thisthick subcategory. We set D b ( MF ( Y, w )) to be this Serre quotient. Perf C nc ( Y, w ) as a category of singularities: This model was proposed originally by D. Orlovas an alternative to the matrix factorization description which is localized near the crit-ical set of w . Orlov proved the equivalence of the two models, various versions of thelocalization theorem, and proved several duality statements relating derived categoriesof singularities to other familiar categories [Orl04, Orl05b, Orl05a].Suppose Z is a quasi-projective complex scheme. The derived category D b Sing ( Z ) ofsingularities of Z is defined as the quotient D b Sing ( Z ) := D b ( Coh ( Z )) / Perf Z of the (dg enhancement of the) bounded derived category D b ( Coh ( Z )) of coherentsheaves on Z by the thick subcategory of perfect complexes on Z . The syzygy theoremimplies that D b Sing ( Z ) = 0 whenever Z is smooth and so D b ( Coh ( Z )) can be thoughtof as an invariant of the singularities of Z .If now w : Y → C is a holomorphic Landau-Ginzburg model we write Y c for the fiber w − ( c )and set Perf C nc ( Y, w ) := D b Sing ( Y ) . ∈ A is not a critical value of w , then with this definition we will get Perf C nc ( Y, w ) = 0. In order to get non-trivial categories we will use the critical values S = { c , . . . , c n } to shift the potentail w / / /o/o/o w − c i and associate with nc ( Y, w ) honest cate-gories Perf i := Perf C nc ( Y, w − c i ) = D b Sing ( Y c i ). Conjecturally, these categories are smooth andcompact.Mirror symmetry suggests that the nc -space nc ( Y, w ) gives rise to the B -model geometricand Hodge theoretic data described in Section 2.5.2, and in particular that the periodiccyclic homology of C nc ( Y, w ) carries a canonical nc -Hodge structure. In fact we have alreadydescribed the geometric part of the data, namely the numbers { c i } and the categories { Perf i } .These data of course fix the regular type (in the sense of Theorem 2.35) of the nc -Hodgestructure but we are still missing the gluing data. Here we propose a construction of theHodge structure on the periodic cyclic homology of C nc ( Y, w ) but similarly to the A -modelwe have to rely on the actual geometry of ( Y, w ) in order to produce the gluing data. Atpresent it is not clear if the gluing data can be reconstructed from the category C nc ( Y, w ) ormore generally from its one parameter deformation.First we discuss the appropriate cohomologies of the Landau-Ginzburg model. Let H • for := H • DR (( Y, w ); C )= H • mod 2Zar ( Y, (Ω • Y [[ u ]] , ud DR + d w ∧ ))be the Z / C [[ u ]]-module of algebraic de Rham cohomology of the potential w . Inthe case when crit( w ) is compact, the C [[ u ]]-module H • for is known to be free by the workof Barannikov and the second author (unpublished), Sabbah [Sab99], or Ogus-Vologodsky[OV05]. This implies the following Lemma 3.6 Assume that Y is quasi-projective and the critical locus of w is compact. Thenwe have: (i) The fiber of H • for at u = 0 is the algebraic Dolbeault cohomology H • Zar ( Y, (Ω • Y , d w ∧ )) ∼ = H • an ( Y, (Ω • Y , d w ∧ )) of the potential w . ii) There is a canonical isomorphism H • Zar ( Y, (Ω • Y [[ u ]] , ud DR + d w ∧ )) ∼ = H • an ( Y, (Ω • Y [[ u ]] , ud DR + d w ∧ )) (iii) If the map w is proper then H • for is the formal germ at u = 0 of an algebraic vectorbundle on the affine line H • alg := H • mod Zar ( Y, (Ω • Y [ u ] , ud DR + d w ∧ )) Proof. The cohomology sheaves of the complex (Ω • Y , d w ∧ ) are supported on the criticallocus of w and so, by our compactness assumption, must be coherent sheaves on Y bothin the analytic and in the Zariski topology. The hypercohomology spectral sequence thenimplies that the hypercohomology of the complex (Ω • Y , d w ∧ ) is finite dimensional and thespectral sequence associated with the filtration induced by multiplication by u implies that H • Zar/an ( Y, (Ω • Y [[ u ]] , ud DR + d w ∧ )) is a finite rank C [[ u ]]-module. Furthermore, the samespectral sequence implies thatdim C (( u )) H • Zar/an ( Y, (Ω • Y (( u )) , ud DR + d w ∧ )) ≤ dim C H • Zar/an ( Y, (Ω • Y , d w ∧ )) . The freeness statement of Barannikov and the second author (see e.g. [Sab99]) now givesthat these two dimensions are equal and so H • Zar ( Y, (Ω • Y [[ u ]] , ud DR + d w ∧ )) is a free finiterank module over C [[ u ]]. This proves part (i) of the lemma.For part (ii) we only need to notice that the two spaces in question are computed byspectral sequences associated with the filtrations by the powers of u and that these spectralsequences have E -sheets whose entries are finite sums of copies of H • Zar ( Y, (Ω • Y , d w ∧ )) and H • an ( Y, (Ω • Y , d w ∧ )) respectively. Each of these can in turn be computed from the hyper-cohomology spectral sequence for the complex (Ω • Y , d w ∧ ) of (Zariski or analytic) coherentsheaves. But the cohomology sheaves of this complex are supported on the zero locus of d w which by assumption is projective. Hence by GAGA the Zariski and analytic cohomologiesof this complex are naturally isomorphic. This gives isomorphisms of the hypercohomologyand filtration spectral sequences in the Zariski and the analytic setup respectively and sothe two types of hypercohomologies are isomorphic.Finally, part (iii) was also proven by Barannikov and the second author, and by Sabbah[Sab99]. (cid:3) emark 3.7 The isomorphism in part (ii) of the previous lemma is not convergent for u → u = 0 is a complex number, then the complex vector space H • an ( Y, (Ω • Y , ud DR + d w ∧ )) is the same as the usual de Rham cohomology H • DR ( Y, C ) of Y .Indeed, for such a fixed u = 0, the complex (Ω • Y , ud DR + d w ∧ )) ∼ = (Ω • Y , d DR + u − d w ∧ ))is the holomorphic de Rham complex of the local system ( O Y , d DR + u − d w ). But themultiplication by exp( − u − w ) is an analytic automorphism of the line bundle O Y whichgauge transforms the connection d DR + u − d w into the trivial connection d DR . Henceexp( − u − w ) identifies (Ω • Y , ud DR + d w ∧ ) with the holomorphic de Rham complex (Ω • Y , d DR )and H • an ( Y, (Ω • Y , ud DR + d w ∧ )) with H • DR ( Y, C ). On the other hand, the space H • Zar ( Y, (Ω • Y , ud DR + d w ∧ )) depends on the potential in an essential way. For instance,if w : Y → A is a Lefschetz fibration, then the complex (Ω • Y , d w ∧ ) is just the Koszulcomplex associated with the regular section d w ∈ Ω Y . In particular the space H • Zar ( Y, (Ω • Y , d DR + d w ∧ )) ∼ = H • Zar ( Y, (Ω • Y , d w ∧ )) has dimension equal to the number of criti-cal points of w . More generally H • Zar ( Y, (Ω • Y , d DR + d w ∧ )) can be identified (see e.g. [Kap91])with the cohomology of the perverse sheaf of vanishing cycles of w . Remark 3.8 Under our assumptions, the algebraic de Rham and Dolbeault cohomologies H • DR (( Y, w ); C ) and H • Dol (( Y, w ); C ) of the potential w can be identified respectively withthe periodic cyclic and Hochschild homologies HP • ( C nc ( Y, w ) ) and HH • ( C nc ( Y, w ) ) of the nc -space C nc ( Y, w ) (more precisely, of the collection of categories Perf i labeled by numbers { c i } ).This can be done, e.g. by choosing strong generators E i of Perf i , and then identifying HP • ( C nc ( Y, w ) ) and HH • ( C nc ( Y, w ) ) with the periodic cyclic and Hochschild homologies of thecurved d( Z / Z / R Hom( E , E ) and a centralcurvature given by w . A detailed proof of the comparison theorem giving the identifications H • DR (( Y, w ); C ) ∼ = HP • ( C nc ( Y, w ) ) and H • Dol (( Y, w ); C ) ∼ = HH • ( C nc ( Y, w ) ) can be found in therecent work of Junwu Tu [Tu08].We will construct a nc -Hodge structure on H • DR (( Y, w ); C ) by using the dual description of nc -Hodge structures given in Theorem 2.35. Here we will assume that we choose an opensubset (in the analytic topology) Y ′ ⊂ Y such that • crit( w ) ⊂ Y ′ , 69 w ( Y ′ ) is an open disc in C , • the closure Y ′ of Y ′ is a manifold with corners, • the restriction of w to the part of the boundary of Y ′ lying over w ( Y ′ ) is a smoothfibration.In the case when w is already proper one can choose Y ′ to be the pre-image under w ofan open disc in C containing all the critical values c i .Label the critical values of w : S = { c , . . . , c n } , and let c ∈ w ( Y ′ ) − S . Choose a systemof paths { a i } ni =1 and discs D i as in the proof of Theorem 2.35. Choose c -based loops γ , . . . , γ n , so that γ i goes once around c i in the counterclockwise direction, all γ i intersectonly at c , and each γ i encloses the path a i and the disc D i (see Figure 4). Let Γ i denote theclosed region in C enclosed by γ i . Adjusting if necessary the choice of the γ i we can ensurealso the each Γ i is convex. From now on we will always assume that this is the case. (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) c i c Γ i γ i Figure 4: A system of thickened loops for S ⊂ C .For i = 1 , . . . , n set Y i := w − (Γ i ) ∩ Y ′ and consider the Q -vector spaces of relative cohomology U i := H • ( Y i , Y c ; Q ) , U := ⊕ ni =1 U i = H • ( w − ( ∪ ni =1 Γ i ) , Y c ; Q )= H • ( Y, Y c ; Q ) . Let T i : U → U be the monodromy along γ i . By definition T i satisfies( T i − |⊕ j = i U j = 0and so we get operators T ji : U i → U j , such that T i | U i = P nj =1 T ji . By construction theoperator T ii is the monodromy along γ i of the local system on Γ i of local relative cohomology,i.e. the local system of Q -vector spaces whose fiber at c ∈ Γ i is H • ( Y i , Y c ; Q ). Hence T ii isan isomorphism, and so the data ( S, { U i } ni =1 , { T ij } ) are nc -Betti data of type nc B(iii) . Remark 3.9 (a) By Lemma 2.30 the data ( S, { U i } ni =1 , { T ij } ) are the same thing as aconstructible sheaf F of Q -vector spaces on C , satisfying R Γ( C , F ) = 0. The sheaf F canbe described directly in terms of the geometry of ( Y, w ): for a c ∈ C the stalk F c of F at c is the relative cohomology H • ( Y, Y c ; Q ).(b) The geometric construction of F makes sense for every cohomology theory K . Indeedfor every such K we can form a constructible sheaf of abelian groups K F whose stalk at c ∈ C is K ( Y, Y c ) and which again satisfies R Γ( C , K F ) = 0. The vanishing of cohomology here isnot obvious but can be proven as follows. Given a disk D ⊂ w ( Y ′ ) ⊂ C s.t. ∂ D ∩ S = ∅ ,and given any point c ∈ ∂ D consider the abelian group A ( D , c ) := K ( w − ( D ) , Y c ). Thecollection of abelian groups A ( D , c ) satisfies: • A ( D , c ) are locally constant under small perturbations of ( D , c ), and • for every decomposition ( D , c ) = ( D , c ) ∪ ( D , c ) of D obtained by cutting D alonga chord starting at c , we have A ( D , c ) = A ( D , c ) ⊕ A ( D , c ).This immediately gives us an equivalent description of K F via data of type nc B(iii) , whichin turn yields the vanishing of cohomology of K F .Next, in order to complete the data nc B(iii) to a full-fledged nc -Hodge structure of expo-nential type, we need to construct: 71 a collection { ( R i , ∇ i ) } mi =1 of holomorphic bundles R i over C { u } equipped with mero-morphic connections ∇ i with at most second order pole and regular singularities, and • for each i = 1 , . . . , m , an isomorphism f i between the local system on S induced from( R i , ∇ i ) and the local system on S corresponding to the vector space U i ⊗ C and themonodromy operator T ii .As explained above the local system on the circle corresponding to the vector space U i ⊗ C and the monodromy operator T ii can be described geometrically as the sheaf of complexvector spaces on the loop γ i , whose stalk at c ∈ γ i is H • (( Y i , Y c ); C ). We will exploit thisgeometric picture to produce ( R i , ∇ i ) and the isomorphism f i . The most convenient way todefine the ∇ i is by using a Betti-to-de Rham cohomology isomorphism given by oscillatingintegrals.Fix i ∈ { , . . . , m } and let Z := Y i , ∆ := Γ i − c i ⊂ C , f := w − c i . By construction wehave: Z is a C ∞ -manifold with boundary which is the closure of an open (in the classical topology)subset in the quasi projective complex manifold Y . ∆ ⊂ C is a closed disc containing zero. f : Z → ∆ is an analytic surjective map whose only critical value is zero and whose criticallocus crit( f ) ⊂ Z is compact.Consider now the Z / C [[ u ]]-module H • DR (( Z, f ); C ) of de Rham cohomology of( Z, f ). By lemma 3.6 we know that H • DR (( Z, f ); C ) is a free C [[ u ]]-module which can be com-puted as the cohomology of the complex ( A • ( Z )[[ u ]] , d tot ), where A • ( Z )[[ u ]] are the global C ∞ complex valued differential forms on Z , and d tot := ¯ ∂ + u∂ + d f ∧ . The C [[ u ]]-module H • DR (( Z, f ); C ) carries a natural meromorphic connection ∇ differentiating in the u -directionand having a second order pole at u = 0. This connection is induced from a connection ∇ on the C [[ u ]]-module A • ( Z )[[ u ]] which also has a second order pole and is defined by theformula ∇ u ddu := u ddu − f · ( • ) + u Gr : A • ( Z )[[ u ]] / / A • ( Z )[[ u ]] , where Gr |A p,q ( Z )[[ u ]] := q − p · id A p,q ( Z )[[ u ]] 72s the grading operator coming from nc -geometry (compare with 2.1.7).With this definition we have Lemma 3.10 The operator ∇ u ddu satisfies: (a) h ∇ u ddu , d tot i = u · d tot . (b) ∇ u ddu preserves ker( d tot ) and im( d tot ) and so induces a meromorphic connection ∇ with a second order pole on the C [[ u ]] -module H • DR (( Z, f ); C ) . Proof. We compute h ∇ u ddu , d tot i = (cid:20) u ddu − f + u Gr , ¯ ∂ + u∂ + d f ∧ (cid:21) = (cid:20) u ddu , u∂ (cid:21) − [ f , u∂ ] + (cid:2) u Gr , ¯ ∂ + u∂ + d f ∧ (cid:3) = u ∂ + ud f ∧ + u ¯ ∂ − ud f ∧ − u ∂ u · d tot . Part (b) follows immediately from (a) (cid:3) Suppose now that α = α + α u + α u + · · · ∈ A • ( Z )[[ u ]], α i = P α p,qi , α p,qi ∈ A p,q ( Z ) is a d tot -cocycle. Then the differential d + u − d f ∧ = ¯ ∂ + ∂ + u − d f ∧ = u − / u Gr d tot u − Gr will killthe element u Gr α := X i ≥ ≤ p,q ≤ dim Z α p,qi u i + q − p ∈ A • ( Z )(( u / )) . Therefore the expression e f u u Gr α satisfies formally d (cid:16) e f u u Gr α (cid:17) = 0 , i.e. is d -closed. Moreover, the action of the operator ∇ u ddu on α translates to the action of u ddu on the above expression modulo formally exact forms.Consider now a closed connected arc δ ⊂ ∂ ∆ = γ i and let Sec( δ ) ⊂ ∆ be the corre-sponding open sector (see Figure 5) with vertex at 0 ∈ ∆ , and boundary made out of the73rc δ and the segments connecting 0 with the end points of δ . Note that the convexity of ∆ assures that Sec( δ ) ⊂ ∆ . Denote by Sec( δ ) ∨ ⊂ C the dual angle sector consisting of u ∈ C such that Re( w/u ) < w ∈ Sec( δ ). (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) δ Sec( δ ) ∆ ∂ ∆ Figure 5: A sector in ∆ .Clearly, for each class in the relative integral homology H • ( Z, f − ( δ ); Z ) we can choosea relative chain c representing it, so that c satisfies:( † ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) • c is piece-wise real analytic; • f (supp( c )) ⊂ Sec( δ ); • f (supp( ∂ c )) ⊂ δ .For every such relative chain c we now have: Lemma 3.11 For every d tot -closed formal power series of forms α ∈ A • Z ( Z )[[ u ]] and everyrelative chain c ∈ C • ( Z, f − ( δ ); Z ) satisfying ( † ) the oscillating integral Z c e f u u Gr α is well defined as an asymptotic series in u Q (log u ) N in the sector Sec( δ ) ∨ . Proof. Let N ≥ e f /u u Gr X ≤ i ≤ N α i u i ! is a well defined analytic function on Z × Sec( δ ) ∨ . Using the fact that ( d + u − d f ∧ ) u Gr α = 0and the Malgrange-Sibuya theory of asymptotic sectorial solutions to analytic differential74quations, we get that(3.2.1) Z c e f /u u Gr X ≤ i ≤ N α i u i ! ≃ X j ∈ Q ,k ∈ N c j,k u j (log( u )) k is asymptotic to a series in u Q (log u ) N in which the logarithms enter with bounded powers.Thus the limit of (3.2.1) as N → ∞ is asymptotic to a series in u Q (log u ) N on Sec( δ ) ∨ . (cid:3) The previous lemma shows that the C [[ u ]]-module with connection ( H • DR (( Z, f ); C ) , ∇ ) isformally isomorphic to a meromorphic local system of the form E f /u ⊗ ( R i , ∇ i ), where R i is afree C [[ u ]]-module, and ∇ i has regular singularities. Furthermore the lemma shows that theoscillating integrals above identify the local system on γ i given by ( c ∈ γ i ) H • (( Y i , Y c ) , Q )with a rational structure on ( R i ⊗ C [[ u ]] C (( u )) , ∇ i ). In particular the data { ( R i , ∇ i ) } mi =1 and( S, { U i } , { T ij } ) constitute the regular type and gluing data (in the sense of Theorem 2.35)of a nc -Hodge structure of exponential type.Usually if one tries to make a Landau-Ginzburg model with proper map w from non-properexamples above, one gets new parasitic critical points. Choosing an appropriate domain Y ′ ⊂ Y one can define the gluing data for the relevant critical points. Finally, in order to give a general idea of the mirror correspondence, we briefly discussthree examples of Landau-Ginzburg models mirror dual to symplectic manifolds of positive,vanishing, and negative anti-canonical class respectively. • For X = CP n one of the possible mirror dual Landau-Ginzburg models is given by Y = ( C × ) n endowed with potential w ( z , . . . , z n ) = z + · · · + z n + qz . . . z n where q ∈ C × is a parameter. In this model the map w is not proper. This can berepaired by compactifying the fibers of w to ( n − Y . In general, for symplecticmanifolds ( X, ω ) with ω representing the anticanonical class, one can combine equationsfor the connection in q and u directions and get a beautiful variation of Hodge structureswith strong arithmetic properties as predicted by our considerations in section 3.1 (seealso Golyshev’s work [Gol01, Gol07]). • For a smooth projective Calabi-Yau variety X one can take for Y the product ( X ∨ × A N , w ) where X ∨ is a Calabi-Yau variety mirror dual to X , N ≥ w is the pullback from A N of a non-degenerate quadratic form. In general, thecomplex dimension of the Landau-Ginzburg model is equal to half of the real dimensionof X modulo 2. • For X being a complex curve of genus g ≥ Y, w ) whichis a complex algebraic 3-dimensional manifold with non-vanishing algebraic volumeelement, such that locally (in the analytic topology) near each point the pair ( Y, w ) isisomorphic to w : C → C , ( x, y, z ) xyz The set of critical point of w is the union of 3 g − CP glued along points 0 , ∞ meeting 3 curves at a point. The graph obtained by contracting each copy of C × to anedge is a connected 3-valent graph with g loops, representing a maximal degenerationpoint in the Deligne-Mumford moduli stack of stable genus g curves. In this section we will examine more closely the other direction of the mirror symmetrycorrespondence, i.e. the situation in which symplectic Landau-Ginzburg models appear asmirrors of complex manifolds with a fixed anti-canonical section. In order to understand theHodge theoretic implications of this process we first revisit a classical concept in the subject:the notion of canonical coordinates. 76 .1 Canonical coordinates for Calabi-Yau variations of nc -Hodgestructures We begin with a reformulation of the definition of variations of nc -Hodge structures (Defi-nition 2.7) to allow for bases that are supermanifolds: Definition 4.1 For a complex analytic supermanifold S , a variation of nc -Hodge struc-tures over S (respectively a variation of nc -Hodge structures over S of exponentialtype ) is a triple ( H, E B , iso ) , where • H is a holomorphic Z / -graded vector bundle on A × S which is algebraic in the A -direction; • E B is a local system of Z / -graded Q -vector spaces on ( A − { } ) × S ; • iso is an analytic isomorphism of holomorphic vector bundles iso : E B ⊗ O ( A −{ } ) × S ∼ = → H | ( A −{ } ) × S ; so that: ♦ the induced meromorphic connection ∇ on H | ( A −{ } ) × S satisfies: locally on S , for everysection ξ of T S , the operators ∇ u ∂∂u , ∇ uξ extend to operators on A × S , and ♦ the triple ( H, E B , iso ) satisfies the ( Q -structure axiom) and the (Opposedness axiom) (respectively ( H, ∇ ) is of exponential type and ( H, E B , iso ) satisfies the ( Q -structureaxiom) exp and the (Opposedness axiom) exp ). Remark 4.2 From now on we will suppress the Q -structure and the opposedness axiomssince they will not play any special role in our analysis. At any given stage of the discussionthey can be added without any harm or alteration to the arguments.77 .1.2 Calabi-Yau variations Suppose now ( H, E B , iso ) is a variation of nc -Hodge structures over a supermanifold S . Forany point x ∈ S let H ,x denote the fiber of H at (0 , x ) ∈ A × S . We get a canonical map µ x : T x S → End ( H ,x ) , defined as follows: Extend the tangent vector v ∈ T x S to some analytic vector field ξ defined in a neighborhood of x . Consider the holomorphic first order differential operator ∇ uξ : H → H . By construction this operator has symbol ( uξ ) ⊗ id H . In particular, therestriction of ∇ uξ to the slice { } × S ⊂ A × S will have zero symbol, and so will be an O -linear endomorphism of H |{ }× S . We define µ x ( v ) to be the action of this O -linear map onthe fiber H (0 ,x ) . It is straightforward to check that this action is independent of the extension ξ and depends on v only. Definition 4.3 Let S be a complex analytic supermanifold. We say that a variation ( H, E B , iso ) of nc -Hodge structures on S is of Calabi-Yau type at a point x ∈ S if there exists an(even or odd) vector h ∈ H (0 ,x ) , so that the linear map T x S / / H (0 ,x ) v (cid:31) / / ( µ x ( v ))( h ) is an isomorphism. Such a vector h will be called a generating vector for H at x . It follows from the definition that if S is the base of a variation of nc -Hodge structures whichis of Calabi-Yau type at a point x ∈ S , then the tangent space T x S is a unital commutativeassociative algebra acting on H ,x via the map µ x and such that H ,x is a free module ofrank one. The condition on a variation to have a Calabi-Yau type (even or odd) is anopen condition on x ∈ S . Variations of nc -Hodge structures of Calabi-Yau type shouldarise naturally on the periodic cyclic homology of smooth and compact d( Z / Example 4.4 Let ( X, ω ) be a compact symplectic manifold with dim R X = 2 d . Conjec-turally there exists a non-empty open subset S ⊂ H • ( X, C ) so that the big quantum product ∗ x is absolutely convergent for all x ∈ S (the product is given by a formula similar to one on78age 109). The manifold S has a natural structure of a supermanifold being an open subsetin the affine super space H • ( X, C ). As in section 3.1 we define a variation of nc -Hodgestructures ( H, E B , iso ) on S by taking H to be the trivial vector bundle on A × S with fiber H • ( X, C ), and defining the connection ∇ on H by the formulas: ∇ ∂∂u := ∂∂u + u − ( κ X ∗ x • ) + u − Gr , ∇ ∂∂ti := ∂∂t i − q − u − ( t i ∗ x • ) , where the ( t i ) form a basis on H • ( X, C ), and ( t i ) are the dual linear coordinates.Clearly, if we restrict ( H, ∇ ) to S ∩ H ( X, C ) we will get back the bundle with connectionwe defined in section 3.1. We now define the integral lattice E B and isomorphism iso on S as the ∇ -horizontal extensions of the integral lattice and isomorphism we had defined on S ∩ H ( X, C ). Finally, in order to match the framework of nc -geometry, we should changethe parity of the bundle H in the case d = 1 mod 2. The variations of nc -Hodge structures of Calabi-Yau type need to be decorated by a fewadditional pieces of data before we can extract canonical coordinates from them. To motivateour choice of such data we first recall the Deligne-Malgrange classification of logarithmicholomorphic extension of regular connections.Let S be a complex analytic supermanifold, let D be a one dimensional complex disc,and let E be a complex local system on ( D − { pt } ) × S and let ( E , ∇ ) be the associatedholomorphic bundle E := E ⊗O ( D −{ pt } ) × S on ( D −{ pt } ) × S with the induced flat connection ∇ . Suppose e E is a holomorphic bundle on D × S which extends E and on which ∇ has alogarithmic pole. The restriction e E |{ pt }× S is a holomorphic bundle on S and ∇ induces: aholomorphic connection e E ∇ and an O S -linear residue endomorphism Res e E ( ∇ ) on e E |{ pt }× S .Furthermore the integrability of ∇ on ( D − { pt } ) × S implies that e E ∇ is also integrableand that the endomorphism Res e E ( ∇ ) is covariantly constant with respect to e E ∇ [Sab02,Section 0.14b].Recall next that by Deligne’s extension theorem (see e.g. [Del70, Chapter II.5] or [Sab02,Corollary II.2.21]) meromorphic bundles with connections with regular singularities always79dmit functorial holomorphic extensions across the pole divisor. Deligne’s extension proce-dure is not unique and depends on the choice of a set-theoretic section of the quotient map C → C / Z . We fix V to be the unique Deligne extension of E for which ∇ has a logarithmicpole at { pt } × S and a residue with eigenvalues whose real parts are in the interval ( − , Holomorphic extensions of E to D × S for which ∇ has a logarithmic singularityalong { pt } × S o o / / Decreasing filtrations of E by C -local subsystems on( D − { pt } ) × S . The equivalence depends on the chosen Deligne extension and is explicitly given as follows.Let t be a complex coordinate on D which vanishes at pt ∈ D . Consider the restriction V /t V of V to { pt } × S . This is a holomorphic bundle on S equipped as above with theholomorphic connection V ∇ and the covariantly constant residue endomorphism Res V ( ∇ ).Suppose now that e E is another holomorphic bundle on D × S which extends E and on which ∇ has a logarithmic pole. For any k ∈ Z we define a subbundle ( V /t V ) k ⊂ V /t V by setting( V /t V ) k := V ∩ t k e Et V ∩ t k e E where V and e E are viewed as subsheaves in the meromorphic bundle E .By construction the sub-bundles ( V /t V ) k are preserved both by V ∇ and by the residueendomorphism Res V ( ∇ ) and so give rise to ∇ -covariantly constant meromorphic subbundlesof E , or equivalently to C -local subsystems of E .Alternatively we can use a more intrinsic description of holomorphic extensions of ( E , ∇ )which is beter adapted to our examples and in particular to Example 4.8. Namely, instead ofrelying on the Deligne extension and the induced filtration we can use decreasing filtrations E ≤ λ of E labeled by real numbers λ ∈ R and such that on the associated graded pieces themonodromy on D − { pt } has eigenvalues in R + × exp(2 πiλ ).We can now introduce the additional data that one needs for the canonical coordinates Definition 4.5 Let S be a complex supermanifold and let ( H, E B , iso ) be a variation of nc -Hodge structures of Calabi-Yau type on S . A decoration on ( H, E B , iso ) is a pair ( e H, ψ ) where: H is an extension of H to ( Z / -graded vector bundle on P × S so that ∇ has a regularsingularity at {∞} × S . ψ is a e H ∇ -covariantly constant section of e H {∞}× S .A decoration is called rational iff the R -filtration on the local system E B ⊗ C is compatiblewith the rational structure, and if the vector ψ ( x ) ∈ e H {∞}×{ x } = gr( E B ⊗ C ) x is rational, i.e.if ψ ( x ) ∈ gr( E B ) x . The previous discussion applied to the local system E B ⊗ C , the disc D = {| u | > } ∪{∞} and the point pt = ∞ shows that the data of a decoration are equivalent to thedata ( E B ⊗ C ) ≤• , ψ ), where ( E B ⊗ C ) ≤• is a decreasing filtration of E B ⊗ C (labeled by realnumbers) and ψ is a covariantly constant section (along S ) of the corresponding logarithmicholomorphic extension of H . We will freely go back and forth between these two points ofview.Any decorated variation ( H, E , iso ; e H, ψ ) of nc -Hodge structures of Calabi-Yau type givesrise to a natural open domain U ⊂ S defined by U := x ∈ S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e H P ×{ x } is holomorphically trivial and if s ∈ Γ (cid:0) P × { x } (cid:1) is such that s x ( ∞ ) = ψ ( ∞ , x ),then s x (0) is a generating vector for ( H, E , iso ). Furthermore for every x ∈ U we get a natural map can x : T x S → e H ∞ ,x defined as thecomposition T x S µ x ( • )( s x (0)) / / can x H ,x ev − ,x ) / / Γ (cid:16) e H | P ×{ x } (cid:17) ev ( ∞ ,x ) / / e H ∞ ,x . where ev ( t,x ) : Γ (cid:16) P , e H | P ×{ x } (cid:17) → e H t,x denotes the natural evaluation of sections, which isinvertible by the triviality assumption on e H | P ×{ x } .The pullback of the flat connection e H ∇ by the map can induces a flat connection on T S | U .The canonical coordinates on S come from the following easy claim whose proof we omit81 laim 4.6 The flat connection can ∗ (cid:16) e H ∇ (cid:17) on T S | U is torsion free and so gives rise to anatural affine structure and affine coordinates on U . If the decoration is rational then thetangent bundle T S | U carries a natural rational structure. Remark 4.7 (i) The canonical coordinates on U corresponding to a decorated nc -variationof Hodge structures are only affine coordinates and are defined only up to a translation. (ii) For any u ∈ A − { } we can introduce another affine structure which is a vectorstructure . In fact, we get an analytic isomorphism between U and a domain in H u, • =( E B ) u, • ⊗ C : x ∈ U ev ( u,x ) ev − ∞ ,x ) ( ψ ( x )) ∈ H ( u,x ) . One can use this to show that the local Torelli theorem holds for decorated Calabi-Yauvariations of nc -Hodge structures. Example 4.8 The setup of Example 4.4 gives not only a variation of nc -Hodge structuresbut in fact gives a rationally decorated nc -Hodge structure of Calabi-Yau type. Indeed bydefinition the fibers of H are identified with Π d H • ( X, C ). The monodromy of the connectionaround ∞ ∈ P is the operator acting by ( − i + d exp( κ X ∧ ( • )) on H i ( X, C ). Consider themonodromy invariant filtration on H • ( X, C ) whose step in degree d − i is H ≥ i ( X, C ). Let e H be the corresponding logarithmic extension of H and let ψ be the section of e H correspondingto the image of 1 ∈ H ( X, C ) ⊂ H • ( X, C ). The bundle e H |{∞}× S is trivialized and ∇ ∂∂ti = ∂∂t i in this trivialization. This gives the desired decoration ( e H, ψ ) and the associated canonicalcoordinates are the standard canonical coordinates in Gromov-Witten theory. The notion of a decorated Calabi-Yau variation of nc -Hodge structures can be generalized invarious ways. For instance, instead of specifying a covariantly constant filtration on H givingthe extension e H we can start with any holomorphic bundle H ′ defined on { u ∈ P | | u | ≥ R } ,82nd an identification of C ∞ -bundles p ∗ (cid:0) H ′|{| u | = R } (cid:1) ∼ = ( E B ⊗ C ) {| u | = R }× S , where p : {| u | = R } × S → {| u | = R } is the projection on the first factor.Furthermore (locally in S ) the holomorphic bundle p ∗ H ′ on { u ∈ P | | u | ≥ R } × S carriesa flat connection defined along S only. We can use the above identification to glue thistogether with H along {| u | = R } × S to get a bundle e H on P × S equipped with a flatconnection ∇ /S along S . This generalizes the first part of the decoration. For the secondpart we will take a ∇ /S -covariantly constant section ψ of e H |{∞}× S . Now the same definitionof the set U and the canonical map can make sense in this context. The resulting connectionon T S | U is again torsion free. The notion of a Calabi-Yau variation extends readily to the formal context. Suppose S =Spf C [[ x , . . . , x N , ξ , . . . , ξ M ]] be a formal algebraic supermanifold, where x i are even and ξ j are odd formal variables. The de Rham part of formal variation of nc -Hodge structures on S is a pair ( H, ∇ ) where H is a ( Z / D × S , where D is the one dimensional formal disc D := Spf( C [[ u ]]). Here ∇ is a meromorphic connectionon H such that ∇ u ∂∂u , ∇ u ∂∂xi , ∇ u ∂∂ξj are regular differential operators on H .We say that such a pair ( H, ∇ ) has the Calabi-Yau property if we can find a vector h ∈ H , , so that the natural linear map T S → H , , v µ ( v )( h ) is an isomorphism.Finally a decoration of a formal Calabi-Yau de Rham data ( H, ∇ ) is a pair ( e , h ), where e is a trivialization e : H | D ×{ } → H , ⊗ O D ×{ } , and h ∈ H , is a generating vector for theCalabi-Yau property.Again a decorated de Rham data of Calabi-Yau type gives an affine structure and canon-ical formal coordinates on S . In this section we discuss the aspects of algebraic deformation theory relevant to the studyof nc -Hodge structures. We will work over C but all algebraic considerations in this sectionmake sense over any field of characteristic zero.83 .2.1 Preliminaries on L ∞ algebras Our main objects of interest here will be differential Z / C or moregenerally Z / L ∞ -algebras over C . We begin with a definition: Definition 4.9 A complex differential Z / -graded Lie algebra g (or a Z / -graded L ∞ -algebra)is called homotopy abelian if it is L ∞ quasi-isomorphic to an abelian d ( Z / g Lie algebra. Remark 4.10 Homotopy abelian differential Z / • A differential Z / g is homotopy abelian if and only if all the higheroperations m n vanish on its L ∞ minimal model g min = H • ( g , d g ), i.e. m n = 0 for n ≥ • A differential Z / g is homotopy abelian if and only if there existd( Z / g and g , and morphisms of d( Z / g ∼ = (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127) ∼ = ! ! BBBB g g so that g is an abelian d( Z / g → g and g → g are quasi-isomorphisms. • A differential Z / g is homotopy abelian if and only if the Liealgebra cohomology algebra H • ( g , C ) is free, i.e. is isomorphic to the algebra of formalpower series on some (possibly infinitely many) supervariables. Here the Lie algebracohomology is defined as H • ( g , C ) := H • Y n ≥ Hom ( C − Vect ) (Sym n Π g , C ) • , d ! where d is the cochain Cartan-Eilenberg differential.After the prioneering work of Deligne and Drinfeld in the 80’s, it is by now a commonwisdom (see e.g. [Man99a, Chapter III.9]) that dg Lie algebras give rise to solutions of84oduli problems. In particular a homotopy abelian d( Z / g gives rise to amoduli space - the formal supermanifold M od ( g ,d g ) := Spf( H • ( g , C )).The property of being homotopy abelian is preserved by suitably non-degenerate deforma-tions and various other natural operations: Proposition 4.11 (i) Let g be a flat family of d ( Z / g Lie algebras (or ( Z / -graded L ∞ algebras) over C [[ u ]] . That is g is a flat ( Z / -graded C [[ u ]] -module, and the Lie bracket anddifferential on g are C [[ u ]] -linear. Assume further that(A) g gen := g ⊗ C [[ u ]] C (( u )) is homotopy abelian over C (( u )) , and(B) H • ( g , d g ) is a flat C [[ u ]] -module.Then the special fiber g := g b ⊗ C [[ u ]] C is also a homotopy abelian d ( Z / g Lie algebra over C . (ii) If g is a homotopy abelian d ( Z / g Lie algebra over C , and g → g is a morphismof L ∞ -algebras inducing a monomorphism H • ( g , d g ) ֒ → H • ( g , d g ) , then g is homotopyabelian as well. (iii) If g is a homotopy abelian d ( Z / g Lie algebra over C , and g → g is a morphism of L ∞ -algebras inducing an epimorphism H • ( g , d g ) ։ H • ( g , d g ) , then g is homotopy abelianas well. Proof. The proof is standard so we only mention some of the highlights of the argument.First note that parts (ii) and (iii) follow immediately by passing to minimal models. Forpart (i) we note first that the assumption (B) implies (and is in fact equivalent to) theexistence of C [[ u ]]-linear quasi-isomorphisms p , p of complexes:( H • ( g , d g ) [[ u ]] , ∼ = ( H • ( g , d g ) , p / / ( g , d g ) , p o o and a C [[ u ]]-linear homotopy h so that p ◦ p = id p ◦ p = id + [ d g , h ] . Next note that the homological perturbation theory of [KS01] carries over verbatim to the L ∞ -context and gives explicit expressions for the higher products m n on ( H • ( g , d g ) [[ u ]] , p , p and h . In particular the operations m n are all C [[ u ]]-linear and are given by universal expressions. But by assumption (A) we know that thehigher operations are zero after tensoring with ⊗ C [[ u ]] C (( u )) and so m n = 0 as formal powerseries in u for all n ≥ 1. This implies that m n | u =0 = 0 for all n ≥ (cid:3) Recall [Man99a, Chapter III.10] the notion of a dg BV algebra: Definition 4.12 A differential Z / -graded Batalin-Vilkovisky algebra over C is the data ( A, d, ∆) , where A is a Z / -graded suppercommutative associative unital algebra, and d : A → A , ∆ : A → A are odd C -linear maps satisfying: • d (1) = ∆(1) = 0 , • d is a differential operator of order ≤ on A , • ∆ is a differential operator of order ≤ on A , • d = ∆ = d ∆ + ∆ d = 0 . Note that the first two properties in the definition imply that d is a derivation of A . Also g := Π A together with [ a, b ] := ∆( ab ) − ∆( a ) b − ( − deg( a ) a ∆( b ) is a Lie superalgebra withtwo anti-commuting differentials d and ∆. Definition 4.13 We will say that a d ( Z / g Batalin-Vilkovisky algebra ( A, d, ∆) has the degeneration property if for every N ≥ we have that H • ( A [ u ] / ( u N ) , d + u ∆) is a free C [ u ] / ( u N ) -module. Equivalently ( A, d, ∆) has the degeneration property iff H • ( A [[ u ]] , d + u ∆) is a topologi-cally free (flat) C [[ u ]]-module. This in turn is equivalent to the existence of a (non-unique)isomorphism of topological C [[ u ]]-modules:(4.2.1) T : H • ( A [[ u ]] , d + u ∆) ∼ = / / H • ( A, d )[[ u ]] . 86n this situation we will always normalize T so that T | u =0 = id H • ( A,d ) .The degeneration property for dg Batalin-Vilkovisky algebras defined above is weaker thanthe ∂ ¯ ∂ -lemma used Barannikov and the second author in [BK98] and by Manin in [Man99a,Man99b]. In particular it has potentially a wider scope of applications - a feature that we willexploit next. We begin with a general smoothness result which was also proven by J.Terilla[Ter07]. Theorem 4.14 Suppose ( A, d, ∆) is a d ( Z / g Batalin-Vilkovisky algebra which has thedegeneration property. Let g := Π A be the associated super Lie algebra with anti-commutingdifferentials d and ∆ . Then: (1) The d ( Z / g Lie algebra ( g , d ) is homotopy abelian, i.e. is quasi-isomorphic to H • ( g , d ) endowed with the trivial bracket and the zero differential. In particular the associatedmoduli space M od ( g ,d ) is (non-canonically) isomorphic to a formal neighborhood of in the super affine space Π H • ( g , d ) . (2) Every choice of a normalized degeneration isomorphism T as in equation (4.2.1) givesan identification of formal manifolds Φ T : M od ( g ,d ) ∼ = / / (cid:18) formal neighborhoodof in Π H • ( g , d ) (cid:19) Proof. Part (1) of the theorem follows immediately from Lemma 4.15 The d ( Z / g Lie algebra ( g (( u )) , d + u ∆) is homotopy abelian over C (( u )) . Proof. Consider the formal completion at zero b A of the vector superspace underlying A = Π g as an algebraic supermanifold, and let as before D = Spf( C [[ u ]]) be the formal onedimensional disc. The d( Z / g [[ u ]] is encoded in an odd vectorfield ξ ∈ Γ( b A × D , T ) on the supermanifold b A × D , defined by˙ a := ξ ( a ) = da + u ∆ a + 12 [ a, a ] . There is a natural automorphism (i.e. a formal change of coordinates) F : b A × D × → b A × D × on the formal supermanifold b A × D × given by F ( a ) := u (cid:16) exp (cid:16) au (cid:17) − (cid:17) = a + 1 u a + 1 u a + · · · , b = F ( a ) the vector field ξ is linear:˙ b = ˙ a · exp (cid:16) au (cid:17) = (cid:18) da + u ∆ a + 12 [ a, a ] (cid:19) · exp (cid:16) au (cid:17) = u · (cid:18) dau + u ∆ (cid:16) au (cid:17) + u h au , au i(cid:19) · exp (cid:16) au (cid:17) = u · ( d + u ∆) exp (cid:16) au (cid:17) = ( d + u ∆) b. So in the b -coordinates, the vector field ξ depends only on the differential d + u ∆ and does notdepend on any higher operations. Passing to the minimal model we see that ( g (( u )) , d + u ∆)is homotopy abelian, which proves the lemma. (cid:3) The lemma implies that the hypothesis (A) of Proposition 4.11 (i) holds. On the other handthe hypothesis (B) holds by the degeneration assumption. Therefore by Proposition 4.11 (i) we conclude that ( g , d ) is homotopy abelian. This proves part (1) of the theorem.Next we construct the identification Φ T . Given a formal path in M od ( g ,d ) , i.e. a family ofsolutions (up to guage equivalence) a ( ε ) = a ε + a ε + a ε + · · · ∈ εA [[ ε ]] d ( a ( ε )) + 12 [ a ( ε ) , a ( ε )] = 0of the Maurer-Cartan equation in ( g , d ), we have to construct the corresponding formal paththrough the origin in H • ( g , d ).As a first step choose a lift of the formal arc a ( ε ) to a formal series in two variables˜ a ( ε, u ) ∈ εA [[ ε, u ]] such that ( d + u ∆)˜ a + 12 [˜ a, ˜ a ] = 0 ,a ( ε, 0) = a ( ε ) . Consider the reparameterization˜ b = F (˜ a ) = u (cid:18) exp (cid:18) ˜ au (cid:19) − (cid:19) ∈ εA (( u ))[[ ε ]] . Arguing as before we see that ˜ b satisfies ( d + u ∆)˜ b = 0. So if we expand˜ b = ˜ b ε + ˜ b ε + · · · , where ˜ b n ∈ A (( u )) , satisfy ( d + u ∆)˜ b n = 0 , 88e can define cohomology classes h ˜ b n i ∈ H • ( A (( u )) , d + u ∆). We can now apply the iso-morphism T ⊗ C [[ u ]] C (( u )) to the series X n ≥ h ˜ b n i ε n ∈ εH • ( A (( u )) , d + u ∆)[[ ε ]] , to obtain an element T X n ≥ h ˜ b n i ε n ! ∈ εH • ( A, d )(( u ))[[ ε ]] . In fact one has the following lemma whose proof we will skip since it is a somewhat tediousapplication of homological perturbation theory: Lemma 4.16 There exists a lift ˜ a ( ε, u ) of a ( ε ) such that the associated class T (cid:16)P n ≥ h ˜ b n i(cid:17) belongs to εH • ( A, d )[[ ε ]] ⊂ εH • ( A, d )(( u ))[[ ε ]] . Any such lift e a produces the same class T (cid:16)P n ≥ h ˜ b n i(cid:17) and this class depends only on the gauge equivalence class of the original arc a , i.e. on the image a ( ε ) of a ( ε ) in M od ( g ,d ) . Now by definition the map Φ T assigns the class T (cid:16)P n ≥ h ˜ b n i(cid:17) ⊂ εH • ( A, d )[[ ε ]] to the formalarc a ( ε ). (cid:3) The previous discussion can be repackaged geometrically as follows. A ( Z / A, d, ∆), gives rise to a family M → D = Spf( C [[ u ]]) of formal manifoldsover the one dimensional formal disc. The family M is the total space of the relative modulispace M od ( g ,d + u ∆) over C [[ u ]]. If ( A, d, ∆) has the degeneration property, then by Lemma 4.15we have an affine structure on the generic fiber M gen := M ⊗ C [[ u ]] C (( u )) of the family (seeFigure 6) given by the map F .Furthermore the map T can be viewed as an extension of the affine bundle M gen → D × toa trivial bundle on P − { } of formal super affine spaces, where the fiber at ∞ is the superaffine space H • ( g , d ). This results into a family M od → P of formal super manifolds, which89 od M od ( g ,d ) u H • ( g , d )0 ∞ Figure 6: The relative moduli M od → P .is a trivial vector bundle outside of zero but has a non-linear fiber at 0 ∈ P . Moreover bypicking the closed point in each fiber we get a section of M od → P , which is just the zerosection of the vector bundle M od | P −{ } → P − { } . The normal bundle to this section in M od is trivial (hence M od is trival as a non-linaer bundle over P ), and the map Φ T gives a(non-linear) trivialization of M od over P . This type of geometry was already discussed in[CKS05]. nc -Hodge structures Suppose ( A, d, ∆) is a d Z / 2g Batalin-Vilkovisky algebra which has the degeneration prop-erty. In this generality one does not expect to find a natural connection on H • ( A, d + u ∆)along u , i.e. one does not expect to have a general formal analogue of a nc -Hodge structure.However, a natural connection along the u -line may exist if we specify some additionaldata on ( A, d, ∆). Following the analogy with the nc -Hodge structure associated with asymplectic manifold and the Gromov-Witten invariants, it is sufficient to specify: • an even element κ ∈ A , with dκ = 0, and • a grading operator Gr : A → A , 90o that if we consider Γ − := Gr : A → A , and Γ − : A → A - the operator of multiplicationby κ , then we have the commutation relations:[Γ − , ∆] = − 12 ∆[Γ − , d ] = 0 d = [Γ − , d ] + [Γ − , ∆] . These commutation relations imply the identity (cid:20) u ∂∂u + u − Γ − + Γ − , d + u ∆ (cid:21) = 12 ( d + u ∆) , which is consistent with the general formulas from Section 2.2.5. In particular, we can definea connection on H • ( A, d + u ∆) along the u -line by setting ∇ ∂∂u := ∂∂u + u − Γ − + u − Γ − . Example 4.17 Let Y be a (possibly non-compact) d -dimensional Calabi-Yau manifold witha fixed holomorphic volume form Ω Y . Let w : Y → C be a proper holomorphic function.This geometry gives rise to a natural dg Batalin-Vilkovisky algebra: A := Γ C ∞ (cid:0) Y, ∧ • T , Y ⊗ ∧ • A , Y (cid:1) ,d := ¯ ∂ + ι d w , ∆ := div Ω Y = ι − Ω Y ◦ ∂ ◦ ι Ω Y , where ι Ω Y : ∧ • T , Y → ∧ d −• Ω , Y denotes the contraction with Ω Y .As discussed in section 3.2 in this situation we get a connection along u which conjec-turally defines a nc -Hodge structure. The connection is defined the above formula withΓ − = the operator of multiplication by − w , and Γ − = Gr : A → A , the grading operatorwhich is equal to q + p − d · id on Γ C ∞ (cid:0) Y, ∧ p T , Y ⊗ ∧ q A , Y (cid:1) .We will elaborate on this geometric picture in the next section. B -model framework: manifolds with anticanonical sections Let X be a compact K¨ahler manifold. By Kodaira-Spencer theory we know that thedeformations of X are controlled by the dg Lie algebra (cid:0) g (1) , d g (1) (cid:1) := (cid:0) Γ C ∞ (cid:0) X, T , X ⊗ C ∞ X A , • X (cid:1) , ¯ ∂ (cid:1) . Theorem 4.18 If X is a compact K¨ahler manifold with c ( X ) = 0 ∈ Pic( X ) , then (cid:0) g (1) , d g (1) (cid:1) is homotopy abelian. In particular the formal moduli space of X is smooth. Proof. Since c ( X ) = 0 ∈ Pic( X ) we can find a unique up to scale holomorphic volumeform Ω X on X . As in example 4.17 the pair ( X, Ω X ) gives rise to a dg Batalin-Vilkoviskyalgebra ( A, d, ∆): A := Γ C ∞ (cid:0) X, ∧ • T , X ⊗ ∧ • A , X (cid:1) d := ¯ ∂ ∆ := div Ω X = ι − Ω X ◦ ∂ ◦ ι Ω X . Consider the associated dg Lie algebra ( g , d g ) := ( Π A, d ). We have a natural inclusion of dgLie algebras (cid:0) g (1) , d g (1) (cid:1) (cid:31) (cid:127) / ( g , d g ) (cid:0) Γ C ∞ (cid:0) X, T , X ⊗ C ∞ X A , • X (cid:1) , ¯ ∂ (cid:1) (cid:31) (cid:127) / Γ C ∞ (cid:0) X, ∧ • T , X ⊗ ∧ • A , X (cid:1) which embeds (cid:0) g (1) , d g (1) (cid:1) as a direct summand in ( g , d g ), and so induces and embedding H • (cid:0) g (1) , d g (1) (cid:1) ⊂ H • ( g , d g ) in cohomology. So by Proposition 4.11 it suffices to check that( g , d g ) is homotopy abelian.On the other hand the contraction map ι Ω X gives an isomorphism of bicomplexes betweenthe dg Batalin-Vilkovisky algebra ( A, d, ∆) and the Dolbeault bicomplex ( A • ( X ) , ¯ ∂, ∂ ). Since X is assumed compact and K¨ahler, the Hodge-to-de Rham spectral sequence degenerateson X which is equivalent to the equality dim H kdR ( X, C ) = dim( ⊕ p + q = k H p ( X, Ω qX )) whichimplies that the Dolbeault bicomplex ( A • ( X ) , ¯ ∂, ∂ ) has the degeneration property. Thus byTheorem 4.14 (1) it follows that ( g , d g ) is homotopy abelian. The theorem is proven. (cid:3) Let X be a Calabi-Yau manifold, i.e. a d -dimensional compact K¨ahler manifold with c ( X ) =0 in Pic( X ). Let ( A, d, ∆) be the dg Batalin-Vilkovisky algebra defined in section 4.3.1. Thecontraction map ι Ω X identifies the C [[ u ]]-module H • ( g [[ u ]] , d + u ∆) with the Rees module92f the nc -Hodge filtration on H • dR ( X, C ) for which H p,q ( X ) ⊂ F p − q . Now choose one of thefollowing equivalent pieces of data: • a filtration G • on H • dR ( X, C ) which is opposed to the nc -Hodge filtration, • a splitting of the nc -hodge filtration, • an extension of the associated nc -Hodge structure to a trivial bundle on P such thatthe connection has at most a first order pole at infinity.Each such choice gives rise to an affine structure on M od ( g ,d g ) . This affine structure is thesame as the one described in section 4.1.3 corresponding to the nc -Hodge structure aboveand the decoration ψ given by the class [ Ω X ] in the associated graded gr G • H • dR ( X, C ).In mirror symmetry considerations a choice of this type arises naturally when X is aCalabi-Yau manifold near a large complex structure limit point. Concretely, suppose X = X z is member in a holomorphic family { X z } of compact d -dimensional Calabi-Yau manifoldsparameterized by z in a polydisc Q Mi =1 { z i ∈ C | < | z i | ≪ } , and such that: • M = dim C H ( X z , T X z ); • for each i = 1 , . . . , M the monodromy operator t i ∈ GL (cid:0) H (cid:0) X z , T X z (cid:1)(cid:1) assigned to thecircle (traced counterclockwise) γ i = (cid:26) z (cid:12)(cid:12)(cid:12)(cid:12) z j = z j , j = i , | z i | = | z i | (cid:27) is unipotent of order d .In this setup, the filtration G • of H • ( X z , C ) invariant under all unipotent operators Q Mi =1 t a i i , a i ∈ Z > will be opposed to the Hodge filtration and will thus give us canonical coordinateson the polydisc. This affine structure corresponds to a rational decoration of a Calabi-Yauvariation of nc -Hodge structures. Here we generalize the previous discussion to the case of varieties with divisors.93 i) Let X be a d -dimensional smooth projective variety over C , and let D ⊂ X be a normalcrossings anti-canonical divisor, i.e O X ( D ) = K − X ∈ Pic( X ). Typically such an X will be aFano or a quasi-Fano. If D is smooth, then by adjunction D will be a Calabi-Yau. Specifyingsuch a divisor is equivalent to specifying a logarithmic volume form on X . This is a uniqueup to scale n -form Ω X log D ∈ Γ (cid:0) X, Ω dX (log D ) (cid:1) on X which has a first order pole along D and does not vanish anywhere on X − D .Let T X,D be the subsheaf of T X of holomorphic vector fields on X which at the pointsof D are tangent to D . This is a locally free subsheaf of T X of rank d which controls thedeformation theory of the pair ( X, D ). The relevant dg Batalin-Vilkovisky algebra ( A, d, ∆)is an obvious generalization of the one in the absolute case: A := Γ C ∞ (cid:0) X, ∧ • T X,D ⊗ C ∞ X ∧ • A , X (cid:1) d := ¯ ∂ ∆ := div Ω X log D = ι − Ω X log D ◦ ∂ ◦ ι Ω X log D , where ι Ω X log D : ∧ • T X,D → Ω d −• X (log D ) is the isomorphism given by contraction with Ω X log D .Again the map ι Ω X log D identifies ( A, d, ∆) with the logarithmic Dolbeault bicomplex (cid:0) A • , • (log D ) , ¯ ∂, ∂ (cid:1) . In particular, for all u = 0 we get an identification of the cohomology ofthe complex ( A, d + u ∆) with the cohomology of the total complex of the double complex (cid:0) Ω • , • X (log D ) , ¯ ∂, ∂ (cid:1) , which is equal [Voi03, Section 6.1] to the cohomology of the open variety X − D . In other words for all u = 0 we have an isomorphism(4.3.1) H • ( A, d + u ∆) ∼ = H • dR ( X − D, C ) . Now mixed Hodge theory implies the following Lemma 4.19 The logarithmic dg Batalin-Vilkovisky algebra ( A, d, ∆) has the degenerationproperty. In particular the formal moduli of the pair ( X, D ) is smooth. We will return to the proof of this lemma in section 4.3.4 but first we will discuss a coupleof variants of this geometric setup. (ii) Suppose X is a smooth projective d -dimensional Calabi-Yau manifold. Let as before Ω X be the holomorphic volume form on X . Let D ⊂ X be a normal crossings divisor.Typically if D is smooth, it will be a variety of general type.94onsider the dg Batalin-Vilkovisky algebra ( A, d, ∆) given by A := Γ C ∞ (cid:0) X, ∧ • T X,D ⊗ C ∞ X ∧ • A , X (cid:1) d := ¯ ∂ ∆ := div Ω X = ι − Ω X ◦ ∂ ◦ ι Ω X , The contraction ι Ω X identifies this algebra with the dg Batalin-Vilkovisky algebra (cid:0) Γ C ∞ (cid:0) X, Ω • X (rel D ) ⊗ C ∞ X ∧ • A , X (cid:1) , ¯ ∂, ∂ (cid:1) , where Ω kX (rel D ) ⊂ Ω kX denotes the subsheaf of all holomorphic k -forms that restrict to0 ∈ Ω kD − sing( D ) . The cohomology of the total complex associated with this double complexis the de Rham cohomology of the pair ( X, D ), and so again we get an identification(4.3.2) H • ( A, d + u ∆) ∼ = H • dR ( X, D ; C )valid for all fixed u = 0. Again using this identification and mixed Hodge theory one deducesthe following Lemma 4.20 The dg Batalin-Vilkovisky algebra ( A, d, ∆) has the degeneration property andhence the formal moduli space of the pair ( X, D ) is smooth. (iii) The setups (i) and (ii) have a natural common generalization. Fix a smooth projectivecomplex variety of dimension d , a normal crossings divisor D = ∪ i ∈ I D i ⊂ X , and a collectionof weights { a i } i ∈ I ⊂ [0 , ∩ Q , so that X i ∈ I a i [ D i ] = − K X ∈ Pic( X ) ⊗ Q . Represent the a i ’s by reduced fractions, take N ≥ X i ∈ I ( N a i )[ D i ] = − N K X ∈ Pic( X ) , and set n i := a i N . In particular we have a unique up to scale section e Ω X ∈ Γ (cid:16) X, K ⊗ ( − N ) X (cid:17) whose divisor is P i ∈ I n i D i . In this situation we can again promote the Dolbeault dg Lie95lgebra which computes the deformation theory of ( X, D ) to a dg Batalin-Vilkovisky algebra( A, d, ∆), where A := Γ C ∞ (cid:0) X, ∧ • T X,D ⊗ C ∞ X ∧ • A , X (cid:1) d := ¯ ∂ ∆ := div e Ω X . The divergence operator div e Ω X is defined as follows. Restricting the section e Ω X to X − D we get a nowhere vanishing section of K ⊗ ( − N ) X − D , i.e. a flat holomorphic connection on K X − D .If U ⊂ X − D is a simply connected open, then we can choose Ω U a holomorphic volumeform on U which is covariantly constant for this flat connection, and define the associateddivergence operator div Ω U := ι − Ω U ◦ ∂ ◦ ι Ω U . But by the flatness of the connection it followsthat any other covariantly constant volume form on U will be proportional to Ω U with aconstant proportionality coefficient. Since by definition div c Ω U = div Ω U for any constant c we get a well defined divergence operator on X − U . Furthermore locally this divergenceoperator is a given by a holomorphic volume form which is a branch of (cid:16) e Ω X (cid:17) − /N and soby continuity it gives a well defined map of locally free sheaves div e Ω X : ∧ i T X,D → ∧ i − T X,D .Again we claim that Lemma 4.21 The dg Batalin-Vilkovisky algebra ( A, d, ∆) has the degeneration property andthe formal moduli space of the pair ( X, D ) is smooth. Proof. The proof of this lemma again reduces to mixed Hodge theory via a map similar tothe isomorphisms (4.3.1) and (4.3.2). However constructing this map is a bit more involvedthan the arguments we used to construct (4.3.1) and (4.3.2).Consider the root stack Z = X D(cid:8) D i N (cid:9) i ∈ I E as defined in e.g. [MO05, IS07]. By construc-tion Z is a smooth proper Deligne-Mumford stack, equipped with a finite and flat morphism π : Z → X .Conceptually the best way to define the stack Z is as a moduli stack classifying (special)log structures associated with X , the divisor D and the number N (see [MO05] for thedetails). Etale locally on X the stack Z can be described easily as a quotient stack. Indeedchoose etale locally an identification of X with a neighborhood of zero in A d with coordinates z , . . . , z d , so that D = D ∪ · · · ∪ D r and D i is identified with the hyperplane z i = 0. Thenthe corresponding etale local patch in Z is canonically isomorphic to the stack quotient (cid:2) A d / µ N × · · · × µ N | {z } r -times (cid:3) , µ N ⊂ C × is the group of N -th roots of unity, and ( ζ , . . . , ζ r ) ∈ µ × rN acts as( z , . . . , z r , z r +1 , . . . , z d ) ( ζ z , . . . , ζ r z r , z r +1 , . . . , z d ).In particular, this description shows (see [MO05, Theorem 4.1]) that: • The map π is an isomorphism over X − D and in general identifies X with the coarsemoduli space of Z ; • There is a strict normal crossings divisor e D = ∪ i ∈ I e D i ⊂ Z , such that O Z (cid:16) − N e D i (cid:17) = π ∗ O X ( − D i )as ideal subsheaves of O Z ; • For all j we have the Hurwitz formula Ω jZ (log e D ) = π ∗ Ω jX (log D ).In particular we have canonical isomorphisms π ∗ K X ∼ = O Z − X i ∈ I n i e D i ! π ∗ K X ∼ = K Z ⊗ O Z (1 − N ) X i ∈ I e D i ! the first given by the section π ∗ e Ω X and the second coming from the Hurwitz formula.There is a natural complex local system of rank one on X − D with monodromy in µ N associated with the choices of N -th root of the section e Ω X . It is easy to see that the pullbackof this local system admits a canonical extension (as a local system) to Z , which we denoteby Ξ . Moreover, we have a canonical meromorphic section Ω Z of K Z ⊗ C Ξ with divisor P i ∈ I ( N − − n i ) e D i . It is easy to check locally by using the etale local description of Z as aquotient stack the contraction ι Ω Z gives a well defined isomorphism of locally free sheaves: ι Ω Z : ∧ j T Z, e D ∼ = / / Ω d − jZ (cid:16) log e D (1) , rel e D (0) (cid:17) ⊗ C Ξ . Here e D (0) := ∪ i ∈ I e D i I = { i ∈ I | a i = 0 } e D (1) := ∪ i ∈ I e D i I = { i ∈ I | a i = 1 } . Now taking into account the Hurwitz isomorphism ∧ j T Z, e D ∼ = π ∗ ∧ j T X,D and using adjunction,we can view ι Ω Z as an isomorphism(4.3.3) ∧ j T X,D ∼ = / / (cid:16) π ∗ Ω d − jZ (cid:16) log e D (1) , rel e D (0) (cid:17) ⊗ C Ξ (cid:17) 97t is immediate from the definition that the isomorphism (4.3.3) (taken for all j ) identifiesthe dg Batalin-Vilkovisky algebra ( A, d, ∆) with the Dolbeault bicomplex (cid:16) Γ C ∞ (cid:16) X, (cid:16) π ∗ Ω • Z (cid:16) log e D (1) , rel e D (0) (cid:17) ⊗ C Ξ (cid:17) ⊗ C ∞ X A , • X (cid:17) , ¯ ∂, ∂ (cid:17) . But the above complex equipped with the differential ∂ + ¯ ∂ is the Dolbeault resolution of thecomplex of sheaves π ∗ (cid:16) Ω • Z (cid:16) log e D (1) , rel e D (0) (cid:17) ⊗ C Ξ , ∂ (cid:17) which is equal to the derived directimage Rπ ∗ (cid:16) Ω • Z (cid:16) log e D (1) , rel e D (0) (cid:17) ⊗ C Ξ , ∂ (cid:17) since π is finite. Now combined with the Lerayspectral sequence for π this gives, for all u = 0 an isomorphism(4.3.4) H • ( A, d + u ∆) ∼ = H • dR (cid:16) Z − e D (1) , e D (0) − e D (1) ; Ξ (cid:17) , which specializes to both isomorphisms (4.3.1) and (4.3.2).Now the fact that Z is a smooth and proper Deligne-Mumford stack and mixed Hodge the-ory (see 4.3.4) for (cid:16) Z − e D (1) , e D (0) − e D (1) (cid:17) endowed with local system Ξ imply that ( A, d, ∆)has the degeneration property. (cid:3) Remark 4.22 The fact that the root stack in the previous proof can be viewed as themoduli stack of special log structures is very interesting. It suggests that the setup we justdiscussed may fit naturally in the recent approach of Gross-Siebert [GS06, GS07] to mirrorsymmetry and instanton corrections via log degenerations of toric Fano manifolds (see also[KS06a, KS01]). The relationship between these two setups is certainly worth studying andwe plan to return to it in the future. (iv) Yet another generalization of the previous picture arises when we take the variety X tobe a normal-crossings Calabi-Yau. More precisely assume that X is a strict normal crossingsvariety with irreducible components X = ∪ i ∈ I X i equipped with a holomorphic volume form Ω X on X − X sing such that the restriction of Ω X on each X i has a logarithmic pole along X i ∩ ( ∪ j = i X j ) and the residues of these restricted forms cancel along each X i ∪ X j . Takinga colimit along the projective system of all finite intersections of components of X we getagain a dg Batalin-Vilkovisky algebra A tot ( X ) = colim J ⊂ I A ( ∩ i ∈ J X i ) and again by usingmixed Hodge theory we can check that this algebra has the degeneration property.98 .3.4 Mixed Hodge theory in a nutshell. In this section we briefly recall the basic arguments from Deligne’s mixed Hodge theory[Del74] that are necessary for proving the degeneration property of the dg Batalin-Vilkoviskyalgebras in section 4.3.3 (i)-(iv) .Suppose we are given: • a finite ordered collection ( X α ) of smooth complex projective varieties; • for every α a choice of a Z × Z -graded complex of sheaves of differential forms which areeither C ∞ or are C −∞ (i.e. currents) and constrained so that their wave front (singularsupport) is contained in a given conical Lagrangian in T ∨ X α which is the conormalbundle to a normal crossings divisor in X α ; • a collection of integers n α ∈ Z .Consider the complex C tot = ⊕ α C • α [ n α ] equipped with three differentials ∂ , ¯ ∂ , δ , where δ = P α<β δ αβ , and the δ αβ come from pullbacks and pushforwards for some maps X β ֒ → X α or X α ֒ → X β . The statement we need now can be formulated as follows: Claim 4.23 For every k ≥ the cohomology H • (cid:0) C tot [ u ] / ( u k ) , ¯ ∂ + δ + u∂ (cid:1) is a free C [ u ] / ( u k ) -module. Proof. If X is smooth projective over C and if (cid:0) A • ( X ) , ¯ ∂ (cid:1) is the ¯ ∂ -complex of (either C ∞ or C −∞ ) differential forms on X , then the inclusion (cid:0) ker ∂, ¯ ∂ (cid:1) ֒ → (cid:0) A • ( X ) , ¯ ∂ (cid:1) is a quasi-isomorphism.This implies that the horizontal arrows in the diagram of complexes (cid:0) ker ∂ [ u ] / ( u k ) , ¯ ∂ + δ + u∂ (cid:1) / / (cid:0) C tot [ u ] / ( u k ) , ¯ ∂ + δ + u∂ (cid:1)(cid:0) ker ∂ [ u ] / ( u k ) , ¯ ∂ + δ (cid:1) / / (cid:0) C tot , ¯ ∂ + δ (cid:1) [ u ] / ( u k ) , are quasi-isomorphisms. Indeed, this follows by noticing that there are natural filtrationson both sides (by the powers of u and the index α ) which give rise to convergent spectral99equences and induce the quasi-isomorphic inclusion (cid:0) ker ∂, ¯ ∂ (cid:1) ֒ → (cid:0) C tot , ¯ ∂ (cid:1) on the associatedgraded. This proves the claim. (cid:3) Remark 4.24 • Note that the same reasoning implies that the natural map (cid:0) ker ∂, ¯ ∂ + δ (cid:1) ։ (cid:0) ker ∂/ im ∂, ¯ ∂ + δ (cid:1) = ( H • ( X α ) , δ ) , is also a quasi-isomorphism, which reduces the problem of computing H • (cid:0) C tot [ u ] / ( u k ) , ¯ ∂ + δ + u∂ (cid:1) to a homological algebra question on a complex of finite di-mensional vector spaces. • There is useful variant of the theory, also discussed in [Del74]: the previous discussion im-mediately generalizes to the case of cochain complexes of a collection of projective manifoldswith coefficients in some unitary local systems.Next we discuss a few examples and applications of the geometric setup from section 4.3.3. As a consequence of section 4.3.3 (iii) we get a new proof and a refinement of the followingresult of Ran [Ran92, Kaw92]: Theorem 4.25 Let X be a complex Fano manifold, that is let X be a smooth proper C -variety for which K − X is ample. Then the versal deformations of X are unobstructed. Proof: Choose N > K ⊗ ( − N ) X is very ample and all the higher cohomology groups H k (cid:16) X, K ⊗ ( − N ) X (cid:17) vanish for k ≥ 1. Choose a generic section e Ω X ∈ H (cid:16) X, K ⊗ ( − N ) X (cid:17) = 0whose zero locus is a smooth and connected divisor D ⊂ X .Consider now g = Π R Γ ( X, ∧ • T X,D ) with the Schouten bracket. By Lemma 4.21 thisd( Z / g (1) = R Γ ( X, T X,D ) is homotopy abelian. Since this d( Z / X, D ) as a variety with a divisor, it follows that the formal germ ofthe deformation space of the pair ( X, D ) is smooth. Next we will need the following simple100 emma 4.26 Suppose ( X ′ , D ′ ) is a small deformation of ( X, D ) as a variety with divisor.Then X ′ is still a Fano with K ⊗ ( − N ) X ′ is very ample and D ′ ∈ (cid:12)(cid:12)(cid:12) K ⊗ ( − N ) X ′ (cid:12)(cid:12)(cid:12) . Proof: The condition of K ⊗ ( − N ) X being very ample is open in the moduli of X . Furthermoreby definition K ⊗ ( − N ) X ⊗ O X ( − D ) = O X and so by the small deformation hypothesis it followsthat K ⊗ ( − N ) X ′ ⊗ O X ′ ( − D ) is in the connected component of the identity of Pic( X ′ ). But X ′ is a Fano and so Lie(Pic ( X ′ )) = H ( X ′ , O X ′ ) = 0. Hence K ⊗ ( − N ) X ′ ⊗ O X ′ ( − D ) = O X aswell. (cid:3) The theorem now follows easily. The versal deformation space of smooth connected D ’s fora given X is smooth and isomorphic to a domain in P h “ X,K ⊗ ( − N ) X ′ ” − . Since the dimension ofthese projective spaces is locally constant in X by Riemann-Roch and vanishing of the highercohomologies, it follows that the map from the versal deformation space of the pairs ( X, D )to the versal deformation stack of X is smooth. In other words the versal deformation stackof X has a presentation in the smooth topology with a smooth atlas - the versal deformationspace for ( X, D ). Hence the versal deformations of X are a smooth stack. (cid:3) Consider again the setup of a holomorphic Landau-Ginzburg model. Suppose Y is smoothand quasi-projective over C and of dimension dim Y = d . Suppose there exists a nowherevanishing algebraic volume form Ω Y ∈ Γ( Y, K Y ), and let w : Y → A be a regular functionwith compact critical locus.This data gives a dg Batalin-Vilkovisky algebra ( A, d, ∆) where A := Γ C ∞ (cid:0) Y, ∧ • T , Y ⊗ C ∞ Y ∧ • A , Y (cid:1) d := ¯ ∂ + ι d w ∆ := div e Ω Y . Again the contraction ι Ω Y identifies ( A, d, ∆) with the twisted Dolbeault bicomplex (cid:0) A • ( Y ) , ¯ ∂ + d w ∧ , ∂ (cid:1) . The latter satisfies the degeneration property by the work of Baran-nikov and the second author, Sabbah [Sab99], or Ogus-Vologodsky [OV05]101 emark 4.27 It will be interesting to combine the previous discussion with the discussion insection 4.3.3 (iii) or with the broken Calabi-Yau geometry from section 4.3.3 (iv) . Supposewe have a quasi-projective smooth complex Y , a regular function w : Y → A with compactcritical locus, and suppose we are given a normal crossings divisor D = ∪ i ∈ I D i and a systemof weights { a i } i ∈ I as in section 4.3.3 (iii) . Then we can write the w -twisted version ofthe dg Batalin-Vilkovisky algebra for ( Y, D ) which by general nonsense will compute thedeformation theory of the data ( Y, D, w ). Similarly we can add a potential to a Y whichitself is a normal-crossings Calabi-Yau, as in section 4.3.3 (iv) . We expect that the resultingalgebras will again have the degeneration property but we have not investigated this question. In this section we briefly discuss some algebraic aspects of the deformation theory of nc -spaces (see section 2.2.1). For simplicity we will discuss the Z -graded case but in fact alldefinitions and statements readily generalize to the Z / nc -spaces. Suppose X = ncSpec ( A ) is a graded nc -affine nc -space over C . If X is smooth, then A ∈ Perf X × X op = Perf ( A ⊗ A op − mod ) and we define the smooth dual of A to be A ! :=Hom A ⊗ A op ( A, A ⊗ A ). Similarly if X is compact, then A ∈ Perf pt and we define the compactdual of A to be A ∗ := Hom C ( A, C ) ∈ ( A ⊗ A op − mod ).If X is both a smooth and compact nc -space, then we have isomorphisms A ! ⊗ A A ∗ ∼ = A ∗ ⊗ A A ! ∼ = A in the category ( A ⊗ A op − mod ). The endofunctor S X : C X → C X given by the A -bimodule A ∗ is called the Serre functor of X . It is an autoequivalence of C X which is central (i.e.commutes with all autoequivalences). Moreover for any two objects E , F ∈ Perf X there is afunctorial identification Hom X ( E , F ) ∨ ∼ = Hom X ( F , S X E ) . With this notation we have the following definition (see also [KS06b]):102 efinition 4.28 We say that a smooth graded nc -affine nc -space X = ncSpec ( A ) is a Calabi-Yau of dimension d ∈ Z if A ! ∼ = A [ − d ] in ( A ⊗ A op − mod ) . We say thata compact nc -affine nc -space X = ncSpec ( A ) is a Calabi-Yau of dimension d ∈ Z if A ∗ ∼ = A [ d ] in ( A ⊗ A op − mod ) . The definition works also in the Z / d is understood asan element of Z / nc -space which is both smooth and compact the two conditions are equivalent and areequivalent to having an isomorphism of endofunctors S X ∼ = [ d ]. Remark 4.29 This definition of a Calabi-Yau structure on a smooth compact nc -spaceis somewhat simplistic and should be taken with a grain of salt. The true definition (see[KS06b]) implies the isomorphism of functors S X ∼ = [ d ] but also involves higher homotopicaldata which is encoded in a cyclic category structure on C X . We will suppress the cyclicstructure here in order to simplify the discussion.We are interested in nc -space analogues of the Tian-Todorov theorem. The unobstructed-ness of graded smooth and compact nc -Calabi-Yau spaces was recently analyzed by Pandit[Pan08] via the T -lifting property of Ran [Ran92] and Kawamata [Kaw92]. Here we formu-late the following general Theorem 4.30 Suppose that X is a smooth and compact nc -Calabi-Yau space of dimension d ∈ Z (or of dimension d ∈ Z / in the Z / -graded case). Assume that X satisfies thedegeneration conjecture (see section 2.2.4). Then: • the Hochschild cochain algebra C • ( X ) of X is a homotopy abelian L ∞ algebra; • the formal moduli space M od X of X is a formal supermanifold, i.e. M od X := M od C • ( A,A ) ∼ = Spf C [[ x , . . . , x N , ξ , . . . , ξ M ]]; • the negative cyclic homology of the universal family over M od X gives a vector bun-dle H → M od X × D which is equipped with a flat meromorphic connection ∇ so that ∇ u∂/∂x i , ∇ u∂/∂ξ j , and ∇ u ∂/∂u are regular; ( H, ∇ ) is the de Rham part of a Calabi-Yau variation of nc -Hodge structures. We will only sketch some of the highlights of the proof of this theorem here since going intofull details will take us too far afield. The proof is based on a mildly generalized versionof Deligne’s conjecture (see e.g. [KS00, Tam03]) which states that the Hochschild cochaincomplex of an affine nc -space is also an algebra over the operad of chains of the little discsoperad. The first step is to show that under the Calabi-Yau assumption the Hochschildcochain complex C • ( X ) is also naturally an algebra over the cyclic operad of chains of theframed little discs operad (i.e. the operad of little discs with a marked point point on theboundary). Next one shows that the validity of the degeneration conjecture for X impliesthat the induced S -action on the cochain complex, is homotopically trivial. Finally by atopological argument one deduces from this the fact that all the higher L ∞ operations on C • ( X ) must vanish. Remark 4.31 It seems certain that from deformation quantization it follows that if X isa smooth and projective Calabi-Yau variety, then the data described in the above theoremis canonically isomorphic to the formal completion of the variation of nc -Hodge structuresdescribed in section 4.3.2.For a general smooth and compact nc -Calabi-Yau space we expect that the formal varia-tion of nc -de Rham data in theorem 4.30 converges to give an analytic de Rham data whichcontains a compatible nc -Betti data E B and so extends to an honest variation of nc -Hodgestructures. In this section we introduce a special version of the general notion of a spherical functor[Ann07] which is tailored to the Calabi-Yau condition. We begin with a definition: We borrowed this delightful expression from [Kap91]. efinition 4.32 Let X and Y be two graded nc -spaces. A morphism f : X → Y is atriple of functors C Xf ∗ (cid:15) (cid:15) C Y f ! O O f ∗ O O so that ( f ∗ , f ∗ ) and ( f ∗ , f ! ) are ( left , right ) pairs of adjoint functors. Suppose now X , Y are smooth and compact graded nc -spaces and let Y be a nc -Calabi-Yauof dimension d . Definition 4.33 A morphism f : X → Y is spherical if: (a) the cone of the natural adjunction morphism id C X → f ! ◦ f ∗ is isomorphic to the shiftedSerre functor of X : cone (cid:0) id C X → f ! ◦ f ∗ (cid:1) ∼ = S X [1 − d ] , (b) the natural map f ! → S X [1 − d ] ◦ f ∗ , induced from the isomorphism in (a) and theadjunction f ! → f ! ◦ f ∗ ◦ f ∗ is an isomorphism of functors. Remark 4.34 (a) If f is spherical, then the associated reflection functor R f := cone (cid:0) f ∗ ◦ f ! → id C Y (cid:1) is an auto-equivalence of C Y [Ann07]. (b) Similarly to the definition of a Calabi-Yau structure the above notion of a sphericalfunctor should be viewed as a weak preliminary version of a stronger more refined notionwhich has to involve higher homotopical data and has yet to be defined carefully. Example 4.35 (i) Let X = pt, and let Y be a d -dimensional smooth and compact nc -Calabi-Yau and let E ∈ C Y be a spherical object, i.e. an object for which the complex of C -vector spaces Hom Y ( E , E ) is quasi isomorphic to ( H • ( S d , C ) , nc -spaces f : pt → Y given by f ∗ ( V ) = E ⊗ V , for any V ∈ C pt = ( Vect C ) is spherical. (ii) Let X be smooth and projective of dimension d + 1, and let i : Y ֒ → X be a smoothanti-canonical divisor in X . The Y is a d -dimensional Calabi-Yau and we have a naturalspherical nc -morphism f : X → Y given by f ∗ := i ∗ , f ! := i ∗ , etc.105 iii) Let Y be a smooth projective d -dimensional Calabi-Yau. Let i : X ֒ → Y be a smoothhypersurface. Then we have a natural spherical nc -morphism f : X → Y given by f ∗ = i ∗ , f ! = i ! , and f ∗ = i ∗ . Remark 4.36 The geometry of Example 4.35 (ii) , where X is taken to be a smooth projec-tive Fano, and i : Y ֒ → X is a smooth anti-canonical divisor, can be encoded algebraicallyin the categories C X = D ( Qcoh ( X )), C Y = D ( Qcoh ( Y )), the functor f ∗ = i ∗ , and anothernatural triple of categories: • the compact category D compactsupport ( Qcoh ( X − Y )) = ker( f ∗ ), • the compact category D supp Y ( Qcoh ( X )) = the subcategory in D ( Qcoh ( X )) generatedby i ∗ D ( Qcoh ( Y )), • the smooth category D ( Qcoh ( X − Y )) = the quotient D ( Qcoh ( X )) /D supp Y ( Qcoh ( X )).There is a similar triple of categories for the setup in Example 4.35 (iii) . It will be veryinteresting to describe the categorical data that encodes anti-canonical divisors with normalcrossings or more generally the fractional anti-canonical divisor setup from section 4.3.3 (iii) .It seems likely that in this situation one gets a system of nested categories and functors witha “spherical” condition imposed on the whole system rather than on individual functors.This is a very interesting question that we plan to investigate in the future. Remark 4.37 It is clear from the examples above that spherical functors give a unifyingframework for handling different type of geometric pairs.Suppose that X and Y are smooth and compact nc -spaces, Y is a nc -Calabi-Yau, f : X → Y is a spherical map, and the degeneration conjecture holds for both X and Y . In thissituation we expect that the deformation theory of f : X → Y is controlled by a homotopyabelian d( Z / L ∞ -quasi-isomorphic to(4.4.1) cone (cid:18) C • ( Y ) f ! / / C • ( X ) (cid:19) [1 − d ] . f ∗ (or f ! ) we can build a new nc -space Z by taking C Z to be the semi-orthogonal extension C Z = h C X , C Y i , where we setHom Z ( C Y , C X ) := 0Hom Z ( E , F ) := Hom Y ( f ∗ E , F ) for all E ∈ C X , F ∈ C Y . We expect that the deformation theory of f : X → Y is equivalent to the deformation theoryof Z and in particular that the L ∞ algebra C • ( Z ) is quasi-isomorphic to the algebra (4.4.1). Remark 4.38 We should point out that even though deformation quantization provides aconceptual bridge between the categorical framework and the geometric framework of theprevious section, the actual connection between the two frameworks is tenuous at best. Thesource of the problem lies in the fact that the deformation quantization of general Poissonmaps can be obstructed [Wil07]. A -model framework: symplectic Landau-Ginzburg models We already noted in Examples 4.4 and 4.8 that there are natural canonical coordinatesand a Calabi-Yau variation of nc -Hodge structures that one can attach to the A -model on acompact symplectic manifold. An interesting open problem is to find an algebraic descriptionof these coordinates and variation in terms of some d ( Z / g Batalin-Vilkovisky algebra thatis naturally attached to the Fukaya category. This question is hard and we will not studyit directly here. Instead we will look at the question of finding canonical coordinates andvariation in another symplectic context, i.e. for symplectic Landau-Ginzburg models, andtry to get an insight into a possible algebraic formulation in that case. It will be interesting tocompare our formalism with the recent work of Fan-Jarvis-Ruan [FJR07] on the symplecticgeometry of quasi-homogeneous Landau-Ginzburg potentials with isolated singularities butat the moment we do not see a direct relationship. The objects we would like to understand are triples ( Y, w , ω ), where • Y is a C ∞ -manifold and ω is a C ∞ -symplectic form on Y .107 w : Y → C is a proper C ∞ -map such that there exists an R > { z ∈ C | | z | ≥ R } the map w is a smooth fibration with fibers symplectic submani-folds in ( Y, ω ).Similarly to the case of compact symplectic manifolds one can associate Gromov-Witteninvariants to such a geometry. Specifically, if we fix n ≥ g ≥ 0, and β ∈ H ( Y, Z ), then wecan use stable pseudo-holomorphic pointed curves in Y to define a natural linear (correlator)map I (1) g,β,n − : H • ( Y, Q ) ⊗ ( n − ⊗ H • (cid:0) M g,n , Q (cid:1) / / H • ( Y, Q ) . Indeed, note that Poincar´e duality gives an identification H • ( Y, Q ) ∼ = H • ( Y, Y R ; Q )[ − dim Y ] , where R > Y R = w − ( { z ∈ C | | z | ≥ R } ) ⊂ Y . Combining this identifi-cation with the isomorphism ( H • ) ∨ = H • we see that I (1) g,β,n − will be given by a class in H • ( Y, Q ) ⊗ ( n − ⊗ H • ( Y, Y R ; Q ) ⊗ H • ( M g,n , Q ).Next consider the usual moduli stack M g,n ( Y, β ) of stable pseudo-holomorphic maps.Here it will be convenient to assume that an almost-complex structure on Y tamed by ω ischosen in such a way that w | Y R is holomorphic. The stack M g,n ( Y, β ) is non compact but nearinfinity it parameterizes only pseudo-holomorphic maps ϕ : C → Y such that w ◦ ϕ : C → C is constant and w ◦ ϕ ( C ) ∈ C is close to infinity. Thus the virtual fundamental class of M g,n ( Y, β ) is well defined as a class in the relative homology (cid:2) M g,n ( Y, β ) (cid:3) vir ∈ H • (cid:0) Y n × M g,n , Y n − × Y R × M g,n ; Q (cid:1) = H • ( Y, Q ) ⊗ ( n − ⊗ H • ( Y, Y R ; Q ) ⊗ H • ( M g,n , Q ) . We define I (1) g,β,n − to be the map given by the relative virtual fundamental class (cid:2) M g,n ( Y, β ) (cid:3) vir .This collection of correlates satisfies analogues of the usual axioms of a cohomologicalfield theory [KM94] but we will not discuss them here. Consider now a cohomology class x = ( x , x =2 ) ∈ H • ( Y, C ) = H ( Y, C ) ⊕ H =2 ( Y, C ) , where H • ( Y, C ) is viewed as a supermanifold over C .Now for every such x we define a quantum product • ∗ x • : H • ( Y, C ) ⊗ H • ( Y, C ) / / H • ( Y, C )108y the formula α ∗ x α := X m ≥ X β ∈ H ( Y, Z ) exp ( h β, x i ) · m ! I (1) g,β,m +1 (cid:0) ( α ⊗ α ⊗ x =2 ⊗ · · · ⊗ x =2 | {z } m times ) ⊗ M ,m +1 (cid:1) . Now this quantum multiplication together with the usual formulas (see Examples 4.4 and 4.8)can be used to define a decorated variation of nc -Hodge structures over the (conjecturallynon-empty) domain in H • ( Y, C ) where the series defining ∗ x is absolutely convergent. Remark 4.39 There are some interesting variants of this construction. For instance we cantake a symplectic manifold ( Y, ω ) with no potential and a pseudo-convex boundary. In thissituation M g,n ( Y, β ) is already compact, as long as β = 0. Also in a symplectic Landau-Ginzburg model ( Y, ω, w ) we can allow for w to be non-proper and instead require that itsfibers have pseudo-convex boundary. Finally one can consider a symplectic Y equipped witha proper map Y → C k , holomorphic at infinity and with k ≥ Let ( Y, ω, w ) be a symplectic geometry with a proper potential. There are two naturalcategories that we can attach to this geometry: the Fukaya category of the general fiber of w , and the Fukaya-Seidel category of w . Understanding the structure properties of thesecategories or even defining them properly is a difficult task which requires a lot of effort andhard work. We will not explain any of these intricate details but will rather use the Fukayaand Fukaya-Seidel categories as conceptual entities. For details of the definitions and arigorous development of the theory we refer the reader to the main sources [FOOO07, FO01],[Sei07b, Sei07a]. The categories that we are interested in are: (1) The Fukaya-Seidel category FS ( Y, ω, w ) of the potential w has objects which are unitarylocal systems V on (graded) ω -Lagrangian submanifolds L ⊂ Y such that: • w ( L ) ⊂ (compact) ∪ R ≤ ; • The restriction of L over the ray R ≤ is a fibration on R ≤− R and when z ∈ R ≤ , and z → −∞ , we have that the fiber L z ⊂ Y z is a Lagrangian submanifold in the symplecticmanifold (cid:0) Y z , ω | Y z (cid:1) . 109he morphisms between two objects ( L , V ) and ( L , V ) are defined as homomorphismsbetween the fibers of the local systems at the intersection points of the two Lagrangians. Asusual to make this work one has to perturb one of the Lagrangians, say L by a Hamiltonianisotopy to ensure transversality of the intersection. A new feature of this setup (comparedto the situation of symplectic manifolds with no potential) is that the allowable isotopiesare tightly controlled - they correspond to small wiggling, see Figure 7, of the tail of thetadpole-like image of the Lagrangian in C and a lift of this wiggling to Y given by a non-linearsymplectic connection identifying the fibers of Y . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) L L Figure 7: Tadpole-like w -images of two Lagrangian submanifolds.The compositions of morphisms are given by correlators counting pseudo-holomorphic discswhose boundary is contained in the given Lagrangian submanifolds. (2) The Fukaya category Fuk ( Y z ) of a fiber (cid:0) Y z , ω | Y z (cid:1) over a point z ∈ C which is not a criticalvalue for w . The objects in this category are again pairs consisting of (graded) Lagrangiansubmanifolds in Y z equipped with unitary local systems, and morphisms and compositionsare defined again by maps between the fibers of the local systems at the intersection pointsand by counting discs. The parallel transport w.r.t. a non-linear symplectic connection on w : Y → C identifies symplectically all fibers (cid:0) Y z , ω | Y z (cid:1) over points z ∈ R ≤ when z → −∞ .We will denote any one such fiber as ( Y −∞ , ω −∞ ).Now observe that by intersecting a Lagrangian L ⊂ Y with the fiber Y −∞ we get an assign-ment L L −∞ := L ∩ Y −∞ . We expect that this assignment can be promoted to a sphericalfunctor (see also [Sei07a] for a similar discussion) F : FS ( Y, w , ω ) / / Fuk ( Y −∞ , ω −∞ )110o that the associated spherical twist R F : Fuk ( Y −∞ , ω −∞ ) → Fuk ( Y −∞ , ω −∞ ) categorifiesthe monodromy around the circle { z ∈ C | | z | = R } .In this situation one can also define relative Gromov-Witten invariants J (1) g,β,n − : H • ( Y, Y −∞ ; Q ) ⊗ H • ( Y, Q ) ⊗ ( n − ⊗ H • (cid:0) M g,n , Q (cid:1) / / H • ( Y, Y −∞ ; Q ) . For we again use the duality ( H • ) ∨ ∼ = H • and the Poincar´e duality H • ( Y, Y −∞ ; Q ) ∼ = H • ( Y, Y + ∞ ; Q ) to rewrite J (1) g,β,n − as a class in H • ( Y, Y −∞ ; Q ) ⊗ H • ( Y, Y + ∞ ; Q ) ⊗ H • ( Y, Q ) ⊗ ( n − ⊗ H • (cid:0) M g,n , Q (cid:1) . This class can again be defined as a virtual fundamental class space M g,n ( Y, β ) of stablepseudo-holomorphic maps. Again we can interpret the virtual class as a relative homologyclass: (cid:2) M g,n ( Y, β ) (cid:3) vir ∈ H • (cid:0) Y n × M g,n , Y n − × (cid:0) ( Y − R,ε × Y ) ∪ ( Y × Y + R,ε ) (cid:1) × M g,n ; Q (cid:1) = H • ( Y, Y − R,ε ; Q ) ⊗ H • ( Y, Y + R,ε ; Q ) ⊗ H • ( Y, Q ) ⊗ ( n − ⊗ H • (cid:0) M g,n , Q (cid:1) = H • ( Y, Y −∞ ; Q ) ⊗ H • ( Y, Y + ∞ ; Q ) ⊗ H • ( Y, Q ) ⊗ ( n − ⊗ H • (cid:0) M g,n , Q (cid:1) , and so it gives the desired map J (1) g,β,n − . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(c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D ε Figure 8: The domain D ε .Here 1 ≫ ε > 0, and Y ± R,ε = w − ( ± D ε ), where D ε ⊂ C is the domain given by (seeFigure 8) D ε := (cid:26) z ∈ C (cid:12)(cid:12)(cid:12)(cid:12) | z | ≥ R and Arg z ∈ (cid:18) π − ε, π ε (cid:19) (cid:27) . J (1) g,β,n − give rise to a quantum multiplication and through theusual formulas from Examples 4.4 and 4.8 we again get a decorated variation of nc -Hodgestructures over a (conjecturally non-empty) domain in H • ( Y, C ) with fiber H • ( Y, Y −∞ ). In conclusion we systematize all the objects introduced above in a mirror table (see also[Aur07]) describing the corresponding A and B -model entities in parallel:112nvariants A -model B -modelgeometry a triple ( Y, w , ω ) where: w : Y → C is a proper C ∞ -map ( Y, ω ) is symplectic with c ( T Y ) = 0 a pair Z ⊂ X where: X is smooth projective, and Z ⊂ X is a smooth anticanon-ical divisorcohomology H • ( Y, C ) H • ( Y, Y −∞ ; C ) H • ( Y −∞ , C ) variations of nc HS overa domain in H • ( Y, C ) H • ( X − Z, C ) H • ( X, C ) H • ( Z, C ) variations of nc HS overa domain in H • ( X − Z, C )categories Fuk ( Y −∞ ) FS ( Y ) F O O : Fuk ( Y −∞ ) is aCY category and F is a sphericalfunctor D ( Z ) D ( X ) F O O : D ( Z ) is a CYcategory and F is a sphericalfunctorThe part of FS ( Y ) consisting ofLagrangians fibered overwhere the circle is of radius R ≫ D supp Z ( X ) : a full compact (nonsmooth) subcate-gory in D ( X )The part of FS ( Y ) consistingof compact Lagrangian sumanifoldsin Y D compactsupport ( X − Z ) : a full compact(non smooth)subcategory in D ( X )The wrapped FS category: the Homspace between ( L , V ) and ( L , V )is the sum of Hom( V , V ) x , x ∈ L ∩ L , and L is deformed so that w ( L ) becomes a spiral:wrapped infinitely many times D ( X − Z ) : a smooth (non com-pact) category References [AKO04] D. 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