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Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models

In this paper we prove the smoothness of the moduli space of Landau-Ginzburg models. We formulate and prove a Tian-Todorov theorem for the deformations of Landau-Ginzburg models, develop the necessary Hodge theory for varieties with potentials, and prove a double degeneration statement needed for the unobstructedness result. We discuss the various definitions of Hodge numbers for non-commutative Hodge structures of Landau-Ginzburg type and the role they play in mirror symmetry. We also interpret the resulting families of de Rham complexes attacted to a potential in terms of mirror symmetry for one parameter families of symplectic Fano manifolds and argue that modulo a natural triviality property the moduli spaces of Landau-Ginzburg models posses canonical special coordinates.


Introduction
In this paper we study the local structure of the moduli space of complex Landau-Ginzburg models. Such a Landau-Ginzburg model is determined by a pair (Y, w), where Y is a complex quasi-projective variety, and w : Y → A 1 is a holomorphic function on Y . Our main objective is to prove the unobstructedness of the deformations of (Y, w) in the case when Y has a trivial canonical class K Y ∼ = O Y . At a first glance, such a statement is not likely to hold since the non-compactness of Y will often cause the moduli space of the pair (Y, w) to be infinite dimensional and to have a complicated and unwieldy local behavior in general.
Before we adress this difficulty it is useful to look at the model example provided by the classical unobstructedness statement for the deformations of compact Calabi-Yau varieties. This statement was proven by different methods by Bogomolov [Bog81], Tian [Tia87], and Todorov [Tod89]. Recall from [Bog81], [Tia87] and [Tod89], that if X is a smooth compact Calabi-Yau manifold of dimension dim C X = d, then the (formal) versal deformation space M X of X is smooth and of dimension h d−1,1 (X). Moreover a choice of a splitting of the Hodge filtration on H d DR (X, C) defines an analytic affine structure on M X . This theorem has many variants establishing the unobstructedness of deformations of log Calabi-Yau varieties or Deligne-Mumford stacks, or of weak Fano varieties or Deligne-Mumford stacks, see e.g. [Ran92,Kaw92,Man04,KKP08,IM13,Iac13,San13]. The log version of the Bogomolov-Tian-Todorov theorem suggests that if we want to attain a good control of the deformations of a Landau-Ginzburg model (Y, w), we should look at a nice, e.g. log Calabi-Yau, compactification Z of Y and consider only deformations that fix the boundary divisor D Z = Z − Y .
To streamline this discussion it will be convenient to distinguish notationally the varying and the fixed parts in any given deformation problem. Our convention in that regard will be that when the deformations of some collection of geometric data are studied, the moving part of the data will be listed in parentheses, while the part of the data that is kept fixed will be listed in a subscript. Thus when we say that we are analyzing the deformations of (Z, f) D Z , we mean that we consider deformations of the pair (Z, f) together with compatible trivial deformations of the divisor D Z .
In this framework prove the following unobstructedness result Theorem A Let Z be a smooth projective variety, f : Z → P 1 a flat morphism, and D Z ⊂ Z a reduced anti-canonical divisor with strict normal crossings. Assume moreover that crit(f) does not intersect the horizontal part of D Z , and that the vertical part of D Z coincides with the scheme theoretic fiber f −1 (∞) of f over ∞ ∈ P 1 . Then the versal deformation space This theorem can be viewed as an unobstructedness result for the Calabi-Yau Landau- Indeed, the theorem asserts that if (Y, w) admits a compactification (Z, f) with normal crossing boundary D Z , then the deformations of (Y, w) that are "anchored at infinity", i.e. the deformations of the compactification that keep the boundary fixed, are unobstructed. To prove Theorem A we identify the L ∞ -algebra that controls the deformation theory of (Z, f) D Z and show in Theorem 2.10 that this L ∞ -algebra is homotopy abelian. We argue that, as in the case of compact Calabi-Yau manifolds, the latter statement can be reduced to a Hodge theoretic property: the double degeneration property for the Hodge-to-De Rham spectral sequence associated with the complex of f-adapted logarithmic forms (see Definition 2.11). This double degeneration is then established in Theorem 2.18.
The setup and conclusion of Theorem A are natural from the point of view of mirror symmetry. To elaborate on this, note first that a Landau-Ginzburg pair (Y, w) as above will typically arise as the mirror of a symplective manifold (X, ω X ) underlying a projective Fano variety. Now the homological mirror symmetry conjecture predicts that the Fukaya category Fuk(X, ω X ) of (X, ω X ) will be equivalent to the category MF(Y, w) of matrix factorizations of the potential w : Y → A 1 . In particular the deformation theories of the Fukaya category and of the category of matrix factorizations will be identified. The heuristics motivating Theorem A comes from the comparison of the corresponding moduli spaces. The versal deformation space of the Fukaya category is manifestly smooth since it is an open cone in the space of harmonic 2-forms on X. Thus mirror symmetry predicts that the versal deformation space of the category of matrix factorizations will also be smooth. Next recall [Orl04,Orl05,Orl12] that MF(Y, w) is the coproduct λ∈crit w D b sing (Y λ ) of the categories of singularities of the singular fibers of w. This interpretation indicates that flat deformations of the geometric data (Y, w) will not necessarily give rise to flat deformations of MF(Y, w).
Indeed, when we deform (Y, w) geometrically, the singularities of fibers of w can coalesce and more importantly can run away to infinity. This will happen for instance if we deform a compactification of (Y, w) so that some interior singular fiber gets absorbed in the fiber at infinity.
Because of this phenomenon we will have flat families of Landau-Ginzburg models which give us families of categories of matrix factorizations whose periodic cyclic homologies jump. This suggests that we should only consider geometric deformations of (Y, w) that are anchored at infinity. Indeed, if ((Z, f), D Z ) is a compactification of (Y, w), then the deformations of (Z, f) that fix the boundary divisor D Z will give deformations of (Y, w) without jumps in the global vanishing cohomology. In this setting the corresponding categories of matrix factorizations will move in a flat family and we expect that the deformations of a compactification with a fixed boundary will provide enough parameters to cover the full versal deformation space M MF(Y,w) of the category MF(Y, w). In fact, in the process of proving the double degeneration property Theorem 2.18 we will check that, under the hypothesis of Theorem A, the natural map of versal deformation spaces M (Z,f) D Z → M MF(Y,w) isétale. More precisely, it is not hard to see that the composition of the isomorphism constructed in Lemma 2.21 with Efimov's comparison isomorphism [Efi12] can be identified with the differential of the map M (Z,f) D Z → M MF(Y,w) at the closed point. In particular this differential is an isomorphism and so the map isétale. Altogether this heuristic reasoning explains why the unobstructedness of the deformation theory of ((Z, f), D Z ) is indeed the expected behavior.
Since the compactified Landau-Ginzburg model ((Z, f), D Z ) plays a central role in the above heuristics it is natural to expect that this compactification should also have a mirror interpretation. A closer look at the associated Hodge/de Rham data and the double degeneration property of ((Z, f), D Z ) suggests that the mirror of ((Z, f), D Z ) is an anticanonical pencil on the symplectic manifold (X, ω X ). In Section 3.2 we discuss this mirror picture in detail and compare the Hodge theoretic data appearing on the two sides of this extended mirror correspondence. We use this analysis to explain how the commutative pure Hodge structure of the compactified Landau-Ginzburg model arising from the double degeneration property can be reconstructed from the non-commutative Hodge structure of the original Landau-Ginzburg model (Y, w). Through the extended mirror symmetry picture we rewrite this reconstruction process for the Fano mirror and use the resulting structure to propose a definition of Hodge numbers for X which is formulated entirely in symplectic terms.

Moduli of Landau-Ginzburg models
In this section we study variations of pure nc Hodge structures that arise from universal families of Landau-Ginzburg models. We focus on the components of the universal variation that encode geometric properties of the Landau-Ginzburg moduli space and investigate the Hodge theoretic input in the Landau-Ginzburg deformation theory.
The relevant class of Landau-Ginzburg models appears naturally in the context of mirror symmetry for Fano manifolds. Since this context is a primary source of examples for us, we recall it next.

Mirrors of Fano manifolds
Mirror symmetry is a duality that identifies seemingly different two dimensional supersymmetric quantum field theories. Geometrically such theories arise as linear or non-linear sigma models with Kähler targets, or as Landau-Ginzburg models with targets given by Kähler manifolds equipped with holomorphic superpotentials. The mirror map matches the target geometries that produce mirror symmetric models into mirror pairs. Typically a sigma model or a Landau-Ginzburg model with a given target geometry admits two topological twists -the A and B twists -each of which gives rise to a category of boundary field theories or D-branes (see e.g. [HKK + 03, ABC + 09]). According to the homological mirror conjecture from [Kon95], the mirror correspondence can be generalized to an identification of the categories of boundary field theories. Specifically the homological mirror conjecture predicts that in a mirror pair, the category of A-branes for one side of the pair must be equivalent to the category of B-branes for the other side. Such an equivalence induces non-obvious isomorphisms between the various invariants that one can extract from the categories. In particular we get a conjectural matching of the cohomology of the two categories; matching of the nc Hodge structures on the cohomology of the two categories; matching of the deformation spaces of the two categories; and matching of the natural variations of nc motives over these deformation spaces. We will exploit these conjectural identifications to deduce interesting predictions for the properties of the moduli spaces and the Hodge theory of the requisite geometric backgrounds and will eventually prove these predictions directly. To set things up we begin by recalling the basic geometric framework for the mirror correspondence.
We will indicate that two geometries (X, · · · ) and (Y, · · · ) are mirror equivalent by writing (X, · · · ) | (Y, · · · ). Mirror pairs of geometries fall naturally into three classes: mirror pairs of Calabi-Yau, Fano, and general type. Here we will discuss in detail only the mirror pairs of Fano type.
By definition a mirror pair of Fano type is a pair where: • X is a projective Fano manifold; • (Y, w) is a holomorphic Landau-Ginzburg model consisting of a quasi-projective Calabi-Yau manifold Y with dim C Y = dim C X = n, and a surjective algebraic function w : Y → A 1 with a compact critical locus crit(w) ⊂ Y ; • ω X ∈ A 2 C (X) and ω Y ∈ A 2 C (Y ) are (complexified) Kähler forms on X and Y ; • s X ∈ H 0 (X, K −1 X ) is an anti-canonical section of X, and vol Y ∈ H 0 (Y, K Y ) is a trivialization of the canonical bundle of Y , i.e. a holomorphic volume form on Y ; The anti-canonical section s X ∈ H 0 (X, K −1 X ) defines a Calabi-Yau hypersurface D X = divisor(s X ) ⊂ X and a nowhere vanishing section s X|X−D X ∈ H 0 (X − D X , K −1 X ). We will write vol X−D X = 1/s X for the corresponding holomorphic volume form on X − D X .
Remark 2.1 Mirror pairs of Fano type can be qualified/refined in different ways: (i) Requiring that D X is smooth is mirrored by the requirement that w is proper.
(ii) Requiring that D X has strict normal crossings is mirrored by the requirement that the fibers of w are Zariski open subsets in projective (n − 1)-dimensional Calabi-Yau manifolds.
It is helpful to examine the shape of the geometry of a mirror pair in examples. Many explicit and detailed descriptions of mirror pairs of Fano type are discussed in e.g. [Giv98, HV00, AKO08, AKO06, Abo09]. Here we just briefly recall Givental's picture [Giv98] of mirrors of projective spaces.
Example 2.2 The first instance of a Fano type mirror pair was described by Givental [Giv98]. In the most basic setting X = P n is a projective space with homogeneous coordinates u 0 , . . . , u n , ω X is the Fubini-Studi form, D X ⊂ P n is the union of the (n + 1) coordinate hyperplanes, and s X is given by the product of the homogeneous coordinate functions. On the mirror side Y = (C × ) n is an n-dimensional affine torus with coordinates z 1 , . . . , z n , the potential w : Y → A 1 is given by the symplectic form is the point a is the the point at infinity, i.e. a = ∞, and the holomorphic volume form is If we change the setting so that on the left hand side of the pair D X is not the toric divisor of P n but rather is a smooth Calabi-Yau hypersurface, the mirror Y is a partial compactification of the torus so that w becomes a proper map with n − 1 dimensional Calabi-Yau fibers.
Accordingly the symplectic form ω Y and holomorphic volume form vol Y have to be extended to the compactification.
The mirror correspondence gives a non-trivial matching [HKK + 03] of the various ingredients of the mirror pair. The complexified Kähler structure ω X is identified with a combination of the complex structure on Y , the potential w, and the volume form vol Y . In the other direction ω Y is identified with a combination of the complex structure on X and the section s X [HV00, HKK + 03].
A Fano type mirror pair gives rise to a pair of mirror non-compact Calabi-Yau manifolds: Under a convergence assumption on the quantum product on X the category of A-branes for the background (X, ω X , s X ) can be identified with the Fukaya category Fuk(X, ω X ) of the symplectic manifold underlying the Fano variety X [ABC + 09]. Fuk(X, ω X ) is a C-linear A ∞ category which is only Z/2-graded [FOOO09a,FOOO09b]. The category of B-branes associated with (X, ω X , s X ) is identified with a dg enhancement of the bounded derived category D b (X) of coherent sheaves on X. We will write D b (X) for this Z-graded C-linear dg category. There are many choices for D b (X), e.g. the homotopy category of complexes of injective O X -modules with coherent cohomology, or Block's category of graded C ∞ complex vector bundles on X with (0, •) superconnections [Blo10]. By a theorem of Lunts and Orlov [LO10] all dg enhancements of D b (X) are quasi-equivalent so one can work with any of those enhancements. By definition D b (X) depends only on the complex structure on X and is independent of the complexified Kähler structure ω X or the section s X . Both D X and s X are of course essential for defining the associated Calabi-Yau pair and its categories of branes.
On the right hand side of the mirror pair the definition of the categories of branes is modified to incorporate the potential w. The category of A-branes associated with the the function w, and on the complexified symplectic structure ω Y while the Z-graded version is a d(Z/2)graded C-linear category which depends only on the complex structure of Y and on the holomorphic function w.
Homological mirror symmetry now predicts several conjectural equivalences of categories of branes for the Fano mirror pair and for the associated Calabi-Yau mirror pair. These equivalences are summarized in Table 1. In this table D b c denotes a dg enhancements of the derived categories of coherent sheaves with compact support and Fuk wr denotes the wrapped version of the Fukaya category [AS10,Abo12]. Additionally, our convention is that whenever the notation for a Fukaya or a Fukaya-Seidel category includes a holomorphic volume form, the objects of this category are graded spin Lagrangians or Lagrangian thimbles, and so the category is Z-graded. In particular, aside from Fuk(X, ω X ) and MF(Y, w) all categories appearing in Table 1 are Z-graded. Table 1 will respect the natural additional structures on these categories of branes. In particular mirror symmetry will respect the natural decompositions of these categories. It is known from the work of Orlov [Orl04,Orl05,Orl12] that the category of matrix factorizations decomposes

Remark 2.3 (i) Homological mirror symmetry predicts that the equivalence
into a sum of categories of singularities of the singular fibers of w.
Similarly (see [KKP08]) the Fukaya category of the Fano manifold X decomposes Fuk(X, ω X ) λ corresponding to the eigenvalues of quantum multiplication 1 with c 1 (T X ) on H • (X, C).
(ii) The equivalences of the categories of branes listed in Table 1 induce respective mirror identifications of cohomology groups. For future reference we collect these identifications in Table 2 below. In this table Y sm ⊂ Y denotes a smooth fiber of w : Y → A 1 taken "near infinity" as explained in [KKP08, Section 4.5.2(2)].
(iii) The B-to-A homological mirror correspondence in Table 1 can be extended to one more case. Let Y −∞ denote the fiber w −1 (z) over z ∈ C with Re z ≪ 0. We will also write ω −∞ for the restriction ω Y |Y −∞ of the symplectic form, and vol −∞ for the induced holomorphic 1 By assumption we are working here with a convergent version of the quantum product * q . The value of q corresponding to the particular complex structure on Y under the mirrror map is normalized in the flat coordinates to be q = 1. This is why we use the * 1 quantum product in the decomposition above.

B-brane charges
Table 2: Matching of cohomology for a mirror pair volume form on the fiber. The parallel transport for the Erhesmann symplectic connection on w : Y → A 1 . identifies symplectically all fibers of w over points z ∈ A 1 with Re z ≪ 0.
So the dg category Fuk(Y −∞ , ω −∞ , vol −∞ ) is well defined up to quasi-equivalence. Now, we can supplement Table 1 by the statement that the category of perfect complexes (= the category of topological B branes) on the Calabi-Yau variety D X is quasi-equivalent to the Fukaya category ( = the category o f A-branes) on the fiber Y −∞ : Again this induces an identification of the associated brane charges, i.e. of the periodic cyclic homologies of the two categories. Since HP • (Perf(D X )) ∼ = H • (D X , C) and conjecturally Note that it is not clear how to extend the A-to-B homological mirror correspondence in a similar manner. If on the A-side we consider the symplectic data D X , ω X|D X , vol D X , there is no obvious complex fiber Y c of w : Y → A 1 for which we can hope to get a quasi-equivalence . The problem is that the mirror of D X , ω X|D X , vol D X is normally understood in terms of the large volume degenration, so for an A-to-B mirror statemnt we will need to understand the large complex structure degeneration of w : Y → A 1 .

Families of Landau-Ginzburg models
As we explained in Section 2.1 the mirror of a symplectic manifold underlying a Fano variety is a quasi-projective Landau-Ginzburg model ((Y, w), vol Y ) equipped with a holomorphic volume form. Such Landau-Ginzburg models admit a natural class of compactifications. where: (a) Z is a smooth projective variety and f : Z → P 1 is a flat projective morphism.
i is a flat projective morphism; -the critical locus crit(f) ⊂ Z does not intersect D h .
(c) vol Z is a meromorphic section of K Z with no zeroes and with poles at most at D Z , i.e.
w : Y → A 1 is defined to be the restriction w := f |Y of f to Y , and vol Y := vol Z|Y is the restriction of vol Z to Y . The condition on vol Z ensures that Y is a quasi-projective variety with a trivial canonical class and that vol Y is a holomorphic volume form on Y . The condition that the critical locus of f does not intersect the horizontal part of D Z in particular implies that crit(w) is proper and so ((Y, w), vol Y ) is exactly the type of Landau-Ginzburg model that we considered in the previous section.
In addition we will often require that the datum ((Z, f), D Z , vol Z ) satisfies the following tameness assumption which bounds the orders of poles of vol Z and f along D Z : for all i and j.

Remark 2.5 (i)
The assumption that vol Z has poles of order exactly 1 along all the components of D Z in particular implies that the reduced divisor D Z is an anti-canonical divisor on Z and so (Z, D Z ) is a log Calabi-Yau pair.
Note also that if we start with a Calabi-Yau quasi-projective Landau-Ginzburg model ((Y, w), vol Y ) and we choose a smooth normal-crossing compactification f : Z → P 1 of w : Y → A 1 , then the condition that the holomorphic volume form has a first order pole along the divisor at infinity is a tight constraint which is rather unnatural since it is not invariant under semi-stable reduction. Nevertheless this condition is often satisfied in mirror symmetry examples and so it is not unreasonable to impose.
(ii) The assumption that w has first order poles at the vertical boundary divivsor D v , i.e.
that the scheme theoretic fiber f −1 (∞) is actually reduced, is more natural and can be justified by mirror symmetry consderations.
Indeed, if f −1 (∞) = j m j D v j , then by Landman's theorem [Lan73] we know that the least common multiple m of the m j 's is the order of the semi-simple part of the local monodromy transformation around infinity. Concretely, choose a small disk ∆ ⊂ P 1 centered at ∞ ∈ P 1 and such that ∞ is the only critical value of f in ∆. Fix a base point c 0 ∈ ∂∆, and orient ∂∆ with the orientation on ∆. Consider the monodromy transformation , ω Y , vol Y )) is the mirror of a Fano datum (X, ω X , s X ), the mirror equivalence (see Table 1) FS((Y, w), ω Y , vol Y ) ∼ = D b (X) identifies the monodromy auto-equivalence with the the Serre functor induces multiplication with exp ((−1) n c 1 (K X )). In other words T is a unipotent operator, and so we must have m = 1. For future reference note that this mirror symmetry description also predicts that under the identification H • (X, becomes identified with the logarithm of monodromy: (iii) It is possible and useful to allow for D v to be a smooth divisor, i.e. for ∞ not to be a critical value of f. Such Landau-Ginzburg models arise naturally as mirrors of quasi-Fano varieties and can be studied in the same manner.
Our goal is to understand the moduli spaces of compactified complex Landau-Ginzburg models satisfying the tameness asumption. If such a model ((Z, f), D Z , vol Z ) compactifies the mirror of a Fano datum (X, ω X , s X ), then its moduli space will be identified with the symplectic moduli of (X, ω X ) and in fact will look like a conical open subset in H 2 (X, C) ⊕ H 0 (X, C). In particular, when ((Z, f), D Z , vol Z ) arises from a mirror situation, we expect its moduli space to be smooth. This motivates the following purely algebraic-geometric statement Theorem 2.6 Let ((Z, f), D Z , vol Z ) be a compactified Landau-Ginzburg model satisfying the tameness assumption (T) and the assumption H 1 (Z, Q) = 0. Then the deformation theory Remark 2.7 The requirement that H 1 (Z, Q) = 0 is a technical requirement that simplifies the Teichmüller theory of Z. It is very likely unnecessary but we will not pursue this here.
Theorem 2.6 extends the classical unobstructedness results of Bogomolov [Bog79,Bog81], Tian [Tia87], and Todorov [Tod89] to the setup of log Calabi-Yau varieties with potentials. It also gives the following more direct generalization of Bogomolov-Tian-Todorov unobstructedness: Corollary 2.8 Let Z be a smooth projective variety of dimension n satisfying H 1 (Z, Q) = 0 and such that the anti-canonical linear system on Z gives a flat projective morphism f : Z → P 1 . Then the deformation theory of Z is unobstructed.
Proof. By assumption the variety Z determines the morphism f : Z → P 1 . Let D Z ⊂ Z be a smooth anti-canonical divisor in Z. and let vol Z be a trivialization of K Z (D Z ). Before we proceed with the proof of Theorem 2.6 we will need to establish some general facts about the deformation theory of varieties with potentials. so the versal deformation space of ((Z, f) D Z , vol Z ) is a principal C × -bundle over the versal deformation space of (Z, f) D Z . Therefore it suffices to prove the unobstructedness of the deformation theory of (Z, f) D Z .

Deformations of compactified Landau-Ginzbug models
As usual the deformation theory of (Z, f) D Z is controlled by an L ∞ algebra [KS05, Hin01, KS09,Lur11]. By standard Kodaira-Spenser theory the deformations of the map f : Z → P 1 are computed [Ill71,Ill72,Hor74,Ser06] by the sheaf of dg Lie algebras ].
Here f * T P 1 denotes the O-module pullback, and this is an complex of locally free coherent sheaves with an O Z -linear differential and C-bilinear (graded) Lie bracket.
Similarly the deformations of f : Z → P 1 which preserve the boundary divisor D Z are computed by the sheaf of dg Lie algebras where for any smooth variety M and any closed reduced subscheme S ⊂ M we write T M,S for the coherent sheaf of vector fields on M that are tangent to S at the points of S. Since D Z ⊂ Z and {∞} ⊂ P 1 are reduced normal crossings divisors, it follows that T Z,D Z ⊂ T Z and T P 1 ,∞ ⊂ T P 1 are locally free subsheaves.
Recall (see e.g. [GM90,KS05,Man04]) that the unobstructedness of the deformation theory defined by an L ∞ algebra follows from the stronger property that this L ∞ algebra is homotopy abelian. Therefore Theorem A and Theorem 2.6 will follow immediately from the following Theorem 2.10 Suppose ((Z, f), D Z , vol Z ) is a compactified Landau-Ginzburg model satisfying the tameness condition (T). Then the L ∞ algebra is homotopy abelian.
As in the compact Calabi-Yau case we will deduce Theorem 2.10 from a Hodge theoretic statement -the degeneration of a "Hodge-to-de Rham spectral sequence" associated with the divisor D Z and the potential f. Our main tool here is a new complex of logarithmic forms adapted to f: Thus where f is viewed as a meromorphic function on Z and df is viewed as a meromorphic one form.
The sheaves Ω a Z (log D Z , f) have several interesting properties. As a first remark we have the following and so is locally free.
Suppose next p ∈ D v . We can find local analytic coordinates z 1 , . . . , z n centered at p so that in a neighborhood of p: • the divisor D v is given by k i=1 z i = 0, the divisor D h is given by k+l 1=k+1 z i = 0; • the potential f is given by f(z 1 , . . . , z n ) = 1 Since df only has poles at the components of D v , the condition that a logarithmic form and so given any ν ∈ ∧ p V , we can write ν and df ∧ ν uniquely as We have df = f · d log f. The logarithmic 1-form d log f also decomposes as In particular we can solve for η in terms of ψ and β. The condition that ν is f-adapted is simply the condition that ϕ ∈ ∧ p+1 W and ψ ∈ ∧ p W . But for any ψ ∈ ∧ p W and any In other words ν is f-adapted if and only if we can find a form ψ ∈ ∧ p W and a form This shows that the subsheaf This completes the proof of part (a) of the lemma. Part (b) follows immediately from the formula (2.3.1), the description of the exceptional divisor of ε : Z → Z as a projectivized normal bundle, and the Euler sequence of this projective bundle.

Remark 2.13
The f-adapted logarithmic forms are equipped with two natural differentials of degree one: • the de Rham differential d : Note that by definition the differential df∧ is O Z -linear, while the de Rham differential satisfies the Leibnitz rule as usual. Note also that for any complex numbers c 1 and c 2 the linear combination c 1 d + c 2 df∧ is also a differential and so we get a family of complexes of f-adapted logarithmic forms The previous discussion connects directly to the L ∞ -algebra RΓ(Z, g • ) since in the Calabi-Yau case we can use the holomorphic volume form to convert f-adapted logarithmic forms to poly-vector fields. Suppose ((Z, f), D Z , vol Z ) is a compactified Landau-Ginzburg model. The contraction with the meromorphic volume form gives a map of O Z -modules Furthermore when ((Z, f), D Z , vol Z ) satisfies the tameness condition (T), the explicit description of the local frames of Ω n−a is a locally free subsheaf of maximal rank in ∧ a T Z , and that ι vol Z induces an isomorphism With this notation we now have the following: Lemma 2.14 Let ((Z, f), D Z , vol Z ) be a compactified Landau-Ginzburg model satisfying the tameness assumption (T). Then the subsheaf Proof. Recall that the Nijenhuis-Schouten bracket on polyvector fields is a degree (−1) bracket that extends the Lie bracket, acts as a graded derivation for the wedge product, and is given on decomposable polyvector fields by for all g ∈ O Z , and all ξ i , η j ∈ T Z .
The statement of the lemma is local on Z and is obvious away from the points of D v .
Indeed, away from D v we have that Ω n−a by the Nijenhuis-Scouten bracket.
Suppose next p ∈ D v . As before we choose local coordinates z 1 , · · · , z n centered at p so that near p we have: Using this formula for vol Z and the description (2.3.1) of the sheaf of f-adapted logarithmic In terms of these sheaves we have the Nijenhuis-Schouten bracket.
Example 2.15 It is instructive to examine more carefully the simplest case of a one dimensional compactified Landau-Ginzburg model. Near a point p of D v we can choose a local coordinate z on Z so that f(z) = z −1 , vol Z = dz/z. Then locally near p we get Using Lemma 2.14 we can organize the f-adapted polyvector fields into a sheaf of dg Lie algebras. For any 1 − n ≤ b ≤ 1 set The sheaves G b fit together with the Nijenhuis-Schouten bracket [•, •] and the differential This sheaf of dg Lie algebras is directly related to our unobstructedness problem. Indeed, note that any stupid truncation of (G • , [f, •]) will be a subsheaf of dg Lie algebras. In particular we have a subsheaf of dg Lie algebras On the other hand this subsheaf maps naturally to the sheaf of dg Lie algebras g • = g 0 df / / g 1 that controls our deformation problem. So we get a diagram x x r r r r r r r r r r r ( ( P P P P P P P P P P P P of sheaves of dg Lie algebras. In fact (2.3.6) is a roof diagram. More precisely we have the following Proposition 2.16 The natural map of sheaves of dg Lie algebras is actually an isomorphism away from D v . Thus it only remains to check the statement locally near a point p ∈ D v .
Choose local coordinates z 1 , . . . , z n as in the proof of Lemma 2.14. In terms of these coordinates we can describe our dg Lie algebras explicitly. For the sheaves of f-adapted poly vector fields we have The differential [f, •] = ι df : G 0 → G 1 is given explicitly by the formula We have an analogous local description of the deformation theory dg algebra: The differential g 0 → g 1 is again given by the formula (2.3.8) and the map of dg Lie algebras given by the natural inclusions G 0 ⊂ g 0 , G 1 ⊂ g 1 . Thus we get a short exact sequence of complexes Since the last vertical map is clearly an isomorphism, this implies that Proposition 2.16 and the roof diagram 2.3.6 suggest that the unobstructedness statement in Theorem 2.10 is related to the unobstructedness of the L ∞ algebra RΓ(Z, In fact the standard formality yoga for L ∞ algebras allows us to deduce Theorem 2.10 from a stronger double degenration statement for the cohomology of a two parameter family of denote the divergence operator associated with vol Z . Note that by definition the differentials [f, •] and div vol Z anticommute and so for any pair of complex numbers (c 1 , c 2 ) we will get a well defined complex . With this notation we now have the following Then RΓ(Z, g • ) is homotopy abelian.
is homotopy abelian; (2) The induced map is injective on cohomology. First note that the stupid filtration σ ≥• (G • , [f, •)) gives rise to a spectral sequence which is independent of c ∈ C and thus this spectral sequence will degenerate at E 1 . This implies that is injective for all k and in particular property (2) holds.
To prove property (1) we consider the flat family of L ∞ algebras over C [[ ]] given by . According to [KKP08, Proposition 4.11(i)] it suffices to check that k satisfies: Condition (B) follows immediately from the flatness assumption (2.3.9). To check condition (A) we will use an observation from [BK98]: C (( )) and an abelian dg algebra over C (( )). The proposition is proven.
Proposition 2.17 finishes the proofs of Theorem 2.10 and Theorem 2.6 modulo the flatness assumption (2.3.9). Converting back to f-adapted logarithmic forms via ι vol Z the assumption (2.3.9) is equivalent to the statement that the dimension of the hypercohomology is independent of (c 1 , c 2 ) ∈ C 2 . We investigate this Hodge theoretic statement in the next section.

The double degeneration property
In this section we complete the proof of the unobstructedness Theorem 2.6 by establishing the double degeneration property for the complex (2.3.2) of f-adapted logarithmic forms associated with a compactified tame Landau-Ginzburg model which is not necessarily of log Calabi-Yau type. An alternative proof and a generalization of this statement can be found in the recent work of Esnault-Sabbah-Yu [ESY13]. For the convenience of the reader we give our original argument here.
Theorem 2.18 Let ((Z, f), D Z ) be geometric datum where (a) Z is a smooth projective variety, and f : Z → P 1 is a flat projective morphism.
Then the following flatness property holds Proof. We will obtain the proof by checking the constancy of dimension of cohomology along various lines in C 2 . First we have the following Lemma 2.19 For every ((Z, f), D Z ) satisfying the hypothesis if the theorem and every a ≥ 0 (2.4.2) In particular, the spectral sequence corresponding to the stupid filtration on Proof. We will use the method of Deligne-Illusie [DI87,EV92,Ill02]. Here we only sketch the necessary modifications that make the method applicable to f-adapted logarithmic forms.
More details can be found in the Esnault-Sabbah-Yu writeup in [ESY13, Appendix D].
By the standard spreading-out argument of [DI87,EV92,Ill02] it suffices to check the E 1 degeneration of the spectral sequence in the case when the geometric datum ((Z, f), D Z ) satisfying the hypotheses of the lemma is defined over a perfect field k of characteristic p > dim X and admits a smooth lift to characteristic 0 (or at least to the second Witt vectors W 2 (k) of k).
Z ′ ) denote the induced relative Frobenius morphism over k. The base change property for algebraic differential forms combined with the fact that Fr is a homeomorphism, and with the local description (2.3.1) of f-adapted forms implies This gives equality of dimensions of these matching cohomology groups and so the E 1 degeneration of (2.4.3) will follow immediately (see e.g. [Ill02, Section 4.8]) if we can show that the complex To that end, recall [Car57,Kat70] that the (inverse) Cartier map defined by γ(Φ * z) = z p and γ(dΦ * z) = [z p−1 dz] on a local function z on Z, extends uniquely by multiplicativity and gives rise to an isomorphism γ : of sheaves of super commutative algebras over O Z ′ .
Using the explicit local description (2.3.1) of the f-adapted logarithmic forms one checks immediately that γ also restricts to an isomorphism of sheaves of super commutative algebras over O Z ′ .
In view of the isomorphism (2.3.1) the formality of Fr * Ω • Z/k (log D Z , f), d as an object in D(Z ′ ) is equivalent to the existence of a morphism in D(Z ′ ): for : a≥0 which induces γ on cohomology sheaves. Following [DI87] the construction of for can be carried out in three stages. Fix a lift ((Z, f), D Z ) over W 2 (k). We abuse notation and again write ϕ : Spec W 2 (k) → Spec W 2 (k) for the absolute Frobenius. Similarly we will write for the pullback of ((Z, f), D Z ) via ϕ and will write Φ : Z ′ → Z for the base change map.
As a first step suppose that the relative Frobenius Fr : Z → Z ′ admits a global lifting to a morphism Fr : Z → Z ′ of W 2 (k)-schemes which furthermore satisfies Fr * (f ′ ) = f p and With such a lifting we associate a formality morphism for Fr as follows: • For a = 0 we set for 0 • For a = 1 we set for 1 • For a > 1 we define for a Fr to be the composition of ∧ a for 1 Fr with the product map The key observation now is that for all a the map for a Fr sends Ω a Once this is checked, the fact that for Fr is a quasi-isomorphism inducing γ on all cohomology sheaves follows tautologically. To show that we argue locally on Z. By (2.3.1) we know that Ω a Z/k (log D Z , f) is a locally free sheaf which near D v Z is equal to the sum of d log f∧Ω a−1 Z/k (log D Z ) and (1/f)Ω a Z/k (log D Z ) inside Ω a Z/k (log D Z ). Choosing an appropriate Zariski local etale map to an affine space we obtain local coordinates z = (z 1 , . . . z n ) as in the proof of Lemma 2.12. In particular we have that the divisor D Z is given by the union of z i = 0 for i = 1, . . . , k + l, and f = 1/(z 1 · . . . · z k ).
From these formulas we now see that for the forms in Ω a Z/k (log D Z , f) of type d log f ∧ Ω a−1 Z/k (log D Z ) the pullback via (1/p a ) Fr * modulo p does not depend on the choice of v 1 , . . . , v k , and is therefore again a log form multiplied by d log f. For the forms of second type, i.e. forms and is therefore again of second type.
In the second step one notes that locally in the Zariski topology we can always choose etale maps to an affine space and then use local coordinates as above to construct a lift of the relative Frobenius over W 2 (k). Thus we have to analyze the relation between the formality isomorphisms associated to different local liftings of the relative Frobenius. Following [DI87] we want to show that for any two liftings Fr 1 : Z 1 → Z ′ and Fr 2 : Z 2 → Z ′ of Fr, we can find a canonical map of sheaves Fr 2 ). Furthermore for a third lifting Fr 3 : Z 3 → Z ′ these maps should satisfy the cocycle condition h(Fr 1 , Fr 2 ) + h(Fr 2 , Fr 3 ) = h (Fr 1 , Fr 3 ).
To construct the maps h we repeat verbatim the reasoning in [DI87,EV92]. To show that the corresponding dh(Fr 1 , Fr 2 ) belongs again to Fr * Ω 1 Z ′ /k log D ′ Z ′ , f ′ one notes that by construction h(Fr 1 , Fr 2 ) is given by substitutions with vector fields of the form k+l i=1 u i (z) · z p i · (∂/∂z i ) + n i=k+l+1 u i (z) · (∂/∂z i ) satisfying k i=1 u i = 0. Such substitution vanish for forms divisible by d log f. For forms α = (1/f) · β with β being log form, the substitution of such vector field in β is again a log form, hence its pullback is a log form. Also note that the pullback of 1/f ′ is 1/f p which is divisible by 1/f. Thus we again get a form of the second type. The same argument should work for a triple of lifts. The key point here is that the forms Ω a Z/k (log D Z , f) are closed under contractions with vector fields preserving f. From this point on the argument proceeds exactly as in [DI87]. First we cover Z by Zariski open sets U i on which we can choose Frobenius lifts Fr i : U i → U ′ i as above. Then on overlaps we use the maps h(Fr i , Fr j ) on overlaps to glue the formality morphisms for 1 Fr i into a morphism in the derived category which induces γ on H 1 .
In the last step we use the condition n = dim Z < p and multiplicative structure on the de Rham complex to define a map given by α 1 ⊗ · · · ⊗ α a → 1 a! σ∈Sa sgn(σ)α σ(1) ⊗ · · · ⊗ α σ(a) . This completes the proof of the lemma. Re z ≪ 0. Then for every a ≥ 0 we have Proof. Before we address the statement of the lemma, it is instructive to look at the anal- is naturally quasi-isomorphic to this constructible complex. Indeed, a direct local calculation [Gri69,Del71] shows that the natural map of complexes  C). The idea is to modify this reasoning to take into account relative cohomology and fadapted forms. To that end consider the real oriented blow-up ε : Z → Z of Z along the reduced normal crossing divisor D Z , and the real oriented blow-up π : P 1 → P 1 of P 1 at ∞ ∈ P 1 . The morphism f : Z → P 1 lifts naturally to a real semi-algebraic map f : Z → P 1 . The spaces P 1 and Z are manifolds with boundary, and ∂ P 1 = π −1 (∞) ∼ = S 1 and The boundary circle ∂ P 1 = π −1 (∞) ∼ = S 1 is the circle of radial directions at ∞ ∈ P 1 . If as before z denotes the affine coordinate on A 1 = P 1 − {∞}, then this circle is parametrized by arg(1/z). Choose a point θ 0 ∈ ∂ P 1 for which Re(z) ≥ 0 and let the map is a quasi-isomorphism. This implies that the natural map from Ω Instead of repeating this calculation here we will give an alternative proof of the lemma which is be of independent interest.
To simplify the discussion let us first assume that D h Z is empty. Consider the de Rham cohomology of the pair (Y, w −1 (ρ)) where ρ is real and ρ ≪ 0. Using de Rham's theorem and the Gauss-Manin parallel transport along the ray ρ ∈ R <0 we can identify the cohomology (b) The Gauss-Manin parallel transport along the ray ρ ∈ R <0 is well defined at the limit Proof. The statement is local on Z and is obvious at points of the open set Y = Z − D Z .
As in the proof of Lemma 2.12 we can choose a local coordinate system z 1 , . . . , z n centered at p so that D Z is given by the equation k i=1 z i = 0 and f = k i=1 z −1 i . Now as in the proof of Lemma 2.12 we write W ⊂ Ω 1 Z (log D Z ) for the sub O Z -module spanned by d log z 1 , . . . , d log z k−1 , and R ⊂ Ω 1 Z (log D Z ) for sub O Z -module spanned by dz k+1 , . . . , dz n .
In these terms we have Write ǫ = 1/ρ, and let Y ǫ = f −1 (ǫ). Then for ǫ close to 0 we can use z 1 , . . . , z k−1 , z k+1 , . . . , z n as coordinates along the divisor Y ǫ . In particular the sheaf of holomorphic forms Ω 1 Yǫ is the O Yǫ -span of the forms dz 1 , . . . , dz k−1 , dz k+1 , . . . , dz n , and so From here we get and so when ǫ → 0 this sheaf specializes to the sheaf of f-adapted logarithmic forms (see This shows that as ǫ → 0 the complex (Ω • Z (log D Z , rel Y ǫ ) , d) will converge to the complex In other words we have a family of complexes on Z parametrized by a small complex number ǫ, where (Ω • Z (log D Z , rel Y ǫ ) , d) is the complex corresponding to ǫ = 0, while at ǫ = 0 we have the complex (Ω • Z (log D Z , f) , d). More invariantly, let ∆ ⊂ P 1 be a small disk centered at ∞ with coordinate ǫ. Let Z := Z × ∆, and let p : Z → ∆ be the natural projection. The proper family p : Z → ∆ is equipped with two relative divisors By construction Γ is smooth, D Z has strict normal crossings, both Γ and D Z are flat over ∆, and the union of Γ ∪ D Z also has strict normal crossings. Write D Γ for the normal crossing divisor in Γ given by D Γ = D Z ∩ Γ.
Consider now the sheaves of relative meromorphic forms, i.e. forms along the fibers of p, having logarithmic poles along D Z and vanishing along Γ: By definition these are locally free sheaves of (certain) relative logarithmic forms along the fibers of p : Z → ∆, and the graded subsheaf Ω • Z/∆ (log D Z , rel Γ) ⊂ Ω • Z/∆ (log D Z ) is clearly preserved by the relative de Rham differential. The calculation in local coordinates above shows that the complex interpolates between relative logarithmic forms vanishing on Y ǫ and f-adapted relative logarithmic forms. In other words we have This proves the first part of the claim.
The statement about the Gauss-Manin parallel transport follows form the homological description [KO68] of the Gauss-Manin connection. To spell this out one needs to describe the local system of relative cohomology via differential forms. To fix notation, we will use a superscript (•) × to indicate the removal of the fiber over ǫ = 0 in the various geometric and sheaf-theoretic objects we are dealing with. Thus we will write Let E a B denote the local system of C-vector spaces on ∆ × whose fiber over ǫ ∈ ∆ × is the relative Betti cohomology H a (Y, Y ǫ ; C). The underlying coherent sheaf E a B ⊗ C O ∆ × can be identified with the sheaf H a DR Y × /∆ × , Γ × /∆ × ; C of relative de Rham cohomology and is thus computed as the hyperderived image Note that the hyper-derived image R a p × * E • Z × /∆ × is naturally an O ∆ × -module since the de Rham differential on relative forms is linear over p −1 O ∆ × . In these terms the Gauss-Manin connection is given by a C-linear map of sheaves satisfying the Leibnitz rule. The analysis of [KO68] applies verbatim to this setting and identifies ∇ GM with the connecting homomorphism in the long exact sequence of hyperderived direct images associated with the short exact sequence of complexes (2.4.5) In order to check that the parallel transport with respect to ∇ GM has a well defined limit when ǫ → 0 it suffices to show that the complex Ω of logarithmic de Rham complexes 3 on Z.
View (2.4.7) as a morphism ξ Z/∆ : We claim that the composition is the zero morphism in D b (C Z ). This follows immediately by noting that ξ Z/∆ fits in a commutative diagram in D b (C Z ): in which the columns are parts of exact triangles and corresponding to the short exact sequence of complexes Here all terms are equipped with the obvious absolute or relative de Rham differentials, so we have omitted them from the notation. The differential in the first term is defined via the identification ∆ and the fact that de Rham differential on relative forms along the fibers of p : of logarithmic forms on Γ. The vanishing of (q ⊗ 1) • ξ Z/∆ • i follows immediately now since from (2.4.8) we see that (q ⊗ 1) • ξ Z/∆ • i = ı Γ * ξ Γ/∆ • q • i = 0. This in turn implies that ξ Z/∆ comes from a morphism cone(q) → cone(q ⊗ 1) in D b (C Z ). In other words we can find a map ξ E : This completes the proof of the claim.
The statement of Claim 2.22 proves Lemma 2.21 in the case when the divisor D Z does not have a horizontal part. In fact we can incorporate the horizontal divisor into the proof of Claim 2.22 without any modification. The local calculation for the limit, and the extension argument repeat verbatim, only the notation becomes more cumbersome. We will not spell this out here and leave it to the interested reader to fill in the details.
We can now complete the proof of the double degeneration property by combining Lemma 2.21 with the following well known facts: Lemma 2.23 For every a ≥ 0 we have Proof. The first equality is the usual identification of de Rham nearby cycles with twisted de Lemma 2.24 For every a ≥ 0 we have Proof. This follows from the usual Grothendieck argument [Gro66]. The local calculation comparing logarithhmic forms with meromorphic forms transfers immediately to the f-adapted complex and combined with the local description of adapted forms given in Lemma 2.12 implies that the natural inclusion of complexes is a quasi-isomorphism.
Taken together Lemmas 2.21, 2.23, and 2.24 imply that the hypercohomology of the complex is constant on the line c 1 = c 2 . This completes the proof of Theporem 2.18.

Invariants of nc Hodge structures of geometric origin
In this section we use the deformation theory developed in Section 2 to elucidate the motivic and Hodge theoretic data naturally present on the cohomology of a compactifiable Landau-Ginzburg model. Using considerations from mirror symmetry we propose various new refined invariants of nc Hodge structures of Landau-Ginzburg type, and discuss in particular the subtleties involved in understaning Hodge numbers and decorations.

Hodge numbers of Landau-Ginzburg models
Each of these sets of Hodge numbers has a different origin. The numbers i p,q (Y, w) come from ordinary mixed Hodge theory, the numbers h p,q (Y, w) come from mirror considerations, and the numbers f p,q (Y, w) come from the sheaf cohomology of the f-adapted logarithmic forms. The specific definitions are as follows.
3.1.1. The numbers f p,q (Y, w). Let ((Z, f), D Z , vol Z ) be a tame compactification of Definition 3.1 The Landau-Ginzburg Hodge numbers f p,q (Y, w) are defined by The fact that dim C H a (Y, Y −∞ ; C) = p+q=a f p,q (Y, w) follows from Theorem 2.6 and Lemma 2.24.
3.1.2. The numbers h p,q (Y, w). Before we explain the definition we need to recall a basic construction from linear algebra. Let V be a finite dimensional complex vector space, N : V → V be a nilpotent linear operator satisfying N m+1 = 0 for some m ≥ 0. The (monodromy) weight filtration of N centered at m is the unique increasing filtration The rationale behind this definition is the geometric mirror symmetry prediction explained in Remark 2.5(ii). Specifically, if (Y, w) is part of a mirror pair of Fano type with dim C X = dim C Y = n, then the homological mirror equivalence induces an isomorphism on period cyclic and on Hochschild homologies of these categories.
In particular we expect a mirror isomorphism for all a where the Hochschild homology can possibly be non-zero, i.e. for all a such that −n ≤ a ≤ n. It is also expected that these categorical homology groups have geometric incarnations: (3.1. 2) The first of the above identification follows from the work [Wei96] of Weibel, while the second has been conjectured in general, and proven in special cases in the works of Seidel (see e.g. [Sei08,Sei09]).
Thus combining the conjectural mirror isomorphism (3.1.1) with this geometric interpretation of Hochschild homology we will get a conjectural isomorphism which after tensoring with C((u)) induces an isomorphism of C((u))-vector spaces From this point of view the isomorphism (3.1.1) is recovered as the induced isomorphism of specializations These mirror isomorphisms translate readily into the geometric language. Recall that similarly to (3.1.2) we have identifications   (FS((Y, w), ω Y , vol Y )) is identified with the Rees module of the filtration The de Rham version of the Dolbeault mirror statement (3.1.3) then becomes the statement that mirror symmetry induces an isomorphism of the filtered complex vector spaces The key to reconstructing the bigradings and thus extracting Hodge numbers on both sides lies in the observation that, in the Fano case, the Dolbeault bigrading also has a categorical interpretation. Indeed, the nilpotent operator c 1 (K X ) ∪ (•) acts on each HH a (D b (X)) = ⊕ p−q H p (X, Ω q X ), and so induces a monodromy weight filtration centered at a. Since the canonical class is anti-ample this filtration is given by the forms of degree ≤ (p + q). In particular the dimensions of the graded pieces for this monodromy weight filtration are precisely the Hodge numbers h p,q (X) of the Fano variety X. Up to a sign, the nilpotent C). Thus, this monodromy weight filtration has a categorical interpretation. But, as we noted in Remark 2.5(ii), the Serre functor of FS((Y, w), ω Y , vol Y ) can be identified with the inverse of the monodromy autoequivalence T . The logarithm of the action of T on HH • (FS((Y, w), is just the nilpotent operator N we considered above. Therefore the monodromy weight filtration corresponding to N is expected to have categorical origin, and homological mirror symmetry predicts, the mirror matching of Hodge numbers: for all p, q. This prediction is still conjectural in general but the case (p, q) = (1, 1) was recently proven by Przyjalkowski and Shramov [PS13] for all smooth Fano varieties.
3.1.3. The numbers i p,q (Y, w). To simplify the discussion we will first assume that D h Z = ∅, i.e. that w : Y → A 1 is proper.
It is well known (see e.g. [Sab99]) that the dimension of the (Zariski) hypercohomology of the w-twisted de Rham complex on Y can be computed from the dimensions of the vanishing cohomology for w: where as usual φ w−λ C Y denotes the perverse sheaf of vanishing cocycles for the fiber Y λ .
From the works of Schmid and Steenbrink (see e.g. [Sch73] [PS08, Section 11.2]) and Saito [Sai90] it is classically known that the constructible complex φ w−λ C Y carries a structure of a mixed Hodge module and so its cohomology is furnished with a functorial mixed Hodge structure.
Given a mixed Hodge structure V we will write i p,q V for the (p, q) Hodge number of the p + q weight graded piece gr W p+q V. We now have the following Definition 3.4 For a proper potential w : Y → A 1 on a quasi-projective variety Y the Landau-Ginzburg Hodge numbers i p,q (Y, w) are defined by where each vanishing cohomology H a Y λ , φ w−λ C Y is taken with its Schmid-Steenbrink mixed Hodge structyure.

Remark 3.5 (i) The combination of Hodge numbers of different weight pieces in this
definition is motivated by mirror symmetry. In the paper [GKR12] it was argued that for a Landau-Ginzburg mirror of a general type complete-intersection S in a toric variety, the above definition of Hodge numbers reproduces the rotated Hodge diamond of S.

(ii)
The assumption that D h Z = ∅ above was introduced solely for technical convenience and is not really needed. If D h Z = ∅, we can still define i p,q (Y, w) by setting (iii) It is very interesting to try and understand the categorical meaning of the numbers i p,q (Y, w). At a first glance, the definition of i p,q (Y, w) relies heavily on the geometry since the information of the variety Y and the potential w enter in an essential way in the construction of the pertinent mixed Hodge structures. On the other hand, from the works of Shklyarov [Shk11] and Efimov [Efi12] it is known that the space H • DR (Y, w; C) together with its nc Hodge filtration admits a purely categorical interpretation. Specifically, in [Efi12] it ) is isomorphic to the periodic cyclic homology HP • (MF(Y, w)) of the d(Z/2)g category of matrix factorizations of w, and that this isomorphism can be chosen so that the irregular connection ∇ DR d/du = d/du + u −1 Gr + u −2 w · (•) codifying the nc Hodge filtration on H • DR (Y, w; C) gets identified with the connection ∇ cat d/du from [KKP08, Section 2.2.5] used to define the categorical nc Hodge filtration on HP • (MF(Y, w)).
In other words, the nc Hodge filtration of a Landau-Ginzburg model (Y, w) admits a purely categorical interpretation. In the case when ∇ cat satisfies the nc-opposedness axiom of [KKP08] we can hope for more. In this case we expect that the pure complex nc Hodge structure on HP • (MF(Y, w)) is polarizable and that it admits a natural limit mixed twistor structure (in the sense of [Sab05]) which in turn is isomorphic to the Z/2-folding of the ordinary mixed Hodge structure on the vanishing cohomology we have a one parameter deformation {A t } t∈A 1 of the d(Z/2)g category A 1 := MF(Y, w), where A t has the same objects and hom sets as MF(Y, w) but the composition, differential, and units in A t are scaled as m The periodic cyclic homology of these categories equipped with the connection ∇ cat in the u-direction and with the Getzler-Gauss-Manin connection [Get93] in the t-direction is a variation of twistor D-modules. When the opposedness and polarizability properties hold, e.g.
for Landau-Ginzburg mirrors of toric Fano varieties, see [RS10], we can form the limit mixed twistor D-module for t → ∞ and we conjecture that this mixed twistor D-module is the one corresponding to the ordinary mixed Hodge structure ⊕ λ∈A 1 H •−1 Y λ , φ w−λ C Y . In the case of potentials given by tame Laurent polynomials this conjecture is verified in [Sab10b].
The conjecture gives a categorical interpretation of the mixed Hodge structure on vanishing cohomology (modulo Tate twists) and as a consequence gives a categorical interpretation of the Z/2-folding of the numbers i p,q (Y, w). Combined with the mirror matching (3.1.5) the previous conjecture predicts Conjecture 3.7 If (X, ω X , s X ) | ((Y, w), ω Y , vol Y ) is a mirror pair of Fano type, and if

Comparison conjectures
for all p, q.

Mirrors of compactified Landau-Ginzburg models
In this section we look more closely at the role that compactified Landau-Ginzburg models play in mirror symmetry. In the setting where a complex Landau-Ginzburg model (Y, w) is the mirror of a symplectic Fano variety (X, ω X ), we give a mirror A-model interpretation of the Hodge information encoded in a compactification ((Z, f), D Z ). This suggests that the mirror symmetry bewteen (Y, w) and (X, ω X ) can be extended to a mirror symmetry between ((Z, f), D Z ) and a one parameter symplectic deformation of (X, ω X ) which interpolates between the Fukaya category of the symplectic Fano variety (X, ω X ) and the Fukaya category of the symplectic non-compact Calabi-Yau datum X − D X , ω X|X−D X , vol X−D X . We discuss such an extension and give some evidence for its validity. This picture is not new and has already been proposed and analyzed in one form or another in the works of Seidel 3.2.1. One parameter families of symplectic Fano varieties Let (X, ω X ) be a symplectic manifold underlying a smooth compact Fano variety of dim C X = n. Let k X be a closed 2-form representing the canonical class K X and let κ X ∈ H 2 (X, Z) denote the first Chern class of K X , i.e. κ X = [k X ] = c 1 (K X ). Consider the (multivalued) family {ω q } q∈C of complex 2-forms ω q := ω X + log(q)k X on X. In the regime when |q| → 1 these are complexified Kähler forms. This is an affine-linear one-parameter family of symplectic structures on X which gives rise to a one-parameter variation of pure nc Hodge structures parametrized by the q-line.
As discussed in [KKP08, Section 3.1] the de Rham part of such variation is encoded in a pair ( a H, a ∇), where is a trivial algebraic Z/2-graded vector bundle on the affine plane with coordinates (u, q), with Z/2-grading given by • a ∇ is a meromorphic connection on a H, with poles along the divisor uq = 0, given by where * q denotes the quantum product corresponding to ω q , and Gr : a H → a H is the grading operator defined to be Gr |H k (X,C) := k−n 2 id H k (X,C) .
Remark 3.8 More generally we have a variation of nc Hodge structures over the whole complexified Kähler cone. The meromorphic connection defining the de Rham part of the variation is the Dubrovin first structure connection [Dub98,Man99] which on each affine line ω + log(q)α is given by the formula with * q being the quantum product corresponding to ω + log(q)α. Traditionally in mirror symmetry one works with the line of slope ω passing through the large volume limit, i.e.
the line 0 + log(q)ω. This is the situation considered in [FOOO09a,FOOO09b] and in [KKP08, Section 3.1]. In contrast, here we need to work with a line of slope k X , i.e. the line ω + log(q)k X which leads to the formula (3.2.1).
The particular affine linear deformation of the symplectic structure (X, ω + log(q)k X ) that we are considering has many special properties even when compared to other affine linear families. For instance, it is expected that the affine one parameter deformation of the symplectic structure (X, ω + log(q)k X ) does not change the Fukaya category. From the point of view of nc Hodge theory this family is significant because of the following simple observation Lemma 3.9 The restriction of ( a H, a ∇) to any non-vertical line L through the origin in A 2 is a meromorphic connection on the trivial Z/2-graded vector bundle H • (X, C) ⊗ O L , which has a first order pole at 0, and monodromy around 0 equal to (−1) k on the graded piece Proof. From the formulas (3.2.1) we see that the a ∇-covariant derivative in the direction of the Euler vector field u∂ ∂u + q∂ ∂q is given by Since u∂ ∂u + q∂ ∂q is tangent to any line through the origin and is equal to the Euler vector field on any such line, we get the statement of the lemma.
To explicate, choose a slope v = 0 and let L v ⊂ A 2 be the line given by u = vq. The variable q is the natural coordinate on L v , and so on L v we have du = vdq. To shorten the notation, write M := κ X * q (•) for the operator of quantum multiplication by κ X . Then on Thus a ∇ |Lv is logarithmic at 0 and has half integer residues. This completes the proof of the lemma.
Recall from [KKP08, Section 2.2.7] that when viewed as nc Hodge structures, ordinary pure Since by Theorem 2.18 the cohomology H • (Z, (Ω • Z (log D Z , f), ud + qdf)) has constant dimension for all (u, q) ∈ A 2 the sheaf b H is locally free.
). As explained in [KKP08, Section 2.2.7], this means that the vector bundle b H |q=c is the Rees module associated with the Hodge filtration , and that b ∇ is a connection with logarithmic singularity at u = 0 and monodromy ±1 on graded pieces. In particular b H, b ∇ has a logarithmic pole at u = 0.
Remark 3.10 It will be useful to have an explicit formula for the Gauss-Manin connection b ∇, similar to the formula (3.2.1). We can write such a formula for the Gauss-Manin connection acting on the complex (Ω • Z ( * D Z ) [u, q], ud − qdf∧): where G is the grading operator defined to be G := − p 2 on Ω p Z ( * D Z ). This is the same formula that appears in the works of Shklyarov [Shk11] and Efimov [Efi12]. Note however that this formula does not preserve the subcomplex of f-adapted logarithmic forms since if α ∈ Ω • Z (log D Z , f), the form f α will not necessarily be in Ω • Z (log D Z , f). Therefore, we can not use these formulas directly to describe the action of b ∇ on b H. This latter action is a combination of the formulas (3.2.2) and the complicated limiting quasiisomorphism in the proof of Lemma 2.21.

Mirror symmetry for one parameter families
The formal similarity between the two connections ( a H, a ∇) and b H, b ∇ is very suggestive. We expect that when the geometric data defining these connections is part of a mirror pair, we should be able to go beyond a mere similarity and identify the pairs ( a H, a ∇) and b H, b ∇ . More precisely we propose the following conjecture.

Then
(i) The one parameter symplectic family (X, ω X + log(q)k X ) is mirrored into a one parameter complex family (Y, q · w)) of deformations of (Y, w).
(ii) The homological mirror correspondence induces an isomorphism The attentive reader will notice that the part (i) of this conjecture relies on a geometric H • (Y, Y sm ; C) respectively. To that end, in Section 3.2.5 we describe a general method for constructing a two parameter meromorphic connection from a pure nc Hodge structure.

Mirrors of tame compactifications of Landau-Ginzburg models.
Before we proceed with the construction in Section 3.2.5, it is instructive to examine more closely the apparent mismatch in the information contained in the one parameter mirror symmetry (X, ω X + log(q)k X ) | (Y, q · w) and in the tame compactification of the Landau-Ginzburg model. This mismatch is ultimately a reflection of the fact that the one parameter deformations (X, ω X + log(q)k X ) and (Y, q · w) only perturb one direction of the mirror symmetry: going from the A-model on the Fano side to the B-model on the Landau-Ginzburg side.
Since the choice of a tame compactification ((Z, f) , D Z ) is a choice additional data on the Landau-Ginzburg side, its mirror partner will necessarily depend on the choice of some additional data on the Fano side. A clue of what this additional data should be, appears in the works of Seidel [Sei08,Sei11] where the one parameter deformation (X, ω X + log(q)k X ) is interpretted intrinsically in categorical terms. The relevant key fact from [Sei08,Sei11] is the statement that the family of Fukaya categories Fuk (X, ω X + log(q)k X ) has a well defined limit as q → 0, namely the Fukaya category of the symplectic manifold underlying the non-compact Calabi-Yau X − D X .
To simplify notation write U := X − D X for the complement of D X , ω U := ω X|U for the restriction of the symplectic structure to U, and vol U = 1/s X for the holomorphic volume form corresponding to the anticanonical section s X . As explained in [Sei08,Sei11] (see also [Aur07]) the Z-graded A ∞ category F 0 = Fuk(U, ω U , vol U ) admits a natural one-parameter deformation {F q } as a Z/2-graded A ∞ category. By construction F q has the same objects and morphisms as F 0 but the A ∞ operations m q k in F q are q-perturbations of the A ∞ operations m 0 k in F 0 where the correction term q a comes with a coefficient counting not pseudo-holomorphic discs in U but rather disks in X that intersect the boundary divisor D X at a points 4 . Now, a comparison with the standard construction [FOOO09a,FOOO09b] of the Fukaya category identifies F q for q = 0 the with the Z/2 category Fuk(X, ω X +log(q)k X ).
Thus we get a streamlined categorical (or nc geometric) interpretation of the A-model data associated with the Fano geometry (X, ω X , s X ). In summary Seidel's analysis shows that: • from the point of view of nc geometry, the primordial object is the Z-graded Fukaya • the data of a symplectic compactification (U, ω U ) ⊂ (X, ω X ) with anti-canonical boundary D X = X − U corresponds to a q-deformation F q = Fuk(X, ω X + log(q)k X ) of F 0 as a Z/2-graded Calabi-Yau category.
To put it differently, the symplectic anti-canonical compactification (U, ω U ) ⊂ (X, ω X ) is encoded in a one parameter degeneration of the Fukaya category F 1 = Fuk(X, ω X ) of the compact symplectic Fano (X, ω X ) to the Fukaya category F 0 = Fuk (U, ω U , vol U ) of the symplectic non-compact Calabi-Yau (U, ω U ).
This categorical interpretation of the compactification of U has a natural mirror incarnation. The non-compact symplectic Calabi-Yau (U, ω U , vol U ) has a complex non-compact Calabi-Yau mirror Y , constructed say by the SYZ prescription as in [Aur07]. Homological mirror symmetry predicts that F 0 is equivalent to the category D b c (Y ). The one parameter deformation F q of F 0 = D b c (Y ) then correpsonds to a class in the Hochschild cohomology HH • (F 0 ) = HH • (D b c (Y )). Since F q is only a Z/2-graded deformation, this Hochschild cohomology class will have a non-trivial component in HH 0 , i.e. will give us a well defined element w ∈ H 0 (Y, O Y ). If we assume for symplicity that the boundary divisor D X 4 Making this precise is quite subtle (see [WW10,Sei11]) and requires a version of the Fukaya category which is linear over C (rather than a Novikov field). In [Sei11] such a version is built out of balanced (rather than arbitrary) Lagrangians in U . We thank Denis Auroux for illuminating explanations of this subtlety. is smooth, then the Fano/Landau-Ginzburg homological mirror symmetry conjecture will identify F q with MF(Y, q · w) for q = 0. If we interpret MF(Y, q · w) as a coproduct of the derived categories of singularities of the singular fibers of q · w, we see that this identification will specialize correctly when q → 0. The category F q specializes to F 0 while MF(Y, q · w) specializes to the compactly supported derived category of singularities of the derived fiber of the zero function on Y , which is readily identified with D b c (Y ). The upshot of the previous discussion is that the mirror one parameter families arising from the Fano mirror pair (X, ω X , s X ) | ((Y, w), ω Y , vol Y ) have a natural homolgical interpretation as families of (term by term equivalent) categories where the family on the left hand side is the Seidel Z/2-graded deformation of Fuk(U, ω U , vol U ) corresponding to the compactification (U, ω U ) ⊂ (X, ω X ).
This interpretation allows us to reverse the process and identify the mirror information corresponding to a tame compactification of Y . If we choose a tame compactification ((Z, f), D Z ) and also choose an extension ω Z of the symplectic form ω Y , then we can apply Seidel's analysis to the symplectic anti-canonical compactification (Y, ω Y ) ⊂ (Z, ω Z ). Since by the tameness assumption D Z is an anti-canonical divisor, the same reasoning shows that this compactification is encoded in a one parameter deformation of the Z-graded category Fuk(Y, ω Y , vol Y ) to the Z/2-graded category Fuk (Z, ω Z + log(r)k Z ). Again the degree zero piece of the Hochschild cohomology class governing this deformation will give us a holomorphic function v : U → A 1 . In fact the description of v in terms of the weighted disk counting on Z relative to the boundary D Z also predicts that v has first order poles along D X and so v = s/s X for some anti-canonical section s ∈ H 0 (X, K −1 X ). This can be packaged in the following conjecture.
where v is a meromorphic function on X with a first order pole along D X .
(ii) The Fano/Landau-Ginzburg homological mirror correspondence induces equivalences of one parameter families of categories.
Note that the geometric ingredients of the compactified mirror pair from Conjecture 3.12(i) appear now in a symmetric fashion in the two sides of the pair.
In particular we expect A-model data on one side to be mirror to the B-model data on the other.
It is clear also that the statement of Conjecture 3.12(ii) is but one facet of the homological mirror correspondence one should associate with the compactified mirror pair. The full homological mirror conjecture will involve various equivalences of categories generalizing the equivalences described in Table 1 for an ordinary (non-compactified) Fano mirror pair. It is possible to list all these equivalences but the list is somewhat cumbersome. Very recently Seidel found [Sei14b] a uniform conceptual way for capturing the homological content of either side of the compactified mirror pair and gave a clean formulation of the complete homological mirror conjecture for the pair Conjecture 3.12(i) in terms of an equivalence of categories equipped with nc anti-canonical pencils.
It is very interesting to understand how the two meromorphic connections from Conjecture 3.11(ii) arise directly as Hodge theoretic data associted with these nc pencils but we will leave this for future investigations. To keep track of the various copies of the affine line and the affine plane appearing in the construction, we will indicate the coordinates on these lines and planes as subscripts.
Thus A 1 u will denote the affine line with coordinate u, A 2 (u,q) will denote the affine plane with coordinates (u, q), etc. By definition (see [ ∇ is a meromorphic connection on H • , which has at most a regular singularity at u = ∞, at most a second order pole at u = 0, and no other singularities in A 1 u . View A 1 u as the u-axis in the plane A 2 (u,q) . Our goal is to extend H • to a holomorphic bundle ‡ H • on all of A 2 (u,q) , and ∇ to a meromorphic connection ‡ ∇ on ‡ H • over A 2 (u,q) so that ‡ ∇ has poles only at uq = 0 and has logarithmic singularities along q = 0. We will carry this out in two steps: Step 1. Start with the connection (H • , ∇) on A 1 u . Write (H, ∇) for the underlying ungraded algebraic vector bundle with connection. Since by assumption ∇ has a regular singularity at u = ∞ we can consider the Deligne extension (H, ∇) of (H, ∇) (see e.g. [ The restrictions F i H := F i H |A 1 u give a Z-labeled filtration of H by holomorphic subbundles.
For any complex number v ∈ C consider the Rees bundle ξ( is a C × -equivariant vector bundle on A 2 q for the scaling action of C × on the q-line. Allowing v to vary we get a Rees bundle ξ(H, F • H) → A 2 (v,q) which is algebraic and equivariant for the C × -action λ · (v, q) := (v, λq). By construction we have canonical identifications Since the filtration F • H was compatible with the grading on H, we get a natural Z/2- (v,q) , and will write † ∇ for the extension of the connection ∇.
Step 2. In this step we will modify the Z/2-graded bundle with connection † H, † ∇ to ensure that its monodromy is ±1 not on vertical lines but rather on lines through the origin.
Consider an affine plane A 2 (u,q) with coordinates (u, q). Let s : S → A 2 (u,q) be the blow-up of A 2 (u,q) at the origin (u, q) = (0, 0). The surface S is glued out of two affine charts A 2 (v,q) and A 2 (u,w) via the gluing map u = vq, w = 1/v. In particular A 2 (v,q) embeds as a Zariski open subset in S and we have a commutative diagram of surfaces where i : A 2 (v,q) ֒→ S denotes the inclusion, and p is the map Note that S − A 2 (v,q) = A 1 u and that a point u = 0 ∈ A 1 u ⊂ S is a limiting point in S completing the hyperbola {(v, q) |vq = u} ⊂ A 2 (v,q) to a copy C u of A 1 embedded in S. Now observe that if we restrict ( † H, † ∇) to the hyperbola vq = u, the restricted connection has a regular singularity as v → ∞. Therefore we have a canonical Deligne extension of † H |vq=u to an algebraic vector bundle on C u . This process depends algebraically on u and so gives an extension of † H to an algebraic vector bundle Ξ on the punctured surface S − {x}, where x ∈ A 2 (u,w) is the point with coordinates u = 0, w = 0. Next observe that since the surface S is smooth, any vector bundle V on S − {x} will extend to a (necessarily unique) vector bundle on S. Indeed, choose a torsion free coherent sheaf F on S which restricts to V on S − {x}. For instance, if  denotes the inclusion of S − {x} in S, we can take F to be the intersection ∩K of all coherent subsheaves K ⊂  * V such that V ⊂  * K. Since V is locally free, the double dual F ∨∨ will also restrict to V . Being the dual of a coherent sheaf F ∨∨ is automatically reflexive, and by Auslander-Buchsbaum theorem can only fail to be locally free in codimension three. Thus F ∨∨ is a locally free sheaf which extends V to S. The uniqueness of the extension follows again from the fact that x is a smooth point and so the local ring O S,x satisfies the Serre condition S 2 .
Let Ξ be the unique extension of Ξ to S. To complete the construction we will need to know that Ξ satisfies a descent property for the morphism s : S → A 2 (u,q) . Let E ⊂ S denote the exceptional P 1 of the blow-up morphism s : S → A 2 (u,q) . With this notation we have the following Definition 3.13 We will say that an nc Hodge filtration (H • , ∇) is extendable if the restriction of the algebraic vector bundle Ξ to E is holomorphically trivial.
Note that if (H • , ∇) is extendable, then by the projection formula this extension Ξ is canonically a pullback of a vector bundle on A 2 (u,q) , namely s * Ξ is a vector bundle and Ξ ∼ = s * s * Ξ. In particular, if the nc Hodge filtration (H • , ∇) is extendable, we get a well defined holomorphic bundle ‡ H := s * Ξ on A 2 (u,q) . The meromorphic connection † ∇ on † H is holomorphic on the open set vq = 0 and so can be viewed as a meromorphic connection ‡ ∇ on ‡ H with poles on uq = 0. Altogether we have proven the following Lemma 3.14 Let (H • , ∇) be an extendable nc Hodge filtration, then (H • , ∇) gives rise to a Z/2-graded meromorphic connection ‡ H • , ‡ ∇ on A 2 (u,q) , such that • ‡ ∇ is holomorphic away from uq = 0; • ‡ ∇ has at most a logarithmic pole along u = 0, and a pole of order ≤ 2 along q = 0; • The restriction of ‡ H 0 , ‡ ∇ to a line through the origin has trivial monodromy, while the restriction of ‡ H 1 , ‡ ∇ to a line through the origin has monodromy (−1).
The discussion in Section 3.2.1 shows that the extendability assumption in the previous lemma holds for the de Rham part of the nc Hodge structure associated with a symplectic Fano variety: Corollary 3.15 Let (X, ω X ) be a symplectic manifold underlying a smooth Fano variety of complex dimension n. Let * 1 denote the quantum product corresponding to the symplectic form ω X . Then the A-model nc Hodge filtration for the nc Hodge structure on the cohomology of (X, ω X ) is extendable and ‡ H • , ‡ ∇ reconstructs the standard q-variation of nc Hodge structures for the symplectic manifold (X, ω X ).
That is, we have a canonical identification where ( a H • , a ∇) is the connection defined in (3.2.1).
Proof. Follows immediately from the two step construction above and by Lemma 3.9 and Lemma 3.14.
In particular Corollary 3.15 shows that the one parameter mirror symmetry Conjecture 3.11(ii) is equivalent to the extendability property for the B-model nc Hodge filtration. More precisely, suppose (Y, w) is a complex Landau-Ginzburg model. Consider the B-model nc Hodge filtration for the cohomology of (Y, w): Here G is the grading operator of multiplication by −p/2 on Ω p Y . When (Y, w) is the mirror of a symplectic Fano variety (X, ω X ), homological mirror symmetry identifies the nc Hodge filtrations on the cohomology, i.e. gives an isomorphism Remark 3.17 In a very interesting recent work Sabbah and Yu [SY14] consider a different but related notion of extendability arising from a nilpotent orbit for the pure complex Hodge structure attached to a compactification of a Landau-Ginzburg model. Moreover they prove that the scaling variation of this pure Hodge structure is polarizable and satisfies their extendability condition. This result seems closely related to Conjecture 3.16 but we have not investigated the precise relation between the two statemnts.

Canonical decorations
In this section we take a closer look at the data needed to write special coordinates on the versal deformation space M of tame compactified Landau-Ginzburg models (Z, f), D Z ) of log Calabi-Yau type. Recall from [Tia87] and [Tod89], that when Y is a smooth compact Calabi- In the remainder of this section we will describe a conjectural construction which will produce a natural decoration in this setting, i.e. will lead to canonical special coordinates that do not If we choose for concreteness V to be the unique Deligne extension on which ∇ has a logarithmic pole at 0 and a residue with eigenvalues whose real parts are in (−1, 0], then the above bijection can be described explicitly as follows. Let V be another extension of V on which ∇ has a logarithmic pole. Fix an analytic trivialization of V near t = 0 and let || • || denote the Hermitian norm of a section of V computed in this trivialization. For any t ∈ D and any v ∈ V t we have a well defined ∇-horizontal section s v (r) of V over the segment (0, 1] · t uniquely determined by the initial condition s v (1) = v. With this notation we have The ∇-horizontal section s v (r) satisfies ||s v (r)|| = O r −i Remark 3.19 • The growth condition defining V ≤i depends on the extension V but not on the choice of a local holomorphic frame of V near 0.
• In [KKP08, Section 4.1.3]) we discussed the classification of logarithmic extensions of (V, ∇) in terms of biregular decreasing filtartions of V . The above description of V ≤i is just a relabeling of the filtrations described in [KKP08, Section 4.1.3]).
3.3.2. The A-model decoration. Let (X, ω X ) be a compact symplectic manifold of real dimension 2n. Under the convergence assumption from [KKP08, Section 3.1] for the quantum multiplication * q , the nc Hodge filtration on the de Rham cohomology of X is encoded in the meromorphic connection on A 1 u : where * 1 denotes the quantum product for ω X . Note that by definition A ∇ has a regular singularity at u = ∞.
Remark 3.20 Conjecturally the convergence assumption on * q is closely related to the properties of the nc geometry attached to the pair (X, ω X ). In particular if convergence for q = 1 holds we expect that (i) the Fukaya category Fuk(X, ω X ) is smooth and compact; (ii) the geometrically defined nc Hodge filtration A H, A ∇ coincides with the nc Hodge filtration on HP • (Fuk(X, ω X )) defined in [KKP08, Section 2.2.5]; (iii) the monodromy of A ∇ around u = ∞ is unipotent and conjugate to the classical multiplication κ X ∧ (•) by the canonical class.
As we explained in Section 3.3.1, logaritmic extensions of A H, A ∇ across u = ∞ will correspond to a A ∇-horizontal filtrations of A H. In particular from the definition of Gr we see that extensions of A H as a trivial bundle over P 1 u = A 1 u ∪ {∞}, will correspond to filtrations A H ≤• whose associated graded is isomorphic to H •+n (X, C) ⊗ O A 1 u as a Z-graded bundle on A 1 u . Therefore, in order to get a decoration for the A-model data A H, A ∇ we need to specify a canonical A ∇-covariant filtration on A H whose associated graded pieces have dimensions h •+n (X, C). If in addition this filtration depends holomorphically on ω X , then it will automatically give a decoration not only for the fixed nc Hodge filtration A H, A ∇ but also for the universal variation A H, A ∇ over the cone of complexified symplectic structures.
Such a canonical decoration arises naturally in the Fano case. Indeed, suppose that (X, ω X ) underlies a complex Fano manifold of complex dimension n, and that property (iii) from Remark 3.20 holds for (X, ω X ). In this case the operator κ X ∧(•) satisfies the Lefschetz property on H • (X, C) and in particular has Jordan blocks which are symmetrically situated around the middle dimension n. In particular the Lefschetz filtartion (= the monodromy weight filtration for the nilpotent operator κ X ∧ (•)) will have associated graded pieces with dimensions h •+n (X, C). Thus the extension of A H across u = ∞ corresponding to this filtration will be holomorphically trivial on P 1 u . This shows that for a symplectic Fano the universal Calabi-Yau variation of nc Hodge structures A H, A ∇ → A 1 u × K over the complexified Kähler cone K will have a canonical decoration data: A H is the holomorphic extension of A H to P 1 u × K which ocrresponds to the monodromy weight filtration for the monodromy around u = ∞.
For ease of reference it will be useful to introduce terminology that describes this extendability behavior. Again fix a small complex disk D and a meromorphic connection (V, ∇) on D × with a regular singularity and unipotent monoromy around zero. Fix the unique Deligne extension V → D of V on which ∇ has a logarithmic pole with nilpotent residue.
As we saw in Section 3.3.1 this data establishes a 1-to-1 correspondence between logarithmic extensions of V to P 1 u and covariantly constant biregular increasing filtrations of V → A 1 u .
Definition 3.21 • The skewed canonical extension of V is the holomorphic vector bundle V which corresponds to the monodromy weight filtration for the monodromy operator around u = ∞.
• An abstract nc Hodge filtration (H • → A 1 u , ∇) will be called special if ∇ has unipotent monodromy around u = ∞ and the corresponding skewed canonical extension H of H is holomorphically trivial.
In these terms the discussion about the extendability behaviour of the A-model nc Hodge structure above can be rephrased as the statement that the A-model nc Hodge filtration A H, A ∇ associated with a symplectic Fano variety is special. Speciality is the main property needed to define the canonical decoration for the universal A-model variation A H, A ∇ .
Since the skewed extension and the speciality property are intrinsically determined by the monodromy, it is straightforward to transfer them through the mirror correspondence and to formulate the B-model extendability that will give rise to a canonical decoration of B H, B ∇ and canonical special coordinates on the moduli M of compactified Landau-Ginzburg models.  w); C)⊗O whose value at (0, ((Z, f), D Z )) is 1 ∈ H 0 (Z, Ω n Z (log D Z , f)).