Log smooth deformation theory via Gerstenhaber algebras

We construct a kQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\left[ \!\left[ Q\right] \!\right] $$\end{document}-linear predifferential graded Lie algebra LX0/S0∙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\bullet }_{X_0/S_0}$$\end{document} associated to a log smooth and saturated morphism f0:X0→S0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_0: X_0 \rightarrow S_0$$\end{document} and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction in Chan et al. (Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties, 2019. arXiv:1902.11174) whereof LX0/S0∙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\bullet }_{X_0/S_0}$$\end{document} is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields; this method is closely related to recent developments in mirror symmetry.


Introduction
Given a smooth variety X over an algebraically closed field k ⊃ Q, smooth deformation theory associates a functor Def X : Art → Set of Artin rings to it such that Def X (A) is the set of isomorphism classes of smooth deformations of X over Spec A.
To understand the properties of such functors of Artin rings, differential graded Lie algebras (dglas) are well-established. Given a dgla L • , there is an associated functor Def L • : Art → Set of Artin rings by taking an Artin ring A to solutions η ∈ L 1 ⊗ m A of the Maurer-Cartan equation up to gauge equivalence. We say a dgla L • controls deformations of X if Def X ∼ = Def L • . Such an isomorphism reduces the problem of understanding deformations of X to the problem of understanding L • . This has been fruitful in various places, e.g. when proving the Bogomolov-Tian-Todorov theorem with purely algebraic means in [12].

Our setup: log smooth deformation theory
In this paper, we establish an analog of the above dgla in logarithmic geometry, replacing the smooth variety X by a log smooth morphism. Log smoothness is the analog of the classical notion of smoothness and has been introduced in [16]. It allows to treat certain singularities as if they were smooth, a technique which has been proven fruitful in smoothing normal crossing spaces, e.g. by Kawamata-Namikawa in [19], and in mirror symmetry, e.g. by Gross-Siebert in [7]. Infinitesimal log smooth deformation theory was established by F. Kato in [14]. It studies deformations of a log smooth morphism f 0 : X 0 → S 0 of fs log schemes. Here, the base is a log point S 0 = Spec (Q → k) for a sharp toric monoid Q, most commonly Q = 0 or Q = N. The map Q → k is 0 → 1 and 0 = q → 0. Log smooth deformation theory replaces Art by Art Q , the category of local Artin k [[Q]]-algebras with residue field k. Every A ∈ Art Q gives rise to a log ring Q → A via the k [[Q]]-algebra structure, and thus to a log scheme S = Spec (Q → A), for which we write S ∈ Art Q by abuse of notation. A log smooth deformation over S ∈ Art Q is a Cartesian diagram where f is log smooth. Because S 0 → S is strict, it does not matter in which category of log schemes-of all, of fine, or of fine saturated log schemes-we consider Cartesianity. For technical reasons, we restrict to morphisms f 0 : X 0 → S 0 which are saturated, cf. Remark 10.2. This notion is also known as being of Cartier type; most strikingly, saturated and log smooth implies that the fibers of f 0 are reduced and Cohen-Macaulay, cf. [28] for more of its elementary properties. In case f 0 is saturated, the deformation f is saturated as well. Étale locally, such deformations exist uniquely up to (non-unique) isomorphism, cf. [14,Prop. 8.4].
Taking isomorphism classes of deformations yields the log smooth deformation functor which has a representable hull in case f 0 : X 0 → S 0 is proper by [14,Thm. 8.7].
The main result: a pre-DGLA controls L D X 0 /S 0 We provide the dgla in the log setting by translating ideas of Chan-Leung-Ma in [1] to our setting and bridging the remaining gap to log smooth deformation theory. To achieve this, k [[Q]]-linear dglas do not suffice. Namely, if L D X 0 /S 0 were controlled by such a dgla, then for every S ∈ Art Q , we would have a deformation over S corresponding to the trivial Maurer-Cartan solution η = 0, so there would exist a log smooth deformation over every base (which is in the classical setting the fiber product X × S). This is not the case-the spaces in [29,Thm. 3] are d-semistable normal crossing spaces all of whose flat deformations are locally trivial. Thus, by [14,Thm. 11.7], they can be endowed with a log smooth log structure over Spec (N → k), but they admit no log smooth deformation over e.g. Spec (N → k[t]/(t 2 )). Instead of dglas, we employ k [[Q]]-linear predifferential graded Lie algebras, a notion which essentially coincides with almost dgla in [1]. We define them and study systematically their basic properties in the Appendix 10. Since Def L • is a deformation functor with tangent space isomorphic to H 1 (X 0 , 1 X 0 /S 0 ), we also recover the existence of a hull in case f 0 is proper. In case Q = 0, the pdgla L • X 0 /S 0 is in fact a dgla. Here Theorem 1.1 is only a slight generalization of existing results. E.g. if D ⊂ X is a normal crossing divisor in a smooth variety, a dgla controlling divisorial deformations (which correspond to log smooth deformations here) has been studied in [17] and [10]. A variant of this are the compactified Landau-Ginzburg models of [18]. This paper uses the L ∞ -approach, which is related to our dgla by homotopy transfer results like [12,Thm. 3.2] going back to [13].

The strategy of the proof
A purely algebraic construction of a dgla controlling smooth deformations is used in [12], but the method is not sufficient to prove Theorem 1.1 because it relies on regluing the trivial smooth deformation. Instead, we prove the theorem in the following three steps: LD ∼ = GD. We study Gerstenhaber algebras of polyvector fields. Given a deformation f : X → S of f 0 : X 0 → S 0 , the exterior powers of the relative log derivations 1 X/S can be endowed with a Schouten-Nijenhuis bracket, giving rise to the Gerstenhaber algebra G • X/S . We view it as a deformation of the Gerstenhaber algebra G • X 0 /S 0 associated to the central fiber f 0 . After fixing an affine cover {V α } of X 0 , every deformation of G • X 0 /S 0 must be locally isomorphic to the Gerstenhaber algebra G • V α /S of the unique log smooth deformation of V α /S 0 (which we denote by V α /S by abuse of notation). It turns out that gluings of the G • V α /S (encoded by a deformation functor G D X 0 /S 0 ) are equivalent to actual log smooth deformations f : X → S, setting up an equivalence of deformation functors L D X 0 /S 0 ∼ = G D X 0 /S 0 .
The equivalence heavily depends on a careful study of automorphisms of log smooth deformations, which we carry out in Sect. 2. GD ∼ = TD. In the next step, the abstract setup of [1] enters. The Gerstenhaber algebra G • X 0 /S 0 corresponds to 0-th order data in [1,Defn. 2.9], and the G • V α /S correspond to the higher order data in [1,Defn. 2.13]. The patching data in [1,Defn. 2.15] correspond to isomorphisms on V α ∩V β which are induced by geometric isomorphisms of log smooth deformations. There is no canonical gluing of the G • V α /S (that would correspond to something like a trivial deformation). Instead, we perform a Thom-Whitney resolution; it yields a bigraded Gerstenhaber algebra TW •,• (G • V α /S ) with a non-trivial differential d : TW p,q → TW p,q+1 . Once we forget this differential, there is-up to isomorphism-a unique gluing PV X 0 /S of the affine patches TW •,• (G • V α /S ), essentially corresponding to [1,Lemma 3.21]. Our proof shows the cohomological principles behind the explicit-constructive proof in [1,Lemma 3.21].
Given a gluing of the G • V α /S , its Thom-Whitney resolution endows PV X 0 /S with a differential d : PV Conversely, every differential on PV X 0 /S gives a gluing of the G • V α /S by taking cohomology. This sets up an equivalence of deformation functors G D X 0 /S 0 ∼ = T D X 0 /S 0 where the latter classifies differentials. TD ∼ = Def. Relaxing the condition d 2 = 0 on a differential, we obtain the notion of predifferential. We prove that there is a compatible way to endow the PV X 0 /S with a predifferential, corresponding to∂ α + [d α , −] in [1,Thm. 3.34]. Once we have a predifferential, the defect of another predifferential to be a differential is measured by a Maurer-Cartan equation, corresponding to the so-called classical Maurer-Cartan equation in [1,Defn. 5.10]. This sets up the final equivalence T D X 0 /S 0 ∼ = Def L • . The geometricČech gluing in [1, 5.3] loosely corresponds to going back from a Maurer-Cartan solution to glue the G • V α /S . Our Lie algebra L • X 0 /S 0 is essentially the (−1, * )-part of the differential graded Batalin-Vilkovisky algebra PV * , * (X ) in [1, Thm. 1.1]. What is new in our paper, is that it actually controls log smooth deformations. The main technical difference is that whereas [1] uses a countable covering {U i } of X 0 and sticks to sections over these opens, we consequently stick to sheaves on X 0 . Therefore, we hope to help the algebraic geometer to get through the ideas of [1].

An example
In Sect. 9, we compute and discuss the k [[N]]-linear pdgla L • C 0 /S 0 of a proper log smooth curve f 0 : C 0 → S 0 over S 0 = Spec (N → k). The computation suggests that, in general, it is very difficult to compute L • X 0 /S 0 explicitly.

Outlook
Beyond the scope of this paper, it should be possible to use the techniques of [1,Thm. 1.2] to prove that in case f 0 : X 0 → S 0 is log Calabi-Yau, i.e., d it is possible to take = 0, i.e., L • X 0 /S 0 is actually a dgla. This is unobstructedness in the sense of [1]. Namely, then η = 0 is a Maurer-Cartan element, corresponding to a distinguished deformation over every S ∈ Art Q . In particular, one can construct a limit over k [[Q]] and ask for an algebraization of that formal scheme. However, this does not imply unobstructedness in the sense of a Bogomolov-Tian-Todorov theorem. Namely, this would require L D X 0 /S 0 (S ) → L D X 0 /S 0 (S) to be surjective for every thickening S → S in Art Q .
When we finished the first version of this article, we hoped to apply Theorem 1.1 to obtain a Bogomolov-Tian-Todorov theorem in log geometry. Namely, in [12], the classical Bogomolov-Tian-Todorov theorem is obtained by proving that the dgla controlling Def X is homotopy abelian. In fact it is possible-under suitable hypotheses-to adapt their methods and use the abstract Bogomolov-Tian-Todorov theorem of [11] to prove the central fiber L ] k homotopy abelian. This is now proven in our preprint [34] and shows a log Bogomolov-Tian-Todorov theorem in case Q = 0; it generalizes results on the deformation of pairs (X, D) in [17,Lemma 4.19] and [10]. Similarly, the Bogomolov-Tian-Todorov theorem for compactified Landau-Ginzburg models in [18] is a variant of this theme. The case Q = 0, which was completely open when we finished the first version of this article, is now settled in [34] with an argument in local algebra; it is essentially a consequence of the unobstructedness results by Chan-Leung-Ma in [1].
Our method is not bound to the log smooth case. E.g. it should also apply to the deformation theory of log toroidal families of [2], which are a generalization of log smooth families which does not need to be log smooth everywhere. The key missing step for the general situation is local uniqueness of deformations, which is only known for some types of log toroidal families, see [2,Thm. 6.13]. In particular, once the theory is established, we would get the existence of a hull of the log toroidal deformation functor. In less generality, the existence of a hull has been recently achieved by Ruddat-Siebert in [30,Thm. C.6] for divisorial deformations of toric log Calabi-Yau spaces. In this paper, we restrict ourselves to the log smooth setting for simplicity.
There has been recent interest in Gerstenhaber algebras of polyvector fields also in tropical geometry. In [23], integral polyvector fields on algebraic tori are studied in order to compute Gromov-Witten invariants via tropical curves. Moreover, it turns out that wall-crossing transformations, i.e., the gluing isomorphisms of the Gross-Siebert program, induce particularly simple operations on these polyvector fields. This gives a new tropical encoding of the Gross-Siebert gluing on the level of Gerstenhaber algebras analogous to our first isomorphism L D X 0 /S 0 ∼ = G D X 0 /S 0 .
We expect this to be related also to [21], where the refined scattering diagrams of [22] are related to Maurer-Cartan solutions in an appropriate dgla.
In [15], F. Kato introduces functors of log Artin rings as a more natural framework for infinitesimal log smooth deformation theory. The question which generalized dglas control these functors is subject to future studies.
Finally, it would be interesting how deformation quantization relates to log smooth deformation theory, relating the base k [[ ]] to k [[N]]. Deformation quantization of affine toric varieties, which are the building blocks of log smooth morphisms, has been achieved recently in [3].

Infinitesimal automorphisms
The relationship between automorphisms of first order deformations and derivations is well-known. In this section, we determine higher infinitesimal automorphisms of deformations in terms of derivations, i.e., automorphisms of higher order deformations. In [12], higher infinitesimal deformations play a crucial role in proving that classical smooth deformations are controlled by a dgla, and we need them as well. Assume we have a surjection A → A in Art Q with kernel I ⊂ A inducing a thickening S → S , and assume a Cartesian diagram of saturated log smooth morphisms. The ideal sheaf of X ⊂ X is because f is flat (as is every integral log smooth morphism). Elements of the sheaf of automorphisms Aut X / X of X which are compatible with f : X → S and induce the identity on X are pairs (φ, ) where are the constituting homomorphisms. As we will see below, the sheaf of groups Aut X / X is isomorphic to the sheaf Der X /S (I) of relative log derivations (D, ) with values in I, i.e., D : O X → I is a derivation and : M X → I is its log part. This is a sheaf of Lie algebras as a subalgebra of 1 X /S which is filtered by for k ≥ 1, the sheaf of derivations with values in I k ⊂ I. The ideal I is generated by elements in the image of then D(I ) ⊂ I k+ . This shows [F k , F ] ⊂ F k+ , so the lower central series of Der X /S (I) is eventually 0, and it is a sheaf of nilpotent Lie algebras. In particular, following e.g. [25], the Baker-Campbell-Hausdorff formula starting with turns Der X /S (I) into a sheaf of groups. The sheaves Der X /S (I) and Aut X / X have classical analogs, which we denote by Der X /S (I) and Aut X / X . They are the classical relative derivations with values in I respective the automorphisms of the underlying scheme X over S . Using the Baker-Campbell-Hausdorff formula again, we find Der X /S (I) a sheaf of groups. There are group isomorphisms Aut X / X log given by plugging in the derivation D : O X → I into the power series expansion of the exponential (at 0) respective the automorphism φ : O X → O X into the expansion of the logarithm (at 1).
Remark 2.1. This pair of inverse group isomorphisms seems to be folklore; it is used e.g. implicitly in [12,Thm. 5.3]. However, we are not aware of any explicit reference where it has been proven in the setting of sheaves of automorphisms of flat deformations. That the construction above yields maps between Der X /S (I) and Aut X / X follows from our explicit computation below; that they are inverse to each other follows from the fact that the two power series exp(T ) and log(1 + T ) are inverse to each other in Q [[T ]]-this argument has been used e.g. in [32,Thm. 7.2]. The maps are homomorphisms, heuristically, because the Baker-Campbell-Hausdorff formula just makes explicit the product we get when composing the exponentials as endomorphisms.
We extend the above picture to the diagram where the vertical arrows are the forgetful maps. Given (D, ) ∈ Der X /S (I), we define (φ, ) = exp X / X (D, ) by the formulas where the series are actually finite sums. The symbol (m) denotes the multiplication operator with this element.
That is a monoid homomorphism follows from the identity which is proven by induction, and the identity (1) shows that α • = φ • α, i.e., (φ, ) defines a log morphism The map is an isomorphism because it is one on underlying schemes and on ghost sheaves, and because M X is fine.
We construct an ansatz for the inverse log X / X . On the classical part, given φ, our ansatz n is the formula of the logarithm. The sum is finite since [φ − Id](I k ) ⊂ I k+1 ; this identity holds by the explicit formula for φ − Id because D(I k ) ⊂ I k+1 . Given , by induction where k (m) − m ∈ M * X is the unique invertible by which these elements differ, so we make an ansatz n for the log part. Setting (m) − m =: v ∈ M * X , we find α(v) ∈ 1 + I; indeed, since | X = Id X , we have v| X = 0, and thus α(v)| X = 1. Now inductively σ n (α(v)).

Lemma 2.3.
We have log X / X (φ, ) ∈ Der X /S (I) where log X / X is defined by the above formulas.
Proof. Given (φ, ) let (D, ) be defined by the formulas. Setting and thus e l s e and therefore by induction starting with σ 0 = 1, so we get The forgetful maps ι D , ι A in diagram (1) above are injective group homomorphisms. For, if V := (X ) str ⊂ X is the strict locus defined in the "Appendix", then ι D and ι A are isomorphisms on V . Because and I ⊂ O X has injective restrictions to V (which is a direct consequence of Proposition 10.1), we get ι D injective. If we have two automorphisms (φ, ) and (φ, ), then and coincide on ghost sheaves-this is because the restriction map M X → M X → M X 0 of a log deformation is an isomorphism-so there is an invertible u with (m) = u + (m). Because ι A | V is an isomorphism, we have u| V = 0, and thus u = 0. We conclude:

Proposition 2.4. The maps
Remark 2.5. The case of first order deformations, i.e., I 2 = 0, has been described in [6,Lemma 2.10]. Indeed, this result has been our main inspiration to find the description of infinitesimal automorphisms of higher order. Automorphisms of all orders of log rings have been studied in [7, Prop. 2.14].

Gerstenhaber algebra of polyvector fields
Given S ∈ Art Q , let f : X → S be log smooth and saturated. The polyvector fields G • X/S := −• 1 X/S in negative grading, i.e., concentrated in degrees −d ≤ • ≤ 0 for d the relative dimension, form a Gerstenhaber algebra. This means it is endowed with two bilinear operations ∧ and [−, −], which are graded in the sense that (for They satisfy the relations where |x| denotes the degree of the homogeneous element x. For the bracket [−, −] we take the negative −[−, −] sn of the Schouten-Nijenhuis bracket. Recall that the latter one is the unique bracket satisfying the above relations and such that is the Lie bracket for vector fields θ, ξ ∈ 1 X/S , and such that [θ, g] sn = dg, θ for the natural pairing ·, · of vector fields and differential forms. Our grading convention as well as the (−1)-sign in the bracket follow [1].
In the spirit of [1, Defn. 2.9], we say a Gerstenhaber algebra is injective. Using Proposition 10.1 of the "Appendix", we find that G • X/S is (−1)injective. Indeed, on the strict locus X str ⊂ X the sheaf G −1 X/S represents classical relative derivations on O X .

The Gerstenhaber algebra and deformations
Let S → S be a thickening given by a surjection A → A in Art Q , and let f : X → S be a deformation of f : X → S. Let I X / X = I · O X be the ideal sheaf. We obtain an exact sequence of Gerstenhaber algebras. Here the right hand homomorphism is induced by pulling back homomorphisms h : 1 which are well-defined Gerstenhaber algebra automorphisms over G • X/S since I X / X is nilpotent. If exp θ = exp ξ , then θ = ξ due to (−1)-injectivity. Indeed, with the Baker-Campbell-Hausdorff product on I X / X · G −1 X /S the map θ → exp θ becomes a group homomorphism, and if exp θ = Id, then θ = 0 by induction since it holds for small extensions S → S .
Automorphisms of G • X /S can be induced by geometric automorphisms as well. Namely, for two deformations f i : X i → S (i = 1, 2) and an isomorphism ϕ : and thus to an element Given a deformation f : X → S of f : X → S, an automorphism ϕ ∈ Aut X / X induces-on the one hand side-a map T ϕ : so it remains to prove equality on for the action of ϕ on differential forms. Writing ϕ = (φ, ) as in Sect. 2, we find for ξ ∈ G −1 which is proven easily by induction. Since G • X /S is (locally) generated by G −1 X /S as a ring with respect to ∧, we see T ϕ = exp −θ . For the second statement, we use that log X / X is a bijection and that G • X /S is (−1)-injective.

Deformations of Gerstenhaber algebras
Given f 0 : X 0 → S 0 , we study deformations of the Gerstenhaber algebra G • X 0 /S 0 , i.e., the deformation functor G D X 0 /S 0 , which we introduce below. We fix an affine cover V = {V α } of X 0 and we denote the-up to non-unique isomorphism-unique log smooth deformation of V α over S by V α → S (by abuse of notation with the same symbol for all S). On overlaps, we choose isomorphisms Gerstenhaber algebras contravariantly (by our construction in Sect. 3). Whenever we can compose these isomorphisms to an automorphism, it is a gauge transform by Lemma 3.2.
A log smooth deformation f : X → S induces a Gerstenhaber deformation G • = G • X/S . For χ α we take the maps (contravariantly) induced by any geometric isomorphism X | V α ∼ = V α . The different choices give rise to isomorphic Gerstenhaber deformations with the identity on G • as isomorphism. This induces a natural Conversely, let G • be a Gerstenhaber deformation over S, and let γ αβ = φ αβ • χ α •χ −1 β be the gauge transform of Definition 4.1. It is induced by an automorphism Note that the (contravariantly) induced map on Gerstenhaber algebras is χ β • χ −1 α , so they satisfy the cocycle condition by Lemma 3.2. Gluing yields a log smooth deformation whose image in G D X 0 /S 0 (S) is G • .

Thom-Whitney resolutions
In this section, we briefly review the Thom-Whitney resolution. These resolutions are acyclic resolutions of complexes that are adapted to preserve additional algebraic structures such as ∧ and [−, −] of a Gerstenhaber algebra; therefore, they have been employed in [12] and [1] to study deformations. In their present form they first occur in [26]. Their construction starts from a semicosimplicial complex V . Recall that, denoting mon the category of sets [n] = {0, 1, . . . , n} for n ≥ 0 with morphisms the order-preserving injective maps, a semicosimplicial object in a category C is a covariant functor C : mon → C. Thus, a semicosimplicial object is a diagram with the usualČech maps, and with F( is a Gerstenhaber algebra, and theČech maps are morphisms thereof.
For each n, the differential forms on {t 0 + · · · + t n = 1} ⊂ A n+1 form a differential graded commutative algebra is a complex-we use the maps δ k,n together with the coface maps ∂ k,n of V to define homomorphisms δ k,n ⊗ Id : of vector spaces. Following [4], the Thom-Whitney bicomplex has graded pieces with differentials given by The Thom-Whitney complex Tot TW (V ) = Tot(C •,• TW (V )) is its total complex. The construction is functorial for homomorphisms of semicosimplicial dg vector spaces, and it is exact by [26,Lemma 2.4 For later use, we prove a base change result on the Thom-Whitney construction. We say V is bounded if V • n = 0 for n > > 0.

Lemma 5.2. Let R → S be a finite type homomorphism of k-algebras, and let V be a bounded semicosimplicial complex of R-modules. Then the canonical map
x n ] is a polynomial ring and T → S is surjective-it suffices to prove the statement for R → S either flat or surjective. For the flat case, first note that the product in the definition of C i, j TW is actually a finite direct sum because V is bounded. Now C i, j so its formation commutes with flat tensor products. For the surjective case, let I ⊂ R be the kernel. Because C i, j TW is exact, we find a diagram with exact rows. Then p S is surjective, and a similar argument using a surjection R ⊕m → I shows p I surjective as well. In particular, we find p S bijective.

Thom-Whitney resolutions on schemes
We globalize the Thom-Whitney construction to schemes. This has been done in [9], but we recall it here for convenience.
is a sheaf, products of sheaves are sheaves, and taking elements that satisfy an equation preserves the sheaf condition.
As above, we say F is bounded if F • n = 0 for n > > 0. In case F is a bounded semicosimplicial complex of quasi-coherent O X -modules, also C i, j TW (F ) is a quasi-coherent O X -module because the infinite product in the construction is actually finite.
for a thickening c : Y → X , and hence the natural map is an isomorphism.
For a complex of sheaves F • , the map induces a quasi-isomorphism F • → Tot TW (F • (U)); this is shown by postcomposing with the integration quasi-isomorphism to theČech complexČ • (U, F • ).
where the product runs over all (finitely many) ordered tuples (i 0 , . . . , i n ). Given for theČech differential, so we can work in (A P L ) i n ⊗Č • (U, F j (U i 0 ...i n )) (notation from Example 5.1) for each tuple (i 0 , . . . , i n ) separately. Because the complex does not have cohomology in degree = 0, we can find elements (4) (when we plug it in on the left). For the induction step, let K n+1 be the kernel ofČ Lemma 3.5]). We assemble these spaces into a diagram with exact columns and surjective rows. Assuming (v • ) to be constructed up to order n, we need to find Proof. We have a factorization A → A [x 1 , . . . , x n ]/(x m 1 1 , . . . , x m n n ) → A in Art Q with the second map surjective, so it suffices to prove the statement for A → A either flat or a small extension. The flat case is by flat base change. In the small extension case, let I ⊂ A be the ideal. Then we have I · C ∼ = (C ⊗ A k) ⊗ k I due to flatness, so Proposition 5.6 shows H 1 (X , I · C) = 0; thus, the result follows from the long exact sequence in sheaf cohomology.

The Thom-Whitney Gerstenhaber algebra
We perform a Thom-Whitney resolution of the Gerstenhaber algebras G • V α /S of polyvector fields and glue them canonically to a global sheaf of (bigraded) Gerstenhaber algebras PV X 0 /S . This sheaf depends on X 0 and the base S, but not on a deformation f : X → S, thus the notation "X 0 /S". The gluing is inspired by the gluing construction in [1, 3.3].
Let G • be a Gerstenhaber k-algebra on a (separated) finite type k-scheme X , and let U be an open affine cover. To perform a Thom-Whitney resolution, we turn G • into a complex of coherent sheaves by endowing it with the differential 0. We set TW p,q (G • ) := C q, p TW (G • (U)); the switch of indices is on purpose to fit the conventions of [1]. The operations turn it into a bigraded Gerstenhaber algebra TW •,• (G • ) (or TW •,• for brevity), i.e., we have TW p,q ∧ TW i, j ⊂ TW p+i,q+ j and [TW p,q , TW i, j ] ⊂ TW p+i+1,q+ j as well as the usual relations with respect to the total degree p + q. The differential d : (5) and d 2 = 0 whereas the other differential TW p,q → TW p+1,q of this Thom-Whitney bicomplex is 0. We say that its total complex TW • (G • ) := Tot TW (G • (U)) is a (strongly) differential Gerstenhaber algebra. The quasi- is a functorial acyclic resolution of differential Gerstenhaber algebras. In particular, we can recover G • as the cohomology sheaves H • (TW • (G • )) which form a Gerstenhaber algebra for every differential Gerstenhaber algebra. Sometimes we write H • (TW •,• ) for the cohomology; then taking the total complex is implicit-cohomologies are, for us, always singly graded. However, in bidegree ( p, q) with q > 0, the cohomology of d : TW p,q → TW p,q+1 is 0 anyway. Remark 6.1. Do not confuse our notion of differential Gerstenhaber algebra with the one of e.g. [20] which is more closely related to Batalin-Vilkovisky algebras. Example 6.2. Let f : X → S be a log smooth and saturated deformation of the space f 0 : X 0 → S 0 . Then we have a resolution G • X/S → TW • (G • X/S ). If we deform further to f : X → S , then applying TW(−) to the exact sequence (2) we obtain the exact sequence is flat over S and compatible with base change. If exp θ : G • X /S → G • X /S is a gauge transform (relative to f : X → S), then the induced automorphism TW(exp θ ) is the gauge transform defined by the element via the formula in (3).
(−1)-injectivity is preserved by the Thom-Whitney construction. This is important for Lemma 6.6 below.

Lemma 6.3. Let G • be a (−1)-injective Gerstenhaber algebra. Then for a non-zero
Proof. We work over an arbitrary open V ⊂ X . Following the description of C i, j TW in Lemma 5.6, the element θ ∈ TW −1, j (G • ) is given by a family (θ i 0 ...i n ). After denoting {t μ } some basis of (A P L ) j n , we can decompose θ i 0 ...i n = μ a μ t μ ⊗θ We apply this construction to the Gerstenhaber algebras G • V α /S and obtain differential Gerstenhaber algebras TW • (G • V α /S ) as well as isomorphisms TW(φ αβ ) on overlaps. Every cocycle

is a gauge transform by an element in
When gluing them as differential Gerstenhaber algebras, this induces a Gerstenhaber deformation (Definition 4.1) by taking cohomology. As a step towards that, our goal is to glue them as a bigraded Gerstenhaber algebra, i.e., without differential (and keeping the bigrading). In general, gauge transforms are not compatible with the differentials, so there is no canonical differential on T •,• coming out of the data. The canonical example of a TWG deformation is the following.
• Given a lifting T •,• on S , the isomorphism classes of liftings are in • The obstructions to the existence of a lifting are in Since TW −1,0 (G • X 0 /S 0 ) is acyclic by Lemma 5.6, there is indeed a unique lifting up to isomorphism. This means for every S, there is up to isomorphism a unique TWG deformation PV X 0 /S (depending on S and the morphism f 0 : X 0 → S 0 , but no further data) which we call the Thom-Whitney Gerstenhaber algebra. In fact, PV X 0 /S has many automorphisms, so there is no canonical choice. Thus we fix once and for all one PV X 0 /S for every S ∈ Art Q . The notation PV is taken from [1]; presumably, PV stands for "polyvector fields" since it is obtained by resolving the Gerstenhaber algebra of polyvector fields. into the sheaf of lifting automorphisms. It is injective due to Lemma 6.3. By the very definition of an automorphism of TWG deformations, every ϕ ∈ Aut T /T is (locally) induced by some θ ∈ m · T −1,0 where m ⊂ A is the maximal ideal.
Because the induced automorphism on T •,• is the identity, we have indeed that θ ∈ I ·T −1,0 , so exp T /T is an isomorphism. Finally we have TW −1,0 (G • X 0 /S 0 )⊗ k I ∼ = I · T −1,0 , so the result follows by the standard methods that are developed for smooth deformations in [8,III]. Proof. This follows by induction on the length of A, breaking the extension into small extensions.
Remark 6.8. If S → S is a map in Art Q corresponding to a homomorphism A → A of Artin rings, then PV X 0 /S ⊗ A A is a TWG deformation on S. In particular, there is an isomorphism to PV X 0 /S . Moreover, we have an induced homomorphism PV X 0 /S → PV X 0 /S ⊗ A A. However, there is no canonical homomorphism PV X 0 /S → PV X 0 /S since there is no preferred isomorphism PV X 0 /S ⊗ A A ∼ = PV X 0 /S . This is because PV X 0 /S has many automorphisms. We see that S → PV X 0 /S is not functorial in a strict sense.

Differentials on TWG deformations
After constructing a unique TWG deformation PV X 0 /S , we now study differentials d : PV where T ([θ, −]) means to plug in the operator [θ, −] into the power series expansion of T (x) = exp(x)−1 x . This means under a gauge transform, the differential d is transformed into something of the form d + [η, −] for η ∈ I X/ X 0 · TW −1,1 (G • X/S ). Hence every differential on PV X 0 /S should be locally of this form. More formally we define: Definition 7.1. Let T •,• be a TWG deformation. Then a predifferential is a map d : T p,q → T p,q+1 that satisfies (5), that is compatible with the differential on TW •,• (G • X 0 /S 0 ), and such that for some η α ∈ I · T −1,1 | V α where d α is the differential on TW •,• (G • V α /S ). It is a differential if d 2 = 0. We denote the set of predifferentials by PDiff(T •,• ) and the set of differentials by Diff(T •,• ). Two predifferentials d 1 , d 2 are gauge equivalent, if there is an automorphism ψ : Example 7.2. Let G • be a Gerstenhaber deformation. Then the differential of TW •,• (G • ) is a differential in the above sense. Since TW •,• (G • ) ∼ = PV X 0 /S , it induces a differential on PV X 0 /S whose gauge equivalence class does not depend on the chosen isomorphism.
Every restriction PV X 0 /S → PV X 0 /S induces a restriction Diff(PV X 0 /S ) → Diff(PV X 0 /S ) on differentials. On gauge equivalence classes, this restriction is independent of the chosen map PV X 0 /S → PV X 0 /S , so we obtain a functor If η α ∈ I S · T −1,1 such that d α +[η α , −] is a differential, then the restrictions to T are gauge equivalent. The gauge transform can be lifted to T , giving a gauge equivalence of d α +[η α , −] and d α + [η α , −] for some η α with η α | S = 0, the latter being gauge equivalent to d α by the above argument.
Given a differential d ∈ Diff(PV X 0 /S ), the isomorphism is compatible with differentials. By Lemma 7.3 we can further compose with a gauge transform to (TW •,• (G • V α /S ), d α ). Thus taking cohomology yields a Gerstenhaber deformation H • (PV X 0 /S , d) whose isomorphism class does not depend on the chosen gauge transform (namely, every gauge transform that leaves the differential invariant descends to a gauge transform on cohomology). Likewise, if d 1 and d 2 are gauge equivalent differentials, the induced Gerstenhaber deformations are isomorphic. We find a natural transformation h : T D X 0 /S 0 ⇒ G D X 0 /S 0 which is inverse to the natural transformation tw constructed above. Proof. Given a Gerstenhaber deformation G • , we have a resolution G • → TW • (G • ), so h•tw = Id. It thus suffices to prove that h is injective. Let d 1 , d 2 be two differentials and let ψ : H • (PV X 0 /S , d 1 ) ∼ = H • (PV X 0 /S , d 2 ) be an isomorphism. Fix the structure of Gerstenhaber deformation on the cohomologies by choosing isomorphismsχ α,i : as in the construction of h. The map ψ induces a gauge transform exp θ α : G • V α /S → G • V α /S which has a unique lift to a gauge transform expθ α : Again by uniqueness they glue to a global isomorphism, so d 1 , d 2 are gauge equivalent.

Maurer-Cartan elements
We construct a k [[Q]]-linear pdgla L • X 0 /S 0 which controls T D X 0 /S 0 and thus log smooth deformations. To this end, we relate differentials on PV X 0 /S to an appropriate Maurer-Cartan equation.
The predifferentials PDiff(PV X 0 /S )-considered as a sheaf on X 0 -form an I S · PV −1,1 X 0 /S -torsor for its additive group structure. Indeed, for η ∈ I S · PV −1,1 Given a differential d ∈ Diff(PV X 0 /S ) and η ∈ I S · PV −1,1 X 0 /S , we find The element ∈ PV −1,2 X 0 /S is unique with that property by Lemma 6.3, so indeed for every predifferential d there is a unique = (d) ∈ PV −1,2 X 0 /S with d 2 = [ , −] (because locally we can compare it to a differential). Now if d is a predifferential, then Our next goal is to construct the k , let m Q ⊂ A be the maximal ideal, let A k := A/m k+1 Q , and let S k := Spec (Q → A k ). Then S k → S k+1 is a small extension, so after choosing restrictions PV X 0 /S k+1 → PV X 0 /S k we choose inductively compatible predifferentials d k on PV X 0 /S k by Lemma 8.1. For S ∈ Art Q , there is at most one morphism S → S k , and there is a minimal k such that Hom(S, S k ) = ∅. For this minimal k, we choose a restriction PV X 0 /S k → PV X 0 /S and we endow PV X 0 /S with the restricted predifferential d k | S . The construction yields an O(S)-pdgla which is functorial for morphisms S → S (where ≥ k). In particular, using Corollary 5.7, we get exact sequences on the level of global sections where the right hand map is compatible with d and . The limit is an A-pdgla since its pieces are complete by [33, 09B8] (note the index shift).
is an isomorphism, so after restriction along S → S k and using Corollary 5.7, we obtain an isomorphism which is functorial with respect to S → S . Maurer-Cartan elements η ∈ I S · L 1 (S) induce differentials d η ∈ Diff(PV X 0 /S ) by Corollary 8.2. If η, η ∈ I S · L 1 (S) are gauge equivalent via θ ∈ I S · L 0 (S), i.e., we have d η • exp θ = exp θ • d η on L • (S), then d η , d η are gauge equivalent on PV X 0 /S . Indeed, by Corollary 8.2 we can find a unique ξ ∈ I S · L 1 (S) such that on PV X 0 /S . Restricting the equation to L • (S) ⊂ PV X 0 /S , we find η = ξ by Lemma 6.3. We get a well-defined transformation mc on the level of objects. It is injective because every automorphism of PV X 0 /S is a gauge transform, and it is surjective by Corollary 8.2 and the fact that d η is a differential if and only if η is Maurer-Cartan. mc is a natural transformation, i.e., compatible with maps S → S , because we can always find a restriction PV X 0 /S → PV X 0 /S which fits into a commutative diagram with the chosen horizontal maps. Finally, recall that on T D X 0 /S 0 the restriction is independent of the choice for PV X 0 /S → PV X 0 /S . Using Proposition 10.5, we recover that L D X 0 /S 0 is a deformation functor, which was first proved in [14]. We find that the tangent space t L D is isomorphic to , so in case f 0 : X 0 → S 0 is proper, we recover that L D X 0 /S 0 has a hull by Schlessingers' criterion [31,Thm. 2.11].

Example: a log smooth curve
In this section, we compute the pdgla L • C 0 /S 0 for a log smooth curve f 0 : C 0 → S 0 ; though not all intricacies of L • X 0 /S 0 show up here, it gives a first impression of how the pdgla looks like. As underlying space, we take i.e., three pairwise intersecting lines. It carries an obvious log structure of embedding type (see [14, 11.4]) via the embedding C 0 ⊂ P 2 k ; however, it does not admit a global section (at least we cannot find it) which would yield a log smooth log morphism f 0 : C 0 → S 0 := Spec (N → k). Instead, we turn C 0 into a log smooth curve as follows: The standard open covering of P 2 k induces an open covering of C 0 with three opens Each open V α has a deformation C α → S := A 1 t ; more precisely, we have three log smooth families We define C x | xy := {y = 0} ⊂ C x etc.-then we have isomorphisms of families of log schemes. These are not the standard isomorphisms which yield P 2 ; in fact, they do not satisfy the cocycle condition on the triple intersection, and thus, they cannot be glued to a family. However, unlike the standard isomorphisms, they are compatible with the map to the base. Thus, after restriction to C 0 , they glue to a log smooth curve f 0 : C 0 → S 0 because on C 0 , the cocycle condition is empty-there is no triple intersection.
We compute the Gerstenhaber algebra G • C 0 /S 0 of polyvector fields; it is concentrated in degrees 0 and −1. The sheaf of absolute log differentials 1 C 0 is on the open cover {V α } given by is the direct sum on formal generators specified in the bracket; the transition maps are given by The image of f * 0 1 S 0 → 1 C 0 is freely generated by the sum of the two generators on every V α , e.g. dy y + dz z on V x ; setting we get a global section ω ∈ (C 0 , 1 C 0 /S 0 ) of the quotient, which defines a trivialization O C 0 ∼ = 1 C 0 /S 0 , 1 → ω, of the log canonical bundle. In particular, the relative tangent bundle G −1 C 0 /S 0 = 1 C 0 /S 0 is trivial as well; under the natural embed- (which forgets the log part of the derivation and the fact that it is relative), the dual of ω is the derivation given by where O V x /S is the structure sheaf of C x × S S-i.e., of the deformation V x /S-and where ∂ yz must be interpreted in the respective ring. Our isomorphisms C x | xy ∼ = C y | xy etc. above induce the required isomorphisms αβ : Because the cocycle condition is empty again, we can actually glue the pieces to a Gerstenhaber deformation G • C 0 /S := O C 0 /S · 1 ⊕ O C 0 /S · ∂ in the sense of Definition 4.1.
Next, we perform the Thom-Whitney resolution of G • C 0 /S with respect to the cover U := {V α }; this yields a TWG deformation together with a differential on it, not only a predifferential. In particular, this computes PV C 0 /S ∼ = TW •,• (G • C 0 /S (U)). Since the Gerstenhaber deformations G • C 0 /S are compatible for all S ∈ Art N , we also get restriction maps between the different PV C 0 /S and compatible differentials. Thus, in order to obtain we only need to compute the global sections of the Thom-Whitney resolution.
After ordering the index set as x < y < z, the global sections of the semicosimplicial sheaf of Gerstenhaber algebras δ 0 (z, x, y) = (1/z, t/x, 1/y), δ 1 (z, x, y) = (t/z, 1/x, t/y), and the same for the 1 -row. The relevant pieces of the semisimplicial dgca A P L are the two non-trivial relevant face maps (A P L ) 0 1 → (A P L ) 0 0 are t 0 → 0 and t 0 → 1. We find the last isomorphism is due to the fact that we can choose the G 0 0 -part and the t i 0 -parts for i ≥ 2 arbitrarily whereafter the remaining two parts are uniquely determined by the equations. To write down the limit TW 0,0 ∞ := lim ← − TW 0,0 k , let us introduce notation then we have Since (A P L ) 1 0 = 0, the next piece is finally, By construction, (TW 0,0 ∞ , ∧) is canonically a subring of and ( p 1 , q 1 , In particular, the unit of the ring is (1, 1, 1, 0, 0, 0); now, TW −1,0 ∞ is a free TW 0,0 ∞module of rank 1 with generator ∂ := 1 · ∂. The operator [∂, −] acts as a derivation on TW 0,0 ∞ , and it acts on TW 0,1 ∞ . Via this action, we describe the Lie bracket on the graded Lie algebra L • C 0 /S 0 = TW −1,• ∞ ; namely, for a, b ∈ TW 0,0 ∞ , we have Since d is a differential, we have = 0; the k [[t]]-linear pdgla L • C 0 /S 0 is actually a dgla in this example. . Namely, f 0 : C 0 → S 0 is log smooth and log Calabi-Yau, so we expect the deformation functor to be represented by a smooth k [[t]]-algebra; its fiber over k should have as many variables as the tangent space Acknowledgements This paper is a byproduct which arose from studying [1] in order to apply it in [2]. I owe many ideas of the present paper to [1]. I thank my collaborator Matej Filip of [2] for pointing me to the subject and especially the idea to control logarithmic deformations by some sort of dgla. I thank my PhD advisor Helge Ruddat for encouraging this work and for constant support. I thank Ziming Ma for valuable comments on a first draft. I thank Bernd Siebert for help with the introduction. I thank the anonymous referee for careful reading and many helpful recommendations. Finally, I thank JGU Mainz for its hospitality and for funding the open access publication, and Carl-Zeiss-Stiftung for financial support.
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The strict locus of log smooth morphisms
We include the result below for which we have, for the general case, no published reference. For an s-injective morphism f : X → S of fs log schemes-s-injective means that the induced map M S, f (x) → M X,x on ghost stalks is injectiveconsider the strict locus X str ⊂ X , i.e., the union of all opens U ⊂ X such that f | U : U → S is strict. It is the maximal open subset with that property. Since X, S are fine, f is strict on U if and only if φ : f −1 (M S ) → M X is an isomorphism on U . For a geometric pointx ∈ X where φx is an isomorphism on stalks, φ is surjective in an étale neighbourhood ofx due to coherence (since there every stalk is a quotient of M X,x ). s-injectivity shows it is an isomorphism. Thusx ∈ X str , and the converse is clear. We see that the formation of X str commutes with base change along strict morphisms T → S. Remark 10.2. The density statement fails if we assume f : X → S only integral, but not saturated. Consider e.g. Spec (N → C[t]/(t 2 )) → Spec (N → C) induced by N → N, 1 → 2, which is log smooth and integral, but nowhere strict. Moreover, we do not know if the injectivity statement remains true. This is the reason why we restrict the whole paper to saturated log smooth morphisms.

Remark 10.3.
In the case S = Spec (0 → k), Proposition 10.1 is (in a weak sense) equivalent to [27,Prop. 2.6]. Namely, if f : X → S is log smooth and saturated, then it is log-regular, and the log-trivial locus X tr equals the strict locus X str . The former is dense by [27,Prop. 2.6], so the latter is dense; since X is reduced, X str is scheme-theoretically dense as well. Conversely, if X is a log-regular fs log scheme of finite type over S, then the (structure) morphism f : X → S is saturated and log smooth by [28, IV, Thm. 3.5.1]. Proposition 10.1 implies that X str ⊂ X is scheme-theoretically dense, so X tr = X str is dense.

-linear predifferential graded lie algebras
Fix a complete local Noetherian ring with maximal ideal m ⊂ . We briefly introduce the type of Lie algebra which controls log smooth deformations. This type of Lie algebra is called almost dgla in [1, Thm. 1.1] where the existence of such a dgla is proven under some abstract conditions. For a local Artin -algebra A (with residue field k := /m ), the tensor product L • ⊗ A is an A-pdgla since L i ⊗ A is complete. We consider them as some sort of infinitesimal deformation of the dgla which we call the central fiber. In fact, since ∈ m · L 2 , it is a dgla (d 2 = 0). We say L • is faithful if for all A ∈ Art and all ξ ∈ L • ⊗ A, we have that [ξ, −] = 0 implies ξ = 0. In a faithful L • , the equality [ξ, −] = 0 implies ξ = 0 also for ξ ∈ L • . Elements θ ∈ m · L 0 give rise to gauge transforms which are well-defined since L • is complete. We find the identities Proof. The proof does not employ new ideas beyond the classical situation (see e.g. [24]), so we give only a brief indication. Let A → A and A → A be surjections in Art . We consider the canonical map for any functor F of Artin rings. Recall that homogeneous means σ is an isomorphism. It suffices to prove this for a small extension A → A. In this case, B := A × A A → A is a small extension as well with the same kernel I , so we obtain a diagram 0 with exact rows. When we use the fact that the two kernels are equal, a straightforward diagram chase yields the result. To prove MC L • ⇒ Def L • smooth, we use that every gauge transform can be lifted to some gauge transform because L 0 ⊗ A → L 0 ⊗ A is surjective. To prove that Def L • is a deformation functor, we use the map MC L • ⇒ Def L • and that, when A = k, the restriction of a gauge transform on L • ⊗ A to A is trivial.
Remark 10.6. The functor MC L • is a deformation functor (like every homogeneous functor), but Def L • is not homogeneous. Namely, a deformation functor is prorepresentable if and only if it is homogeneous and has finite-dimensional tangent space, but e.g. the flat deformation functor of some surfaces is not prorepresentable (which is certainly an example of our theory).
If we replace d by d η and by the corresponding η , then the deformation functor remains unchanged. Taking this into account, we propose the following notion of homomorphism for future study of pdglas. A homomorphism induces a map Def(ψ) sending η → ψ(η) − κ ψ .