Log Smooth Deformation Theory via Gerstenhaber Algebras

We construct a $k[[Q]]$-linear predifferential graded Lie algebra $L^*_{X/S}$ associated to a log smooth and saturated morphism $f: X \rightarrow S$ and prove that it controls the log smooth deformation functor. This provides a geometric interpretation of a construction by Chan-Leung-Ma whereof $L^*_{X/S}$ is a purely algebraic version. Our proof crucially relies on studying deformations of the Gerstenhaber algebra of polyvector fields and interestingly does not need to keep track of the log structure. The method of using Gerstenhaber algebras 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 [14].
In this paper, we establish an analogue of the above dgla in logarithmic geometry, replacing the smooth variety X by a log smooth morphism. Log smoothness is the analogue of the classical notion of smoothness and has been introduced in [18]. It allows to treat certain singularities as if they were smooth, a technique which has proven fruitful in smoothing normal crossing spaces, e.g. by Kawamata-Namikawa in [21], and in mirror symmetry, e.g. by Gross-Siebert in [9]. Infinitesimal log smooth deformation theory was established by F. Kato in [16]. 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 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 fs log schemes -we consider Cartesianity. For technical reasons we restrict to f 0 ∶ X 0 → S 0 which are saturated, cf. Remark 9.2. This notion is also known as being of Cartier type; most strikingly, saturated and log smooth implies reduced Cohen-Macaulay fibers, cf. [27] 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. [16,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 [16,Thm. 8.7]. We provide the dgla in the log setting by translating ideas of Chan-Leung-Ma in [2] 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 LD X0 S0 were controlled by such an dgla, then for every S ∈ Art Q we had 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 [28,Thm. 3] are d-semistable normal crossing spaces which admit only locally trivial flat deformations. Thus by [16,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 [2]. We define them and study systematically their basic properties in the Appendix 9.2. They are graded Lie algebras (L • , [−, −]) over k Q endowed with a derivation d which need not be a differential, but which admit an element ℓ ∈ L 2 with d 2 = [ℓ, −]. This allows us to define an associated deformation functor  Since Def L • is a deformation functor with tangent space isomorphic to H 1 (X 0 , Θ 1 X0 S0 ), we also recover the existence of a hull in case f 0 is proper. In case Q = 0, the pdgla L • X0 S0 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 [19] and [12]. A variant of this are the compactified Landau-Ginzburg models of [20]. This paper uses the L ∞ -approach which is related to our dgla by homotopy transfer results like [14,Thm. 3.2] going back to [15].
A purely algebraic construction of a dgla controlling smooth deformations is used in [14], but the method is not sufficient to prove Theorem 1.1 because it relies on regluing the trivial smooth deformation. Instead 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 Θ 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 • X0 S0 associated to the central fiber f 0 . Fixing an affine cover {V α } of X 0 , any deformation of G • X0 S0 must be locally isomorphic to the Gerstenhaber algebra G • Vα 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 • Vα S (encoded by a deformation functor GD X0 S0 ) are equivalent to actual log smooth deformations f ∶ X → S, setting up an equivalence of deformation functors LD X0 S0 ≅ GD X0 S0 . The equivalence heavily depends on a careful study of automorphisms of log smooth deformations which we carry out in Section 2.
In the next step the abstract setup of [2] enters. The Gerstenhaber algebra G • X0 S0 corresponds to 0-th order data in [2,Defn. 2.9], and the G • Vα S correspond to the higher order data in [2,Defn. 2.13]. The patching data in [2,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 • Vα S (that would correspond to something like a trivial deformation). Instead we perform a Thom-Whitney resolution yielding a bigraded Gerstenhaber algebra TW(G • Vα S ) with a non-trivial differential d ∶ TW p,q → TW p,q+1 . Once we forget that differential, there is a unique gluing PV X0 S of the TW(G • Vα S ) up to isomorphism, essentially corresponding to [2,Lemma 3.21]. Our proof shows the cohomological principles behind the explicit-constructive proof in [2,Lemma 3.21].
Given a gluing of the G • Vα S , its Thom-Whitney resolution endows PV X0 S with a differential d ∶ PV p,q X0 S → PV p,q+1 X0 S . Conversely any differential on PV X0 S gives a gluing of the G • Vα S by taking cohomology. This sets up an equivalence of deformation functors GD X0 S0 ≅ T D X0 S0 where the latter classifies differentials.
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 X0 S with a predifferential, corresponding to∂ α + [d α , −] in [2,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 [2,Defn. 5.10]. This sets up the final equivalence T D X0 S0 ≅ Def L • . The geometricČech gluing in [2, §5.3] loosely corresponds to going back from a Maurer-Cartan solution to glue the G • Vα S . Our Lie algebra L • X0 S0 is essentially the (−1, * )-part of the differential graded Batalin-Vilkovisky algebra P V * , * (X) in [2,Thm. 1.1]. What is new in this paper is that it actually controls log smooth deformations. The main technical difference is that whereas [2] use a countable covering {U i } of X 0 and stick to sections over these opens, we consequently stick to sheaves on X 0 . We hope therefore to help the algebraic geometer to get through the ideas of [2].
Outlook. Beyond the scope of this paper, it should be possible to use the techniques of [2,Thm. 1.2] is actually a dgla. This is unobstructedness in the sense of [2]. 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 LD X0 S0 (S ′ ) → LD X0 S0 (S) to be surjective for any thickening S → S ′ in Art Q .
Nonetheless, Theorem 1.1 might be applied to obtain a Bogomolov-Tian-Todorov theorem in log geometry. In [14], the classical BTT theorem is obtained by proving the dgla controlling Def X homotopy abelian. In fact it is possible -under suitable hypotheses -to adapt their methods and use the abstract Bogomolov-Tian-Todorov theorem of [13] to prove the central fiber L • X0 S0 ⊗ Q k homotopy abelian. This shows a log BTT theorem in case Q = 0 and generalizes results on the deformation of pairs (X, D) in [19,Lemma 4.19] and [12]. Similarly, the BTT theorem for compactified LG models in [20] is a variant of this theme. However, the case Q = 0 remains completely open so far.
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 [4] which are a generalization of log smooth families that need not 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 [4,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 [29, Thm. C.6] for divisorial deformations of toric log Calabi-Yau spaces. In this paper we restrict 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 LD X0 S0 ≅ GD X0 S0 .
In [17] 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 ̵ h to k N . Deformation quantization of affine toric varieties which are the building blocks of log smooth morphisms has been achieved recently in [5].
Acknowledgement. This paper is a byproduct which arose from studying [2] in order to apply it in [4]. I owe many ideas of the present paper to [2]. I thank my collaborator Matej Filip of [4] 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. Finally I thank JGU Mainz for its hospitality and Carl-Zeiss-Stiftung for financial support.

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 [14] 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 I = I X ′ X = I ⋅ O X ′ because f ′ is flat (as is any 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, 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 Θ 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 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 analogues 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 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). We extend this picture to the diagram where the vertical arrows are the forgetful maps.
where the series are actually finite sums. The symbol ∆(m) denotes the multiplication operator with this element.
which is proven by induction, and the identity The map Ψ is an isomorphism because it is one on underlying schemes and on ghost sheaves and because M V ′ is fine.
We construct an ansatz for the inverse log X ′ X . On the classical part, given φ, our ansatz where Φ k (m) − m ∈ M * X ′ is the unique invertible by which these elements differ, so we make an ansatz where log X ′ X is defined by the above formulas.

Gerstenhaber Algebra of Polyvector Fields
Given S ∈ Art Q , let f ∶ X → S be log smooth and saturated. The polyvector fields G • X S ∶= ⋀ −• Θ 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 [g, h] sn = 0 for functions g, h ∈ O X , such that [θ, ξ] sn = [θ, ξ] is the Lie bracket for vector fields θ, ξ ∈ Θ 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 [2].
In the spirit of [2, Defn. 2.9] we say a Gerstenhaber algebra is injective. Using Proposition 9.1 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 of Gerstenhaber algebras. Here the right hand map is induced by pulling back 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 ′ . For so it remains to prove equality on G −1 for the action of ϕ on differential forms. Writing ϕ = (φ, Φ) as in Section 2 we find for ξ ∈ G −1 as a ring with respect to ∧, we see T ϕ = exp −θ . For the second statement, 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 • X0 S0 , i.e. the deformation functor GD X0 S0 which we introduce below. We fix an affine cover V = {V α } of X 0 and we denote the 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 have isomorphisms Vα∩V β of Gerstenhaber algebras contravariantly (by our construction in Section 3). Whenever we can compose these isomorphisms to an automorphism, it is a gauge transform by Lemma 3.1.
and such that the cocycle χ 2α ○ψ○χ −1 1α is a gauge transform. Isomorphism classes define a functor GD X0 S0 ∶ Art Q → Set of Artin rings.
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 Vα ≅ V α . The different choices give rise to isomorphic Gerstenhaber deformations with the identity on G • as isomorphism. This induces a natural transformation LD X0 S0 ⇒ GD X0 S0 .
according to Definition 4.1 which are induced by automorphisms Ψ α ∶ V α → V α by Lemma 3.1. They give isomorphisms Y Vα → X Vα that glue to a global isomorphism Ψ ∶ Y → X, so the transformation is injective. 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 Γ αβ ∶ V β Vα∩V β → V β Vα∩V β which we use to define isomorphisms Note that the (contravariantly) induced map on Gerstenhaber algebras is χ β ○ χ −1 α , so they satisfy the cocycle condition by Lemma 3.1. Gluing yields a log smooth deformation whose image in GD X0 S0 (S) is G • .

Thom-Whitney Resolutions
In this section we briefly review the Thom-Whitney resolution. They are acyclic resolutions of complexes that are well adapted to preserve additional algebraic structures such as ∧ and [−, −] of a Gerstenhaber algebra, so they have been employed in [14] and [2] 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 where for each n ≥ 1, we have n + 1 morphisms ∂ k,n ∶ A n−1 → A n satisfying ∂ ℓ,n+1 ∂ k,n = ∂ k+1,n+1 ∂ ℓ,n .
Example 5.1. Let X be a k-scheme, let U = {U i } be an affine cover of X, and let F be a sheaf of k-vector spaces. TheČech semicosimplicial sheaf F (U) is the semicosimplicial sheaf with the usualČech maps, and with F (U ) ∶= j * F U . Similarly, if G • is a sheaf of Gerstenhaber algebras on X, then G • (U) is a semicosimplicial sheaf of Gerstenhaber algebras, i.e. each term G • (U) n = ∏ i0<...<in G • (U i0...in ) is a Gerstenhaber algebra, and theČech maps are morphisms thereof.
For each n, the differential forms on {t 0 +...+t n = 0} ⊂ A n+1 form a differential graded commutative algebra (A P L ) n = k[t 0 , ..., t n , dt 0 , ..., dt n ] (1 − t i , dt i ) The inclusions A n → A n+1 of coordinate hyperplanes induce face maps δ k,n ∶ (A P L ) n → (A P L ) n−1 which turn A P L into a semisimplicial dgca. Given a semicosimplicial complex V ∆ of k-vector spaces, we use the maps δ k,n together with the coface maps ∂ k,n of V ∆ to define homomorphisms of vector spaces. Following [6], 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 [ 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 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.
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 [11], but we recall it here for convenience.
Construction 5.3. Let X be a k-scheme and F ∆ be a semicosimplicial complex of sheaves of k-vector spaces. Then applying the above construction on every open, we obtain a presheaf C i,j TW (F ∆ ) which is a sheaf. Indeed, for a sheaf E of k-vector spaces, U ↦ (A P L ) i n ⊗ k E(U ) 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.
The inverse image c * F ∆ is a bounded semicosimplicial complex of quasicoherent O Y -modules, so everything is quasi-coherent. It thus suffices to compare on some affine cover of Y where it follows from Lemma 5.2.
Example 5.5. Let X be a separated k-scheme of finite type, let U = {U i } be a finite affine cover and let F • be a complex of quasi-coherent O X -modules. Then for a thickening c ∶ Y → X and hence the natural map For a complex of sheaves F • , the map induces a quasi-isomorphism F • → Tot TW (F • (U)) which is shown postcomposing with the integration quasi-isomorphism to theČech complexČ • (U, F • ).
Corollary 5.7. Let A ′ → A be a morphism in Art Q , let f ′ ∶ X ′ → S ′ be separated, let U ′ be a finite affine cover of X ′ and let X = X ′ × S ′ S. Let F • be a complex of quasi-coherent O X ′ -modules which are flat over A ′ , and let 1 , ..., x mn 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 and 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 • Vα S of polyvector fields and glue them canonically to a global sheaf of (bigraded) Gerstenhaber algebras PV X0 S . This sheaf depends on X 0 and the base S, but not on a deformation f ∶ X → S, thus the notation. The gluing is inspired by the gluing construction in [2, 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. We set TW p,q (G • ) ∶= C q,p TW (G • (U)) with the switch of indices on purpose to fit the conventions of [2]. The operations and turn it into a bigraded Gerstenhaber algebra, 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 differ- (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( 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 any differential Gerstenhaber algebra. Remark 6.1. Do not confuse our notion of differential Gerstenhaber algebra with the one of e.g. [22] which are more closely related to Batalin-Vilkovisky algebras.
Example 6.2. Let f ∶ X → S be a log smooth and saturated deformation of 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. (3). (−1)-injectivity is preserved by the Thom-Whitney construction. This is important for Lemma 6.6 below.
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 i0...in ). Denoting {t µ } some basis of (A P L ) j n we can decompose θ i0.
We apply this construction to the Gerstenhaber algebras G • Vα S and obtain differential Gerstenhaber algebras TW(G • Vα S ) as well as isomorphisms TW(φ αβ ) on overlaps. Any cocycle is a gauge transform by an element in I ⋅ TW −1,0 (G • Vα S Vα∩V β ∩Vγ ) where I is the kernel of O X → O X0 . Gluing them as differential Gerstenhaber algebras induces a Gerstenhaber deformation by taking cohomology. As a step towards that our goal is to glue them as a bigraded Gerstenhaber algebra, i.e. without differential.
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. Example 6.5. Let G • be a Gerstenhaber deformation. Then TW(G • ) is a TWG deformation upon forgetting the differential.
Given a morphism S → S ′ in Art Q and a TWG deformation T •,• on S ′ , Lemma 5.4 shows that T •,• ⊗ A ′ A is a TWG deformation as well (where A = O(S)). Moreover, TWG deformations have a deformation theory just like classical flat deformations: Lemma 6.6. Let 0 → I → A ′ → A → 0 be a small extension in Art Q and let T •,• be a TWG deformation on S = Spec A. Then: • Given a lifting T ′•,• on S ′ , the relative automorphisms are in • Given a lifting T ′•,• on S ′ , the isomorphism classes of liftings are in • The obstructions to the existence of a lifting are in ) 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 X0 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 X0 S has many automorphisms, so there is no canonical choice. Thus we fix once and for all one PV X0 S for every S ∈ Art Q .
Proof of the Lemma. Given a lifting T ′•,• , we consider the choice of a morphism T ′•,• → T •,• that is the given one on TW(G • Vα S ′ ) → TW(G • Vα S ) as part of the datum. Since the gauge transforms φ αβ ○ χ α ○ χ −1 β are A ′ -linear, the TWG deformation T ′•,• consists of sheaves of A ′ -modules. Thus the sequence makes sense and it is exact because it is locally the sequence of Example 6.2. Every element θ ∈ I ⋅ T ′−1,0 induces a gauge transform exp θ by the formula in Section 4. It is an automorphism of T ′•,• in the sense of Definition 6.4 because it is a gauge transform on TW(G • Vα S ′ ) as well, so we obtain a sheaf map into the sheaf of lifting automorphisms. It is injective due to Lemma 6.3. By the very definition of an automorphism of TWG deformations, any 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 θ ∈ I ⋅ T ′−1,0 , so exp T ′ T is an isomorphism. Finally we have TW −1,0 (G • X0 S0 ) ⊗ k I ≅ I ⋅ T ′−1,0 , so the result follows by the standard methods that are developed for smooth deformations in [10,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 X0 S ′ ⊗ A ′ A is a TWG deformation on S. In particular, there is an isomorphism to PV X0 S . Moreover, we have an induced homomorphism PV X0 S ′ → PV X0 S ′ ⊗ A ′ A. However, there is no canonical homomorphism PV X0 S ′ → PV X0 S since there is no preferred isomorphism PV X0 S ′ ⊗ A ′ A ≅ PV X0 S . This is because PV X0 S has many automorphisms. We see that S ↦ PV X0 S is not functorial in a strict sense.

Differentials on TWG Deformations
After constructing a unique TWG deformation PV X0 S we now study differ- where . This means under a gauge transform, the differential d is transformed into something of the form d+[η, −] for η ∈ I X X0 ⋅TW −1,1 (G • X S ). Hence any differential on PV X0 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 • X0 S0 ), and such that Vα where d α is the differential on TW(G • Vα 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 , 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 X0 S , it induces a differential on PV X0 S whose gauge equivalence class does not depend on the chosen isomorphism.
Any restriction PV X0 S ′ → PV X0 S induces a restriction Diff(PV X0 S ′ ) → Diff(PV X0 S ) on differentials. On gauge equivalence classes, this restriction is independent of the chosen map PV X0 S ′ → PV X0 S , so we obtain a functor T D X0 S0 ∶ Art Q → Set of Artin rings. To construct an inverse of the natural transformation tw ∶ GD X0 S0 ⇒ T D X0 S0 given by Example 7.2, we need a lemma.
is a differential. Then the two differentials d α and d α + [η α , −] are gauge equivalent.
Given a differential d ∈ Diff(PV X0 S ), the isomorphism is compatible with differentials. By Lemma 7.3 we can further compose with a gauge transform to (TW(G • Vα S ), d α ). Thus taking cohomology yields a Gerstenhaber deformation H • (PV X0 S , d) whose isomorphism class does not depend on the chosen gauge transform (for any 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 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 h injective. Let d 1 , d 2 be two differentials and let ψ ∶ H • (PV X0 S , d 1 ) ≅ H • (PV X0 S , d 2 ) be an isomorphism. Fix the structure of Gerstenhaber deformation on the cohomologies by choosing isomorphismsχ α,i ∶ (PV X0 S , d i ) Vα ≅ (TW(G • Vα S ), d α ) as in the construction of h. The map ψ induces a gauge transform exp θα ∶ G • Vα S → G • Vα S which has a unique lift to a gauge transform expθ Using theχ α,i we obtain an isomorphism (PV X0 S , d 1 ) Vα → (PV X0 S , d 2 ) Vα inducing ψ in cohomology. 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 • X0 S0 which controls T D X0 S0 and thus log smooth deformations. To this end we relate differentials on PV X0 S to an appropriate Maurer-Cartan equation.

X0 S
and d ∈ PDiff(PV X0 S ), also d + [η, −] is a predifferential. Every predifferential is (locally) of this form and Lemma 6.3 shows that for Using Lemma 5.6 inductively over small extensions, we find H ℓ (X 0 , PV i,j X0 S ) = 0 for ℓ ≥ 1. This suggests H 1 (X 0 , I S ⋅ PV −1,1 X0 S ) = 0 (which does not follow immediately), so there would be always a predifferential. In fact more is true: Lemma 8.1. Let S → S ′ be a small extension and let d ∈ PDiff(PV X0 S ). Choose a restriction PV X0 S ′ → PV X0 S (which is unique up to automorphisms of PV X0 S ). Then there is a predifferential d ′ ∈ PDiff(PV X0 S ′ ) with d ′ S = d. Proof. The sheaf of predifferentials on PV X0 S ′ that restrict to d over S is an Given a differential d ∈ Diff(PV X0 S ) and η ∈ I S ⋅ PV −1,1 X0 S , we find The element ℓ ∈ PV −1,2 X0 S is unique with that property by Lemma 6.3, so indeed for every predifferential d there is a unique ℓ = ℓ(d) ∈ PV −1,2 X0 S with d 2 = [ℓ, −] (because locally we can compare it to a differential Our next goal is to construct the k Q -pdgla L • X0 S0 . Let A ∶= k Q , 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 X0 S k+1 → PV X0 S k we choose inductively compatible predifferentials d k on PV X0 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 X0 S k → PV X0 S and we endow PV X0 S with the restricted predifferential d k S . The construction yields an O(S)-pdgla which is functorial for morphisms S → S ℓ (where ℓ ≥ 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 ℓ. The limit is an A-pdgla since its pieces are complete by [1, 09B8] (note the index shift).
Proposition 8.5. There is a natural equivalence mc ∶ Def L • ⇒ T D X0 S0 .
Proof. By [1, 09B8] the canonical map 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 ℓ . Maurer-Cartan elements η ∈ I S ⋅ L 1 (S) induce differentials d η ∈ Diff(PV X0 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 X0 S . Indeed, by Corollary 8.2 we can find a unique ξ ∈ I S ⋅ L 1 (S) such that on PV X0 S . Restricting the equation to L • (S) ⊂ PV X0 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 X0 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 X0 S ′ → PV X0 S which fits into a commutative diagram with the chosen horizontal maps. Finally recall that on T D X0 S0 the restriction is independent of the choice for PV X0 S ′ → PV X0 S .
Using Proposition 9.4 we recover that LD X0 S0 is a deformation functor which was first proved in [16]. We find that the tangent space t LD is isomorphic to H 1 (L • 0 ) ≅ H 1 (X 0 , Θ 1 X0 S0 ), so in case f 0 ∶ X 0 → S 0 is proper we recover that LD X0 S0 has a hull by Schlessingers' criterion [30,Thm. 2.11].

Λ-Linear Predifferential Graded Lie Algebras
Fix a complete local Noetherian ring Λ with maximal ideal m ⊂ Λ. We briefly introduce the type of Lie algebra which controls log smooth deformations. This type of Lie algebra is called almost dgla in [2, Thm. 1.1] where the existence of such a dgla is proven under some abstract conditions. is a graded Λ-linear Lie algebra such that every L i is complete, d ∶ L • → L •+1 is a Λ-linear derivation of degree 1 and ℓ ∈ m ⋅ L 2 is such that d 2 = [ℓ, −]. In particular, d is not a differential in general.
For a local Artin Λ-algebra A (with residue field k ∶= Λ mΛ), the tensor product L • ⊗ Λ A is an A-pdgla since L i ⊗ Λ A is complete. We consider them as some sort of infinitesimal deformation of the dgla L • 0 ∶= L • ⊗ Λ k which we call the central fiber. In fact, since ℓ ∈ mL 2 it is a dgla (d 2 = 0). We say L • is faithful, if for all A ∈ Art Λ and all ξ ∈ L • ⊗ Λ 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 exp θ (ξ + χ) = exp θ (ξ) + exp θ (χ), exp θ ([ξ, χ]) = [exp θ (ξ), exp θ (χ)] but exp θ is not compatible with d, i.e. in general exp θ ○ d = d ○ exp θ . If L • is faithful, then exp θ = exp θ ′ implies θ = θ ′ . We consider gauge transforms as some sort of infinitesimal automorphism. In fact, exp θ induces the identity on L • 0 . Given an element η ∈ m⋅L 1 we define a derivation d η ∶= d+[η, −] and consider it as a deformation of the differential on L • 0 . We have d 2 η = 0 if and only if η satisfies the Maurer-Cartan equation In this case we say η is Maurer-Cartan. If L • is faithful, then d η = d η ′ implies η = η ′ . We say two elements η, η ′ are gauge equivalent if there is θ ∈ m ⋅ L 0 with d η ○ exp θ = exp θ ○ d η ′ . In this case, η is Maurer-Cartan if and only if η ′ is. We consider gauge equivalence classes of Maurer-Cartan elements as deformations of L • 0 . We define the Maurer-Cartan functor MC L • ∶ Art Λ → Set by taking A ∈ Art Λ to the set of Maurer-Cartan elements in L • ⊗ Λ A and the deformation functor Def L • ∶ Art Λ → Set by taking gauge equivalence classes thereof. In case Λ = k (in which L • is an actual dgla) these functors reduce to the classical Maurer-Cartan and deformation functor. In general, they share a number of properties with them: