Abstract
We construct a kleft[ !left[ Qright] !right] -linear predifferential graded Lie algebra L^{bullet }_{X_0/S_0} associated to a log smooth and saturated morphism f_0: X_0 rightarrow S_0 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 L^{bullet }_{X_0/S_0} 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.
Highlights
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 are well-established
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
If L DX0/S0 were controlled by such a dgla, for every S ∈ ArtQ, 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
Summary
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. If L DX0/S0 were controlled by such a dgla, for every S ∈ ArtQ, 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 This gives a new tropical encoding of the Gross–Siebert gluing on the level of Gerstenhaber algebras analogous to our first isomorphism L DX0/S0 ∼= G DX0/S0 We expect this to be related to [21], where the refined scattering diagrams of [22] are related to Maurer–Cartan solutions in an appropriate dgla. The question which generalized dglas control these functors is subject to future studies 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]
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