Log smooth deformation theory via Gerstenhaber algebras

Abstract

We construct a $$k\left[ \!\left[ Q\right] \!\right] $$ k Q -linear predifferential graded Lie algebra $$L^{\bullet }_{X_0/S_0}$$ L X 0 / S 0 ∙ associated to a log smooth and saturated morphism $$f_0: X_0 \rightarrow S_0$$ f 0 : X 0 → 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 $$L^{\bullet }_{X_0/S_0}$$ L 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.

The logarithmic Bogomolov–Tian–Todorov theorem

“…Now we assume furthermore that 𝑘 has characteristic 0. In [7], the first author proves that LD 𝑋 0 ∕𝑆 0 is controlled by a 𝑘…”

Smoothing, scattering, and a conjecture of Fukaya

“…Using well-known algebraic techniques [42,27], we can solve the Maurer-Cartan equation and construct the thickening k X. In this subsection, we will summarize the whole procedure, incorporating the nice reformulation by Felten [13] in terms of deformations of Gerstenhaber algebras.…”

“…The next step is to consider gluing of the local sheaves k P V α 's for higher orders k. Similar constructions have been done in [5,13] using the combinatorial Thom-Whitney resolution for the sheaves k G α 's. We make suitable modifications of those arguments to fit into our current setting.…”

“…Using the argument in [5,Thm. 3.34] or [13,Lem. 8.1], we can always lift any k th -order predifferential k ∂ to a (k + 1) st -order predifferential, and any two such liftings differ by a global element d ∈ 0 P V −1,1 ⊗ m k+1 /m k .…”

“…This was originally called an almost dgBV algebra in[5], but we later found the name pre-dgBV algebra from[13] more appropriate.…”