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.

中文翻译：

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通过 Gerstenhaber 代数对数平滑变形理论

我们构造了一个 $k[[Q]]$-线性前微分分级李代数 $L^*_{X/S}$ 与对数平滑和饱和态射 $f: X \rightarrow S$ 相关联，并证明它控制对数平滑变形函子。这提供了 Chan-Leung-Ma 构造的几何解释，其中 $L^*_{X/S}$ 是纯代数版本。我们的证明主要依赖于研究多向量场的 Gerstenhaber 代数的变形，有趣的是不需要跟踪日志结构。使用 Gerstenhaber 代数的方法与镜像对称的最新发展密切相关。