论文标题
通过Gerstenhaber代数来实现平滑变形理论
Log Smooth Deformation Theory via Gerstenhaber Algebras
论文作者
论文摘要
我们构造了$ k [[q] $ - 线性预性分级lie代数$ l^*_ {x/s} $与log平滑且饱和的形态相关的$ f：x \ rightarrow s $，并证明它控制了log平滑变形函数。这提供了Chan-Leung-Ma对结构的几何解释，其中$ l^*_ {x/s} $是纯粹的代数版本。我们的证据至关重要的是研究多矢量场的Gerstenhaber代数的变形，有趣的是，不需要跟踪对数结构。使用Gerstenhaber代数的方法与镜像对称性的最新发展密切相关。
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.
