Deformation-obstruction theory for diagrams of algebras and applications to geometry

TL;DR

This paper develops an explicit deformation-obstruction framework for diagrams of algebras, specifically for the structure sheaf of a smooth algebraic variety, linking to complex structure and quantization deformations.

Contribution

It constructs an explicit $L_$ algebra controlling higher deformations of the structure sheaf on covers of two acyclic open sets, extending deformation theory for algebraic varieties.

Findings

Explicit $L_$ algebra structure on Gerstenhaber-Schack complex.

Deformation-obstruction calculus for specific algebraic structures.

Connections to complex structure and quantization deformations.

Abstract

Let $X$ be a smooth complex algebraic variety and let $\operatorname{Coh} (X)$ denote its Abelian category of coherent sheaves. By the work of W. Lowen and M. Van den Bergh, it is known that the deformation theory of $\operatorname{Coh} (X)$ as an Abelian category can be seen to be controlled by the Gerstenhaber-Schack complex associated to the restriction of the structure sheaf $\mathcal O_X \vert_{\mathfrak U}$ to a cover of affine open sets. We construct an explicit $L_\infty$ algebra structure on the Gerstenhaber-Schack complex controlling the higher deformation theory of $\mathcal O_X \vert_{\mathfrak U}$ in case $X$ can be covered by two acyclic open sets, giving an explicit deformation-obstruction calculus for such deformations. Deformations of complex structures and deformation quantizations of $X$ are recovered as degenerate cases, as is shown by means of concrete examples.