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

Abstract

Let $X$ be a smooth complex algebraic variety and let $\mathrm{Coh}(X)$ 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 $\mathrm{Coh}(X)$ as an Abelian category can be seen to be controlled by the Gerstenhaber-Schack complex associated to the restriction of the structure sheaf ${\mathcal{O}}_X{|}_{\mathfrak{U}}$ 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 ${\mathcal{O}}_X{|}_{\mathfrak{U}}$ 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.

27 pages, final version to appear in J. Noncommut. Geom