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

Abstract

Let $X$ be an algebraic variety over an algebraically closed field of characteristic $0$ 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 ${\mathcal{O}}_X{|}_{\mathfrak{U}}$ or $\mathrm{Coh}(X)$ in case $X$ can be covered by two acyclic open sets, giving an explicit deformation-obstruction calculus for such deformations. For smooth $X$, such deformations recover the deformation of complex structures and deformation quantizations of $X$ as degenerate cases, as we show by means of concrete examples.