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 $\Coh (X)$ 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 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 $\mathcal O_X \vert_{\mathfrak U}$ or 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.