Deformations of Affine Varieties and the Deligne Crossed Groupoid

TL;DR

This paper establishes a canonical equivalence between the Deligne crossed groupoid of polydifferential operators and the groupoid of associative or Poisson deformations of the structure sheaf of a smooth affine algebraic variety, advancing deformation theory.

Contribution

It introduces a canonical equivalence of crossed groupoids linking deformation structures with polydifferential operators, extending prior work and aiding twisted deformation quantization.

Findings

Canonical equivalence of crossed groupoids established

Extension of deformation theory to affine algebraic varieties

Framework supports twisted deformation quantization

Abstract

Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of O_X. The proof relies on a careful study of adically complete sheaves. In the associative case we also have to use ring theory (Ore localizations) and the properties of the Hochschild cochain complex. The results of this paper extend previous work by various authors. They are needed for our work on twisted deformation quantization of algebraic varieties.