Derived equivalences by quantization

TL;DR

This paper establishes an equivalence between the derived category of coherent sheaves on a symplectic resolution and modules over a non-commutative algebra, with applications to derived equivalences and cohomology of fibers.

Contribution

It proves that symplectic resolutions can be described via non-commutative algebras, extending Van den Bergh's framework, and explores their geometric and categorical implications.

Findings

Derived categories of sheaves are equivalent to modules over non-commutative resolutions.

Resolutions are derived-equivalent under certain conditions.

Cohomology of fibers is generated by algebraic cycle classes.

Abstract

We assume given a smooth symplectic (in the algebraic sense) resolution $X$ of an affine algebraic variety $Y$, and we prove that, possibly after replacing $Y$ with an etale neighborhood of a point, the derived category of coherent sheaves on $X$ is equivalent to the dervied category of finitely generated left modules over a non-commutative algebra $R$, a non-commutative resolution of $Y$ in a sense close to that of M. Van den Bergh. We also prove some applications, such as: two resolutions are derived-equivalent; every resolution $X$ admits a "resolution of the diagonal"; the cohomology groups of the fibers of the map $X \to Y$ are spanned by fundamental classes of algebraic cycles.