NC-smooth thickenings and Jacobians

TL;DR

This paper demonstrates that certain complexes on abelian varieties extend to noncommutative smooth thickenings and deformation quantizations, and explores a conjecture linking Jacobians to NC-smooth thickenings.

Contribution

It establishes the extension of complexes to NC-smooth thickenings and deformation quantizations, and proposes a conjectural characterization of Jacobians via NC-smooth structures.

Findings

Extension of complexes to NC-smooth thickenings proven.

Extension to deformation quantizations established.

Conjectural link between Jacobians and NC-smooth thickenings discussed.

Abstract

We prove that a bounded complex of coherent sheaves on an abelian variety A, whose Fourier-Mukai transform has support of dimension at most 1, extends to a perfect complex on the standard NC-smooth thickening of A and on the deformation quantization of any constant Poisson bracket on A. We discuss a related conjectural characterization of Jacobians in terms of NC-smooth thickenings.