NC-smooth thickenings and Jacobians

TL;DR

The paper develops a framework connecting NC-smooth thickenings, especially the standard thickening $A^{NC}$ of an abelian variety $A$, with deformation quantization in the form of Moyal algebras for constant Poisson bivectors. It proves that objects on the dual abelian variety with Fourier-Mukai transforms supported in dimension at most $1$ extend to perfect complexes on $A^{NC}$ and constructs a corresponding NC-Fourier transform; it also analyzes extensions of the theta divisor to the NC-setting and formulates Jacobian-characterization conjectures. A dg-approach is used to relate NC-thickenings and deformation quantizations, producing a natural homomorphism ${\mathcal{O}}_A^{NC}\to \mathcal{D}_{P,\nabla}$ and establishing partial results toward a criterion (Conjecture B) for extendability via FM-spectral conditions. The work demonstrates that first-order deformation quantizations impose Andreotti–Mayer-type constraints on Jacobians, provides concrete examples showing the limitations of first-order truncations, and highlights a nontrivial bridge between noncommutative thickening theory and deformation-quantization theory on abelian varieties.

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.

Theorems & Definitions (30)

• Theorem 1.2.1
• Proposition 1.2.2
• proof
• Corollary 1.2.3
• proof
• Theorem 1.3.1
• proof
• Remark 1.3.2
• Theorem 1.3.3
• Proposition 1.3.4