Moduli of sheaves and deformation to the normal cone

TL;DR

This paper proves that the operations of constructing moduli spaces of stable sheaves and deforming to the normal cone commute, with applications to degenerations of Kummer varieties within Hitchin systems.

Contribution

It establishes a precise compatibility between moduli of sheaves and deformation to the normal cone for smooth projective varieties, extending previous work on curves in symplectic surfaces.

Findings

Moduli of stable sheaves and deformation to the normal cone commute.

Degeneration of generalized Kummer varieties to symplectic subvarieties of Hitchin systems.

Application to curves of genus at least 2.

Abstract

Given a closed immersion between arbitrary smooth complex projective varieties, we prove that the two operations: (1) taking the moduli space of stable sheaves, and (2) taking the deformation to the normal cone, commute in a precise sense. In the case of curves inside symplectic surfaces, previously studied by Donagi-Ein-Lazarsfeld, the corresponding deformation to the normal cone space is an open subset of the relative moduli space of sheaves. As an application, we show generalized Kummer varieties degenerate to natural symplectic subvarieties of the Hitchin system for curves of genus at least 2.