Moduli of sheaves and deformation to the normal cone

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.