Moduli of sheaves and deformation to the normal cone
Yifan Zhao
TL;DR
The paper proves that taking the moduli space of stable sheaves and performing deformation to the normal cone commute for a closed immersion $i:Y\hookrightarrow X$ of smooth projective varieties, via an injective morphism $\psi: D_{\mathcal{M}_Y/\mathcal{M}_X} \to \mathcal{M}_{D_{Y/X}/\mathbb{C}}$ compatible with the $\mathbb{C}^*$-action. The construction leverages Rees algebras to model $D_{Y/X}$ and builds a flat family of stable sheaves $\mathcal{F}$ over the deformation so that $\psi$ arises from a universal-to-relative push-forward, with a detailed spectral-Atiyah analysis on the special fibre. In the curve-on-surface setting, the degeneration $\mathcal{M}_{D_{C/S}/\mathbb{C}}$ yields a Poisson variety whose symplectic leaves interpolate between Beauville--Mukai and Hitchin systems; for abelian surfaces, Kum degenerates to a symplectic subvariety $\mathcal{N}$ in the $GL(r)$ Higgs moduli, with a dual fibration $\mathcal{N}^{\vee}$. The work also identifies the reduced Atiyah class $\operatorname{At}_p$ with the deformation-theoretic map $\psi_p$, reinforcing the link between moduli-theoretic deformations and Higgs- or spectral-theoretic data.
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.