Quasimaps to moduli spaces of sheaves on a $K3$ surface

TL;DR

This work develops a reduced obstruction-theory framework for quasimaps to moduli spaces of sheaves on a $K3$ surface and proves a reduced wall-crossing formula that links reduced Gromov–Witten theory of $S^{[n]}$ with reduced Donaldson–Thomas theory of $S imes C$, where $C$ is nodal. The authors extend semiregularity via twisted Atiyah classes and non-commutative first-order deformations to achieve surjectivity of the cosection, enabling reduced invariants and their wall-crossing. Applying the reduced wall-crossing, they derive genus-zero and genus-one correspondences, including the Igusa cusp form conjecture and DT/PT relations with relative insertions on relative geometries like $S imes P^1$ and $S imes E$, and they relate higher-rank DT theory to rank-one DT via deformation arguments. These results unify GW/DT/PT theories in a non-Calabi–Yau, relative setting and yield explicit generating-function identities tied to modular forms, with implications for enumerative geometry on Hilbert schemes and moduli of sheaves on $K3$ surfaces.

Abstract

In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface $S$. We construct a surjective cosection of the obstruction theory of moduli spaces of quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov-Witten theory of moduli spaces of sheaves on $S$ and the reduced Donaldson-Thomas theory of $S\times C$, where $C$ is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson-Thomas correspondence with relative insertions on $S\times C$, if $g(C)\leq1$; Donaldson-Thomas/Pandharipande-Thomas correspondence with relative insertions on $S\times \mathbb{P}^1$.

Figures (2)

• Figure 1: Higher-rank/rank-one DT correspondence
• Figure 2: DT/PT correspondence

Theorems & Definitions (24)

• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• Remark 2.4
• Remark 3.1
• Corollary 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Lemma 3.6