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

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 $g(C)\le 1$; Donaldson-Thomas/Pandharipande-Thomas correspondence with relative insertions on $S\times {\mathbb{P}}^1$.

arXiv comment30 pages, revision following the referee's suggestions, published version

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• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The intrinsic normal cone
• Curve counting via stable pairs in the derived category
• Curves on K3 surfaces and modular forms
• Reduced classes and curve counting on surfaces I: theory
• Holomorphic anomaly equations and the Igusa cusp form conjecture
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Cited by in corpus (3)

• Holomorphic anomaly equations for the Hilbert scheme of points of a K3 surface
• Quasimaps to moduli spaces of sheaves
• Gromov-Witten/Hilbert versus AdS3/CFT2 Correspondence