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

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 $S\times \mathbb{P}^1$.