Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes

Let $S$ be a K3 surface. We study the reduced Donaldson-Thomas theory of the cap $(S \times \mathbb{P}^1) / S_{\infty}$ by a second cosection argument. We obtain four main results: (i) A multiple cover formula for the rank 1 Donaldson-Thomas theory of $\mathrm{K3} \times E$, leading to a complete solution of this theory. (ii) Evaluation of the wall-crossing term in Nesterov's quasi-map wallcrossing between the punctual Hilbert schemes and Donaldson-Thomas theory of $\mathrm{K3} \times \text{Curve}$. (iii) A multiple cover formula for the genus $0$ Gromov-Witten theory of punctual Hilbert schemes. (iv) Explicit evaluations of virtual Euler numbers of Quot schemes of stable sheaves on K3 surfaces.