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

TL;DR

This paper develops multiple cover formulas and wall-crossing evaluations for Donaldson-Thomas and Gromov-Witten theories on K3 surfaces, providing explicit calculations for Quot schemes and related invariants.

Contribution

It introduces new multiple cover formulas for DT and GW theories on K3 surfaces and evaluates wall-crossing terms and virtual Euler numbers explicitly.

Findings

Complete solution for rank 1 DT theory of K3 times E.

Explicit wall-crossing evaluations between Hilbert schemes and DT theory.

Formulas for virtual Euler numbers of Quot schemes on K3 surfaces.

Abstract

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.