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

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 $K3\times E$, 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 $K3\times \text{Curve}$. (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.

31 pages. v3: changes to Section 2.3 and added Section 4.6

References in corpus (3)

• Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on $K3$ Surfaces
• Gromov-Witten classes of K3 surfaces
• A blowup formula for virtual enumerative invariants on projective surfaces

Cited by in corpus (4)

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