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

Oberdieck, Georg

doi:10.2140/gt.2024.28.3221

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ISSN (electronic): 1364-0380

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

Georg Oberdieck

Geometry & Topology 28 (2024) 3221–3256

DOI: 10.2140/gt.2024.28.3221

Abstract

Let $S$ be a K3 surface. We study the reduced Donaldson–Thomas theory of the cap $(S \times ℙ^{1}) ∕ S_{\infty}$ by a second cosection argument. We obtain four main results: (i) A multiple cover formula for the rank 1 Donaldson–Thomas theory of $S \times E$ , leading to a complete solution of this theory. (ii) Evaluation of the wall-crossing term in Nesterov’s quasimap wall-crossing between the punctual Hilbert schemes and Donaldson–Thomas theory of $S \times Curve$ . (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.

Keywords

Donaldson–Thomas theory, K3 surfaces, Hilbert schemes, wall-crossing, quasimaps, multiple cover formulas

Mathematical Subject Classification

Primary: 14J28, 14N35

References

Publication

Received: 11 February 2022

Revised: 27 July 2023

Accepted: 25 August 2023

Published: 25 November 2024

Proposed: Richard P Thomas

Seconded: Dan Abramovich, Jim Bryan

Authors

Georg Oberdieck
Department of Mathematics KTH Royal Institute of Technology Stockholm Sweden

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Volume 28, issue 7 (2024)