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Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics

Dragos Oprea and Rahul Pandharipande

Geometry & Topology 25 (2021) 3425–3505

We compute tautological integrals over Quot schemes on curves and surfaces. After obtaining several explicit formulas over Quot schemes of dimension-0 quotients on curves (and finding a new symmetry), we apply the results to tautological integrals against the virtual fundamental classes of Quot schemes of dimension 0 and 1 quotients on surfaces (using also universality, torus localization and cosection localization). The virtual Euler characteristics of Quot schemes of surfaces, a new theory parallel to the Vafa–Witten Euler characteristics of the moduli of bundles, is defined and studied. Complete formulas for the virtual Euler characteristics are found in the case of dimension-0 quotients on surfaces. Dimension-1 quotients are studied on K3 surfaces and surfaces of general type, with connections to the Kawai–Yoshioka formula and the Seiberg–Witten invariants, respectively. The dimension-1 theory is completely solved for minimal surfaces of general type admitting a nonsingular canonical curve. Along the way, we find a new connection between weighted tree counting and multivariate Fuss–Catalan numbers, which is of independent interest.

Quot schemes, virtual classes, tautological integrals, virtual Euler characteristics
Mathematical Subject Classification 2010
Primary: 14C17, 14D20
Secondary: 14C05, 14J28, 14J80
Received: 19 October 2019
Revised: 11 June 2020
Accepted: 6 November 2020
Published: 25 January 2022
Proposed: Lothar Göttsche
Seconded: Mark Gross, Gang Tian
Dragos Oprea
Department of Mathematics
University of California, San Diego
La Jolla, CA
United States
Rahul Pandharipande
Department of Mathematics
ETH Zürich