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

Abstract

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.

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Cited by in corpus (12)

• Higher rank K-theoretic Donaldson-Thomas theory of points
• The virtual K-theory of Quot schemes of surfaces
• Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes
• Virtual ${\chi }_{-y}$-genera of Quot schemes on surfaces
• Hyperquot schemes on curves: virtual class and motivic invariants
• Sheaves on surfaces and virtual invariants
• Rank-one sheaves and stable pairs on surfaces
• Cosection localization and the Quot scheme ${\mathrm{Quot}}_S^l(\mathcal{E})$
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• K-Theoretic Descendent Series for Hilbert Schemes of Points on Surfaces
• Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points
• Equivariant Segre and Verlinde invariants for Quot schemes