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

## TL;DR

This paper develops formulas for tautological integrals over Quot schemes on curves and surfaces, introduces a new theory for virtual Euler characteristics of surface Quot schemes, and explores connections to various invariants and combinatorial structures.

## Contribution

It provides explicit formulas for virtual Euler characteristics of Quot schemes on surfaces and establishes new links to combinatorics and geometric invariants.

## Key findings

- Explicit formulas for Quot schemes of dimension 0 on curves.
- Complete solutions for virtual Euler characteristics of surface Quot schemes.
- New connections between weighted tree counting and Fuss-Catalan numbers.

## 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.