1-point Gromov-Witten invariants of the moduli spaces of sheaves over the projective plane

TL;DR

This paper computes specific genus-0 Gromov-Witten invariants for moduli spaces of rank-2 sheaves on the complex projective plane, utilizing the Gieseker-Uhlenbeck morphism and virtual fundamental class techniques.

Contribution

It determines all extremal 1-point genus-0 Gromov-Witten invariants for these moduli spaces, extending understanding of their enumerative geometry.

Findings

Explicit formulas for extremal Gromov-Witten invariants.

Application of the Gieseker-Uhlenbeck morphism in enumerative geometry.

Use of a meromorphic 2-form to analyze the virtual fundamental class.

Abstract

The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points. When the surface is the complex projective plane, we determine all the 1-point genus-0 Gromov-Witten invariants extremal with respect to the Gieseker-Uhlenbeck morphism. The main idea is to understand the virtual fundamental class of the moduli space of stable maps by studying the obstruction sheaf and using a meromorphic 2-form on the Gieseker moduli space.