Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces

TL;DR

This paper establishes a correspondence between stability conditions for sheaves with sections and solutions to the Vafa-Witten equations on smooth projective surfaces, providing an alternative proof using a Mehta-Ramanathan style argument.

Contribution

It offers a new proof of the Hitchin-Kobayashi correspondence for the Vafa-Witten equations, connecting stability and solutions via an approach inspired by Donaldson's method.

Findings

Proves the stability-equation correspondence for sheaves with twisted sections.

Provides an alternative proof using Mehta-Ramanathan style arguments.

Suggests potential extensions to higher-dimensional cases and related equations.

Abstract

This article describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair $(\mathcal{E}, \varphi)$, where $\mathcal{E}$ is a locally-free sheaf over a surface $X$ and $\varphi$ is a section of $\text{End} (\mathcal{E}) \otimes K_{X}$; and the existence of a solution to certain gauge-theoretic equations, the Vafa-Witten equations, for a Hermitian metric on $\mathcal{E}$. It turns out to be a special case of results obtained by Alvarez-Consul and Garcia-Prada. In this article, we give an alternative proof which uses a Mehta-Ramanathan style argument originally developed by Donaldson for the Hermitian-Einstein problem, as it relates the subject with the Hitchin equations on Riemann surfaces, and surely indicates a similar proof of the existence of a solution under the assumption of stability for the Donaldson-Thomas instanton equations described in arXiv:0805.2192 on smooth projective threefolds; and more broadly that for the quiver vortex equation on higher dimensional smooth projective varieties.