Vortex equation and reflexive sheaves

TL;DR

This paper extends the vortex equation solution from stable holomorphic pairs to pairs involving reflexive sheaves on compact Kähler manifolds, broadening the scope of the theory.

Contribution

It generalizes the existence of Hermitian metrics solving the vortex equation to reflexive sheaves, beyond smooth vector bundles.

Findings

Hermitian metrics exist for stable pairs with reflexive sheaves

Extension of vortex equation solutions to singular sheaves

Broader applicability of vortex equation theory

Abstract

It is known that given a stable holomorphic pair $(E ,\phi)$, where $E$ is a holomorphic vector bundle on a compact K\"ahler manifold $X$ and $\phi$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(\E ,\phi)$, where $\E$ is a reflexive sheaf on $X$.