On the curvature of vortex moduli spaces

TL;DR

This paper investigates the local curvature properties of vortex moduli spaces on Riemann surfaces, showing that certain nonnegative curvature conjectures do not hold in higher genus cases.

Contribution

It computes the homotopy type of the universal cover of vortex moduli spaces and disproves a conjecture about their curvature properties.

Findings

Holomorphic bisectional curvature cannot always be nonnegative for genus g>1.

The homotopy type of the universal cover is that of symmetric powers of the surface.

The results extend to all Kähler metrics on certain symmetric powers.

Abstract

We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric powers of the surface), we prove that, for genus g>1, the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kaehler metrics on certain symmetric powers. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.