Coupled vortex equations and Moduli: Deformation theoretic Approach and Kaehler Geometry

TL;DR

This paper explores the geometric structure of moduli spaces of coupled vortex solutions on compact Kähler manifolds, establishing a natural Kähler metric and analyzing its curvature, with special results for Riemann surfaces and projective varieties.

Contribution

It constructs a Hermitian, Kähler structure on the moduli space of coupled vortex solutions using deformation theory and fiber integrals, linking it to Quillen metrics.

Findings

The moduli space admits a natural Kähler structure.

The curvature tensor of the Kähler form is explicitly computed.

The bisectional curvature is semi-positive for Riemann surfaces.

Abstract

We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kaehler manifold X. These solutions are known to be related to polystable triples via a Kobayashi-Hitchin type correspondence. Using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex, we construct a Hermitian structure on these moduli spaces. This Hermitian structure is proved to be Kaehler. The proof involves establishing a fiber integral formula for the Hermitian form. We compute the curvature tensor of this Kaehler form. When X is a Riemann surface, the holomorphic bisectional curvature turns out to be semi--positive. It is shown that in the case where X is a smooth complex projective variety, the Kaehler form is the Chern form of a Quillen metric on a certain determinant line bundle.