Coupled vortex equations and Moduli: Deformation theoretic Approach and Kaehler Geometry
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.
Cite
@article{arxiv.0808.3260,
title = {Coupled vortex equations and Moduli: Deformation theoretic Approach and Kaehler Geometry},
author = {Indranil Biswas and Georg Schumacher},
journal= {arXiv preprint arXiv:0808.3260},
year = {2008}
}