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

Indranil Biswas; Georg Schumacher

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arXiv · 0808.3260

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

Indranil Biswas·Georg Schumacher

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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.

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Indranil Biswas, Georg Schumacher. 2008-08-24. Coupled vortex equations and Moduli: Deformation theoretic Approach and Kaehler Geometry. https://arxiv.org/abs/0808.3260

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