Differential geometry of moduli spaces of quiver bundles

TL;DR

This paper studies the differential geometric structure of moduli spaces of quiver bundle representations on Kähler manifolds, constructing a Kähler form and computing its curvature, with applications to determinant line bundles.

Contribution

It introduces a Hermitian form on moduli spaces of quiver bundles, proves it is Kähler via a fiber integral formula, and relates it to determinant line bundle curvature.

Findings

Hermitian form on moduli spaces is Kähler

Fiber integral formula for the Hermitian form

Curvature of the Kähler form computed and related to determinant line bundles

Abstract

Let P be a parabolic subgroup of a simple affine algebraic group G defined over C and X a compact connected K\"ahler manifold. L. \'Alvarez-C\'onsul and O. Garc\'ia-Prada associated to these a quiver Q and representations of Q into holomorphic vector bundles on X. Our aim here is to investigate the differential geometric properties of the moduli spaces representations of Q into vector bundles on X. In particular, we construct a Hermitian form on these moduli spaces. A fiber integral formula is proved for this Hermitian form; this fiber integral formula implies that the Hermitian form is K\"ahler. We compute the curvature of this K\"ahler form. Under an assumption which says that X is a complex projective manifold, this K\"ahler form is realized as the curvature of a certain determinant line bundle equipped with a Quillen metric.