Differential geometry of moduli spaces of quiver bundles
arXiv:1606.01517 · doi:10.1016/j.geomphys.2016.11.013
Abstract
Let P be a parabolic subgroup of a simple affine algebraic group G defined over C and X a compact connected Kähler manifold. L. Álvarez-Cónsul and O. García-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 Kähler. We compute the curvature of this Kähler form. Under an assumption which says that X is a complex projective manifold, this Kähler form is realized as the curvature of a certain determinant line bundle equipped with a Quillen metric.
arXiv admin note: text overlap with arXiv:math/0112160 by other authors. 1.Introduction, 2. Quiver bundles - Notation and fundamental properties (Summary of known results), 3. Deformation theory, 4. Fiber integral formula and Kähler property, 5. Determinant bundle and Quillen metric, 6. Curvature of the metric. To appear in Journal of Geometry and Physics