The Weil-Petersson current for moduli of vector bundles and applications to orbifolds

We investigate stable holomorphic vector bundles on a compact complex K\"ahler manifold and more generally on an orbifold that is equipped with a K\"ahler structure. We use the existence of Hermite-Einstein connections in this set-up and construct a generalized Weil-Petersson form on the moduli space of stable vector bundles with fixed determinant bundle. We show that the Weil-Petersson form extends as a (semi-)positive closed current for degenerating families that are restrictions of coherent sheaves. Such an extension will be called a Weil-Petersson current. When the orbifold is of Hodge type, there exists a determinant line bundle on the moduli space; this line bundle carries a Quillen metric, whose curvature coincides with the generalized Weil-Petersson form. As an application we show that the determinant line bundle extends to a suitable compactification of the moduli space.Comment: To appear in Annales de la Facult\'e des Sciences de Toulouse. Math\'ematiques. Abstract added, typos correcte

Geometry of moduli spaces of Higgs bundles

We construct a Petersson-Weil type K\"ahler form on the moduli spaces of Higgs bundles over a compact K\"ahler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Petersson-Weil K\"ahler form is computed. We also show that, under certain assumptions, a moduli space of Higgs bundles supports of natural hyper-K\"ahler structure.Comment: To appear in Communications in Analysis and Geometr

Estimates of Weil-Petersson volumes via effective divisors

We study the asymptotics of the Weil-Petersson volumes of the moduli spaces of compact Riemann surfaces of genus $g$ with $n$ punctures, for fixed $n$ as $g \to \infty$