Kahler structure on moduli spaces of principal bundles

Abstract

Let M be a moduli space of stable principal G-bundles over a compact Kahler manifold (X,ω _X), where G is a reductive linear algebraic group defined over C. Using the existence and uniqueness of a Hermite-Einstein connection on any stable G-bundle P over X, we have a Hermitian form on the harmonic representatives of H¹(X,ad(P)), where ad (P) is the adjoint vector bundle. Using this Hermitian form a Hermitian structure M on is constructed; we call this the Petersson-Weil form. The Petersson-Weil form is a Kahler form, a fact which is a consequence of a fiber integral formula that we prove here. The curvature of the Petersson-Weil Kahler form is computed. Some further properties of this Kahler form are investigated.