Kähler structure on moduli spaces of principal bundles

Let MM be a moduli space of stable principal G  -bundles over a compact Kähler manifold (X,ωX)(X,ωX), where G   is a reductive linear algebraic group defined over CC. 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))H1(X,ad(P)), where ad(P)ad(P) is the adjoint vector bundle. Using this Hermitian form a Hermitian structure on MM is constructed; we call this the Petersson–Weil form. The Petersson–Weil form is a Kähler form, a fact which is a consequence of a fiber integral formula that we prove here. The curvature of the Petersson–Weil Kähler form is computed. Some further properties of this Kähler form are investigated.