CURVATURE OF THE L-2-METRIC ON THE DIRECT IMAGE OF A FAMILY OF HERMITIAN-EINSTEIN VECTOR-BUNDLES

For a holomorphic family of simple Hermitian-Einstein holomorphic vect or bundles over a compact Kahler manifold, the locally free part of th e associated direct image sheaf over the parameter space forms a holom orphic vector bundle, and it is endowed with a Hermitian metric given by the L-2 pairing using the Hermitian-Einstein metrics. Our main resu lt in this paper is to compute the curvature of the L-2-metric. In the case of a family of Hermitian holomorphic line bundles with fixed pos itive first Chern form and under certain curvature conditions, we show that the L-2-metric is conformally equivalent to a Hermitian-Einstein metric. As applications, this proves the semi-stability of certain Pi card bundles, and it leads to an alternative proof of a theorem of Kem pf.