In his famous Annals paper Bo Berndtsson proved the positivity of direct images of the relative canonical bundle twisted by a semi-positive holomorphic line bundle. Berndtsson's method involves direct computation of the curvature, followed by some Hodge Theory to prove the positivity of this curvature. In this talk we consider twisting the relative canonical bundle by a holomorphic vector bundle of higher rank. There are several notions of curvature of such vector bundles, and we show that the resulting family of finite-dimensional Bergman spaces has the same positivity as the vector bundle. Our method of proof is different from Berndtsson's; it uses a generalization of the Gauss-Griffiths formula relating the curvature of the ambient bundle to the curvature of the subbundle and the second fundamental form of the subbundle.
