Curvature of smooth proper direct images by way of a holomorphic Gauss formula

The direct image of the relative canonical bundle of a smooth proper holomorphic family, twisted by a holomorphic vector bundle E with smooth Hermitian metric h with relatively Nakano-positive curvature, is locally trivial. The underlying vector bundle has a natural Hermitian metric called the L² metric. In his 2009 Annals paper Berndtsson computed the curvature of this metric when E has rank 1 and h has semi-positive curvature form. Later, Liu and Yang computed the curvature of the relative canonical bundle twisted by any Hermitian holomorphic vector bundle, under the assumption that this direct image is locally trivial. We give a new proof of the results of Berndtsson-Liu-Yang. Our proof uses a generalization of the holomorphic Gauss formula to the setting of BLS fields, introduced by the author.