Chern Forms of Hermitian Metrics with Analytic Singularities on Vector Bundles

We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular Hermitian metric h with analytic singularities on a holomorphic vector bundle E. The currents are constructed as pushforwards of generalized Monge-Ampere products on the projectivization of E. The Chern and Segre currents represent the Chern and Segre classes of E, respectively, and coincide with the Chern and Segre forms of E and h where h is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases when these are defined