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 $h$ analytic singularities on holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Amp\`ere products the projectivization represent classes $E$, respectively, coincide forms $E$ $h$, where is smooth. Moreover, our by first three authors Ruppenthal in cases when these defined.