  Given a Hermitian holomorphic vector bundle over a complex manifold, consider
its flag bundles with the associated universal vector bundles endowed with the
induced metrics. We prove that the universal formula for the push-forward of a
polynomial in the Chern classes of all the possible universal vector bundles
also holds pointwise at the level of Chern forms. A key step in our proof is
the explicit computation, at a point of any flag bundle, of the Chern curvature
of the universal vector bundles with the induced metrics. As an application, we
provide an alternative version of the Jacobi-Trudi identity at the level of
differential forms. We also show the positivity of a family of polynomials in
the Chern forms of Griffiths semipositive vector bundles. This latter result
partially confirms the Griffiths' conjecture on positive characteristic forms,
which has raised considerable interest in recent years.
