Positivity of characteristic forms via pointwise universal push-forward formulae

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 show that the universal formula for the push-forward of a homogeneous polynomial in the Chern classes of the universal vector bundles also holds pointwise at the level of Chern forms in this Hermitianized situation. As an application, we obtain the positivity of several polynomials in the Chern forms of Griffiths semipositive vector bundles not previously known. This gives new evidences towards a conjecture proposed by Griffiths, which has raised interest in the past as well as in recent years. This conjecture can be interpreted as a pointwise Hermitianized version of the Fulton-Lazarsfeld theorem on numerically positive polynomials for ample vector bundles.