Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

Abstract

Let X be a complex projective manifold and (E, h) → X be a rank r holomorphic hermitian vector bundle. Let Qj, j = 1, . . . , r, be the tautological quotient line bundles over the flag bundle F(E) → X, endowed with the natural metric induced by that of E, with Chern curvature Ξj. We show that the universal Gysin formula à la Darondeau-Pragacz for the push forward of a homogeneous polynomial in the Chern classes of the Qj's also hold pointwise at the level of Chern forms in this hermitianized situation.As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which is in turn a pointwise hermitianized version of the Fulton-Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles.

Mean curvature negativity and HN-negativity of holomorphic vector bundles

“…Recently, there are several interesting works on the above Griffiths' question ( [47,24,17,52,38,21,59,18]). On semi-stable ample vector bundles over complex surfaces, Pingali proved that there must exist a Hermitian metric H such that its second Chern form is positive.…”