Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $ρ$ be a sequence of dimensions $0 = ρ_0 < ρ_1 < \cdots < ρ_m = r$. Let $Q_{ρ,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_ρ(E) \to X$ associated to $ρ$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $Ξ_{ρ,j}$. We show that the universal Gysin formula \textsl{à la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{ρ,j}$'s also hold pointwise at the level of the Chern forms $Ξ_{ρ,j}$ in this hermitianized situation. As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles.

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