Pointwise universal Gysin formulae and applications towards Griffiths’ conjecture

Let $X$ be a complex manifold, $(E,h)\to X$ rank $r$ holomorphic hermitian vector bundle, and $\rho$ sequence of dimensions $0 = \rho_0 < \rho_1 \cdots \rho_m r$. $Q_{\rho,j}$, $j=1,\dots,m$, the tautological line bundles over (possibly incomplete) flag bundle $\mathbb{F}_{\rho}(E) \to associated to $\rho$, endowed with natural metrics induced by that $E$, Chern curvatures $\Xi_{\rho,j}$. We show universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for push-forward homogeneous polynomial in classes $Q_{\rho,j}$'s also hold pointwise at level forms $\Xi_{\rho,j}$ this hermitianized situation. As an application, we positivity several polynomials Griffiths (semi)positive not previously known, thus giving some new evidences towards conjecture Griffiths, which turn can seen as version Fulton--Lazarsfeld Theorem on numerically positive ample bundles.