Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

TL;DR

This paper extends Gysin formulae to pointwise Chern form computations for hermitian vector bundles, providing new positivity results that support Griffiths' conjecture on positive polynomials.

Contribution

It establishes pointwise Gysin formulae for Chern forms of hermitian bundles, enabling new positivity results related to Griffiths' conjecture.

Findings

Proves pointwise Gysin formulae for Chern forms in hermitian bundles.

Demonstrates positivity of previously unverified Chern form polynomials.

Provides evidence supporting Griffiths' conjecture on positive polynomials.

Abstract

Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $\rho$ be a sequence of dimensions $0 = \rho_0 < \rho_1 < \cdots < \rho_m = r$. Let $Q_{\rho,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_{\rho}(E) \to X$ associated to $\rho$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $\Xi_{\rho,j}$. We show that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{\rho,j}$'s also hold pointwise at the level of the Chern forms $\Xi_{\rho,j}$ 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.