Accepted Paper
Inserted: 9 feb 2022
Last Updated: 9 feb 2022
Journal: Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5)
Year: 2021
Doi: 10.2422/2036-2145.202011_021
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 à la Darondeau-Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{\rho,j}$'s also holds pointwise at the level of the Chern forms $\Xi_{\rho,j}$ in this Hermitianized situation. As an application, we show the strong 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.
Keywords: Griffiths' conjecture, Flag bundles, Gysin's formulae, Positive polynomials, Fulton-Lazarsfeld theorem