Universal vector bundles, push-forward formulae and positivity of characteristic forms

TL;DR

This paper proves a universal push-forward formula for Chern forms of flag bundles, computes their curvature explicitly, and demonstrates positivity of certain characteristic forms, advancing understanding in complex differential geometry.

Contribution

It establishes a pointwise push-forward formula for Chern forms, computes universal bundle curvatures explicitly, and confirms positivity of specific characteristic forms related to Griffiths' conjecture.

Findings

Push-forward formula holds pointwise at the level of Chern forms.

Explicit curvature computation of universal bundles.

Positivity of polynomials in Chern forms of Griffiths semipositive bundles.

Abstract

Given a Hermitian holomorphic vector bundle over a complex manifold, consider its flag bundles with the associated universal vector bundles endowed with the induced metrics. We prove that the universal formula for the push-forward of a polynomial in the Chern classes of all the possible universal vector bundles also holds pointwise at the level of Chern forms. A key step in our proof is the explicit computation, at a point of any flag bundle, of the Chern curvature of the universal vector bundles with the induced metrics. As an application, we provide an alternative version of the Jacobi-Trudi identity at the level of differential forms. We also show the positivity of a family of polynomials in the Chern forms of Griffiths semipositive vector bundles. This latter result partially confirms the Griffiths' conjecture on positive characteristic forms, which has raised considerable interest in recent years.