A note on Griffiths' conjecture about the positivity of Chern-Weil forms

TL;DR

This paper proves a specific positivity property of Chern-Weil forms for Griffiths semipositive vector bundles, providing evidence for Griffiths' conjecture and establishing new inequalities among Chern forms.

Contribution

It demonstrates the positivity of a particular differential form related to Chern classes for rank 3 bundles, supporting Griffiths' conjecture and extending positivity results to higher ranks.

Findings

Proves positivity of c_1 ∧ c_2 - c_3 for rank 3 bundles.

Establishes new inequalities between Chern forms.

Provides insights and open questions on Griffiths' conjecture.

Abstract

Let $ (E,h) $ be a Griffiths semipositive Hermitian holomorphic vector bundle of rank $ 3 $ over a complex manifold. In this paper, we prove the positivity of the characteristic differential form $ c_1(E,h) \wedge c_2(E,h) - c_3(E,h) $, thus providing a new evidence towards a conjecture by Griffiths about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, we establish a new chain of inequalities between Chern forms. Moreover, we point out how to obtain the positivity of the second Chern form $ c_2(E,h) $ in any rank, starting from the well-known positivity of such form if $ (E,h) $ is just Griffiths positive of rank $ 2 $. The final part of the paper gives an overview on the state of the art of Griffiths' conjecture, collecting several remarks and open questions.