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

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.

16 pages, no figures, comments are very welcome! v4: some minor changes. A remark added in Section 2 following referee's comments. A reference has been added. Version accepted for publication in Differ. Geom. Appl

References in corpus (3)

• On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities
• On the positivity of high-degree Schur classes of an ample vector bundle
• Pointwise Universal Gysin formulae and Applications towards Griffiths' conjecture

Cited by in corpus (3)

• Chern numbers on positive vector bundles and combinatorics
• Positivity of Schur forms for strongly decomposably positive vector bundles
• Universal vector bundles, push-forward formulae and positivity of characteristic forms