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

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 c1(E,h)∧c2(E,h)−c3(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 c2(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.