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 $c_1(E,h) \wedge c_2(E,h) - c_3(E,h)$, 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.