*Published Paper*

**Inserted:** 9 feb 2022

**Last Updated:** 9 feb 2022

**Journal:** Differ. Geom. Appl.

**Volume:** 81

**Year:** 2022

**Doi:** 10.1016/j.difgeo.2022.101848

**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.

**Keywords:**
Chern-Weil forms, Griffiths' conjecture, Flag bundles, Push-forward formulae, Schur forms