Let $E$ be an arbitrary rank holomorphic vector bundle on a projective manifold $X$ with ample determinant bundle. There are several notions of positivity of the Chern curvature tensor of a Hermitian metric $h$ on $E$, namely Nakano positivity, Griffiths positivity, and Ampleness. A conjecture due to Griffiths asks about the equivalence between ampleness and Griffiths positivity. In a recent paper by J.-P. Demailly proposes a system of differential equations of Hermitian-Yang-Mills type for the curvature tensor to settle this conjecture. In this talk, we shall discuss Demailly's approach to this conjecture.

[Dem21]     J.-P. Demailly. Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. Mat. Sb., 212(3):39-53, 2021. (arxiv version)