An approach to Griffiths conjecture

The Griffiths conjecture asserts that every ample vector bundle \(E\) over a compact complex manifold \(S\) admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on \(\mathcal{O}_{\mathbb{P}(E^*)}(1)\) to induce a Griffiths positive \(L^2\)-metric on the vector bundle \(E\). This result suggests to study the relative K\"ahler-Ricci flow on \(\mathcal{O}_{\mathbb{P}(E^*)}(1)\) for the fibration \(\mathbb{P}(E^*)\to S\). We define a flow and give arguments for the convergence.