An approach to the Griffiths conjecture

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