An approach to Griffiths conjecture

TL;DR

This paper provides a sufficient condition for positive hermitian metrics on certain line bundles to induce Griffiths positive metrics on vector bundles, and explores a related Kähler-Ricci flow for potential convergence.

Contribution

It introduces a new criterion linking hermitian metrics on line bundles to Griffiths positivity of vector bundles and studies a related Kähler-Ricci flow for this purpose.

Findings

A sufficient condition for Griffiths positivity via hermitian metrics.

Definition and analysis of a relative Kähler-Ricci flow on the projectivization.

Arguments suggesting the convergence of the proposed flow.

Abstract

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.