An approach to the 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 first give a sufficient condition for a positive hermitian metric on $\mathcal{O}_{\mathbb{P}(E^\ast)} (1)$ to induce a Griffiths positive $L^2$-metric on the vector bundle $E$. This result suggests to study the relative Kähler-Ricci flow on $\mathcal{O}_{\mathbb{P}(E^\ast)} (1)$ for the fibration $\mathbb{P}(E^\ast) \to S$. We define this flow and prove its convergence.

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Received 22 April 2020

Accepted 5 September 2020

Published 16 August 2022