An approach to the Griffiths conjecture

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 first give a sufficient condition for a positive hermitian metric on O P(E * ) (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 O P(E * ) (1) for the fibration P(E * ) → S. We define this flow and prove its convergence.

Dual Nakano positivity and singular Nakano positivity of direct image sheaves

“…Griffiths conjecture is known when Y is a compact curve (cf. [Ume73]), and it was recently shown to hold under a certain condition for the L 2 metric (see [Nau21]). Since the Kodaira-Spencer forms vanishes under certain condition in [Nau21], we obtain the following theorem for dual Nakano positivity of the canonical Hermitian metric which is a different metric in [LSY13].…”