Positively curved Finsler metrics on vector bundles, II

Kuang-Ru Wu

Vol. 326 (2023), No. 1, 161–186

DOI: 10.2140/pjm.2023.326.161

Abstract

We show that if $E$ is an ample vector bundle of rank at least two with some curvature bound on $O_{P(E^{\ast })}(1)$, then $E^{\ast }\otimes \det <!--FUNCTION\; APPLICATION-->E$ is Kobayashi positive. The proof relies on comparing the curvature of ${(\det <!--FUNCTION\; APPLICATION-->E^{\ast })}^k$ and $S^{k}E$ for large $k$ and using duality of convex Finsler metrics. Following the same thread of thought, we show if $E$ is ample with similar curvature bounds on $O_{P(E^{\ast })}(1)$ and $O_{P(E\otimes \det <!--FUNCTION\; APPLICATION-->E^{\ast })}(1)$, then $E$ is Kobayashi positive. With additional assumptions, we can furthermore show that $E^{\ast }\otimes \det <!--FUNCTION\; APPLICATION-->E$ and $E$ are Griffiths positive.

Keywords

Kobayashi positivity, Griffiths positivity, $L^2$-metrics

Mathematical Subject Classification

Primary: 32F17, 32J25

Milestones

Received: 24 March 2023

Revised: 3 August 2023

Accepted: 21 October 2023

Published: 14 December 2023