Positively curved Finsler metrics on vector bundles II

TL;DR

This paper investigates conditions under which ample vector bundles with certain curvature bounds exhibit Kobayashi and Griffiths positivity, extending previous results on Finsler metrics and curvature comparisons.

Contribution

It establishes new positivity results for vector bundles using curvature bounds and duality of convex Finsler metrics, advancing the understanding of positivity in complex geometry.

Findings

E^* ⊗ det E is Kobayashi positive under curvature bounds.

E is Kobayashi positive if similar curvature bounds hold on associated line bundles.

Under additional assumptions, E^* ⊗ det E and E are Griffiths positive.

Abstract

We show that if $E$ is an ample vector bundle of rank at least two with some curvature bound on $O_{P(E^*)}(1)$, then $E^*\otimes \det E$ is Kobayashi positive. The proof relies on comparing the curvature of $(\det E^*)^k$ and $S^kE$ 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 $O_{P(E^*)}(1)$ and $O_{P(E\otimes \det E^*)}(1)$, then $E$ is Kobayashi positive. With additional assumptions, we can furthermore show that $E^*\otimes \det E$ and $E$ are Griffiths positive.