Positively curved Finsler metrics on vector bundles II

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 $E^{\ast }\otimes \det E$ is Kobayashi positive. The proof relies on comparing the curvature of $(\det E^{\ast }{)}^k$ 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 $O_{P(E\otimes \det E^{\ast })}(1)$, then $E$ is Kobayashi positive. With additional assumptions, we can furthermore show that $E^{\ast }\otimes \det E$ and $E$ are Griffiths positive.

24 pages