Chern forms of holomorphic Finsler vector bundles and some applications

TL;DR

This paper develops new Chern and Segre forms for holomorphic Finsler vector bundles, providing insights into positivity, stability, and flatness conditions with applications to complex geometry.

Contribution

It introduces novel total Chern and Segre forms expressed via Finsler metrics, addressing a question of Faran and linking Finsler flatness to Hermitian flatness.

Findings

Signed Segre forms are positive under positive Kobayashi curvature

Finsler-Einstein bundles are semi-stable under certain conditions

A new weaker notion of Finsler flatness is equivalent to Hermitian flatness

Abstract

In this paper, we present two kinds of total Chern forms $c(E,G)$ and $\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $\pi:(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of J. Faran (\cite{Faran}) to some extent. As some applications, we show that the signed Segre forms $(-1)^ks_k(E,G)$ are positive $(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler-Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou's one (\cite{Aikou}) and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.