Some Studies On Holomorphic Finsler Vector Bundles

This thesis consists of four chapters.In Chapter 1,we give an introduction on our research background and the moti-vations of the problem discussed in this thesis.We also present our main results in this thesis.In Chapter 2,associated to a holomorphic vector bundle(E,G)over a complex manifold M,we will construct two kinds Chern-Weil representations c(E,G)and C(E,G)of the Chern classes,as well as Chern-Weil representations s_k(E,G)of the Segre classes,of E by using the Finsler metric G on E,which answers a question of J.Faran([1])to some extent.As an application,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 also prove in this chaper,under an extra assumption,that a Finsler-Einstein vector bundle is semi-stable in the sense of Kobayashi.Finally,we introduce a new definition of a flat Finsler metric,which is weaker than Aikou's one([2])and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.In Chapter 3,we solve a problem of Kobayashi posed in[3]by introducing a Donaldson type functional on the space F~+(E)of strongly pseudo-convex complex Finsler metrics on E–a holomorphic vector bundle over a closed K?hler manifold M.This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has Finsler-Einstein metrics on E as its only critical points,at which this functional attains the absolute minimum.In Chapter 4,by using Siu-Yau's method[4],we give a simple and direct proof of the theorem that a compact K?hler manifold with positive orthogonal bisectional curvature must be biholomorphic to P~n,which avoids the heavily use of the K?hler-Ricci flow techniques in X.Chen and Gu-Zhang's approach to this problem.