The study of the semi-stability of Einstein-Finsler vector bundles

对于紧致Kahler流形M上的全纯向量丛E，我们用P（E）表示与E相关联的投射丛。则P（E）可以被认为是一个Kahler态射，它的相对切丛允许一个部分联络D，称为复Bott联络。这个部分联络由它的伪Kahler度量唯一确定。P（E）上的任意伪Kahler度量确定E上的凸Finsler结构F.这样的Finsler结构是唯一的，直到M上的正函数的乘法。我们称（E，F）为Einstein-Finsler，如果D的平均曲率满足爱因斯坦条件。本文研究了Einstein-Finsler向量丛在Mumford（或Mumford-Takemoto）意义下的（半）稳定性。我们的研究主要分为三个部分：（1）Bochner型消失定理及其应用，（2）满足一定条件的Einstein-Finsler向量丛的半稳定性，（3）射影（或共形）不变量与射影平坦Finsler结构，当Einstein-Finsler向量丛满足一定条件时，它的（半）稳定性是肯定的。从我们的结果中，我们陈述了以下两个与（半）稳定性有关的定理。(1)设（E，F）是Einstein-Finsler向量丛.如果（E，F）是在复Minkowski空间上建模的，则（E，F）在Mumford-Takemoto意义下是半稳定的。(2)设E是紧Rieman曲面上的全纯向量丛。则E是Mumford意义下稳定的当且仅当它允许一个射影平坦的Finsler结构，或等价地，它的射影丛P（E）→M是平坦的Kahler态射。

For a holomorphic vector bundle E over a compact Kahler manifold M, we denote by P(E) the projective bundle associated with E. Then P(E) may be considered as a Kahler morphism, and its relative tangent bundle admits a partial connection D called complex Bott connection. This partial connection is uniquely determined from its pseudo Kahler metric. Any pseudo Kahler metric on P(E) determines a convex Finsler structure F on E. Such a Finsler structure is unique up to the multiplication by a positive function on M. We say (E, F) a Einstein-Finsler if the mean curvature of D satisfies the Einstein condition. In this research, we have studied the (semi-) stability in the sense of Mumford (or Mumford-Takemoto) of Einstein-Finsler vector bundles. Our research is divided mainly in three parts:(1) The vanishing theorem of Bochner type and its applications,(2) The semi-stability of Einstein-Finsler vector bundle satisfying some conditions,(3) Projective (or conformal) invariants and projectively flat Finsler structures.The (semi-) stability of Einstein-Finsler vector bundle is affirmative if it satisfy some conditions. From among our results, we state the following two theorems which are concerned with (semi-) stability.(1) Let (E, F) be an Einstein-Finsler vector bundle. If (E, F) is modeled on a complex Minkowski space, then (E, F) is semi-stable in the sense of Mumford-Takemoto.(2) Let E be a holomorphic vector bundle over a compact Rieman surface. Then E is stable in the sense of Mumford if and only if it admits a projectively flat Finsler structure, or equivalently, its projective bundle P(E)→M is a flat Kahler morphism.