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

点击翻译按钮获取中文摘要

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.