| Co-Investigator(Kenkyū-buntansha) |
ATSUMI Tsuyoshi KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (20041238)
OHMOTO Toru KAGOSHIMA UNIVERSITY, Faculty of Science, Associate Professor, 理学部, 助教授 (20264400)
MIYAJIMA Kimio KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (40107850)
SHINMORI Shuichi KAGOSHIMA UNIVERSITY, Faculty of Science, Associate Professor, 理学部, 助教授 (40226353)
SAKAI Koukichi KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (20041759)
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| Research Abstract |
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.
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