Project/Area Number  10640083 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Geometry

Research Institution  KAGOSHIMA UNIVERSITY 
Principal Investigator 
AIKOU Tadashi KAGOSHIMA UNIVERSITY, Faculty of Science, Associate Professor, 理学部, 助教授 (00192831)

CoInvestigator(Kenkyūbuntansha) 
OHMOTO Toru KAGOSHIMA UNIVERSITY, Faculty of Science, Associate Professor, 理学部, 助教授 (20264400)
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)
ATSUMI Tsuyoshi KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (20041238)
MIYAJIMA Kimio KAGOSHIMA UNIVERSITY, Faculty of Science, Professor, 理学部, 教授 (40107850)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥2,200,000 (Direct Cost : ¥2,200,000)

Keywords  EinsteinFinsler bundle / Kahler morphism / Complex Bott connection / Conformal flatness / Projective flatness / Vanishing theorem / EinsteinFinslerベクトル束 / Kahler射 / 複素Bott接続 / 共形的平坦性 / 射影的平坦性 / 消滅定理 / EinsteinFinsler構造 / 凖安定性 / 準安定性 
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 EinsteinFinsler 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 MumfordTakemoto) of EinsteinFinsler vector bundles. Our research is divided mainly in three parts: (1) The vanishing theorem of Bochner type and its applications, (2) The semistability of EinsteinFinsler vector bundle satisfying some conditions, (3) Projective (or conformal) invariants and projectively flat Finsler structures. The (semi) stability of EinsteinFinsler 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 EinsteinFinsler vector bundle. If (E, F) is modeled on a complex Minkowski space, then (E, F) is semistable in the sense of MumfordTakemoto. (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.
