1996 Fiscal Year Final Research Report Summary
Cross-Subject Study of Geometry
| Project/Area Number |
07640108
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Nagoya University |
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Principal Investigator |
KOBAYASHI Ryoichi Nagoya University, Graduate School of Polymathematics Professor, 大学院・多元数理科学研究科, 教授 (20162034)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIKAWA Ken-ichi Graduate School of Polymathematics, Research Associate, 大学院・多元数理科学研究科, 助手 (20242810)
SATO Takeshi Graduate School of Polymathematics, Research Associate, 大学院・多元数理科学研究科, 助手 (60252219)
MUKAI Shigeru Graduate School of Polymathematics, Professor, 大学院・多元数理科学研究科, 教授 (80115641)
SATO Hajime Graduate School of Polymathematics, Professor, 大学院・多元数理科学研究科, 教授 (30011612)
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| Project Period (FY) |
1995 – 1996
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| Keywords | Nevanlinna theory / Diophuntine approximation / Weil funition / Riiu-flat Kahlir metrus / Monge-Ampere equation / weightel isoperimetric inequalty / Blaschke conjecture / adapted complex structure |
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| Research Abstract |
The purpose of this project is to study problems in geometry in which several different areas are involved. As a first example, we investigated the possibility of establishing the so far unknown "geometry" which unifies Nevanlinna theory and Diophantine approximations. The main difficulty is the lack of the notion of differentiation of rational points of arithmetically defined projective varieties. We obtained functional equations among Weil functions of jets of holomorphic curves which is expected to serve as "defining equations" of differentials of rational points in Spec Z direction.
Secondly we investigated problems concerning the existence of Ricci-flat Kahler structures on the tangent bundle of compact symmetric spaces. For instance, to show the existence of a Ricci-flat Kahler structure on tangent bundles of compact symmetric spaces of rank at least 2, we must show a priori estimates for solutions of certain Monge-Ampere eqations under appropriate boundary conditions. We overcome this analytic problem by introducing weighted isoperimetric inequality. An interesting question in rank 1 case is to characterize such spaces. The Blaschke conjecture asks to recover the symmetry hidden behind the periodic behavior of geodesics. We showed the existence of a unique complete non-compact Ricci-flat Kahler structure on the complexified Blaschke manifold. By using this, we showed that symmetry (which is shown to exist) at infinity propagates to symmetry of the original Blaschke manifold.
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