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Cross-Subject Study of Geometry

几何的跨学科研究

基本信息

批准号：
07640108
负责人：
KOBAYASHI Ryoichi
金额：
$ 1.66万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07640108/
关键词：
Nevanlinna theory Diophuntine approximation Weil funition Riiu-flat Kahlir metrus Monge-Ampere equation weightel isoperimetric inequalty Blaschke conjecture adapted complex structure Blaschke多様体 Monge-Ampere方程式正則曲線 Diophantine appro

项目摘要

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.

这个项目的目的是研究涉及几个不同领域的几何问题。作为第一个例子，我们研究了建立迄今未知的“几何”，统一Nevanlinna理论和丢番图近似的可能性。主要的困难是缺乏算术定义的射影簇的有理点的微分的概念。得到了全纯曲线喷流的Weil函数之间的函数方程，该方程有望作为Spec Z方向上有理点微分的“定义方程”。其次，研究了紧致对称空间切丛上Ricci平坦Kahler结构的存在性问题。例如，为了证明秩至少为2的紧致对称空间的切丛上的Ricci平坦Kahler结构的存在性，我们必须在适当的边界条件下证明某些Monge-Ampere方程的解的先验估计。我们通过引入加权等周不等式克服了这个解析问题。在秩为1的情况下，一个有趣的问题是刻画这样的空间。Blaschke猜想要求恢复隐藏在测地线周期行为背后的对称性。证明了复化Blaschke流形上存在唯一的完全非紧Ricci平坦Kahler结构。利用这一点，我们证明了在无穷远处的对称性（已证明存在）传播到原Blaschke流形的对称性。