Cross-Subject Study of Geometry

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，我们研究了建立到目前为止未知的“几何”的可能性，它统一了内万林纳理论和丢番图近似。主要的困难是缺乏算术定义的射影簇的有理点的区分的概念。我们得到了全纯曲线的喷流的Weil函数之间的函数方程，它有望作为有理点在Spec Z方向的微分的“定义方程”。其次，我们研究了紧对称空间切丛上的Ricci-Flat Kahler结构的存在性问题。例如，为了证明在秩数至少为2的紧对称空间的切丛上存在Ricci-Flat Kahler结构，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来克服这个解析问题。在等级1的情况下，一个有趣的问题是刻画这样的空间。布拉施克猜想要求恢复隐藏在测地线周期行为背后的对称性。证明了复化的Blaschke流形上存在唯一的完全非紧Ricci-Flat Kahler结构。利用这一点，我们证明了在无穷远处的对称性(证明存在)传播到原始Blaschke流形的对称性。

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.