Value Distribution Theeory and Algebraic Geometry

本课题的最终目的是建立充分解释全纯曲线在射影代数变量上的值分布理论与在算术定义的射影变量上的丢芬图近似的相似性的几何。本研究的主要困难在于缺乏对算术变量有理点在Spec Z方向上的微分概念的定义。通过这个项目下的调查，我们得出了基本的想法，可以描述如下。而不是试图在Spec Z本身的方向上定义有理点的微分的概念，我们试图在值分布理论函数（例如，Weil函数）中找到一个泛函方程系统。然后，利用Vojta字典，我们将泛函方程转化为丢芬图近似方程。这些方程将“定义”有理点的微分。在证明这一思想的过程中，我们得到了以下结果：(1)我们发明了Radon变换，将高维全纯曲线变换为亚纯函数系统。(2)如果将“有理函数”用全纯映射替换成双曲黎曼曲面，则可以将Abelian变体的群结构纳入Radon变换的框架中。结果表明，对于阿贝尔变簇的全纯曲线，第二个主要猜想成立。(3)我们证明了如果所考虑的曲线不包含在一个特殊的固有代数子集中，对于所有的全纯曲线，射流的Weil函数的渐近行为是相同的，这是对第二个主要猜想的阻碍。这就是我们要找的泛函方程组。

The ultimate purpose of this project is to establish the geometry which fully explain the similarity between value distribution theory of holomorphic curves in projective algebraic varieties and the diophantine approximations on arithmetically defined projective varicties. The major difficulty in this research is the lack of definition of the notion of differentiation (in the direction of Spec Z of rational points of arithmetic varieties. Through the investigation under this project, we arrived at the fundamental idea which may be described as follows. Instead of trying to define the notion of differentiation of rational points in the direction of Spec Z itself, we try to find a system of functional equations among value-distrivution-theoretic functions (e.g., Weil functions). Then, using Vojta's dictionary, we translate the functional equations to equations in diophantine approximations. These equations will "define" the diffrentials of rational points. In the course of justifying this idea, we obtained the following results : (1) We invented the Radon transform transforming a holomorphic curve in higher dimensional varieties into a system of meromnorphic functions. (2) We can include the group structure of Abelian varieties into the framework of Radon transform, if we replace "rational functions" by holomorphic maps into hyperbolic Riemann surfaces. As a result, we can show that the second main conjecture holds for holomorphic curves in Abelian varieties. (3) We showed that the asymptotic behavior of the Weil functions of jets of a given holomorphic curve is the same for all jets, if a curve under consideration is not contained in a special proper algebraic subset, which is the obstruction to the second main conjecture. This is considered to be system of functional equations we are looking for.