Radon transformation in Nevanlinina theory and Diophantire approximation

第二个主要猜想是内瓦林纳理论的最终目的。在这项研究中，我通过对第二个主要猜想的尝试，提出了几个关于纳瓦林纳理论和丢番图近似统一的“猜想几何”的想法。Pn中全纯曲线的Castan-Ahlfors-Weyl理论与Schmidt关于射影空间上Diophantino逼近的Sulspace定理之间的相似之处不仅存在于它们的陈述中，而且也存在于它们的教授中。这种类比的本质在于积分几何中的“Radon变换”的思想，在本研究中，我能够建立起一种同时隐藏在Nevanlinna理论和丢番图近似背后的“Radon变换”。这种“Radon变换”的一个优点是：通过这种变换，我们将能够分离丢番图几何中完全类似于关于对数导数的引理起本质…作用的部分更原始的角色。丢番图几何中的这一部分是最具数论性质的，Roth-Schmidt-Faltings型的丢番图逼近定理以及Mihkowski-Bombieri-Vasler的“数的几何”起着最重要的作用。这与全纯情形有很大不同，但其结果与loganithic导数上的lemnia完全相似。另一方面，对第二个主要猜想的尝试有了很大的进展。也就是说，我发现了n个独立的交换全纯向量场的几何框架，以及在“非交换”射影簇上引入这种向量场所引起的奇异性。根据这n个交换向量场，错误的形式主义是可能的。因此，困难局限于“奇点”，并且我能够建立一个新的关于逻辑导数的引理(“射影版”)来分析存在“奇点”的全纯曲线。希望这将为复代数几何提供一种新的方法。较少

The second main conjective is the ultimate purpose of the Nevanlinna theory. In this research, I proposed several ideas of the "conjectural geometry" unifying Nevanlinna theory and Diophantive approximation through the attempt toward the second main conjecture. The analogy between Castan-Ahlfors-Weyl theory of holomorphic curves in Pn and the sulpace theorem of Schmidt on Diophantino approximation on projective spaces is ohsewed not only in their statements but also in their profs. The essence of the analogy lies in the idea of the "Radon-transformation" in integral geomeyty, In this research, I was able to establish a kind of "Radon-transform" which lies behind both Nevanlinna theory and Diophantine approximation. One advantage of this "Radon-transform" is the following : through this transformation we will be able to separate the part of the Diophantine geometry which is completely analogous to the part of Nevanlinna theory in which the lemma on logarithmic derivative plays an essent … More ial role. This part in Diophantine geometry is most number-theoretic and the theorems of Diophantine approximation of Roth-Schmidt-Faltings type, as well as Mihkowski-Bombieri-Vasler "geometry of numbers" play the most enential role. This is most different from holomorphic case, but never the less the result is completely analogous to the lemnia on loganithmic derivature. On the other hand, there was a big progress on the attempt toward 2nd main conjecture. Namely I discovered the geometric framework of n independent commutative holomorphic vector fields and the singularity caused by introducing such vector fields on "noncommutative" projective varieties. The wronskian formalism is possible in terms of these n commutative vector fields. As a result, the difficulty is localized in "singularities" and I was able to establish a new lemma on loganthmic derivative ("projective version") to analyze holomorphic curves in the presence of "singularities". I hope this will provide a new method in complex algebraic geometry. Less