Radon transformation in Nevanlinina theory and Diophantire approximation

第二个主要的连词是内万林纳理论的最终目的。在本研究中，我通过对第二个主要猜想的尝试，提出了几种统一奈万林纳理论和透隐近似的“猜想几何”思想。在他们的陈述和他们的理论中，我们发现了Pn中全纯曲线的Castan-Ahlfors-Weyl理论与Schmidt关于投影空间上丢芬提诺近似的sulpace定理的相似之处。类比的本质在于积分几何中的“radon变换”思想，在本次研究中，我建立了一种“radon变换”，它隐藏在Nevanlinna理论和丢番图近似的背后。这个“radon变换”的一个优点是：通过这个变换，我们将能够分离丢番图几何的一部分，这部分完全类似于奈万林纳理论的部分，其中对数导数的引理起着重要的作用。丢番图几何的这一部分是最具数论性的，其中Roth-Schmidt-Faltings型丢番图近似定理以及Mihkowski-Bombieri-Vasler“数的几何”起着最重要的作用。这与全纯情况非常不同，但其结果完全类似于对数导数的定理。另一方面，对第二个主要猜想的尝试取得了很大的进展。即我发现了n个独立的可交换全纯向量场的几何框架，以及在“非可交换”射影变体上引入这些向量场所引起的奇异性。在这n个可交换向量场中，朗斯基的形式是可能的。因此，问题的难点被局限在“奇点”上，我建立了一个新的关于对数导数的引理（“投影版”）来分析存在“奇点”的全纯曲线。我希望这将为复杂代数几何提供一种新的方法。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