Nevanlinna Theory in Several Complex Variables and Diophantine Approximation
Nevanlinna Theory in Several Complex Variables and Diophantine Approximation
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DOI：
10.1007/978-4-431-54571-2
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发表时间：
2013-12
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影响因子：
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通讯作者：
野口 潤次郎;J. Winkelmann
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作者：
野口 潤次郎;J. Winkelmann
本书的目的是为研究生和感兴趣的研究人员提供高维 Nevanlinna 理论及其与丢番图近似理论的关系的全面说明。本书共九章，系统地阐述了代数簇或复空间之间的亚纯映射 Nevanlinna 理论，该理论建立在复平面上亚纯函数的经典理论的基础上，并在第 1 章中得到了充分的证明。 1 研究现状。第二章以非常一般的形式介绍了相干理想滑轮的第一主定理。通过多次谐波函数的制备，描述了如何将理论推广到更高维度。在第 1 章中。 3 Griffiths 等人提出的可微分非简并亚纯映射第二大定理被证明是高维 Nevanlinna 理论的原型。为一般复杂代数簇中的整个曲线建立这样的第二大定理是一个悬而未决的问题。在第 1 章中。 4、证明了线性射影情况下的Cartan-Nochka第二大定理和一般代数簇情况下的对数Bloch-Ochiai定理。然后，半阿贝尔簇的整个曲线理论，包括野口-温克尔曼-山野第二大定理，在第 1 章中进行了详细讨论。 6. 为此目的，第 1 章第 5 章致力于半阿贝尔簇的概念。结果导致了许多应用。根据这些结果，小林双曲问题在第 1 章中进行了讨论。 7.最后两章从高维Nevanlinna理论的角度讨论了丢番图近似理论，并在某些情况下证实了Lang-Vojta猜想。在第 1 章中。 8 讨论了函数域理论。最后，在第 1 章中。在第 9 章中，根据 Nevanlinna 理论提出并表述了 Roth、Schmidt、Faltings 和 Vojta 在数域上的定理，其结果受到第 1 章的启发。 4、6 和 7。
The aim of this book is to provide a comprehensive account of higher dimensional Nevanlinna theory and its relations with Diophantine approximation theory for graduate students and interested researchers. This book with nine chapters systematically describes Nevanlinna theory of meromorphic maps between algebraic varieties or complex spaces, building up from the classical theory of meromorphic functions on the complex plane with full proofs in Chap. 1 to the current state of research. Chapter 2 presents the First Main Theorem for coherent ideal sheaves in a very general form. With the preparation of plurisubharmonic functions, how the theory to be generalized in a higher dimension is described. In Chap. 3 the Second Main Theorem for differentiably non-degenerate meromorphic maps by Griffiths and others is proved as a prototype of higher dimensional Nevanlinna theory. Establishing such a Second Main Theorem for entire curves in general complex algebraic varieties is a wide-open problem. In Chap. 4, the Cartan-Nochka Second Main Theorem in the linear projective case and the Logarithmic Bloch-Ochiai Theorem in the case of general algebraic varieties are proved. Then the theory of entire curves in semi-abelian varieties, including the Second Main Theorem of Noguchi-Winkelmann-Yamanoi, is dealt with in full details in Chap. 6. For that purpose Chap. 5 is devoted to the notion of semi-abelian varieties. The result leads to a number of applications. With these results, the Kobayashi hyperbolicity problems are discussed in Chap. 7. In the last two chapters Diophantine approximation theory is dealt with from the viewpoint of higher dimensional Nevanlinna theory, and the Lang-Vojta conjecture is confirmed in some cases. In Chap. 8 the theory over function fields is discussed. Finally, in Chap. 9, the theorems of Roth, Schmidt, Faltings, and Vojta over number fields are presented and formulated in view of Nevanlinna theory with results motivated by those in Chaps. 4, 6, and 7.