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 nondegenerate 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 wideopen problem. In Chap. 4, the CartanNochka Second Main Theorem in the linear projective case and the Logarithmic BlochOchiai Theorem in the case of general algebraic varieties are proved. Then the theory of entire curves in semiabelian varieties, including the Second Main Theorem of NoguchiWinkelmannYamanoi, is dealt with in full details in Chap. 6. For that purpose Chap. 5 is devoted to the notion of semiabelian 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 LangVojta 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.
About Junjiro Noguchi
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Author  :  Junjiro Noguchi 
Publisher  :  Springer Verlag, Japan 
Data Published  :  09 December 2013 
ISBN  :  4431545700 
EAN  :  9784431545705 
Format Book  :  PDF, Epub, DOCx, TXT 
Number of Pages  :  416 pages 
Age +  :  15 years 
Language  :  English 
Rating  : 
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