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Nevanlinna Theory in Several Complex Variables and Diophantine Approximation (Hardcover, 2014 ed.)
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Nevanlinna Theory in Several Complex Variables and Diophantine Approximation (Hardcover, 2014 ed.)
Series: Grundlehren der mathematischen Wissenschaften, 350
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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.
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