Nevanlinna Theory in Several Complex Variables and Diophantine Approximation
 Author/Creator
 Noguchi, Junjirō, 1948 author.
 Language
 English.
 Publication
 Tokyo : Springer, [2014]
 Copyright notice
 ©2014
 Physical description
 xiv, 416 pages ; 25 cm.
 Series
 Grundlehren der mathematischen Wissenschaften ; 350.
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Available online

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QA331 .N63 2014

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QA331 .N63 2014
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Contributors
 Contributor
 Winkelmann, Jörg, 1963 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 393410) and index.
 Contents

 Nevanlinna theory of meromorphic functions
 The first main theorem
 The second main theorem
 Examples of functions of finite order
 The first main theorem
 Plurisubharmonic functions
 One variable
 Several variables
 PoincaréLelong formula
 The first main theorem
 Meromorphic mappings, divisors and line bundles
 Differentiable functions on complex spaces
 Metrics and curvature forms of line bundles
 The first main theorem for coherent ideal sheaves
 Proximity functions for coherent ideal sheaves
 The case of m= 1
 Order functions
 Metrics
 Cartan's order function
 A family of rational functions
 Characterization of rationality
 Nevanlinna's inequality
 Ramified covers over Cm
 Differentiably nondegenerate meromorphic maps
 Lemma on logarithmic derivatives
 The second main theorem
 Applications and generalizations
 Applications
 NonKähler counterexample
 Generalizations
 Entire curves in algebraic varieties
 Nochka weights
 The CartanNochka theorem
 Entire curves omitting hyperplanes
 Generalizations and applications
 Derived curves
 Generalization to higher dimensional domains
 Finite ramified covering spaces
 The EremenkoSodin second main theorem
 The second main theorem of CorvajaZannier, EvertseFerretti and Ru
 Krutin's theorem
 Moving targets
 Yamanoi's second main theorem
 Applications
 Logarithmic forms
 Logarithmic jet bundles
 Jet bundles in general
 Jet spaces
 Logarithmic jet bundles and logarithmic jet spaces
 Lemma on logarithmic forms
 Inequality of the second main theorem type
 Entire curves omitting hypersurfaces
 The fundamental conjecture of entire curves
 Semiabelian varieties
 Semitori
 Definition
 Characteristic subgroups of complex semitori
 Holomorphic functions
 Semiabelian varieties
 Presentations
 Presentations of semiabelian varieties
 Inequivalent algebraic structures
 Choice of presentation
 Construction of semitori via presentations
 Morphisms and gaga
 Reductive group actions
 Semitoric varieties
 Toric varieties
 Semitoric varieties
 Key properties of semitoric varieties
 Quasialgebraic subgroups
 Compactifiable groups and kähler condition
 Examples of nonsemitoric varieties
 Jet bundles over semitoric varieties
 Line bundles on toric varieties
 Ample line bundles
 Leray spectral sequence
 Decomposition of line bundles
 Global span and very ampleness
 Stabilizer and bigness
 Good position and stabilizer
 Good position
 Good position and choice of compactification
 Regular subgroups
 More facts on semitori
 Entire curves in semiabelian varieties
 Order functions
 Structure of jet images
 Image of f (case k = 0)
 Jet projection method
 A counterexample
 Compact complex tori
 Entire curves
 Applications to differentiably nondegenerate maps
 Semitori : truncation level ko
 Semiabelian varieties : truncation level 1
 Truncation level 1
 The second main theorem for jet lifts
 Higher codimensional subvarieties of Xk(f)
 Proof of theorem 6.5.1
 Applications
 Algebraic degeneracy of entire curves
 Kobayashi hyperbolicity
 Complements of divisors in projective space
 Strong GreenGriffiths conjecture
 Lang's questions on theta divisors
 Algebraic differential equations
 Kobayashi hyperbolicity
 Kobayashi pseudodistance
 Brody's theorem
 Brody's reparametrization
 Hyperbolicity as an open property
 Kobayashi hyperbolic manifolds
 Kobayashi hyperbolic projective hypersurfaces
 Hyperbolic embedding into complex projective space
 Brody curves and Yosida functions
 Growth conditions and Yosida functions
 Characterizing Brody maps into Tori
 Brody curves with prescribed points in the image
 Ahlfors' currents
 Nevanlinna theory over function fields
 Lang's conjecture
 NevanlinnaCartan theory over function fields
 Borel's identity and unit equations
 Generalized borel's theorem and applications
 Diophantine approximation
 Valuations
 Definition and the basic properties
 Extensions of valuations
 Normalized valuations
 Heights
 Theorems of Roth and Schmidt
 Unit equations
 The abcconjecture and the fundamental conjecture
 The FallingsVojta theorem
 Distribution of rational points
 References
 Index
 Symbols.
 Nevanlinna Theory of Meromorphic Functions
 First Main Theorem
 Differentiably NonDegenerate Meromorphic Maps
 Entire Curves into Algebraic Varieties
 SemiAbelian Varieties
 Entire Curves into SemiAbelian Varieties
 Kobayashi Hyperbolicity
 Nevanlinna Theory over Function Fields
 Diophantine Approximation
 Bibliography
 Index
 Symbols.
 Summary
 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.
Bibliographic information
 Publication date
 2014
 Copyright date
 2014
 Responsibility
 Junjiro Noguchi, Jörg Winkelmann.
 Series
 Grundlehren der mathematischen Wissenschaften, 00727830 ; 350
 ISBN
 9784431545705
 4431545700