Includes bibliographical references (pages 393-410) 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
Poincaré-Lelong 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 non-degenerate meromorphic maps
Lemma on logarithmic derivatives
The second main theorem
Applications and generalizations
Applications
Non-Kähler counter-example
Generalizations
Entire curves in algebraic varieties
Nochka weights
The Cartan-Nochka theorem
Entire curves omitting hyperplanes
Generalizations and applications
Derived curves
Generalization to higher dimensional domains
Finite ramified covering spaces
The Eremenko-Sodin second main theorem
The second main theorem of Corvaja-Zannier, Evertse-Ferretti 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
Semi-abelian varieties
Semi-tori
Definition
Characteristic subgroups of complex semi-tori
Holomorphic functions
Semi-abelian varieties
Presentations
Presentations of semi-abelian varieties
Inequivalent algebraic structures
Choice of presentation
Construction of semi-tori via presentations
Morphisms and gaga
Reductive group actions
Semi-toric varieties
Toric varieties
Semi-toric varieties
Key properties of semi-toric varieties
Quasi-algebraic subgroups
Compactifiable groups and kähler condition
Examples of non-semi-toric varieties
Jet bundles over semi-toric 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 semi-tori
Entire curves in semi-abelian varieties
Order functions
Structure of jet images
Image of f (case k = 0)
Jet projection method
A counter-example
Compact complex tori
Entire curves
Applications to differentiably non-degenerate maps
Semi-tori : truncation level ko
Semi-abelian 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 Green-Griffiths 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
Nevanlinna-Cartan 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 abc-conjecture and the fundamental conjecture
The Fallings-Vojta theorem
Distribution of rational points
References
Index
Symbols.
Nevanlinna Theory of Meromorphic Functions
First Main Theorem
Differentiably Non-Degenerate Meromorphic Maps
Entire Curves into Algebraic Varieties
Semi-Abelian Varieties
Entire Curves into Semi-Abelian 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 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.