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Nevanlinna Theory in Several Complex Variables and Diophantine Approximation / Junjiro Noguchi, Jörg Winkelmann.



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Noguchi, Junjirō, 1948- author.
Publication date:
Copyright date:
Tokyo : Springer, [2014]
Copyright notice:
  • Book
  • xiv, 416 pages ; 25 cm.
Includes bibliographical references (pages 393-410) and index.
  • 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.
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.
Winkelmann, Jörg, 1963- author.
Grundlehren der mathematischen Wissenschaften, 0072-7830 ; 350
Grundlehren der mathematischen Wissenschaften ; 350.

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