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El. knyga: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation

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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.



This book reviews higher dimensional Nevanlinna theory and its relationship with Diophantine approximation theory. Coverage builds up from the classical theory of meromorphic functions on the complex plane with full proofs, to the current state of research.

Recenzijos

This intriguing monograph describes the state of the art of large parts of Nevanlinna theory in several complex variables. an important reference for graduate students and researchers in Nevanlinna theory for many years. (CH. Baxa, Monatshefte für Mathematik, Vol. 184, 2017)



I consider this book as extremely helpful both for research and teaching in this area, since it does not only give a self-contained access to some of the deep original research results of the two authors, but it also always discusses the limits and the side aspects of these results by relevant examples, respectively counter-examples, and puts them into its historic contexts, which allows a much deeper understanding than if one justwould see the main results. (Gerd Dethloff, zbMATH 1337.32004, 2016)



The book under review gives a panoramic survey of Nevanlinnas theory of valued distribution and Diophantine approximation in several complex variables. The book is addressed to researchers and graduate students, and an interested reader will be rewarded with a clear exposition of classical and recent progress in this active area of mathematics. (Felipe Zaldivar, MAA Reviews, April, 2014)

1 Nevanlinna Theory of Meromorphic Functions
1(24)
1.1 The First Main Theorem
1(11)
1.2 The Second Main Theorem
12(7)
1.3 Examples of Functions of Finite Order
19(6)
2 The First Main Theorem
25(66)
2.1 Plurisubharmonic Functions
25(17)
2.1.1 One Variable
25(8)
2.1.2 Several Variables
33(9)
2.2 Poincare-Lelong Formula
42(8)
2.3 The First Main Theorem
50(16)
2.3.1 Meromorphic Mappings, Divisors and Line Bundles
50(4)
2.3.2 Differentiable Functions on Complex Spaces
54(4)
2.3.3 Metrics and Curvature Forms of Line Bundles
58(8)
2.4 The First Main Theorem for Coherent Ideal Sheaves
66(7)
2.4.1 Proximity Functions for Coherent Ideal Sheaves
66(5)
2.4.2 The Case of m = 1
71(2)
2.5 Order Functions
73(13)
2.5.1 Metrics
73(4)
2.5.2 Cartan's Order Function
77(2)
2.5.3 A Family of Rational Functions
79(4)
2.5.4 Characterization of Rationality
83(3)
2.6 Nevanlinna's Inequality
86(2)
2.7 Ramified Covers over Cm
88(3)
3 Differentiably Non-degenerate Meromorphic Maps
91(22)
3.1 Lemma on Logarithmic Derivatives
91(2)
3.2 The Second Main Theorem
93(9)
3.3 Applications and Generalizations
102(11)
3.3.1 Applications
102(3)
3.3.2 Non-Kahler Counter-Example
105(5)
3.3.3 Generalizations
110(3)
4 Entire Curves in Algebraic Varieties
113(48)
4.1 Nochka Weights
113(10)
4.2 The Cartan-Nochka Theorem
123(8)
4.3 Entire Curves Omitting Hyperplanes
131(2)
4.4 Generalizations and Applications
133(5)
4.4.1 Derived Curves
133(1)
4.4.2 Generalization to Higher Dimensional Domains
134(1)
4.4.3 Finite Ramified Covering Spaces
134(1)
4.4.4 The Eremenko-Sodin Second Main Theorem
135(1)
4.4.5 The Second Main Theorem of Corvaja-Zannier, Evertse-Ferretti and Ru
136(1)
4.4.6 Krutin's Theorem
136(1)
4.4.7 Moving Targets
136(1)
4.4.8 Yamanoi's Second Main Theorem
137(1)
4.4.9 Applications
137(1)
4.5 Logarithmic Forms
138(6)
4.6 Logarithmic Jet Bundles
144(4)
4.6.1 Jet Bundles in General
144(2)
4.6.2 Jet Spaces
146(1)
4.6.3 Logarithmic Jet Bundles and Logarithmic Jet Spaces
146(2)
4.7 Lemma on Logarithmic Forms
148(2)
4.8 Inequality of the Second Main Theorem Type
150(7)
4.9 Entire Curves Omitting Hypersurfaces
157(2)
4.10 The Fundamental Conjecture of Entire Curves
159(2)
5 Semi-abelian Varieties
161(54)
5.1 Semi-tori
161(15)
5.1.1 Definition
161(3)
5.1.2 Characteristic Subgroups of Complex Semi-tori
164(2)
5.1.3 Holomorphic Functions
166(1)
5.1.4 Semi-abelian Varieties
167(2)
5.1.5 Presentations
169(1)
5.1.6 Presentations of Semi-abelian Varieties
170(1)
5.1.7 Inequivalent Algebraic Structures
171(1)
5.1.8 Choice of Presentation
171(1)
5.1.9 Construction of Semi-tori via Presentations
172(1)
5.1.10 Morphisms and GAGA
173(3)
5.2 Reductive Group Actions
176(4)
5.3 Semi-toric Varieties
180(11)
5.3.1 Toric Varieties
180(1)
5.3.2 Semi-toric Varieties
181(1)
5.3.3 Key Properties of Semi-toric Varieties
182(3)
5.3.4 Quasi-algebraic Subgroups
185(2)
5.3.5 Compactifiable Groups and Kahler Condition
187(3)
5.3.6 Examples of Non-semi-toric Varieties
190(1)
5.4 Jet Bundles over Semi-toric Varieties
191(1)
5.5 Line Bundles on Toric Varieties
192(11)
5.5.1 Ample Line Bundles
192(3)
5.5.2 Leray Spectral Sequence
195(1)
5.5.3 Decomposition of Line Bundles
196(2)
5.5.4 Global Span and Very Ampleness
198(3)
5.5.5 Stabilizer and Bigness
201(2)
5.6 Good Position and Stabilizer
203(12)
5.6.1 Good Position
203(1)
5.6.2 Good Position and Choice of Compactification
204(5)
5.6.3 Regular Subgroups
209(1)
5.6.4 More Facts on Semi-tori
210(5)
6 Entire Curves in Semi-abelian Varieties
215(74)
6.1 Order Functions
215(5)
6.2 Structure of Jet Images
220(5)
6.2.1 Image of f (Case k = 0)
220(1)
6.2.2 Jet Projection Method
220(4)
6.2.3 A Counter-Example
224(1)
6.3 Compact Complex Tori
225(10)
6.3.1 Entire Curves
225(8)
6.3.2 Applications to Differentiably Non-degenerate Maps
233(2)
6.4 Semi-tori: Truncation Level k0
235(13)
6.5 Semi-abelian Varieties: Truncation Level 1
248(22)
6.5.1 Truncation Level 1
248(1)
6.5.2 The Second Main Theorem for Jet Lifts
249(5)
6.5.3 Higher Codimensional Subvarieties of Xk(f)
254(14)
6.5.4 Proof of Theorem 6.5.1
268(2)
6.6 Applications
270(19)
6.6.1 Algebraic Degeneracy of Entire Curves
270(8)
6.6.2 Kobayashi Hyperbolicity
278(1)
6.6.3 Complements of Divisors in Projective Space
279(2)
6.6.4 Strong Green-Griffiths Conjecture
281(2)
6.6.5 Lang's Questions on Theta Divisors
283(2)
6.6.6 Algebraic Differential Equations
285(4)
7 Kobayashi Hyperbolicity
289(52)
7.1 Kobayashi Pseudodistance
289(4)
7.2 Brody's Theorem
293(8)
7.2.1 Brody's Reparametrization
293(7)
7.2.2 Hyperbolicity as an Open Property
300(1)
7.3 Kobayashi Hyperbolic Manifolds
301(8)
7.4 Kobayashi Hyperbolic Projective Hypersurfaces
309(6)
7.5 Hyperbolic Embedding into Complex Projective Space
315(6)
7.6 Brody Curves and Yosida Functions
321(20)
7.6.1 Growth Conditions and Yosida Functions
322(9)
7.6.2 Characterizing Brody Maps into Tori
331(1)
7.6.3 Brody Curves with Prescribed Points in the Image
332(1)
7.6.4 Ahlfors' Currents
333(8)
8 Nevanlinna Theory over Function Fields
341(20)
8.1 Lang's Conjecture
341(4)
8.2 Nevanlinna-Cartan Theory over Function Fields
345(5)
8.3 Borel's Identity and Unit Equations
350(5)
8.4 Generalized Borel's Theorem and Applications
355(6)
9 Diophantine Approximation
361(32)
9.1 Valuations
361(7)
9.1.1 Definition and the Basic Properties
361(3)
9.1.2 Extensions of Valuations
364(1)
9.1.3 Normalized Valuations
364(4)
9.2 Heights
368(9)
9.3 Theorems of Roth and Schmidt
377(6)
9.4 Unit Equations
383(2)
9.5 The abc-Conjecture and the Fundamental Conjecture
385(3)
9.6 The Faltings-Vojta Theorem
388(1)
9.7 Distribution of Rational Points
389(4)
References 393(18)
Index 411(4)
Symbols 415
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