Nevanlinna theory in several complex variables and diophantine approximation:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Tokyo ; Heidelberg ; New York ; Dordrecht ; London
Springer
[2014]
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
350 |
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xiv, 416 Seiten Illustrationen |
| ISBN: | 9784431545705 9784431562139 |
Internformat
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| 100 | 1 | |a Noguchi, Junjiro |d 1948- |0 (DE-588)172286816 |4 aut | |
| 245 | 1 | 0 | |a Nevanlinna theory in several complex variables and diophantine approximation |c Junjiro Noguchi, Jörg Winkelmann |
| 264 | 1 | |a Tokyo ; Heidelberg ; New York ; Dordrecht ; London |b Springer |c [2014] | |
| 264 | 4 | |c © 2014 | |
| 300 | |a xiv, 416 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 350 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 0 | 7 | |a Nevanlinna-Theorie |0 (DE-588)4239989-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Diophantische Approximation |0 (DE-588)4135760-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nevanlinna-Theorie |0 (DE-588)4239989-0 |D s |
| 689 | 0 | 1 | |a Diophantische Approximation |0 (DE-588)4135760-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Winkelmann, Jörg |d 1963- |0 (DE-588)133717615 |4 aut | |
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| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026998449&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-026998449 | ||
Datensatz im Suchindex
| _version_ | 1804151662863974400 |
|---|---|
| adam_text | Contents
Nevanlinna
Theory of Meromorphic Functions
........... 1
1.1
The First Main Theorem
......................
I
1.2
The Second Main Theorem
.................... 12
1.3
Examples of Functions of Finite Order
.............. 19
The First Main Theorem
........................ 25
2.1
Plurisubharmonic Functions
.................... 25
2.1.1
One Variable
........................ 25
2.1.2
Several Variables
...................... 33
2.2
Poincaré—
Lelong Formula
..................... 42
2.3
The First Main Theorem
...................... 50
2.3.1
Meromorphic Mappings, Divisors and Line Bundles
... 50
2.3.2
Differentiable Functions on Complex Spaces
....... 54
2.3.3
Metrics and Curvature Forms of Line Bundles
...... 58
2.4
The First Main Theorem for Coherent Ideal Sheaves
....... 66
2.4.1
Proximity Functions for Coherent Ideal Sheaves
..... 66
2.4.2
The Case of
m
= 1..................... 71
2.5
Order Functions
.......................... 73
2.5.1
Metrics
........................... 73
2.5.2
Cartan s Order Function
.................. 77
2.5.3
A Family of Rational Functions
.............. 79
2.5.4
Characterization of Rationality
.............. 83
2.6
Nevanlinna s Inequality
...................... 86
2.7
Ramified Covers over Cm
..................... 88
Differentiably Non-degenerate Meromorphic Maps
......... 91
3.1
Lemma on Logarithmic Derivatives
................ 91
3.2
The Second Main Theorem
.................... 93
3.3
Applications and Generalizations
................. 102
3.3.1
Applications
........................ 102
3.3.2
Non-
Kahler
Counter-Example
............... 105
3.3.3
Generalizations
...................... 110
ix
Contents
Entire
Curves
in
Algebraic Varieties
................. 113
4.1
Nochka Weights
.......................... 113
4.2
The Cartan-Nochka Theorem
................... 123
4.3
Entire Curves Omitting
Hyperplanes............... 131
4.4
Generalizations and Applications
................. 133
4.4.1
Derived Curves
...................... 133
4.4.2
Generalization to Higher Dimensional Domains
..... 134
4.4.3
Finite Ramified Covering Spaces
............. 134
4.4.4
The Eremenko-Sodin Second Main Theorem
....... 135
4.4.5
The Second Main Theorem of Corvaja-Zannier,
Evertse-Ferretti and Ru
.................. 136
4.4.6
Krutin s Theorem
..................... 136
4.4.7
Moving Targets
...................... 136
4.4.8
Yamanoi s Second Main Theorem
............. 137
4.4.9
Applications
........................ 137
4.5
Logarithmic Forms
......................... 138
4.6
Logarithmic Jet Bundles
...................... 144
4.6.1
Jet Bundles in General
................... 144
4.6.2
Jet Spaces
......................... 146
4.6.3
Logarithmic Jet Bundles and Logarithmic Jet Spaces
. . . 146
4.7
Lemma on Logarithmic Forms
.................. 148
4.8
Inequality of the Second Main Theorem Type
.......... 150
4.9
Entire Curves Omitting Hypersurfaces
.............. 157
4.10
The Fundamental Conjecture of Entire Curves
.......... 159
Semi-abelian Varieties
......................... 161
5.1
Semi-tori
.............................. 161
5.1.1
Definition
......................... 161
5.1.2
Characteristic Subgroups of Complex Semi-tori
..... 164
5.1.3
Holomorphic Functions
.................. 166
5.1.4
Semi-abelian Varieties
................... 167
5.1.5
Presentations
........................ 169
5.1.6
Presentations of Semi-abelian Varieties
.......... 170
5.Î.7
Inequivalent Algebraic Structures
............. 171
5.1.8
Choice of Presentation
................... 171
5.1.9
Construction of Semi-tori via Presentations
........ 172
5.1.10
Morphisms and GAGA
.................. 173
5.2
Reductive Group Actions
..................... 176
5.3
Semi-toric Varieties
........................ 180
5.3.1
Tone Varieties
....................... 180
5.3.2
Semi-toric Varieties
.................... 181
5.3.3
Key Properties of Semi-toric Varieties
.......... 182
5.3.4
Quasi-algebraic Subgroups
................ 185
5.3.5
Compactifiable Groups and
Kahler
Condition
...... 187
5.3.6
Examples of Non-semi-toric Varieties
........... 190
5.4
Jet Bundles over Semi-toric Varieties
............... 191
Contents xi
5.5 Line
Bundles on Toric Varieties
.................. 192
5.5.1
Ample Line Bundles
.................... 192
5.5.2 Leray
Spectral Sequence
.................. 195
5.5.3
Decomposition of Line Bundles
.............. 196
5.5.4
Global Span and Very Amplenes
s
............. 198
5.5.5
Stabilizer and Bigness
................... 201
5.6
Good Position and Stabilizer
................... 203
5.6.1
Good Position
....................... 203
5.6.2
Good Position and Choice of
Compactifìcation
...... 204
5.6.3
Regular Subgroups
..................... 209
5.6.4
More Facts on Semi-tori
.................. 210
6
Entire Curves in Semi-abelian Varieties
............... 215
6.1
Order Functions
.......................... 215
6.2
Structure of Jet Images
....................... 220
6.2.1
Image of
ƒ
(Case
к
= 0).................. 220
6.2.2
Jet Projection Method
................... 220
6.2.3
A Counter-Example
.................... 224
6.3
Compact Complex Tori
...................... 225
6.3.1
Entire Curves
....................... 225
6.3.2
Applications to Differentiably Non-degenerate Maps
. . . 233
6.4
Semi-tori: Truncation Level
ko
.................. 235
6.5
Semi-abelian Varieties: Truncation Level
1............ 248
6.5.1
Truncation Level
1..................... 248
6.5.2
The Second Main Theorem for Jet Lifts
.......... 249
6.5.3
Higher Codimensional Subvarieties of Xk(f)
...... 254
6.5.4
Proof of Theorem
6.5.1 .................. 268
6.6
Applications
............................ 270
6.6.1
Algebraic Degeneracy of Entire Curves
.......... 270
6.6.2
Kobayashi Hyperbolicity
................. 278
6.6.3
Complements of Divisors in
Projective
Space
....... 279
6.6.4
Strong Green-Griffiths Conjecture
............ 281
6.6.5
Lang s Questions on Theta Divisors
............ 283
6.6.6
Algebraic Differential Equations
............. 285
7
Kobayashi Hyperbolicity
........................ 289
7.1
Kobayashi
Pseudodistance
..................... 289
7.2
Brody s
Theorem
......................... 293
7.2.1
Brody
s Reparametrization
................ 293
7.2.2
Hyperbolicity as an
Open Property............
300
7.3
Kobayashi
Hyperbolic Manifolds
................. 301
7.4
Kobayashi
Hyperbolic
Projective
Hypersurfaces
......... 309
7.5
Hyperbolic Embedding into Complex
Projective
Space
...... 315
7.6
Brody
Curves and Yosida Functions
................ 321
7.6.1
Growth Conditions and Yosida Functions
......... 322
7.6.2
Characterizing
Brody
Maps into Tori
........... 331
xii Contents
7.6.3
Brody
Curves
with Prescribed Points in the Image
.... 332
7.6.4
Ahlfors Currents
..................... 333
8
Nevanlinna Theory over Function Fields
............... 341
8.1
Lang s Conjecture
......................... 341
8.2
Nevanlinna-Cartan Theory over Function Fields
......... 345
8.3
Boreľs
Identity and Unit Equations
................ 350
8.4
Generalized Borel s Theorem and Applications
.......... 355
9
Diophantine Approximation
...................... 361
9.1
Valuations
............................. 361
9.1.1
Definition and the Basic Properties
............ 361
9.1.2
Extensions of Valuations
.................. 364
9.1.3
Normalized Valuations
................... 364
9.2
Heights
............................... 368
9.3
Theorems of Roth and Schmidt
.................. 377
9.4
Unit Equations
........................... 383
9.5
The «¿»c-Conjecture and the Fundamental Conjecture
...... 385
9.6
The Fairings-
Vojta
Theorem
.................... 388
9.7
Distribution of Rational Points
.................. 389
References
.................................. 393
Index
..................................... 411
Symbols
.................................... 415
Grundlehren der mathematischen Wissenschaften
A Series of Comprehensive Studies in Mathematics
Junjiro Noguchi
· Jörg Winkelmann
Nevanlinna Theory in Several Complex Variables
and Diophantine Approximation
The aim of this book is to provide a comprehensive account of higher dimensional Ne¬
vanlinna theory and its relations with Diophantine approximation theory for graduate
students and interested researchers.
This book with nine chapters systematically describes Nevanlinna theory of meromor-
phic 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,
ι
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 gen¬
eralized 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.
Mathematics
ISSN
0072-7830
ISBN
978-4-431-54570-5
springer.com
|
| any_adam_object | 1 |
| author | Noguchi, Junjiro 1948- Winkelmann, Jörg 1963- |
| author_GND | (DE-588)172286816 (DE-588)133717615 |
| author_facet | Noguchi, Junjiro 1948- Winkelmann, Jörg 1963- |
| author_role | aut aut |
| author_sort | Noguchi, Junjiro 1948- |
| author_variant | j n jn j w jw |
| building | Verbundindex |
| bvnumber | BV041552750 |
| classification_rvk | SK 180 SK 750 SK 780 |
| ctrlnum | (OCoLC)869870483 (DE-599)BVBBV041552750 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041552750 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:58:23Z |
| institution | BVB |
| isbn | 9784431545705 9784431562139 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026998449 |
| oclc_num | 869870483 |
| open_access_boolean | |
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| owner_facet | DE-20 DE-703 DE-188 DE-384 DE-11 DE-83 DE-29T |
| physical | xiv, 416 Seiten Illustrationen |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Noguchi, Junjiro 1948- (DE-588)172286816 aut Nevanlinna theory in several complex variables and diophantine approximation Junjiro Noguchi, Jörg Winkelmann Tokyo ; Heidelberg ; New York ; Dordrecht ; London Springer [2014] © 2014 xiv, 416 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 350 Hier auch später erschienene, unveränderte Nachdrucke Nevanlinna-Theorie (DE-588)4239989-0 gnd rswk-swf Diophantische Approximation (DE-588)4135760-7 gnd rswk-swf Nevanlinna-Theorie (DE-588)4239989-0 s Diophantische Approximation (DE-588)4135760-7 s DE-604 Winkelmann, Jörg 1963- (DE-588)133717615 aut Erscheint auch als Online-Ausgabe 978-4-431-54571-2 Grundlehren der mathematischen Wissenschaften 350 (DE-604)BV000000395 350 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026998449&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026998449&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Noguchi, Junjiro 1948- Winkelmann, Jörg 1963- Nevanlinna theory in several complex variables and diophantine approximation Grundlehren der mathematischen Wissenschaften Nevanlinna-Theorie (DE-588)4239989-0 gnd Diophantische Approximation (DE-588)4135760-7 gnd |
| subject_GND | (DE-588)4239989-0 (DE-588)4135760-7 |
| title | Nevanlinna theory in several complex variables and diophantine approximation |
| title_auth | Nevanlinna theory in several complex variables and diophantine approximation |
| title_exact_search | Nevanlinna theory in several complex variables and diophantine approximation |
| title_full | Nevanlinna theory in several complex variables and diophantine approximation Junjiro Noguchi, Jörg Winkelmann |
| title_fullStr | Nevanlinna theory in several complex variables and diophantine approximation Junjiro Noguchi, Jörg Winkelmann |
| title_full_unstemmed | Nevanlinna theory in several complex variables and diophantine approximation Junjiro Noguchi, Jörg Winkelmann |
| title_short | Nevanlinna theory in several complex variables and diophantine approximation |
| title_sort | nevanlinna theory in several complex variables and diophantine approximation |
| topic | Nevanlinna-Theorie (DE-588)4239989-0 gnd Diophantische Approximation (DE-588)4135760-7 gnd |
| topic_facet | Nevanlinna-Theorie Diophantische Approximation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026998449&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026998449&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT noguchijunjiro nevanlinnatheoryinseveralcomplexvariablesanddiophantineapproximation AT winkelmannjorg nevanlinnatheoryinseveralcomplexvariablesanddiophantineapproximation |