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The Resource Iitaka Conjecture : an introduction, Osamu Fujino, (electronic book)

Iitaka Conjecture : an introduction, Osamu Fujino, (electronic book)

Creator
Summary
The ambitious program for the birational classification of higher-dimensional complex algebraic varieties initiated by Shigeru Iitaka around 1970 is usually called the Iitaka program. Now it is known that the heart of the Iitaka program is the Iitaka conjecture, which claims the subadditivity of the Kodaira dimension for fiber spaces. The main purpose of this book is to make the Iitaka conjecture more accessible. First, Viehweg's theory of weakly positive sheaves and big sheaves is described, and it is shown that the Iitaka conjecture follows from the Viehweg conjecture. Then, the Iitaka conjecture is proved in some special and interesting cases. A relatively simple new proof of Viehweg's conjecture is given for fiber spaces whose geometric generic fiber is of general type based on the weak semistable reduction theorem due to Abramovick-Karu and the existence theorem of relative canonical models by Birkar-Cascini-Hacon-McKernan. No deep results of the theory of variations of Hodge structure are needed. The Iitaka conjecture for fiber spaces whose base space is of general type is also proved as an easy application of Viehweg's weak positivity theorem, and the Viehweg conjecture for fiber spaces whose general fibers are elliptic curves is explained. Finally, the subadditivity of the logarithmic Kodaira dimension for morphisms of relative dimension one is proved. In this book, for the reader's convenience, known arguments as well as some results are simplified and generalized with the aid of relatively new techniques
Language
eng
Work
Publication
Extent
1 online resource (138 p.).
Note
Description based upon print version of record
Contents
  • Intro -- Preface -- Acknowledgements -- Contents -- 1 Overview -- 1.1 What is the Iitaka Program? -- 1.2 The Main Results -- 1.3 Historical Note -- 1.4 Precise Contents -- 1.5 References -- 2 Preliminaries -- 2.1 Notations, Terminology, and Conventions -- 2.2 Prerequisites -- 2.3 Preliminary Results -- 2.3.1 Basic Definitions -- 2.3.2 Iitaka Dimension and Kodaira Dimension -- 2.3.3 Cyclic Covers -- 2.3.4 Weakly Semistable Morphisms -- 2.3.5 Kollár's Torsion-Freeness and Vanishing Theorem -- 2.3.6 On Abelian Varieties -- 2.3.7 Invariance of Plurigenera -- 2.3.8 Related Conjectures
  • 4.3 Fiber Spaces Whose Geometric Generic Fiber is of General Type -- 4.4 Elliptic Fibrations -- 4.5 Some Other Cases -- 4.5.1 Fiber Spaces Whose Base Space Has Maximal Albanese Dimension -- 4.5.2 Fiber Spaces Whose Geometric Generic Fiber Has a Good Minimal Model -- 4.5.3 Iitaka Conjecture in Positive Characteristic -- 5 overlineCn, n-1 Revisited -- 5.1 Background of overlineCn, n-1 -- 5.2 overlineCn, n-1 -- 5.3 Some Related Results -- 5.3.1 Conjecture overlineCn, m for Affine Varieties, and so on -- 5.3.2 Subadditivity Due to Kovács-Patakfalvi and Hashizume -- 5.4 Appendix: A Vanishing Lemma
  • 6 Appendices -- 6.1 C2,1 -- 6.2 Conjectures ()n and (>)n -- Appendix References -- Index
Isbn
9789811533464
Instance
Label
Iitaka Conjecture : an introduction
Title
Iitaka Conjecture
Title remainder
an introduction
Statement of responsibility
Osamu Fujino
Creator
Subject
Language
eng
Summary
The ambitious program for the birational classification of higher-dimensional complex algebraic varieties initiated by Shigeru Iitaka around 1970 is usually called the Iitaka program. Now it is known that the heart of the Iitaka program is the Iitaka conjecture, which claims the subadditivity of the Kodaira dimension for fiber spaces. The main purpose of this book is to make the Iitaka conjecture more accessible. First, Viehweg's theory of weakly positive sheaves and big sheaves is described, and it is shown that the Iitaka conjecture follows from the Viehweg conjecture. Then, the Iitaka conjecture is proved in some special and interesting cases. A relatively simple new proof of Viehweg's conjecture is given for fiber spaces whose geometric generic fiber is of general type based on the weak semistable reduction theorem due to Abramovick-Karu and the existence theorem of relative canonical models by Birkar-Cascini-Hacon-McKernan. No deep results of the theory of variations of Hodge structure are needed. The Iitaka conjecture for fiber spaces whose base space is of general type is also proved as an easy application of Viehweg's weak positivity theorem, and the Viehweg conjecture for fiber spaces whose general fibers are elliptic curves is explained. Finally, the subadditivity of the logarithmic Kodaira dimension for morphisms of relative dimension one is proved. In this book, for the reader's convenience, known arguments as well as some results are simplified and generalized with the aid of relatively new techniques
Member of
Cataloging source
EBLCP
http://library.link/vocab/creatorName
Fujino, Osamu
Dewey number
516.3/53
Index
index present
LC call number
QA564
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
Series statement
SpringerBriefs in Mathematics
http://library.link/vocab/subjectName
Algebraic varieties
Label
Iitaka Conjecture : an introduction, Osamu Fujino, (electronic book)
Instantiates
Publication
Note
Description based upon print version of record
Antecedent source
file reproduced from an electronic resource
Bibliography note
Includes bibliographical references and index
Contents
  • Intro -- Preface -- Acknowledgements -- Contents -- 1 Overview -- 1.1 What is the Iitaka Program? -- 1.2 The Main Results -- 1.3 Historical Note -- 1.4 Precise Contents -- 1.5 References -- 2 Preliminaries -- 2.1 Notations, Terminology, and Conventions -- 2.2 Prerequisites -- 2.3 Preliminary Results -- 2.3.1 Basic Definitions -- 2.3.2 Iitaka Dimension and Kodaira Dimension -- 2.3.3 Cyclic Covers -- 2.3.4 Weakly Semistable Morphisms -- 2.3.5 Kollár's Torsion-Freeness and Vanishing Theorem -- 2.3.6 On Abelian Varieties -- 2.3.7 Invariance of Plurigenera -- 2.3.8 Related Conjectures
  • 4.3 Fiber Spaces Whose Geometric Generic Fiber is of General Type -- 4.4 Elliptic Fibrations -- 4.5 Some Other Cases -- 4.5.1 Fiber Spaces Whose Base Space Has Maximal Albanese Dimension -- 4.5.2 Fiber Spaces Whose Geometric Generic Fiber Has a Good Minimal Model -- 4.5.3 Iitaka Conjecture in Positive Characteristic -- 5 overlineCn, n-1 Revisited -- 5.1 Background of overlineCn, n-1 -- 5.2 overlineCn, n-1 -- 5.3 Some Related Results -- 5.3.1 Conjecture overlineCn, m for Affine Varieties, and so on -- 5.3.2 Subadditivity Due to Kovács-Patakfalvi and Hashizume -- 5.4 Appendix: A Vanishing Lemma
  • 6 Appendices -- 6.1 C2,1 -- 6.2 Conjectures ()n and (>)n -- Appendix References -- Index
Dimensions
unknown
Extent
1 online resource (138 p.).
File format
one file format
Form of item
online
Isbn
9789811533464
Level of compression
unknown
Other control number
10.1007/978-981-15-3
Quality assurance targets
unknown
Reformatting quality
unknown
Specific material designation
remote
System control number
  • on1151190204
  • (OCoLC)1151190204
Label
Iitaka Conjecture : an introduction, Osamu Fujino, (electronic book)
Publication
Note
Description based upon print version of record
Antecedent source
file reproduced from an electronic resource
Bibliography note
Includes bibliographical references and index
Contents
  • Intro -- Preface -- Acknowledgements -- Contents -- 1 Overview -- 1.1 What is the Iitaka Program? -- 1.2 The Main Results -- 1.3 Historical Note -- 1.4 Precise Contents -- 1.5 References -- 2 Preliminaries -- 2.1 Notations, Terminology, and Conventions -- 2.2 Prerequisites -- 2.3 Preliminary Results -- 2.3.1 Basic Definitions -- 2.3.2 Iitaka Dimension and Kodaira Dimension -- 2.3.3 Cyclic Covers -- 2.3.4 Weakly Semistable Morphisms -- 2.3.5 Kollár's Torsion-Freeness and Vanishing Theorem -- 2.3.6 On Abelian Varieties -- 2.3.7 Invariance of Plurigenera -- 2.3.8 Related Conjectures
  • 4.3 Fiber Spaces Whose Geometric Generic Fiber is of General Type -- 4.4 Elliptic Fibrations -- 4.5 Some Other Cases -- 4.5.1 Fiber Spaces Whose Base Space Has Maximal Albanese Dimension -- 4.5.2 Fiber Spaces Whose Geometric Generic Fiber Has a Good Minimal Model -- 4.5.3 Iitaka Conjecture in Positive Characteristic -- 5 overlineCn, n-1 Revisited -- 5.1 Background of overlineCn, n-1 -- 5.2 overlineCn, n-1 -- 5.3 Some Related Results -- 5.3.1 Conjecture overlineCn, m for Affine Varieties, and so on -- 5.3.2 Subadditivity Due to Kovács-Patakfalvi and Hashizume -- 5.4 Appendix: A Vanishing Lemma
  • 6 Appendices -- 6.1 C2,1 -- 6.2 Conjectures ()n and (>)n -- Appendix References -- Index
Dimensions
unknown
Extent
1 online resource (138 p.).
File format
one file format
Form of item
online
Isbn
9789811533464
Level of compression
unknown
Other control number
10.1007/978-981-15-3
Quality assurance targets
unknown
Reformatting quality
unknown
Specific material designation
remote
System control number
  • on1151190204
  • (OCoLC)1151190204

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