SpringerBriefs in Mathematics

Iitaka Conjecture

An Introduction

Authors: Fujino, Osamu

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  • Is the first introductory textbook for the Iitaka conjecture
  • Gives a simple new proof of Viehweg's conjecture for fiber spaces whose geometric generic fiber is of general type
  • Provides a very accessible account for Viehweg's theory of weakly positive sheaves and big sheaves
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  • ISBN 978-981-15-3347-1
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Softcover $59.99
price for USA in USD
  • ISBN 978-981-15-3346-4
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  • Usually ready to be dispatched within 3 to 5 business days, if in stock
About this book

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.

Reviews

“The importance of Iitaka’s conjecture is obvious to anyone who is aware of the notion of Kodaira dimension and its rôle in classification theory. … the literature on positivity of direct image sheaves is vast and highly technical, this text gathers the most fundamental tools and makes Viehweg’s theory more accessible. The use of the more recent contributions to the minimal model program simplifies some arguments, exposed in the author’s precise style of writing.” (Andreas Höring, zbMATH 1452.14002, 2021)


Table of contents (6 chapters)

Table of contents (6 chapters)

Buy this book

eBook $44.99
price for USA in USD
  • ISBN 978-981-15-3347-1
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.99
price for USA in USD
  • ISBN 978-981-15-3346-4
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • Usually ready to be dispatched within 3 to 5 business days, if in stock
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Bibliographic Information

Bibliographic Information
Book Title
Iitaka Conjecture
Book Subtitle
An Introduction
Authors
Series Title
SpringerBriefs in Mathematics
Copyright
2020
Publisher
Springer Singapore
Copyright Holder
Springer Nature Singapore Pte Ltd.
eBook ISBN
978-981-15-3347-1
DOI
10.1007/978-981-15-3347-1
Softcover ISBN
978-981-15-3346-4
Series ISSN
2191-8198
Edition Number
1
Number of Pages
XIV, 128
Number of Illustrations
1 b/w illustrations
Topics