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© 2020

Iitaka Conjecture

An Introduction

Book
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Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Osamu Fujino
    Pages 1-10
  3. Osamu Fujino
    Pages 11-34
  4. Osamu Fujino
    Pages 35-64
  5. Osamu Fujino
    Pages 97-111
  6. Osamu Fujino
    Pages 113-119
  7. Back Matter
    Pages 121-128

About this book

Introduction

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.

Keywords

weakly positive sheaf semipositivity big sheaf Kodaira dimension logarithmic Kodaira dimension weak semistable reduction Iitaka conjecture generalized Iitaka conjecture Viehweg conjecture effective freeness

Authors and affiliations

  1. 1.OsakaJapan

Bibliographic information

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)