Birational Geometry of Foliations

Cover
IMPA Monographs; Volume 1
Birational Geometry of Foliations
Copyright
© Springer International Publishing Switzerland 2015
ISBN 978-3-319-14309-5
ISBN 978-3-319-14310-1 (eBook)
DOI 10.1007/978-3-319-14310-1
Library of Congress Control Number: 2014122930
Preface
Contents

Introduction: From Surfaces to Foliations

1 Local Theory
1 Reduced Singularities and Their Separatrices
2 Blowing-up and Resolution

2 Foliations and Line Bundles
1 Basic Definitions
2 Degrees of the Bundles on Curves
3 Some Examples

3 Index Theorems
1 Baum–Bott Formula
2 Camacho–Sad Formula
3 The Separatrix Theorem and its Singular Generalization
4 An Index Theorem for Invariant Measures
5 Regular Foliations on Rational Surfaces

4 Some Special Foliations
1 Riccati Foliations
2 A Very Special Foliation
3 Turbulent Foliations

5 Minimal Models
1 Minimal Models and Relatively Minimal Models
2 Existence of Minimal Models

6 Global 1-Forms and Vector Fields
1 Holomorphic and Logarithmic 1-Forms
2 A Theorem of Jouanolou
3 Holomorphic Vector Fields

7 The Rationality Criterion
1 Statement and First Consequences
2 Foliations in Positive Characteristic
3 Proof of Theorem 7.1
4 A Proof by Bogomolov and McQuillan
5 Construction of Special Metrics

8 Numerical Kodaira Dimension
1 Zariski Decomposition and Numerical Kodaira Dimension
2 The Structure of the Negative Part
3 Foliations with Vanishing Numerical Kodaira Dimension
4 Contraction of the Negative Part and Canonical Singularities

9 Kodaira Dimension
1 Kodaira Dimension of Foliations
2 Foliations of Kodaira Dimension 1
3 Foliations of Kodaira Dimension 0
4 Foliations with an Entire Leaf
5 Foliations of Negative Kodaira Dimension

References

Index