Birational Geometry of Foliations

Nome: Birational Geometry of Foliations

Autor(es): e Marco Brunella

Páginas: 130

Publicação: IMPA-Springer 2015

ISBN: 978-3-319-14309-5

Edição: 1

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
3 Turbulent Foliations

5 Minimal Models

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

6 Global 1-Form and Vector Fields

1 Holomorphic and Logarithmic l-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 Cosntruction od Special Metrics

8 Numerical Kodaira Dimension

1 Zariski Decomposition and Numerical Kodaira Dimension
2 The Structure of the Nagative 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