The text presents the birational classification of holomorphic foliations of surfaces. It discusses at length the theory developed by L.G. Mendes, M. McQuillan and the author to study foliations of surfaces in the spirit of the classification of complex algebraic surfaces.
| Introduction: From Surfaces to Foliations |
|
ix | |
|
|
|
1 | (8) |
|
1 Reduced Singularities and Their Separatrices |
|
|
1 | (3) |
|
2 Blowing-up and Resolution |
|
|
4 | (5) |
|
2 Foliations and Line Bundles |
|
|
9 | (14) |
|
|
|
9 | (4) |
|
2 Degrees of the Bundles on Curves |
|
|
13 | (3) |
|
|
|
16 | (7) |
|
|
|
23 | (18) |
|
|
|
23 | (3) |
|
|
|
26 | (5) |
|
3 The Separatrix Theorem and its Singular Generalization |
|
|
31 | (2) |
|
4 An Index Theorem for Invariant Measures |
|
|
33 | (4) |
|
5 Regular Foliations on Rational Surfaces |
|
|
37 | (4) |
|
4 Some Special Foliations |
|
|
41 | (20) |
|
|
|
41 | (8) |
|
2 A Very Special Foliation |
|
|
49 | (5) |
|
|
|
54 | (7) |
|
|
|
61 | (8) |
|
1 Minimal Models and Relatively Minimal Models |
|
|
61 | (3) |
|
2 Existence of Minimal Models |
|
|
64 | (5) |
|
6 Global 1-Forms and Vector Fields |
|
|
69 | (12) |
|
1 Holomorphic and Logarithmic 1-Forms |
|
|
69 | (5) |
|
|
|
74 | (1) |
|
3 Holomorphic Vector Fields |
|
|
75 | (6) |
|
7 The Rationality Criterion |
|
|
81 | (10) |
|
1 Statement and First Consequences |
|
|
81 | (2) |
|
2 Foliations in Positive Characteristic |
|
|
83 | (2) |
|
|
|
85 | (2) |
|
4 A Proof by Bogomolov and McQuillan |
|
|
87 | (1) |
|
5 Construction of Special Metrics |
|
|
88 | (3) |
|
8 Numerical Kodaira Dimension |
|
|
91 | (16) |
|
1 Zariski Decomposition and Numerical Kodaira Dimension |
|
|
91 | (5) |
|
2 The Structure of the Negative Part |
|
|
96 | (5) |
|
3 Foliations with Vanishing Numerical Kodaira Dimension |
|
|
101 | (3) |
|
4 Contraction of the Negative Part and Canonical Singularities |
|
|
104 | (3) |
|
|
|
107 | (20) |
|
1 Kodaira Dimension of Foliations |
|
|
107 | (2) |
|
2 Foliations of Kodaira Dimension 1 |
|
|
109 | (1) |
|
3 Foliations of Kodaira Dimension 0 |
|
|
110 | (8) |
|
4 Foliations with an Entire Leaf |
|
|
118 | (4) |
|
5 Foliations of Negative Kodaira Dimension |
|
|
122 | (5) |
| References |
|
127 | (2) |
| Index |
|
129 | |
Marco Brunella was a CNRS researcher working at Institut de Mathematiques de Bourgogne in Dijon, France. He has produced extraordinary mathematical work, focusing on the study of Holomorphic Foliations and Complex Geometry. Dr. Brunella passed away in January 2012, but his profound, creative mathematics continues to have an impact on geometers and analysts.
