Birational Geometry of Foliations - Brunella, Marco (Springer, 2015)
This text was originally written as a lecture note for a course given at the First Latin American Congress of Mathematicians, held at IMPA in August 2000. It is now reprinted with no changes, but some new related results appeared in the last 3 years. First of all, the Hilbert Modular Conjecture appearing at the end of the book, and completing the classification of foliations with Kodaira dimension 1, has been completely proved. One essential ingredient is the plurisubharmonicity of the leafwise Poincaré metric, as vaguely stated here on pages 96–98. This is done in author’s paper subharmonic variation of the leafwise Poincaré metric, Invent. Math. 152 (2003), 119–148. Another important ingredient is the continuity of the same metric, which can be found in a revised version of McQuillan’s preprint Noncommutative Mori Theory (IHES M/01/42, 2001) and even better in the version of the same preprint which will be published under the name Canonical models of foliations (Pure Appl. Math. Q., vol. 4, 2008). Using these two results (plurisubharmonicity and continuity), the Hilbert Modular Conjecture is proved via the Monge–Ampère approach, as indicated here: this is done in both author’s paper and McQuillan’s revised preprint. A survey article on the full classification, with reasonably complete and simple proofs, is Foliations on complex projective surfaces, Proceedings of a Trimester on Dynamical Systems, Pisa SNS 2002 (also available at www.arxiv.org as math.CV/0212082)...