Rigid Germs, the Valuative Tree, and Applications to Kato Varieties
This thesis deals with specific features of the theory of holomorphic dynamics in dimension 2 and then sets out to study analogous questions in higher dimensions, e.g. dealing with normal forms for rigid germs, and examples of Kato 3-folds.The local dynam
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tesi di perfezionamento in Matematica sostenuta il 15 marzo 2011 C OMMISSIONE G IUDICATRICE Luigi Ambrosio, Presidente Marco Abate Filippo Bracci Charles Favre Pietro Majer Giorgio Patrizio Giuseppe Tomassini
Matteo Ruggiero IMJ - Universit´e Paris Diderot 75205 Paris Cedex 13, France Rigid Germs, the Valuative Tree, and Applications to Kato Varieties
Matteo Ruggiero
Rigid Germs, the Valuative Tree, and Applications to Kato Varieties
c 2015 Scuola Normale Superiore Pisa
ISBN 978-88-7642-558-5 ISBN 978-88-7642-559-2 (eBook)
a Laura e zia Anna
Contents
Introduction
ix
1 Background 1.1. Holomorphic dynamics . . . . . . . . . . . . . . . . 1.1.1. Local holomorphic dynamics . . . . . . . . . 1.1.2. Local dynamics in one complex variable . . . 1.1.3. Local dynamics in several complex variables 1.1.4. Parabolic curves . . . . . . . . . . . . . . . 1.1.5. Stable and unstable manifolds . . . . . . . . 1.2. Algebraic geometry . . . . . . . . . . . . . . . . . . 1.2.1. Divisors and line bundles . . . . . . . . . . . 1.2.2. Blow-ups and Modifications . . . . . . . . . 1.2.3. Canonical and Normal Bundles . . . . . . . 1.2.4. Intersection numbers . . . . . . . . . . . . . 1.2.5. Valuations . . . . . . . . . . . . . . . . . . . 1.3. Algebraic topology . . . . . . . . . . . . . . . . . . 1.4. Compact complex varieties . . . . . . . . . . . . . . 1.4.1. Minimal models . . . . . . . . . . . . . . . 1.4.2. Kodaira Dimension . . . . . . . . . . . . . . 1.4.3. Class VII . . . . . . . . . . . . . . . . . . . 1.4.4. Ruled Surfaces . . . . . . . . . . . . . . . .
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2 Dynamics in 2D 2.1. The valuative tree . . . . . . . . . . . . . . 2.1.1. Tree structure . . . . . . . . . . . . 2.1.2. Universal Dual Graph . . . . . . . 2.1.3. Valuations . . . . . . . . . . . . . . 2.1.4. Classification of Valuations . . . . 2.1.5. The Valuative Tree . . . . . . . . . 2.1.6. Skewness, multiplicity and thinness
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viii Matteo Ruggiero
2.2. 2.3.
2.4.
2.5.
2.6. 2.7.
2.1.7. Universal dual graph and valuative tree . . Dynamics on the valuative tree . . . . . . . . . . . Rigidification . . . . . . . . . . . . . . . . . . . . 2.3.1. General result . . . . . . . . . . . . . . . . 2.3.2. Semi-superattracting case . . . . . . . . . Rigid germs . . . . . . . . . . . . . . . . . . . . . 2.4.1. Attracting rigid germs . . . . . . . . . . . 2.4.2. Rigid germs of type (0, C \ D) . . . . . . . Formal classification of semi-superattracting germs 2.5.1. Invariants . . . . . . . . . . . . . . . . . . 2.5.2. Classification . . . . . . . . . . . . . . . . Rigid germs of type (0, 1) . . . . . . . . . . . . . Normal forms . . . . . . . . . . . . . . . . . . . . 2.7.1. Nilpotent case . . . . . . . . . . . . . . . . 2.7.2. Semi-superattracting case . . . . . . . . . 2.7.3. Some remarks and examples . . . . . . . .
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