20 TESI THESES tesidiperfezionamentoinMatematicasostenutail15marzo2011 COMMISSIONEGIUDICATRICE LuigiAmbrosio,Presidente MarcoAbate FilippoBracci CharlesFavre PietroMajer GiorgioPatrizio GiuseppeTomassini MatteoRuggiero IMJ-Universite´ParisDiderot75205ParisCedex13,France RigidGerms,theValuativeTree,andApplicationstoKatoVarieties Matteo Ruggiero Rigid Germs, the Valuative Tree, and Applications to Kato Varieties (cid:2)c 2015ScuolaNormaleSuperiorePisa ISBN978-88-7642-558-5 ISBN978-88-7642-559-2(eBook) aLauraeziaAnna Contents Introduction ix 1 Background 1 1.1. Holomorphicdynamics . . . . . . . . . . . . . . . . . . 1 1.1.1. Localholomorphicdynamics . . . . . . . . . . . 1 1.1.2. Localdynamicsinonecomplexvariable . . . . . 2 1.1.3. Localdynamicsinseveralcomplexvariables . . 5 1.1.4. Paraboliccurves . . . . . . . . . . . . . . . . . 7 1.1.5. Stableandunstablemanifolds . . . . . . . . . . 8 1.2. Algebraicgeometry . . . . . . . . . . . . . . . . . . . . 10 1.2.1. Divisorsandlinebundles . . . . . . . . . . . . . 10 1.2.2. Blow-upsandModifications . . . . . . . . . . . 11 1.2.3. CanonicalandNormalBundles . . . . . . . . . 13 1.2.4. Intersectionnumbers . . . . . . . . . . . . . . . 14 1.2.5. Valuations . . . . . . . . . . . . . . . . . . . . . 16 1.3. Algebraictopology . . . . . . . . . . . . . . . . . . . . 17 1.4. Compactcomplexvarieties . . . . . . . . . . . . . . . . 18 1.4.1. Minimalmodels . . . . . . . . . . . . . . . . . 18 1.4.2. KodairaDimension . . . . . . . . . . . . . . . . 19 1.4.3. ClassVII . . . . . . . . . . . . . . . . . . . . . 20 1.4.4. RuledSurfaces . . . . . . . . . . . . . . . . . . 21 2 Dynamicsin2D 25 2.1. Thevaluativetree . . . . . . . . . . . . . . . . . . . . . 25 2.1.1. Treestructure . . . . . . . . . . . . . . . . . . . 25 2.1.2. UniversalDualGraph . . . . . . . . . . . . . . 27 2.1.3. Valuations . . . . . . . . . . . . . . . . . . . . . 28 2.1.4. ClassificationofValuations . . . . . . . . . . . 29 2.1.5. TheValuativeTree . . . . . . . . . . . . . . . . 31 2.1.6. Skewness,multiplicityandthinness . . . . . . . 31 viii MatteoRuggiero 2.1.7. Universaldualgraphandvaluativetree . . . . . 33 2.2. Dynamicsonthevaluativetree . . . . . . . . . . . . . . 34 2.3. Rigidification . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.1. Generalresult . . . . . . . . . . . . . . . . . . . 39 2.3.2. Semi-superattractingcase . . . . . . . . . . . . 42 2.4. Rigidgerms . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4.1. Attractingrigidgerms . . . . . . . . . . . . . . 48 2.4.2. Rigidgermsoftype(0,C\D) . . . . . . . . . . 51 2.5. Formalclassificationofsemi-superattractinggerms . . . 57 2.5.1. Invariants . . . . . . . . . . . . . . . . . . . . . 57 2.5.2. Classification . . . . . . . . . . . . . . . . . . . 61 2.6. Rigidgermsoftype(0,1) . . . . . . . . . . . . . . . . 64 2.7. Normalforms . . . . . . . . . . . . . . . . . . . . . . . 70 2.7.1. Nilpotentcase. . . . . . . . . . . . . . . . . . . 70 2.7.2. Semi-superattractingcase . . . . . . . . . . . . 75 2.7.3. Someremarksandexamples . . . . . . . . . . . 76 3 Rigidgermsinhigherdimension 79 3.1. Definitions. . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1.1. Notations . . . . . . . . . . . . . . . . . . . . . 79 3.1.2. Invariants . . . . . . . . . . . . . . . . . . . . . 84 3.2. Classification . . . . . . . . . . . . . . . . . . . . . . . 87 3.2.1. Poincare´-Dulactheory . . . . . . . . . . . . . . 87 3.2.2. TopologicalResonances . . . . . . . . . . . . . 91 3.2.3. Affineactions . . . . . . . . . . . . . . . . . . . 97 3.2.4. Remarks . . . . . . . . . . . . . . . . . . . . . 102 3.3. Rigidgermsindimension3 . . . . . . . . . . . . . . . . 104 4 Constructionofnon-Kahler3-folds 109 4.1. Katosurfaces,rigidgermsandHe´nonmaps . . . . . . . 109 4.1.1. Katosurfaces . . . . . . . . . . . . . . . . . . . 109 4.1.2. He´nonmaps . . . . . . . . . . . . . . . . . . . 110 4.1.3. DynamicalpropertiesofsomeKatosurfaces . . 113 4.2. Constructioninthe3Dcase . . . . . . . . . . . . . . . . 116 4.2.1. Theexample . . . . . . . . . . . . . . . . . . . 116 4.2.2. Resolutionof f . . . . . . . . . . . . . . . . . 118 0 4.2.3. Constructionanduniversalcovering . . . . . . . 122 4.3. Algebraicproperties. . . . . . . . . . . . . . . . . . . . 125 4.3.1. Topologyofthedivisors . . . . . . . . . . . . . 125 4.3.2. HomologyGroups . . . . . . . . . . . . . . . . 129 4.3.3. Intersectionnumbers . . . . . . . . . . . . . . . 133 4.3.4. CanonicalbundleandKodairadimension . . . . 138 ix RigidGerms,theValuativeTree,andApplicationstoKatoVarieties 4.3.5. Canonicalandnormalbundles . . . . . . . . . . 140 4.4. Dynamicalproperties . . . . . . . . . . . . . . . . . . . 151 4.4.1. Contractiontoapoint . . . . . . . . . . . . . . 151 4.4.2. Foliations . . . . . . . . . . . . . . . . . . . . . 156 4.4.3. CurvesandSurfaces . . . . . . . . . . . . . . . 157 References 163 Index 169 Introduction Holomorphic dynamics has several points of view: it can be discrete or continuous, and be studied locally or globally, but all these aspects are, sometimes surprisingly and in a very fascinating way, linked to one an- other. Thesettingofglobaldiscreteholomorphicdynamicsisthefollowing: one has a complex space X of dimension d, and a holomorphic map f : X → X,andwantstounderstandthebehavioroftheiterates f◦n of f. For example one can check if the orbit of a point x ∈ X (i.e., the set {f◦n(x)|n ∈ N})changesregularlybymovingthestartingpointx. On the other hand, local discrete holomorphic dynamics still studies the behavior of a map f, but near a given fixed point p, and hence in coordinates one is interested into the behavior of a holomorphic germ f : (Cd,0) → (Cd,0)anditsiterates,existenceofbasinsofattractions, orthestructureofthestableset(wherealltheiteratesof f aredefinedin aneighborhoodof0). One of the main techniques to study the dynamics of a family F of holomorphicgermsislookingfornormalforms. Roughlyspeaking,one looksfora(possiblysmall)familyG ofgerms,whosedynamicsiseasier tostudy, andsuchthat every f ∈ F canbereducedtoagerm g ∈ G by changingcoordinates. Definition. Let f,g : (Cd,0) → (Cd,0) be two holomorphic germs. Weshallsaythat f and g are(holomorphically,topologically,formally) conjugated if there exists a (biholomorphism, homeomorphism, formal invertiblemap)φ : (Cd,0) →(Cd,0)suchthat φ◦ f = g◦φ. Depending on the regularity of the change of coordinates: holomorphic, homeomorphic,formal,wetalkaboutholomorphic,topologicalorformal classification.