Rigid Germs, the Valuative Tree, and Applications to Kato Varieties

194 Pages·2015·1.061 MB·English

by  Matteo Ruggiero (auth.)

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.