Algebraic criteria for Kobayashi hyperbolic projective varieties
82 pages
English

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Algebraic criteria for Kobayashi hyperbolic projective varieties

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Niveau: Supérieur, Doctorat, Bac+8
Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier Lecture Notes of a Course given at Santa Cruz (CA, USA) at the AMS Summer Research Institute in Algebraic Geometry (July 1995) Abstract. These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995. The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi. A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs (X, V ) whereX is a complex manifold and V a holomorphic subbundle of TX . If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C ? X tangent to V (Brody's criterion). We describe a construction of projectivized k- jet bundles PkV , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature. More precisely, πk : PkV ? X is a tower of projective bundles over X and carries a canonical line bundle OPkV (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on OPkV (?1) for k large enough.

  • hyperbolic algebraic

  • projective variety

  • over

  • jet differentials

  • c1 ?

  • drawn

  • used when


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Nombre de lectures 20
Langue English

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AlgebraiccriteriaforKobayashi
hyperbolicprojectivevarieties
andjetdifferentials
Jean-PierreDemailly
Universite´deGrenobleI,InstitutFourier
LectureNotesofaCoursegivenatSantaCruz(CA,USA)
attheAMSSummerResearchInstitute
inAlgebraicGeometry(July1995)

Abstract.
ThesenotesareanexpandedversionoflecturesdeliveredattheAMS
SummerSchoolonAlgebraicGeometry,heldatSantaCruzinJuly1995.The
maingoalofthenotesistostudycomplexvarieties(mostlycompactorprojective
algebraicones),throughafewgeometricquestionsrelatedtohyperbolicityinthe
senseofKobayashi.Aconvenientframeworkforthisisthecategoryof“directed
manifolds”,thatis,thecategoryofpairs(
X,V
)where
X
isacomplexmanifoldand
V
aholomorphicsubbundleof
T
X
.If
X
iscompact,thepair(
X,V
)ishyperbolic
ifandonlyiftherearenononconstantentireholomorphiccurves
f
:
C

X
tangentto
V
(Brody’scriterion).Wedescribeaconstructionofprojectivized
k
-
jetbundles
P
k
V
,whichgeneralizesaconstructionmadebySemplein1954and
allowstoanalyzehyperbolicityintermsofnegativitypropertiesofthecurvature.
Moreprecisely,
π
k
:
P
k
V

X
isatowerofprojectivebundlesover
X
andcarries
acanonicallinebundle
O
P
k
V
(1);thehyperbolicityof
X
isthenconjecturally
equivalenttotheexistenceofsuitablesingularhermitianmetricsofnegative
curvatureon
O
P
k
V
(

1)for
k
largeenough.Thedirectimages(
π
k
)

O
P
k
V
(
m
)can
beviewedasbundlesofalgebraicdifferentialoperatorsoforder
k
anddegree
m
,
actingongermsofcurvesandinvariantunderreparametrization.Followingan
approachinitiatedbyGreenandGriffiths,weestablishabasicAhlfors-Schwarz
lemmainthesituationwhen
O
P
k
V
(

1)hasa(possiblysingular)metricofnegative
curvature,andweinferthateverynonconstantentirecurve
f
:
C

V
tangentto
V
mustbecontainedinthebaselocusofthemetric.Thisbasicresultisthenused
toobtainaproofoftheBlochtheorem,accordingtowhichtheZariskiclosureofan
entirecurveinacomplextorusisatranslateofasubtorus.Ourhope,supported
byexplicitRiemann-Rochcalculationsandothergeometricconsiderations,isthat
theSemplebundleconstructionshouldbeanefficienttooltocalculatethebase
locus.Necessaryorsufficientalgebraiccriteriaforhyperbolicityarethenobtained
intermsofinequalitiesrelatinggeneraofalgebraiccurvesdrawnonthevariety,
andsingularitiesofsuchcurves.Wefinallydescribesometechniquesintroduced
bySiuinvaluedistributiontheory,basedonauseofmeromorphicconnections.
ThesetechniqueshavebeendeveloppedlaterbyNadeltoproduceelegantexamples
ofhyperbolicsurfacesoflowdegreeinprojective3-space;thankstoasuitable
conceptof“partialprojectiveprojection”andtheassociatedWronskianoperators,
substantialimprovementsonNadel’sdegreeestimatewillbeachievedhere.

2J.-P.Demailly,Kobayashihyperbolicprojectivevarietiesandjetdifferentials

Keywords:
Kobayashihyperbolicvariety,directedmanifold,genusofcurves,jet
bundle,jetdifferential,jetmetric,Chernconnectionandcurvature,negativityof
jetcurvature,varietyofgeneraltype.
A.M.S.Classification(1991):
32H20,32L10,53C55,14J40

Contents

§
0.Introduction
..................................................
.
........................
2
§
1.Hyperbolicityconceptsanddirectedmanifolds
..........................................
7
§
2.Hyperbolicityandboundsforthegenusofcurves
......................................
10
§
3.TheAhlfors-Schwarzlemmaformetricsofnegativecurvature
..........................
16
§
4.Projectivizationofadirectedmanifold
.................................................
20
§
5.JetsofcurvesandSemplejetbundles
..................................................
24
§
6.Jetdifferentials
..................................................
.
....................
28
§
7.
k
-jetmetricswithnegativecurvature
..................................................
36
§
8.Algebraiccriterionforthenegativityofjetcurvature
...................................
44
§
9.ProofoftheBlochtheorem
..................................................
.
........
50
§
10.LogarithmicjetbundlesandaconjectureofLang
.....................................
51
§
11.ProjectivemeromorphicconnectionsandWronskians
..................................
54
§
12.Decompositionofjetsinirreduciblerepresentations
...................................
65
§
13.Riemann-Rochcalculationsandstudyofthebaselocus
...............................
68
§
14.Appendix:avanishingtheoremforholomorphictensorfields
..........................
74
References
..................................................
.
............................
78

§
0.Introduction
Inthesenotes,weinvestigatesomegeometricquestionsrelatedtotheconcept
ofhyperbolicvarietyinthesenseofKobayashi[Kob70].Hyperbolicalgebraic
varietieshaveattractedconsiderableattention,inpartbecauseoftheirconjectured
diophantineproperties.Forinstance,[Lang86]hasconjectured(amongother
things)thatanyhyperboliccomplexprojectivevarietyoveranumberfield
K
can
containonlyfinitelymanyrationalpointsover
K
;thisconjecture,whichseemsat
presentfarbeyondreach,mayberegardedasahigherdimensionalanalogueofthe
Mordellconjecture.ThereadercanconsultP.Vojta[Voj87]foraspectsconnected
todiophantineproblems.
Wewillbeconcernedhereonlywiththegeometricaspectsofthetheory
which,althoughapriorimoretractablethanthediophantineaspects,arestill
conjecturalforamajorpart;infactveryfewsatisfactorygeneralpurposetheorems
areavailable.Wehopethatsomeoftheideaspresentedherewillproveusefulto
achievesubstantialprogress.ThereaderisreferredtoS.Lang’ssurvey[Lang86]
andbook[Lang87]foranoverviewofthetheoryuntilthemid80’s,andto
J.Noguchi-T.Ochiai[NoOc90],P.M.Wong[Wong93]andM.Zaidenberg[Zai93]
foragoodexpositionofmorerecentproblems.Ourgoalhereisnottoprovide
anexhaustivecompilationofknownresults,butrathertoemphasizetwoorthree
importantideasaroundtheconceptsofjetbundlesandjetmetrics.Similarideas

§
0.Introduction3

havebeenappliedsuccessfullyinasomewhatspecialsituationintherecentwork
[SiYe96a]bySiuandYeung,wheretheauthorsprovethehyperbolicityofthe
complementofanirreduciblegenericcurveofhighdegree
d
>
10
13
in
P
2
.Letus
fixhereourterminology:theword“generic”willrefertoapropertywhichholds
trueinthecomplementofaglobalalgebraicoranalyticsubsetintheparameter
space,andtheexpression“verygeneric”willbeusedwhentheexceptionalsetof
parametersisacountableunionofalgebraicoranalyticsubsets.Aswewillseein
severalinstances,thegeometryofjetsconveysmanynaturalinterestingproblems
concerningtherelationshipbetweenhyperbolicityandjetcurvaturenegativity.
Wenowgiveashortoutlineofthecontents.Recallthatacomplexvarietyis
hyperbolicinthesenseofKobayashi
ifthefamilyofholomorphicmaps
f


X
fromtheunitdiskinto
X
isanormalfamily.If
X
iscompact(e.g.projective
algebraic),itiswellknownthat
X
isKobayashihyperbolicifandonlyifitis
Brodyhyperbolic
,thatis,iftherearenononconstantentireholomorphiccurve
f
:
C

X
.Inparticular
X
hasnorationalorellipticcurves,andmoregenerally
everyholomorphicmap
f
:
Z

X
fromanabelianvariety(orcomplextorus)to
X
mustbeconstant.Conversely,ithasbeensuggestedbyKobayashi[Kob70]and
[Lang86]thatthesealgebraicpropertiescouldbeequivalenttohyperbolicity.To
provethis,onewouldhavetoconstructatorus
Z
andanontrivialholomorphicmap
f
:
Z

X
whenever
X
isnonhyperbolic.Ahintthatthisshouldbetrueisgiven
bythefollowingobservation:if
X
ishyperbolic,thereisanabsoluteconstant
ε>
0
suchthatthegenusofanycompactcurveof
X
isboundedbelowby
ε
timesthe
degree;conversely,thispropertyfailstobetrueinmanyexamplesofnonhyperbolic
projectivevarieties.Ourbelief,supportedbysomeheuristicarguments,isthat
anysequenceofcompactcurves(
C

)withgenus(
C

)
/
degree(
C

)

0shouldhave
aclustersetsweptoutbytheimageofamap
f
:
Z

X
fromacomplex
torus
Z
,suchthatthelimitofsomesubsequenceofthesequenceofuniversal
coveringmapsΔ

C


X
(suitablyreparametrized)coincideswiththeimage
ofa(nonnecessarilycompact)straightlineof
Z
into
X
.Arelatedconjecture
of[Lang86]statesthataprojectivevarietyishyperbolicifandonlyifallits
irreduciblealgebraicvarietiesareofgeneraltype.Themostelementarystepwould
betoexcludethecaseofmanifoldswith
c
1

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