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Publié par | profil-zyak-2012 |
Nombre de lectures | 20 |
Langue | English |
Extrait
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
:Δ
→
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
ℓ
)withgenus(
C
ℓ
)
/
degree(
C
ℓ
)
→
0shouldhave
aclustersetsweptoutbytheimageofamap
f
:
Z
→
X
fromacomplex
torus
Z
,suchthatthelimitofsomesubsequenceofthesequenceofuniversal
coveringmapsΔ
→
C
ℓ
→
X
(suitablyreparametrized)coincideswiththeimage
ofa(nonnecessarilycompact)straightlineof
Z
into
X
.Arelatedconjecture
of[Lang86]statesthataprojectivevarietyishyperbolicifandonlyifallits
irreduciblealgebraicvarietiesareofgeneraltype.Themostelementarystepwould
betoexcludethecaseofmanifoldswith
c
1