Holomorphic Morse inequalities and the Green Griffiths Lang conjecture
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- jet bundle
- holomorphic morse
- analyse probabiliste de la cohomologie des espaces de jets
- cohomologie
- jet differentials
- tx
- brody-hyperbolic when there
- group hn
- entire curves
HolomorphicMorseinequalitiesand
theGreen-Griffiths-Langconjecture
Jean-PierreDemailly
Universite´deGrenobleI,De´partementdeMathe´matiques
InstitutFourier,38402Saint-Martind’He`res,France
e-mail
:
demailly@fourier.ujf-grenoble.fr
DedicatedtothememoryofEckartViehweg
Abstract.
Thegoalofthisworkistostudytheexistenceandpropertiesofnon
constantentirecurves
f
:
C
→
X
drawninacomplexirreducible
n
-dimensional
variety
X
,andmorespecificallytoshowthattheymustsatisfycertainglobal
algebraicordifferentialequationsassoonas
X
isprojectiveofgeneraltype.
BymeansofholomorphicMorseinequalitiesandaprobabilisticanalysisofthe
cohomologyofjetspaces,weareabletoreachasignificantsteptowardsa
generalizedversionoftheGreen-Griffiths-Langconjecture.
Re´sume´.
Lebutdecetravailestd’e´tudierl’existenceetlesproprie´te´sdescourbes
entie`resnonconstantes
f
:
C
→
X
trace´essurunevare´te´complexeirre´ductiblede
dimension
n
,etpluspre´cise´mentdemontrerquecescourbesdoiventsatisfairea`
certainese´quationsalge´briquesoudiffe´rentiellesglobalesde`sque
X
estprojective
detypege´ne´ral.Aumoyendesine´galite´sdeMorseholomorphesetd’uneanalyse
probabilistedelacohomologiedesespacesdejets,nousatteignonsunepremie`re
e´tapesignificativeendirectiond’uneversionge´ne´ralise´edelaconjecturedeGreen-
Griffiths-Lang.
Keywords.
Cherncurvature,holomorphicMorseinequality,jetbundle,co-
homologygroup,entirecurve,algebraicdegeneration,weightedprojectivespace,
Green-Griffiths-Langconjecture
Mots-cle´s.
CourburedeChern,ine´galite´deMorseholomorphe,fibre´dejets,
groupedecohomologie,courbeentie`re,de´ge´ne´rescencealge´brique,espaceprojectif
a`poids,conjecturedeGreen-Griffiths-Lang.
MSC2010Classification.
32Q45,32L20,14C30
0.Introduction
Let
X
beacomplex
n
-dimensionalmanifold;mostofthetimewewillassume
that
X
iscompactandevenprojectivealgebraic.If
:
X
e
→
X
isamodification
and
f
:
C
→
X
isanentirecurvewhoseimage
f
(
C
)isnotcontainedintheimage
(
E
)oftheexceptionallocus,then
f
admitsauniquelifting
f
e
:
C
→
X
e
.For
thisreason,thestudyofthealgebraicdegenerationof
f
isabirationallyinvariant
2HolomorphicMorseinequalitiesandtheGreen-Griffiths-Langconjecture
problem,andsingularitiesdonotplayanessentialroleatthisstage.Wewill
thereforeassumethat
X
isnonsingular,possiblyafterperformingasuitable
compositionofblow-ups.Weareinterestedmoregenerallyinthesituationwhere
thetangentbundle
T
X
isequippedwitha
linearsubspace
V
⊂
T
X
,thatis,an
irreduciblecomplexanalyticsubsetofthetotalspacesuchthat
(0.1)allfibers
V
x
:=
V
∩
T
X,x
arevectorsubspacesof
T
X,x
.
Thentheproblemistostudyentirecurves
f
:
C
→
X
whicharetangentto
V
,
i.e.suchthat
f
∗
T
C
⊂
V
.Wewillrefertoapair(
X,V
)asbeinga
directedvariety
(or
directedmanifold
).AmorphismofdirectedvarietiesΦ:(
X,V
)
→
(
Y,W
)
isaholomorphicmapΦ:
X
→
Y
suchthatΦ
∗
V
⊂
W
;bytheirreducibility,
itisenoughtocheckthisconditionoverthedenseopensubset
X
r
V
sing
where
V
isactuallyasubbundle(here
V
sing
istheindeterminacysetoftheassociated
meromorphicmap
X
>
G
r
(
T
X
)totheGrassmannianof
r
-planesin
T
X
,
r
=rank
V
).Inthatway,wegetacategory,andwewillbemostlyinterestedin
thesubcategorywhoseobjects(
X,V
)areprojectivealgebraicmanifoldsequipped
withalgebraiclinearsubspaces.
Thecasewhere
V
=
T
X/S
istherelativetangentspaceofsomefibration
X
→
S
isofspecialinterest,andsoisthecaseofafoliatedvariety(thisisthe
situationwherethesheafofsections
O
(
V
)satisfiestheFrobeniusintegrability
condition[
O
(
V
)
,
O
(
V
)]
⊂
O
(
V
));however,itisveryusefultoallowaswellnon
integrablelinearsubspaces
V
.Wereferto
V
=
T
X
asbeingthe
absolutecase
.Our
maintargetisthefollowingdeepconjectureconcerningthealgebraicdegeneracy
ofentirecurves,whichgeneralizessimilarstatementsmadein[GG79](seealso
[Lang86,Lang87]).
(0.2)GeneralizedGreen-Griffiths-Langconjecture.
Let
(
X,V
)
beaprojec-
tivedirectedmanifoldsuchthatthecanonicalsheaf
K
V
isbig
(
intheabsolutecase
V
=
T
X
,thismeansthat
X
isavarietyofgeneraltype,andintherelativecase
wewillsaythat
(
X,V
)
isofgeneraltype
)
.Thenthereshouldexistanalgebraic
subvariety
Y
(
X
suchthateverynonconstantentirecurve
f
:
C
→
X
tangent
to
V
iscontainedin
Y
.
Theprecisemeaningof
K
V
andofitsbignesswillbeexplainedbelow.One
saysthat(
X,V
)isBrody-hyperbolicwhentherearenoentirecurvestangentto
V
.
Accordingto(generalizedversionsof)conjecturesofKobayashi[Kob70,Kob76]
thehyperbolicityof(
X,V
)shouldimplythat
K
V
isbig,andevenpossiblyample,
inasuitablesense.Itwouldthenfollowfromconjecture(0.2)that(
X,V
)is
hyperbolicifandonlyifforeveryirreduciblevariety
Y
⊂
X
,thelinearsubspace
V
Y
e
=
T
Y
e
r
E
∩
∗−
1
V
⊂
T
Y
e
hasabigcanonicalsheafwhenever
:
Y
e
→
Y
isa
desingularizationand
E
istheexceptionallocus.
ThemoststrikingresultknownontheGreen-Griffiths-Langconjectureat
thisdateisarecentrecentofDiverio,MerkerandRousseau[DMR10]inthe
absolutecase,confirmingthestatementwhen
X
⊂
P
C
n
+1
isagenericnonsingular
5hypersurfaceoflargedegree
d
,withanestimatedsufficientlowerbound
d
>
2
n
.
0.Introduction3
TheirproofisbasedinanessentialwayonastrategydevelopedbySiu[Siu02,
Siu04],combinedwithtechniquesof[Dem95].NoticethatiftheGreen-Griffiths-
Langconjectureholdstrue,amuchstrongerandprobablyoptimalresultwould
betrue,namelyallsmoothhypersurfacesofdegree
d
>
n
+3wouldsatisfythe
expectedalgebraicdegeneracystatement.Moreover,byresultsofClemens[Cle86]
andVoisin[Voi96],a(very)generichypersurfaceofdegree
d
>
2
n
+1wouldin
factbehyperbolicforevery
n
>
2.Suchagenerichyperbolicitystatementhas
beenobtainedunconditionallybyMcQuillan[McQ98,McQ99]when
n
=2and
d
>
35,andbyDemailly-ElGoul[DEG00]when
n
=2and
d
>
21.Recently
Diverio-Trapani[DT10]provedthesameresultwhen
n
=3and
d
>
593.By
definition,provingthealgebraicdegeneracymeansfindinganonzeropolynomial
P
on
X
suchthatallentirecurves
f
:
C
→
X
satisfy
P
(
f
)=0.Allknown
methodsofproofarebasedonestablishingfirsttheexistenceofcertainalgebraic
differentialequations
P
(
f
;
f
′
,f
′′
,...,f
(
k
)
)=0ofsomeorder
k
,andthentrying
tofindenoughsuchequationssothattheycutoutaproperalgebraiclocus
Y
(
X
.
Let
J
k
V
bethespaceof
k
-jetsofcurves
f
:(
C
,
0)
→
X
tangentto
V
.One
definesthesheaf
O
(
E
k
G
,
G
m
V
∗
)ofjetdifferentialsoforder
k
anddegree
m
tobe
thesheafofholomorphicfunctions
P
(
z
;
ξ
1
,...ξ
k
)on
J
k
V
whicharehomogeneous
polynomialsofdegree
m
onthefibersof
J
k
V
→
X
withrespecttolocalcoordinate
derivatives
ξ
j
=
f
(
j
)
(0).Thedegree
m
consideredhereistheweighteddegree
withrespecttothenatural
C
∗
actionon
J
k
V
definedby
λ
f
(
t
):=
f
(
λt
),
i.e.byreparametrizingthecurvewithahomotheticchangeofvariable.Since
(
λ
f
)
(
j
)
(
t
)=
λ
j
f
(
j
)
(
λt
),theweightedactionisgivenincoordinatesby
(0
.
3)
λ
(
ξ
1
,ξ<
