Holomorphic Morse inequalities and the Green Griffiths Lang conjecture

Jean-Pierre Demailly

Description

Holomorphic Morse inequalities and the Green-Griffiths-Lang conjecture Jean-Pierre Demailly Universite de Grenoble I, Departement de Mathematiques Institut Fourier, 38402 Saint-Martin d'Heres, France e-mail : Dedicated to the memory of Eckart Viehweg Abstract. The goal of this work is to study the existence and properties of non constant entire curves f : C ? X drawn in a complex irreducible n-dimensional variety X , and more specifically to show that they must satisfy certain global algebraic or differential equations as soon as X is projective of general type. By means of holomorphic Morse inequalities and a probabilistic analysis of the cohomology of jet spaces, we are able to reach a significant step towards a generalized version of the Green-Griffiths-Lang conjecture. Resume. Le but de ce travail est d'etudier l'existence et les proprietes des courbes entieres non constantes f : C ? X tracees sur une varete complexe irreductible de dimension n, et plus precisement de montrer que ces courbes doivent satisfaire a certaines equations algebriques ou differentielles globales des que X est projective de type general. Au moyen des inegalites de Morse holomorphes et d'une analyse probabiliste de la cohomologie des espaces de jets, nous atteignons une premiere etape significative en direction d'une version generalisee de la conjecture de Green- Griffiths-Lang. Key words.

• 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

Sujets

• Demailly
• Jet bundle
• Homologie et cohomologie
• maths
• Mathématiques

Informations

• Publié par: pefav
• Nombre de lectures: 21
• Langue: English

Extrait

HolomorphicMorseinequalitiesand
theGreen-Griffiths-Langconjecture
Jean-PierreDemailly
Universite´deGrenobleI,De´partementdeMathe´matiques
InstitutFourier,38402Saint-Martind’He`res,France
e-mail
:
[e-mail]

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<