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Holomorphic Morse inequalities and the Green Griffiths Lang conjecture

De
36 pages
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


Voir plus Voir moins

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

2
,...,ξ
k
)=(
λξ
1

2
ξ
2
,...,λ
k
ξ
k
)
.
OneofthemajortoolofthetheoryisthefollowingresultduetoGreen-Griffiths
[GG79](seealso[Dem95,Dem97],[SY96a,SY96b],[Siu97]).
(0.4)Fundamentalvanishingtheorem.
Let
(
X,V
)
beadirectedprojective
varietyand
f
:(
C
,T
C
)

(
X,V
)
anentirecurvetangentto
V
.Thenforevery
globalsection
P

H
0
(
X,E
k
G
,
G
m
V


O
(

A
))
where
A
isanampledivisorof
X
,
onehas
P
(
f
;
f

,f
′′
,...,f
(
k
)
)=0
.
Itisexpectedthattheglobalsectionsof
H
0
(
X,E
k
G
,
G
m
V


O
(

A
))areprecisely
thosewhichultimatelydefinethealgebraiclocus
Y
(
X
wherethecurve
f
should
lie.Theproblemisthenreducedtothequestionofshowingthattherearemany
nonzerosectionsof
H
0
(
X,E
k
G
,
G
m
V


O
(

A
)),andfurther,understandingwhat
istheirjointbaselocus.Thefirstpartofthisprogramisthemainresultofthe
presentpaper.
(0.5)Theorem.
Let
(
X,V
)
beadirectedprojectivevarietysuchthat
K
V
isbig
andlet
A
beanampledivisor.Thenfor
k

1
and
δ

Q
+
smallenough,
δ
6
c
(log
k
)
/k
,thenumberofsections
h
0
(
X,E
k
G
,
G
m
V


O
(

mδA
))
hasmaximal
growth,i.e.islargerthat
c
k
m
n
+
kr

1
forsome
m
>
m
k
,where
c,c
k
>
0
,
n
=dim
X
and
r
=rank
V
.Inparticular,entirecurves
f
:(
C
,T
C
)

(
X,V
)
satisfy
(
many
)
algebraicdifferentialequations.

4HolomorphicMorseinequalitiesandtheGreen-Griffiths-Langconjecture
Thestatementisveryelementarytocheckwhen
r
=rank
V
=1,andtherefore
when
n
=dim
X
=1.Inhigherdimensions
n
>
2,onlyverypartialresults
wereknownatthispoint,concerningmerelytheabsolutecase
V
=
T
X
.In
dimension2,Theorem(0.5)isaconsequenceoftheRiemann-Rochcalculation
ofGreen-Griffiths[GG79],combinedwithavanishingtheoremduetoBogomolov
[Bog79]–thelatteractuallyonlyappliestothetopcohomologygroup
H
n
,and
thingsbecomemuchmoredelicatewhenextimatesofintermediatecohomology
groupsareneeded.Inhigherdimensions,Diverio[Div09]provedtheexistence
ofsectionsof
H
0
(
X,E
k
G
,
G
m
V


O
(

1))whenever
X
isahypersurfaceof
P
C
n
+1
ofhighdegree
d
>
d
n
,assuming
k
>
n
and
m
>
m
n
.Morerecently,Merker
[Mer10]wasabletotreatthecaseofarbitraryhypersurfacesofgeneraltype,i.e.
d
>
n
+3,assumingthistime
k
tobeverylarge.Thelatterresultisobtained
throughexplicitalgebraiccalculationsofthespacesofsections,andtheproofis
computationallyveryintensive.Be´rczi[Ber10]alsoobtainedrelatedresultswith
adifferentapproachbasedonresidueformulas,assuming
d
>
2
7
n
log
n
.
Alltheseapproachesarealgebraicinnature,andwhiletheyusesomeform
ofholomorphicMorseinequalities[Dem85],theyonlyrequireaveryspecial
elementaryalgebraiccase,namelythelowerbound
nh
0
(
X,L

m
)
>
m
(
A
n

nA
n

1

B
)

o
(
m
n
)
!nfor
L
=
O
(
A

B
)with
A,B
nef(cf.Trapani[Tra95]).Here,ourtechniques
arebasedonmoreelaboratecurvatureestimatesinthespiritofCowen-Griffiths
[CG76].TheyrequirethestrongeranalyticformofholomorphicMorseinequalities.
(0.6)HolomorphicMorseinequalities
([Dem85])
.
Let
(
L,h
)
beaholomorphic
linebundleonacompactcomplexmanifold
X
,equippedwithasmoothhermitian
metric
h
,andlet
E
beaholomorphicvectorbundle.Denoteby
Θ
L,h
=

2

∂∂
log
h
thecurvatureformof
(
L,h
)
andconsidertheopenset
X
(
L,h,q
)=
x

X

L,h
(
x
)
hassignature
(
n

q,q
)(
q
-indexsetof
Θ
L,h
)
,
`sothatthereisapartition
X
=
0
6
q
6
n
X
(
L,q
)

S
where
S
=
{
detΘ
L,h
(
x
)=0
}
isthedegeneracyset.Then,ifweput
r
=rank
E
,wehaveasymptoticallyas
m
tendstoinfinity
:
(a)(
WeakMorseinequalities
)
nZh
q
(
X,E

L

m
)
6
mr
(

1)
q
Θ
Ln,h
+
o
(
m
n
)
.
n
!
X
(
L,h,q
)
(b)(
StrongMorseinequalities
)
If
X
(
L,h,
6
q
)=
j
6
q
X
(
L,h,j
)
,then
`qnZ(

1)
q

j
h
j
(
X,E

L

m
)
6
mr
(

1)
q
Θ
Ln,h
+
o
(
m
n
)
.
Xj
=0
n
!
X
(
L,h,
6
q
)

0.Introduction5

(c)(
Lowerboundon
h
0
)
nmZh
0
(
X,E

L

m
)

h
1
(
X,E

L

m
)
>
r
Θ
Ln,h

o
(
m
n
)
.
n
!
X
(
L,h,
6
1)
Theproofoftheaboveisbasedonrefinedspectralestimatesforthecomplex
Laplace-Beltramioperators.Observethat(0.6c)isjustthespecialcaseof(0.6b)
when
q
=1.Ithasbeenrecentlyobservedthattheseinequalitiesshouldbe
optimalinthesensethattheasymptoticcohomologyfunctional
h
b
q
(
X,L
):=
limsup
m

+

n
!
n
h
0
(
X,L

m
)satisfies
mZ(0
.
7)
h
b
q
(
X,L
)
6
inf

(

1)
q
Θ
Ln,h
,
h

CX
(
L,h,q
)
andthatconjecturallytheinequalityshouldbeanequality;itisprovedin
[Dem10a],[Dem10b]thatthisisindeedthecaseif
n
6
2or
q
=0,atleast
when
X
isprojectivealgebraic.
NoticethatholomorphicMorseinequalitiesareessentiallyinsensitivetosin-
gularities,aswecanpasstononsingularmodelsandblow-up
X
asmuchaswe
qwant:if

:
X
e

X
isamodificationthen


O
X
e
=
O
X
and
R

O
X
e
issupported
onacodimension1analyticsubset(evencodimension2if
X
issmooth).Itfollows
bytheLerayspectralsequencethattheestimatesfor
L
on
X
orfor
L
e
=


L
on
X
e
differbynegligibleterms
O
(
m
n

1
).Finally,wecanevenworkwithsingularhermi-
tianmetrics
h
whichhave(positiveandrational)analyticsingularities,that
P
is,one
canwritelocally
h
=
e

ϕ
where,possiblyafterblowingup,
ϕ
(
z
)=
c
log
j
|
g
j
|
2
mod
C

,with
c

Q
+
and
g
j
holomorphic.Blowing-uptheideal
J
=(
g
j
)leads
todivisorialsingularities,andthenbyreplacing
L
with
L
e
=


L

O
(

E
)where
E

Div
Q
(
X
e
)isthesingularitydivisor,weseethatholomorphicMorseinequali-
tiesstillholdforthesequenceofgroups
H
q
(
X,E

L

m

I
(
h

m
))where
I
(
h

m
)
isthemultiplieridealsheafof
h

m
(seeBonavero[Bon93]formoredetails).In
thecaseoflinearsubspaces
V

T
X
,weintroducesingularhermitianmetricsas
follows.
(0.8)Definition.
Asingularhermitianmetriconalinearsubspace
V

T
X
is
ametric
h
onthefibersof
V
suchthatthefunction
log
h
:
ξ
7→
log
|
ξ
|
h
2
islocally
integrableonthetotalspaceof
V
.
Suchametriccanalsobeviewedasasingularhermitianmetriconthetauto-
logicallinebundle
O
P
(
V
)
(

1)ontheprojectivizedbundle
P
(
V
)=
V
r
{
0
}
/
C

,
andthereforeitsdualmetric
h

1
definesacurvaturecurrentΘ
O
P
(
V
)
(1)
,h

1
oftype
(1
,
1)on
P
(
V
)

P
(
T
X
),suchthat
ip

Θ
O
P
(
V
)
(1)
,h

1
=
∂∂
log
h,
where
p
:
V
r
{
0
}→
P
(
V
)
.
π2Iflog
h
isquasi-plurisubharmonic(orquasi-psh,whichmeanspshmoduloaddition
ofasmoothfunction)on
V
,thenlog
h
isindeedlocallyintegrable,andwehave

6HolomorphicMorseinequalitiesandtheGreen-Griffiths-Langconjecture
moreover
(0
.
9)Θ
O
P
(
V
)
(1)
,h

>


forsomesmoothpositive(1
,
1)-formon
P
(
V
)andsomeconstant
C>
0;
conversely,if(0.9)holds,thenlog
h
isquasipsh.
(0.10)Definition.
Wewillsaythatasingularhermitianmetric
h
on
V
is
admissible
if
h

1
hasanalyticsingularitieswhenseenasametricon
O
P
(
V
)
(1)
and
ifmoreover
h
isasmoothhermitianmetriconaZariskiopenset
X


X
r
V
sing
.
If
h
isanadmissiblemetric,wedefine
O
h
(
V

)tobethesheafofgerms
ofholomorphicsectionssectionsof
V
|∗
X

whichare
h

1
-boundednear
X
r
X

;
bytheassumptionontheanalyticsingularities,thisisacoherentsheaf,and
actuallyasubsheafofthesheaf
O
(
V

):=
O
h
0
(
V

)associatedwithasmooth
positivedefinitemetric
h
0
on
T
X
.If
r
isthegenericrankof
V
and
m
apositive
integer,wedefinesimilarly
K
Vm,h
tobesheafofgermsofholomorphicsectionsof
(det
V
|∗
X

)

m
=(Λ
r
V
|∗
X

)

m
whicharedet
h

1
-bounded,and
K
Vm
:=
K
Vm,h
0
.With
ourassumptions,therealwaysexistsamodification

:
X
e
andaninteger
m
0
suchthatforallmultiples
m
=
pm
0
thepull-back


K
Vm,h
isaninvertiblesheaf
on
X
e
,anddet
h

1
inducesasmoothnonsingularmetriconit.Wethenthink
ratherof
K
V,h
(resp.
K
V
)asthe“virtual”
Q
-linebundle


(


K
Vm,h
0
)
1
/m
0
(resp.


(


K
Vm
0
)
1
/m
0
),andwesaythat
K
V,h
isbigif
h
0
(
X,K
Vm,h
)
>
cm
n
for
m
>
m
1
,
m∗mwith
c>
0;noticethatbydefinitionwealwayshave
K
V,h
=


(
K
V,h
).
Ourstrategycanbedescribedasfollowing.WeconsidertheGreen-Griffiths
bundleof
k
-jets
X
k
GG
=
J
k
V
r
{
0
}
/
C

,whichby(0.3)consistsofafibrationin
weightedprojectivespaces
,anditsassociatedtautologicalsheaf
L
=
O
X
k
GG
(1)
,
viewedratherasavirtual
Q
-linebundle
O
X
k
GG
(
m
0
)
1
/m
0
with
m
0
=lcm(1
,
2
,...,k
).
Then,if
π
k
:
X
k
GG

X
isthenaturalprojection,wehave
E
k
G
,
G
m
=(
π
k
)

O
X
k
GG
(
m
)and
R
q
(
π
k
)

O
X
k
GG
(
m
)=0for
q
>
1
.
Hence,bytheLerayspectralsequencewegetforeveryinvertiblesheaf
F
on
X
theisomorphism
(0
.
11)
H
q
(
X,E
k
G
,
G
m
V


F
)

H
q
(
X
k
GG
,
O
X
k
GG
(
m
)

π
k

F
)
.
ThelattergroupcanbeevaluatedthankstoholomorphicMorseinequalities.In
factwecanassociatewithanyadmissiblemetric
h
on
V
ametric(orrathera
naturalfamily)ofmetricson
L
=
O
X
GG
(1).Thespace
X
k
GG
alwayspossesses
kquotientsingularitiesif
k
>
2(andevensomemoreif
V
issingular),butwe
donotreallycaresinceMorseinequalitiesstillworkinthissetting.Aswewill
see,itisthenpossibletogetniceasymptoticformulasas
k

+

.They

1.Hermitiangeometryofweightedprojectivespaces7

appeartobeofa
probabilisticnature
ifwetakethecomponentsofthe
k
-jet
(i.e.thesuccessivederivatives
ξ
j
=
f
(
j
)
(0),1
6
j
6
k
)asrandomvariables.
ThisprobabilisticbehaviourwassomehowalreadyvisibleintheRiemann-Roch
calculationof[GG79].Inthisway,assuming
K
V
big,weproducealotofsections
σ
j
=
H
0
(
X
k
GG
,
O
X
GG
(
m
)

π
k

F
),correspondingtocertaindivisors
Z
j

X
k
GG
.
kGreen-Griffiths-Langconjectureistocomputethebaselocus
Z
=
Z
j
andto
Thehardproblemwhichisleftinordertocompleteaproofofthe
T
generalized
showthat
Y
=
π
k
(
Z
)

X
mustbeaproperalgebraicvariety.Althoughwe
cannotaddressthisproblematpresent,wewillindicateafewtechnicalresults
andacoupleofpossibilitiesinthisdirection.

1.Hermitiangeometryofweightedprojectivespaces
ThegoalofthissectionistointroducenaturalKa¨hlermetricsonweighted
projectivespaces,andtoevaluatethecorrespondingvolumeforms.Hereweput
iid
c
=
4
π
(



)sothat
dd
c
=
2
π
∂∂
.Thenormalizationofthe
d
c
operatorischosen
suchthatwehaveprecisely(
dd
c
log
|
z
|
2
)
n
=
δ
0
fortheMonge-Ampe`reoperator
in
C
n
;also,foreveryholomorphicormeromorphicsection
σ
ofahermitianline
bundle(
L,h
)theLelong-Poincare´canbeformulated

(1
.
1)
dd
c
log
|
σ
|
h
2
=[
Z
σ
]

Θ
L,h
,
2iwhereΘ
L,h
=
2
π
D
L,h
isthe(1
,
1)-curvatureformof
L
and
Z
σ
thezerodivisorof
σ
.
Theclosed(1
,
1)-formΘ
L,h
isarepresentativeofthefirstChernclass
c
1
(
L
).Given
a
k
-tupleof“weights”
a
=(
a
1
,...,a
k
),i.e.ofrelativelyprimeintegers
a
s
>
0we
introducetheweightedprojectivespace
P
(
a
1
,...,a
k
)tobethequotientof
C
k
r
{
0
}
bythecorrespondingweighted
C

action:

(1
.
2)
P
(
a
1
,...,a
k
)=
C
k
r
{
0
}
/
C



z
=(
λ
a
1
z
1
,...,λ
a
k
z
k
)
.
Asiswellknown,thisdefinesatoric
k

1-dimensionalalgebraicvarietywith
quotientsingularities.Onthisvariety,weintroducethepossiblysingular(but
almosteverywheresmoothandnondegenerate)Ka¨hlerform
ω
a,p
definedby
1X(1
.
3)
π
a

ω
a,p
=
dd
c
ϕ
a,p

a,p
(
z
)=log
|
z
s
|
2
p/a
s
,
p
1
6
s
6
k

where
π
a
:
C
k
r
{
0
}→
P
(
a
1
,...,a
k
)isthecanonicalprojectionand
p>
0is
apositiveconstant.Itisclearthat
ϕ
p,a
isrealanalyticon
C
k
r
{
0
}
if
p
isan
integerandacommonmultipleofallweights
a
s
.Itisatleast
C
2
is
p
isrealand
p
>
max(
a
s
),whichwillbemorethansufficientforourpurposes(buteverything
wouldstillworkforany
p>
0).Theresultingmetricisinanycasesmooth
andpositivedefiniteoutsideofthecoordinatehyperplanes
z
s
=0,andthese
hyperplaneswillnotmatterheresincetheyareofcapacityzerowithrespectto

8HolomorphicMorseinequalitiesandtheGreen-Griffiths-Langconjecture
allcurrents(
dd
c
ϕ
a,p
)

.Inordertoevaluatethevolume
P
(
a,...,a
)
ω
ak,

p
1
,onecan
Rk1observethat
ZZω
k

1
=
π

ω
k

1

d
c
ϕ
P
(
a
1
,...,a
k
)
a,pz

C
k

a,p
(
z
)=0
aa,pa,p
Z=(
dd
c
ϕ
a,p
)
k

1

d
c
ϕ
a,p
z

C
k

a,p
(
z
)=0
(1
.
4)=1(
dd
c
e

a,p
)
k
.
Zp
kz

C
k

a,p
(
z
)
<
0
Thefirstequalitycomesfromthefactthat
{
ϕ
a,p
(
z
)=0
}
isacirclebundle
o
R
ver
P
(
a
1
,...,a
k
),byusingtheidentities
ϕ
a,p
(
λ

z
)=
ϕ
a,p
(
z
)+log
|
λ
|
2
and
|
λ
|
=1
d
c
log
|
λ
|
2
=1.ThethirdequalitycanbeseenbyStokesformulaappliedto
the(2
k

1)-form
(
dd
c
e

a,p
)
k

1

d
c
e

a,p
=
e

a,p
(
dd
c
ϕ
a,p
)
k

1

d
c
ϕ
a,p
onthepseudoconvexopenset
{
z

C
k
;
ϕ
a,p
(
z
)
<
0
}
.Now,wefind
X
k
Y
p
p

(1
.
5)(
dd
c
e

a,p
)
k
=
dd
c
|
z
s
|
2
p/a
s
=
|
z
s
|
as

1
(
dd
c
|
z
|
2
)
k
,
1
6
s
6
k
1
6
s
6
k
a
s
k(1
.
6)(
dd
c
e

a,p
)
k
=
p
=
p.
ZYz

C
k

a,p
(
z
)
<
01
6
s
6
k
a
s
a
1
...a
k
Infact,(1.5)and(1.6)areclearwhen
p
=
a
1
=
...
=
a
k
=1(thisisjustthe
kstandardcalculationofthevolum
p
e
/a
o
s
ftheunitballin
C
);thegeneralcasefollows
bysubstitutingformally
z
s
7→
z
s
,andusingrotationalinvariancealongwith
a/ptheobservationthattheargumentsofthecomplexnumbers
z
s
s
nowruninthe
interval[0
,
2
πp/a
s
[insteadof[0
,
2
π
[(say).Asaconsequenceof(1.4)and(1.6),
weobtainthewellknownvalue
(1
.
7)
ω
k

1
=1
,
ZP
(
a
1
,...,a
k
)
a,p
a
1
...a
k
forthevolume.Noticethatthisisindependentof
p
(asitisobviousbyStokes
theorem,sincethecohomologyclassof
ω
a,p
doesnotdependon
p
).When
p
tends
to+

,wehave
ϕ
a,p
(
z
)
7→
ϕ
a,

(
z
)=logmax
1
6
s
6
k
|
z
s
|
2
/a
s
andthevolumeform
ω
ak,

p
1
convergestoarotationallyinvariantmeasuresupportedbytheimageofthe
polycircle
{|
z
s
|
=1
}
in
P
(
a
1
,...,a
k
).Thisissobecausenotall
|
z
s
|
2
/a
s
are
Qequaloutsideoftheimageofthepolycircle,thus
ϕ
a,

(
z
)locallydependsonly
on
k

1complexvariables,andso
ω
ak,
−∞
1
=0therebyloghomogeneity.
Ourlatercalculationswillrequireaslightlymoregeneralsetting.Insteadof
lookingat
C
k
,weconsidertheweighted
C

actiondefinedby
(1
.
8)
C
|
r
|
=
C
r
1
×
...
×
C
r
k


z
=(
λ
a
1
z
1
,...,λ
a
k
z
k
)
.

1.Hermitiangeometryofweightedprojectivespaces9
Here
z
s

C
r
s
forsome
k
-tuple
r
=(
r
1
,...,r
k
)and
|
r
|
=
r
1
+
...
+
r
k
.Thisgives
risetoaweightedprojectivespace
P
(
a
[1
r
1
]
,...,a
[
kr
k
]
)=
P
(
a
1
,...,a
1
,...,a
k
,...,a
k
)
,
(1
.
9)
π
a,r
:
C
r
1
×
...
×
C
r
k
r
{
0
}−→
P
(
a
[1
r
1
]
,...,a
[
kr
k
]
)
obtainedbyrepeating
r
s
timeseachweight
a
s
.Onthisspace,weintroducethe
degenerateKa¨hlermetric
ω
a,r,p
suchthat
(1
.
10)
π
a

,r
ω
a,r,p
=
dd
c
ϕ
a,r,p

a,r,p
(
z
)=1log
|
z
s
|
2
p/a
s
Xp
1
6
s
6
k
where
|
z
s
|
standsnowforthestandardhermitiannorm(
P
1
6
j
6
r
|
z
s,j
|
2
)
1
/
2
on
C
r
s
.
sThismetriciscohomologoustothecorresponding“polydisc-like”metric
ω
a,p
alreadydefined,andthereforeStokestheoremimplies
|
r
|−
1
1
Z(1
.
11)
P
(
a
[
r
1]
,...,a
[
rk
]
)
ω
a,r,p
=
a
1
r
1
...a
r
k
.
kk1Since(
dd
c
log
|
z
s
|
2
)
r
s
=0on
C
r
s
r
{
0
}
byhomogeneity,weconcludeasbeforethat
theweaklimitlim
p

+

ω
|
ar,
|
r,

p
1
=
ω
|
ar,
|
r,
−∞
1
associatedwith
(1
.
12)
ϕ
a,r,

(
z
)=logmax
|
z
s
|
2
/a
s
k6s61isameasuresupportedbytheimageoftheproductofunitspheres
Q
S
2
r
s

1
in
P
(
a
[1
r
1
]
,...,a
[
kr
k
]
),whichisinvariantundertheactionof
U
(
r
1
)
×
...
×
U
(
r
k
)
on
C
r
1
×
...
×
C
r
k
,andthuscoincideswiththehermitianareameasureuptoa
constantdeterminedbycondition(1.11).Infact,outsideoftheproductofspheres,
ϕ
a,r,

locallydependsonlyonatmost
k

1factorsandthus,fordimension
reasons,thetoppower(
dd
c
ϕ
a,r,

)
|
r
|−
1
mustbezerothere.Inthenextsection,the
followingchangeofvariableformulawillbeneeded.Forsimplicityofexposition
werestrictourselvestocontinuousfunctions,butastandarddensityargument
wouldeasilyextendtheformulatoallfunctionsthatareLebesgueintegrablewith
respecttothevolumeform
ω
|
ar,
|
r,

p
1
.
(1.13)Proposition.
Let
f
(
z
)
beaboundedfunctionon
P
(
a
[1
r
1
]
,...,a
[
kr
k
]
)
which
iscontinuousoutsideofthehyperpla
Q
nesections
z
s
=0
.Wealsoview
f
asa
C

-invariantcontinuousfunctionon
(
C
r
s
r
{
0
}
)
.Then
Z
|
r
|−
1
P
(
a
[
r
1]
,...,a
[
rk
]
)
f
(
z
)
ω
a,r,p
k1ZY
r
s

1
rk1s=(
|
Q
r
|−
1)!
f
(
x
a
1
/
2
p
u
1
,...,x
a
k
/
2
p
u
k
)
x
s
dxd
(
u
)
s
a
s
(
x,u
)

Δ
k

1
×
S
2
rs

1
1
6
s
6
k
(
r
s

1)!
Q

10HolomorphicMorseinequalitiesandtheGreen-Griffiths-Langconjecture
Pwhere
Δ
k

1
isthe
(
k

1)
-simplex
{
x
s
>
0
,
x
s
=1
}
,
dx
=
dx
1

...

dx
k

1
its
standardmeasure,andwhere
d
(
u
)=
d
1
(
u
1
)
...d
k
(
u
k
)
istherotationinvariant
probabilitymeasureontheproduct
s
S
2
r
s

1
ofunitspheresin
C
r
1
×
...
×
C
r
k
.
QAsaconsequence
|
r
|−
1
1
ZZP
(
a
1
,...,a
k
)
ssS
s
p
l

i
+
m

[
r
1][
rk
]
f
(
z
)
ω
a,r,p
=
Q
a
r
s
Q
2
r

1
f
(
u
)
d
(
u
)
.
2cRProof
.Theareaformulaofthedisc
|
λ
|
<
1
dd
|
λ
|
=1andaconsiderationofthe
unitdiscbundleover
P
(
a
[
r
1
]
,...,a
[
r
k
]
)implythat
k1ZZI
p
:=
f
(
z
)
ω
|
r
|−
1
=
f
(
z
)(
dd
c
ϕ
a,r,p
)
|
r
|−
1

dd
c
e
ϕ
a,r,p
.
p,r,aP
(
a
[1
r
1]
,...,a
[
krk
]
)
z

C
|
r
|

a,r,p
(
z
)
<
0
Now,astraightforwardcalculationon
C
|
r
|
gives
X
|
r
|
(
dd
c
e

a,r,p
)
|
r
|
=
dd
c
|
z
s
|
2
p/a
s
k6s61Y
p

r
s
+1
=
|
z
s
|
2
r
s
(
p/a
s

1)
(
dd
c
|
z
|
2
)
|
r
|
.
1
6
s
6
k
a
s
!|r|Ontheotherhand,wehave(
dd
c
|
z
|
2
)
|
r
|
=
r
1
!
...r
k
!1
6
s
6
k
(
dd
c
|
z
s
|
2
)
r
s
and
Q(
dd
c
e

a,r,p
)
|
r
|
=
pe

a,r,p
(
dd
c
ϕ
a,r,p
+
pdϕ
a,r,p

d
c
ϕ
a,r,p
)
|
r
|
=
|
r
|
p
|
r
|
+1
e
|
r
|

a,r,p
(
dd
c
ϕ
a,r,p
)
|
r
|−
1


a,r,p

d
c
ϕ
a,r,p
=
|
r
|
p
|
r
|
+1
e
(
|
r
|
p

1)
ϕ
a,r,p
(
dd
c
ϕ
a,r,p
)
|
r
|−
1

dd
c
e
ϕ
a,r,p
,
thankstothehomogeneityrelation(
dd
c
ϕ
a,r,p
)
|
r
|
=0.Puttingeverythingtogether,
dnfiewZ=IYpPz

C

a,r,p
(
z
)
<
0
s
|
s
|
s
r
s
!
a
s
|
z
s
|
Astandardcalculationinpolarcoordinateswith
z
s
=
ρ
s
u
s
,
u
s

S
2
r
s

1
,yields
(
dd
c
|
z
s
|
2
)
r
s

s
|
z
|
2
r
s
=2
r
s
ρd
s
(
u
s
)
sswhere

s
isthe
U
(
r
s
)-invariantprobabilitymeasureon
S
2
r
s

1
.Therefore
Z
2
pr
s
/a
s

(
|
r
|−
1)!
p
k

1
f
(
ρ
1
u
1
,...,ρ
k
u
k
)
Y
2
ρ

ss
d
s
(
u
s
)
ϕ
a,r,p
(
z
)
<
0
(
ρ
s
2
p/a
s
)
|
r
|−
1
/ps
(
r
s

1)!
a
s
s
I
p
=
P
r
+1
k6s61sk1Z=(
|
r
|−
1)!
p

1
f
(
t
a
1
/
2
p
u
1
,...,t
a
k
/
2
p
u
k
)
Y
t
r
s

1
dt
s
d
s
(
u
s
)
Psu
s

S
2
rs

1
,
P
t
s
<
1
(
1
6
s
6
k
t
s
)
|
r
|−
1
/p
(
r
s

1)!
a
sr
s

.)sa/p−1(sr21+srsr)2|sz|cdd(p/1−|r|)sa/p2z()z(f1−kp!)1−|r|(|r|