A Kawamata Viehweg Vanishing Theorem on compact Kahler manifolds

AKawamata-ViehwegVanishingTheorem
oncompactKa¨hlermanifolds

Jean-PierreDemailly
⋆
,ThomasPeternell
⋆⋆
⋆
Universite´deGrenobleI,BP74
⋆⋆
Universita¨tBayreuth
InstitutFourier,UMR5582duCNRSMathematischesInstitut
38402Saint-Martind’He`res,FranceD-95440Bayreuth,Deutschland

Abstract.
WeproveaKawamata-Viehwegvanishingtheoremonanormalcompact
Ka¨hlerspace
X
:if
L
isaneflinebundlewith
L
2
6
=0,then
H
q
(
X,K
X
+
L
)=0
for
q
≥
dim
X
−
1.Asanapplicationwecompleteapartoftheabundancetheorem
forminimalKa¨hlerthreefolds:if
X
isaminimalKa¨hlerthreefold,thentheKodaira
dimension
κ
(
X
)isnonnegative.

§
0.Introduction
InthispaperweestablishthefollowingKawamata-Viehwegtypevanishingtheorem
onacompactKa¨hlermanifoldor,moregenerally,anormalcompactKa¨hlerspace.
0.1Theorem.
Let
X
beanormalcompactKa¨hlerspaceofdimension
n
and
L
anef
linebundleon
X
.Assumethat
L
2
6
=0
.Then
H
q
(
X,K
X
+
L
)=0
for
q
≥
n
−
1
.
Ingeneral,oneexpectsavanishing
H
q
(
X,K
X
+
L
)=0
for
q
≥
n
+1
−
ν
(
L
),where
ν
(
L
)isthenumericalKodairadimensionofthenefline
bundle
L
,i.e.
ν
(
L
)isthelargestinteger
ν
suchthat
L
ν
6
=0.
Ofcourse,when
X
isprojective,Theorem0.1iscontainedintheusualKawamata-
Viehwegvanishingtheorem,butthemethodsofproofinthealgebraiccaseclearlyfail
inthegeneralKa¨hlersetting.Insteadweproceedinthefollowingway.Clearlywemay
assumethat
X
issmoothandbySerreduality,onlythecohomologygroup
H
n
−
1
isof
interest.Takeasingularmetric
h
on
L
withpositivecurvaturecurrent
T
withlocal
weightfunction
h
.By[Si74,De93a]thereexistsadecomposition
T
=
λ
j
D
j
+
G,
(
D
)
Xwhere
λ
j
≥
1areirreducibledivisors,and
G
isapseudo-effectivecurrentsuchthat
G
|
D
i
ispseudo-effectiveforall
i
.Considerthemultiplieridealsheaf
I
(
h
).Weassociate
to
h
another,“upperregularized”multiplieridealsheaf
I
+
(
h
)bysetting
I
+
(
h
):=lim
I
(
h
1+
ε
)=lim
I
(1+
ε
)
ϕ.
ε
→
0
+
ε
→
0
+

2AKawamata-ViehwegVanishingTheoremoncompactKa¨hlermanifolds

Itisunknownwhether
I
(
h
)and
I
+
(
h
)actuallydiffer;inallknownexamplestheyare
equal.TheninSection2thefollowingvanishingtheoremisproved.
0.2Theorem.
Let
(
L,h
)
beaholomorphiclinebundleoveracompactKa¨hler
n
-fold
X
.Assumethat
L
isnefandhasnumericalKodairadimension
ν
(
L
)=
ν
>
0
,
i.e.
c
1
(
L
)
ν
6
=0
and
ν
ismaximal.Thenthemorphism
H
q
(
X,
O
(
K
X
+
L
)
⊗
I
+
(
h
))
−→
H
q
(
X,K
X
+
L
)
inducedbytheinclusion
I
+
(
h
)
⊂
O
X
vanishesfor
q>n
−
ν
.
ThestrategyoftheproofofTheorem0.2isbasedonadirectapplicationofthe
BochnertechniquewithspecialhermitianmetricsconstructedbymeansoftheCalabi-
Yautheorem.
Now,comingbacktotheprinciplesoftheproofofTheorem0.1,weintroducethe
divisor
X
D
=[
λ
j
]
D
j
.
ThenTheorem0.2yieldsthevanishingofthemapincohomology
H
n
−
1
(
X,
−
D
+
L
+
K
X
)
−→
H
n
−
1
(
X,L
+
K
X
)
.
Thuswearereducedtoshowthat
H
n
−
1
(
D,L
+
K
X
|
D
)=0,orduallythat
H
0
(
D,
−
L
+
D
|
D
)=0
.
Thisisnowdonebyadetailedanalysisofapotentialnon-zerosectionin
−
L
+
D
|
D
;
makinguseofthedecomposition(
D
)andofaHodgeindextypeinequality.
Thevanishingtheorem0.1ismostpowerfulwhen
X
isathreefold,andinthesecond
partofthepaperweapply0.1-orratheratechnicalgeneralization-toprovethe
followingabundancetheorem.
0.3Theorem.
Let
X
bea
Q
-GorensteinKa¨hlerthreefoldwithonlyterminalsingu-
larities,suchthat
K
X
isnef(aminimalKa¨hlerthreefoldforshort).Then
κ
(
X
)
≥
0
.
ThistheoremwasestablishedintheprojectivecasebyMiyaokaandin[Pe01]for
Ka¨hlerthreefolds,withtheimportantexceptionthat
X
isasimplethreefoldwhich
isnotKummer.Recallthat
X
issaidtobe
simple
ifthereisnopropercompact
subvarietythroughaverygeneralpointof
X
,andthat
X
issaidtobeKummerif
X
is
bimeromorphictoaquotientofatorus.Soourcontributionhereconsistsinshowing
thatsuchasimplethreefold
X
with
K
X
nefhasactually
κ
(
X
)=0.Needlesstosaythat
amongallKa¨hlerthreefoldsthesimplenon-Kummerones(whichconjecturallydonot
exist)aremostdifficulttodealwith,sincetheydonotcarrymuchglobalinformation
besidesthefactthat
π
1
isfiniteandthattheyhaveaholomorphic2-form.
Thefirstmainingredientinourapproachistheinequality
K
X

c
2
(
X
)
≥
0

§
1.Preliminaries3
foraminimalsimplyconnectedKa¨hlerthreefold
X
withalgebraicdimension
a
(
X
)=0.
PhilosophicallythisinequalitycomesfromEnoki’stheoremthatthetangentsheafof
X
is
K
X
-semi-stablewhen
K
2
X
6
=0resp.(
K
X
,ω
)-semi-stablewhen
K
2
X
=0;here
ω
isanyKa¨hlerformon
X
.Nowifthissemi-stabilitywithrespecttoadegenerate
polarizationwouldyieldaMiyaoka-Yauinequality,then
K
X

c
2
(
X
)
≥
0wouldfollow.
HoweverthistypeofMiyoka-Yauinequalitieswithrespecttodegeneratepolarizations
iscompleteyunknown.Intheprojectivecase,theinequalityfollowsfromMiyaoka’s
genericnefnesstheoremandisbasedonchar.
p
-methods.Insteadweapproximate
K
X
(incohomology)byKa¨hlerforms
ω
j
.If
T
X
isstill
ω
j
-semi-stableforsufficiently
large
j
,thenwecanapplytheusualMiyaoka-Yauinequalityandpasstothelimit
toobtain
K
X

c
2
(
X
)
≥
0.Otherwiseweexaminethemaximaldestabilizingsubsheaf
whichessentially(becauseof
a
(
X
)=0)isindependentofthepolarization.
Thesecondmainingredientistheboundedness
h
2
(
X,mK
X
)
≤
1.If
K
2
X
6
=0,thisis
ofcoursecontainedinTheorem0.1.If
K
2
X
=0,weprovethisboundednessunderthe
additionalassumptionthat
a
(
X
)=0andthat
π
1
(
X
)isfinite(otherwisebyaresult
ofCampana
X
isalreadyKummer).Themainpointisthatif
h
2
(
X,mK
X
)
≥
2,then
weobtain“many”non-splitextensions
0
−→
K
X
−→
E
−→
mK
X
−→
0
andweanalyzewhether
E
issemi-stableornot.Theassumptionon
π
1
isusedto
concludethatif
E
isprojectivelyflat,then
E
istrivialafterafinitee´talecover.
FromthesetwoingredientsTheorem0.3immediatelyfollowsbyapplyingRiemann-
Rochonadesingularizationof
X
.
TheonlyremainingproblemconcerningabundanceonKa¨hlerthreefoldsistoprove
thatasimpleKa¨hlerthreefoldwith
K
X
nefand
κ
(
X
)=0mustbeKummer.

§
1.Preliminaries
Westartwithafewpreliminarydefinitions.
1.1Definition.
Anormalcomplexspace
X
issaidtobeKa¨hlerifthereexistsa
Ka¨hlerform
ω
ontheregularpartof
X
suchthatthefollowingholds.Everysingular
point
x
∈
X
admitsanopenneighborhood
U
andaclosedembedding
U
⊂
V
intoan
openset
U
⊂
C
N
suchthatthereisaKa¨hlerform
η
on
V
with
η
|
U
=
ω
.
1.2Remark.
Let
X
beacompactKa¨hlerspaceandlet
f
:
X
ˆ
−→
X
beadesingu-
larizationbyasequenceofblow-u