The pseudo effective cone of a compact Kahler manifold and

Thepseudo-effectiveconeofa
compactKa¨hlermanifoldand
varietiesofnegativeKodairadimension

Se´bastienBoucksom
1
MihaiPa˘un
3
1
Universite´deParisVII
InstitutdeMathe´matiques
175rueduChevaleret
75013Paris,France
3
Universite´deStrasbourg
De´partementdeMathe´matiques
67084Strasbourg,France

Jean-PierreDemailly
2
ThomasPeternell
4
2
Universite´deGrenobleI,BP74
InstitutFourier
UMR5582duCNRS
38402Saint-Martind’He`res,France
4
Universita¨tBayreuth
MathematischesInstitut
D-95440Bayreuth,Deutschland

Abstract.
Weprovethataholomorphiclinebundleonaprojectivemanifoldispseudo-effective
ifandonlyifitsdegreeonanymemberofacoveringfamilyofcurvesisnon-negative.Thisisa
consequenceofadualitystatementbetweentheconeofpseudo-effectivedivisorsandtheconeof
“movablecurves”,whichisobtainedfromageneraltheoryofmovableintersectionsandapproximate
Zariskidecompositionforclosedpositive(1
,
1)-currents.Asacorollary,aprojectivemanifoldhasa
pseudo-effectivecanonicalbundleifandonlyifitisisnotuniruled.Wealsoprovethata4-foldwith
acanonicalbundlewhichispseudo-effectiveandofnumericalclasszeroinrestrictiontocurvesofa
goodcoveringfamily,hasnonnegativeKodairadimension.
(1)

§
0Introduction
Oneofthemajoropenproblemsintheclassificationtheoryofprojectiveorcompact
Ka¨hlermanifoldsisthefollowinggeometricdescriptionofvarietiesofnegativeKodaira
dimension.
0.1Conjecture.
Aprojective
(
orcompactKa¨hler
)
manifold
X
hasKodairadimension
κ
(
X
)=
−∞
ifandonlyif
X
isuniruled.
Onedirectionistrivial,namely
X
uniruledimplies
κ
(
X
)=
−∞
.Also,thecon-
jectureisknowntobetrueforprojectivethreefoldsby[Mo88]andfornon-algebraic
Ka¨hlerthreefoldsby[Pe01],withthepossibleexceptionofsimplethreefolds(recallthat
avarietyissaidtobesimpleifthereisnocompactpositivedimensionalsubvariety
throughaverygeneralpointof
X
).Inthecaseofprojectivemanifolds,theproblem
canbesplitintomoretractableparts:
(1)
TheoriginalversionofthepresentpaperhasbeenbeenforwardedverylongagotoarXiv(inMay
2004),andhasbeenrevisedseveraltimessincethen.Ithadalsobeensubmittedtoajournalin2004,
butthenthesubmissionwascancelledaftertherefereeexpressedconcernsaboutcertainpartsofthe
paper,astheywerewrittenatthattime.Althoughtheresultsofsections1-5havebeenreproduced
severaltimes,e.g.inlecturenotesofthesecondnamedauthororinRobLazarsfeld’sbook[Laz04],a
completeversionneverappearedinarefereedjournal.WethanktheJournalofAlgebraicGeometry
forsuggestingtorepairthisomission.

2Thepseudo-effectiveconeofcompactKa¨hlermanifolds

A.
Ifthecanonicalbundle
K
X
isnotpseudo-effective,i.e.notcontainedintheclosure
oftheconespannedbyclassesofeffectivedivisors,then
X
isuniruled.
B.
If
K
X
ispseudo-effective,then
κ
(
X
)
≥
0
.
IntheKa¨hlercase,thestatementsshouldbeessentiallythesame,exceptthateffective
divisorshavetobereplacedbyclosedpositive(1
,
1)-currents.

PartBagainsplitsintotwopieces:
B1.
If
K
X
ispseudo-effectivebutnotbig,i.e.ontheboundaryofthepseudo-effective
cone,thenthereexistsacoveringfamilyofcurves
(
C
t
)
suchthat
K
X
·
C
t
=0
.
B2.
If
K
X
ispseudo-effectiveandthereexistsacoveringfamily
(
C
t
)
ofcurveswith
K
X
·
C
t
=0
,
then
κ
(
X
)
≥
0
.
Inthispaperwegiveapositiveanswerto(A)forprojectivemanifoldsofany
dimension,anddealwith(B2),mostlyindimension4.Part(A)followsinfactfroma
muchmoregeneralfactwhichdescribesthegeometryofthepseudo-effectivecone.
0.2Theorem.
Alinebundle
L
onaprojectivemanifold
X
ispseudo-effectiveifand
onlyif
L
·
C
≥
0
forallirreduciblecurves
C
whichmoveinafamilycovering
X
.
Inotherwords,thedualconetothepseudo-effectiveconeistheclosureofthecone
of“movable”curves.Thisshouldbecomparedwiththedualitybetweenthenefcone
andtheconeofeffectivecurves.
0.3Corollary
(Solutionof(A))
.
Let
X
beaprojectivemanifold.If
K
X
isnotpseudo-
effective,then
X
iscoveredbyrationalcurves.
Infact,if
K
X
isnotpseudo-effective,thenby(0.2)thereexistsacoveringfamily
(
C
t
)ofcurveswith
K
X
·
C
t
<
0,sothat(0.3)followsbyawell-knowncharacteristic
p
argumentofMiyaokaandMori[MM86](thesocalledbend-and-breaklemmaessentially
amountstodeformthe
C
t
sothattheybreakintopieces,oneofwhichisarational
curve).
IntheKa¨hlercasebothasuitableanalogueto(0.2)andthetheoremofMiyaoka-
Moriareunknown.Itshouldalsobementionedthatthedualitystatementfollowing
(0.2)isactually(0.2)for
R
-divisors.Theproofisbasedonauseof“approximate
Zariskidecompositions”andanestimateforanintersectionnumberrelatedtothis
decomposition.Amajortoolisthevolumeofan
R
-divisorwhichdistinguishesbig
divisors(positivevolume)fromdivisorsontheboundaryofthepseudo-effectivecone
(volume0).
Concerning(B2)weneedtodistinguishbetweencoveringbutnotconnectingfam-
ilies(
C
t
)ononeside,andconnectingfamiliesontheotherside.Thislatterterm
“connecting”meansthatanytwopointcanbejoinedbyachainofcurves
C
t
.
For
technicalpurposesitishoweverbettertoconsider
stronglyconnecting
families,i.e.,
anytwosufficientlygeneralpointscanbejoinedbyachainofirreducible
C
t
′
s.If
X
has
agoodminimalmodelviacontractionsandflips,then
X
clearlyadmitsacoveringnon-
connectingorastronglyconnectingfamily(
C
t
)suchthat
K
X
·
C
t
=0;moreoverif
X
simplyhasagoodminimalmodel,thenatleastafterblowingupthiswillbethecase.
Letussaythat(
C
t
)isagoodcoveringfamily,if(
C
t
)isacovering,non-connecting

§
1Positiveconesinthespacesofdivisorsandofcurves3

familyorastronglyconnectingfamily.ThenourremarksjustifythedivisionofProb-
lem(B)intothetwoparts(B1)and(B2),possiblybyreplacing“coveringfamilies”by
“goodcoveringfamilies”.
0.4Theorem.
Let
X
beasmoothprojective
4
-fold.Assumethat
K
X
ispseudo-
effectiveandthereisagoodcoveringfamily
(
C
t
)
ofcurvessuchthat
K
X
·
C
t
=0
.
Then
κ
(
X
)
≥
0
.
Oneimportantingredientoftheproofof(0.4)isthequotientdefinedbythefamily
(
C
t
).Thereasonfortherestrictiontodimension4isthatweuse
C
n,m
andthe
logminimalmodelprogramonthebaseofthequotientofthefamily(
C
t
)
.
Inone
circumstancehoweverwehaveageneralresult:
0.5Theorem.
Let
X
beaprojectivemanifoldand
(
C
t
)
astronglyconnectingfamily
ofcurves.Let
L
beapseudo-effective
R
−
divisorwith
L
·
C
t
=0
.
Thenthenumerical
dimension
ν
(
L
)=0
.
If
L
isCartier,then
L
isnumericallyequivalenttoalinebundle
L
′
with
κ
(
L
′
)=0
.
If
L
=
K
X
,theninconnectionwith[CP09]weobtain
κ
(
X
)=0
.
Inordertoobtain
theanswertoProblem(B1)(e.g.indimension4),wewouldstillneedtoprovethat
K
X
iseffectiveif
K
X
ispositiveonallgoodcoveringfamiliesofcurves.Infact,inthat
case,
K
X
shouldbebig,i.e.ofmaximalKodairadimension.

§
1Positiveconesinthespacesofdivisorsandofcurves
Inthissectionweintroducetherelevantcones,bothintheprojectiveandKa¨hler
contexts–inthelattercase,divisorsandcurvesshouldsimplybereplacedbypositive
currentsofbidimension(
n
−
1
,n
−
1)and(1
,
1),respectively.Weimplicitlyusethatall
(DeRham,resp.Dolbeault)cohomologygroupsunderconsiderationcanbecomputed
intermsofsmoothformsorcurrents,sinceinbothcaseswegetresolutionsofthesame
sheafoflocallyconstantfunctions(resp.ofholomorphicsections).
1.1Definition.
Let
X
beacompactKa¨hlermanifold.
(i)
TheKa¨hlerconeistheset
K
⊂
H
R
1
,
1
(
X
)
ofclasses
{
ω
}
ofKa¨hlerforms
(
thisis
anopenconvexcone
)
.
(ii)
Thepseudo-effectiveconeistheset
E
⊂
H
R
1
,
1
(
X
)
ofclasses
{
T
}
ofclosedpositive
currentsoftype
(1
,
1)(
thisisaclosedconvexcone
)
.Clearly
E
⊃
K
.
(iii)
TheNeron-Severispaceisdefinedby
NS