The pseudo effective cone of a compact Kahler manifold and

Sebastien Boucksom1

Description

The pseudo-effective cone of a compact Kahler manifold and varieties of negative Kodaira dimension Sebastien Boucksom1 Jean-Pierre Demailly2 Mihai Pa˘un3 Thomas Peternell4 1Universite de Paris VII 2Universite de Grenoble I, BP 74 Institut de Mathematiques Institut Fourier 175 rue du Chevaleret UMR 5582 du CNRS 75013 Paris, France 38402 Saint-Martin d'Heres, France 3Universite de Strasbourg 4Universitat Bayreuth Departement de Mathematiques Mathematisches Institut 67084 Strasbourg, France D-95440 Bayreuth, Deutschland Abstract. We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of “movable curves”, which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1, 1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non negative Kodaira dimension.(1) 0 Introduction One of the major open problems in the classification theory of projective or compact Kahler manifolds is the following geometric description of varieties of negative Kodaira dimension.

• also been submitted


• cone


• divisors


• statement following


• divisors a˜j


• projective manifold


• pseudo-effective cone

Sujets

• Divisor

Informations

• Publié par: pefav
• Nombre de lectures: 11
• Langue: English

Extrait

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