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The pseudo effective cone of a compact Kahler manifold and

De
41 pages
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.

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  • divisors

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  • projective manifold

  • pseudo-effective cone


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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
R
(
X
):=
H
R
1
,
1
(
X
)

H
2
(
X,
Z
)
/
tors

Z
R
.
(iv)
Weset
K
NS
=
K

NS
R
(
X
)
,
E
NS
=
E

NS
R
(
X
)
.
Algebraicgeometerstendtorestrictthemselvestothealgebraicconesgeneratedby
ampledivisorsandeffectivedivisors,respectively.Using
L
2
estimatesfor

,onecan
showthefollowingexpectedrelationsbetweenthealgebraicandtranscendentalcones
(see[Dem90],[Dem92]).

4Thepseudo-effectiveconeofcompactKa¨hlermanifolds

1.2Proposition.
Inaprojectivemanifold
X
,
E
NS
istheclosureoftheconvexcone
generatedbyeffectivedivisors,and
K
NS
istheclosureoftheconegeneratedbynef
R
-divisors.
Byextension,wewillsaythat
K
istheconeof
nef
(1
,
1)-cohomologyclasses(even
thoughtheyarenotnecessarilyintegral).Wenowturnourselvestoconesincohomo-
logyofbidegree(
n

1
,n

1).

1.3Definition.
Let
X
beacompactKa¨hlermanifold.
(i)
Wedefine
N
tobethe
(
closed
)
convexconein
H
R
n

1
,n

1
(
X
)
generatedbyclasses
ofpositivecurrents
T
oftype
(
n

1
,n

1)(
i.e.,ofbidimension
(1
,
1))
.
(ii)
Wedefinethecone
M

H
n

1
,n

1
(
X
)
of
movableclasses
tobetheclosureof
Rtheconvexconegeneratedbyclassesofcurrentsoftheform
µ

(
ω
e
1

...

ω
e
n

1
)
where
µ
:
X
e

X
isanarbitrarymodification
(
onecouldjustrestrictoneselfto
compositionsofblow-upswithsmoothcenters
)
,andthe
ω
e
j
areKa¨hlerformson
X
e
.
Clearly
M

N
.
(iii)
Correspondingly,weintroducetheintersections
N
NS
=
N

N
1
(
X
)
,
M
NS
=
M

N
1
(
X
)
,
inthespaceofintegralbidimension
(1
,
1)
-classes
N
1
(
X
):=(
H
R
n

1
,n

1
(
X
)

H
2
n

2
(
X,
Z
)
/
tors)

Z
R
.
(iv)
If
X
isprojective,wedefine
NE(
X
)
tobetheconvexconegeneratedbyalleffective
curves.Clearly
NE(
X
)

N
NS
.
(v)
If
X
isprojective,wesaythat
C
isa
stronglymovablecurve
if
C
=
µ

(
A
e
1

...

A
e
n

1
)
forsuitableveryampledivisors
A
e
j
on
X
e
,where
µ
:
X
e

X
isamodification.
Welet
SME(
X
)
tobetheconvexconegeneratedbyallstronglymovable
(
effective
)
curves.Clearly
SME(
X
)

M
NS
.
(vi)
Wesaythat
C
isa
S
movablecurve
if
C
=
C
t
0
isamemberofananalyticfamily
(
C
t
)
t

S
suchthat
t

S
C
t
=
X
and,assuch,isareducedirreducible
1
-cycle.We
let
ME(
X
)
tobetheconvexconegeneratedbyallmovable
(
effective
)
curves.
Theupshotofthisdefinitionliesinthefollowingeasyobservation.
1.4Proposition.
Let
X
beacompactKa¨hlermanifold.ConsiderthePoincare´duality
pairing
Z
H
R
1
,
1
(
X
)
×
H
R
n

1
,n

1
(
X
)
−→
R
,
(
α,β
)
7−→
α

β.
X

§
1Positiveconesinthespacesofdivisorsandofcurves5

Thenthedualitypairingtakesnonnegativevalues
(i)
forallpairs
(
α,β
)

K
×
N
;
(ii)
forallpairs
(
α,β
)

E
×
M
.
(iii)
forallpairs
(
α,β
)
where
α

E
and
β
=[
C
t
]

ME(
X
)
istheclassofamovable
curve.
Proof
.(i)isobvious.Inordertoprove(ii),wemayassumethat
β
=
µ

(
ω
e
1

...

ω
e
n

1
)
forsomemodification
µ
:
X
e

X
,where
α
=
{
T
}
istheclassofapositive(1
,
1)-current
on
X
and
ω
e
j
areKa¨hlerformson
X
e
.Then
ZZZXXXα

β
=
T

µ

(
ω
e
1

...

ω
e
n

1
)=
µ

T

ω
e
1

...

ω
e
n

1
>
0
.
Here,wehaveusedthefactthataclosedpositive(1
,
1)-current
T
alwayshasapull-back
µ

T
,whichfollowsfromthefactthatif
T
=
i∂∂ϕ
locallyforsomeplurisubharmonic
functionin
X
,wecanset
µ

T
=
i∂∂
(
ϕ

µ
).For(iii),wesuppose
α
=
{
T
}
and
β
=
{
[
C
t
]
}
.Thenwetakeanopencovering(
U
j
)on
X
suchthat
T
=
i∂∂ϕ
j
with
suitableplurisubharmonicfunctions
ϕ
j
on
U
j
.Ifweselectasmoothpartitionofunity
θ
j
=1subordinateto(
U
j
),wethenget
PZZZXα

β
=
T
|
C
t
=
θ
j
i∂∂ϕ
j
|
C
t
>
0
.
XC
t
jC
t

U
j
Forthistomakesense,itshouldbenoticedthat
T
|
C
t
isawelldefinedclosedpositive
(1
,
1)-current(i.e.measure)on
C
t
foralmostevery
t

S
,inthesenseofLebesgue
measure.Thisistrueonlybecause(
C
t
)covers
X
,thus
ϕ
j
|
C
t
isnotidentically
−∞
for
almostevery
t

S
.Theequalityinthelastformulaisthenshownbyaregularization
argumentfor
T
,writing
T
=lim
T
k
with
T
k
=
α
+
i∂∂ψ
k
andadecreasingsequence
ofsmoothalmostplurisubharmonicpotentials
ψ
k

ψ
suchthattheLeviformshave
auniformlowerbound
i∂∂ψ
k
>


(suchasequenceexistsby[Dem92]).Then,
writing
α
=
i∂∂v
j
forsomesmoothpotential
v
j
on
U
j
,wehave
T
=
i∂∂ϕ
j
on
U
j
with
ϕ
j
=
v
j
+
ψ
,andthisisthedecreasinglimitofthesmoothapproximations
ϕ
j,k
=
v
j
+
ψ
k
on
U
j
.Hence
T
k
|
C
t

T
|
C
t
fortheweaktopologyofmeasureson
C
t
.
If
C
isaconvexconeinafinitedimensionalvectorspace
E
,wedenoteby
C

the
dualcone,i.e.thesetoflinearforms
u

E

whichtakenonnegativevaluesonall
elementsof
C
.BytheHahn-Banachtheorem,wealwayshave
C
∨∨
=
C
.
Proposition1.4leadstothenaturalquestionwhetherthecones(
K
,
N
)and(
E
,
M
)
aredualunderPoincare´duality.Thisquestionisaddressedinthenextsection.Before
doingso,weobservethatthealgebraicandtranscendentalconesof(
n

1
,n

1)
cohomologyclassesarerelatedbythefollowingequalities(similartowhatwealready
noticedfor(1
,
1)-classes,seeProp.1.2).
1.5Theorem.
Let
X
beaprojectivemanifold.Then
(i)NE(
X
)=
N
NS
.
(ii)SME(
X
)=ME(
X
)=
M
NS
.

6Thepseudo-effectiveconeofcompactKa¨hlermanifolds

Proof
.(i)Itisastandardresultofalgebraicgeometry(seee.g.[Har70]),thatthecone
ofeffectiveconeNE(
X
)isdualtothecone
K
NS
ofnefdivisors,hence
N
NS

NE(
X
)=
K

.
Ontheotherhand,(1.4)(i)impliesthat
N
NS

K

,sowemusthaveequalityand(i)
follows.
Similarly,(ii)requiresadualitystatementwhichwillbeestablishedonlyinthenext
sections,sowepostponetheproof.

§
2Mainresultsandconjectures
First,thealreadymentioneddualitybetweennefdivisorsandeffectivecurvesex-
tendstotheKa¨hlercaseandtotranscendentalclasses.Moreprecisely,[DPa04]gives
1,12.1Theorem
(Demailly-Pa˘un,2001)
.
If
X
isKa¨hler,thenthecones
K

H
R
(
X
)
n

1
,n

1
and
N

H
R
(
X
)
aredualbyPoincare´duality,and
N
istheclosedconvexcone
generatedbyclasses
[
Y
]

ω
p

1
where
Y

X
rangesover
p
-dimensionalanalytic
subsets,
p
=1
,
2
,...,n
,and
ω
rangesoverKa¨hlerforms.
Proof
.Indeed,Prop.1.4showsthatthedualcone
K

contains
N
whichitselfcontains
thecone
N

ofallclassesoftheform
{
[
Y
]

ω
p

1
}
.Themainresultof[DPa04]conversely
showsthatthedualof(
N

)

isequalto
K
,sowemusthave
K

=
N

=
N
.

Themainnewresultofthispaperisthefollowingcharacterizationofpseudo-
effectiveclasses(inwhichthe“onlyif”partalreadyfollowsfrom1.4(iii)).
2.2Theorem.
If
X
isprojective,thenaclass
α

NS
R
(
X
)
ispseudo-effectiveif
(
and
onlyif
)
itisinthedualconeofthecone
SME(
X
)
ofstronglymovablecurves.
Inotherwords,alinebundle
L
ispseudo-effectiveif(andonlyif)
L
·
C
>
0for
all
movablecurves
,i.e.,
L
·
C
>
0foreveryverygenericcurve
C
(notcontained
inacountableunionofalgebraicsubvarieties).Infact,bydefinitionofSME(
X
),
itisenoughtoconsideronlythosecurves
C
whichareimagesofgenericcomplete
intersectionofveryampledivisorsonsomevariety
X
e
,underamodification
µ
:
X
e

X
.
Byastandardblowing-upargument,italsofollowsthatalinebundle
L
onanormal
Moishezonvarietyispseudo-effectiveifandonlyif
L
·
C

0foreverymovablecurve
C
.
TheKa¨hleranalogueshouldbe:
2.3Conjecture.
ForanarbitrarycompactKa¨hlermanifold
X
,thecones
E
and
M
aredual.
Therelationbetweenthevariousconesofmovablecurvesandcurrentsin(1.5)is
nowaratherdirectconsequenceofTheorem2.2.Infact,usingideashintedin[DPS96],
wecansayalittlebitmore.Givenanirreduciblecurve
C

X
,weconsideritsnormal
“bundle”
N
C
=Hom(
I
/
I
2
,
O
C
),where
I
istheidealsheafof
C
.If
C
isageneral

§
2Mainresultsandconjectures7

memberofacoveringfamily(
C
t
),then
N
C
isnef.Now[DPS96]saysthatthedual
coneofthepseudo-effectiveconeof
X
containstheclosedconespannedbycurveswith
nefnormalbundle,whichinturncontainstheconeofmovablecurves.Inthiswaywe
:teg2.4Theorem.
Let
X
beaprojectivemanifold.Thenthefollowingconescoincide.
(i)
thecone
M
NS
=
M

N
1
(
X
);
(ii)
theclosedcone
SME(
X
)
ofstronglymovablecurves
;
(iii)
theclosedcone
ME(
X
)
ofmovablecurves
;
(iv)
theclosedcone
ME
nef
(
X
)
ofcurveswithnefnormalbundle.
Proof
.Wehavealreadyseenthat
SME(
X
)

ME(
X
)

ME
nef
(
X
)

(
E
NS
)

dnaSME(
X
)

ME(
X
)

M
NS

(
E
NS
)

by1.4(iii).NowTheorem2.2implies(
M
NS
)

=SME(
X
),and2.4follows.
2.5Corollary.
Let
X
beaprojectivemanifoldand
L
alinebundleon
X
.
(i)
L
ispseudo-effectiveifandonlyif
L
·
C

0
forallcurves
C
withnefnormalsheaf
.NC(ii)
If
L
isbig,then
L
·
C>
0
forallcurves
C
withnefnormalsheaf
N
C
.
2.5(i)strenghtensresultsfrom[PSS99].Itishowevernotyetclearwhether
M
NS
=
M

N
1
(
X
)isequaltotheclosedconeofcurveswith
ample
normalbundle(although
wecertainlyexpectthistobetrue).
ThemostimportantspecialcaseofTheorem2.2is
2.6Theorem.
If
X
isaprojectivemanifoldandisnotuniruled,then
K
X
ispseudo-
effective,i.e.
K
X

E
NS
.
Proof
.ThisismerelyarestatementofCorollary0.3,whichwasprovedintheintro-
duction(asaconsequenceoftheresultsof[MM86]).
Theorem2.6canbegeneralizedasfollows.
2.7Theorem.
Let
X
beaprojectivemanifold
(
oranormalprojectivevariety
)
.Let
F

T
X
beacoherentsubsheaf.If
det
F

isnotpseudo-effective,then
X
isuniruled.
Inotherwords,if
X
isnotuniruledand
Ω
1
X

G
isgenericallysurjective,then
det
G
ispseudo-effective.
Proof
.Infact,sincedet
F

isnotpseudo-effective,thereexistsby(2.2)acovering
family(
C
t
)suchthat
c
1
(
F
)
·
C
t
>
0.Hence
X
isuniruledby[Miy87],[SB92].
2.8.Remark.
(1)In[CP09]Theorem2.7isgeneralizedtosubsheaves
F

T
X

m
.

8Thepseudo-effectiveconeofcompactKa¨hlermanifolds

(2)Supposein2.7thatonly
κ
(det
F

)=
−∞
.Is
X
stilluniruled?Whatcanbesaid
if
c
1
(
F

)isontheboundaryofthepseudo-effectivecone?
Turningtovarietieswithpseudo-effectivecanonicalbundles,wehavethe
2.9Conjecture
(partofthe“abundanceconjecture”)
.
If
K
X
ispseudo-effective,
then
κ
(
X
)
>
0
.
Thisproblemsplitsintotwoparts:
(1)
If
K
X
ispseudo-effectivebutnotbig,i.e.ontheboundaryofthepseudo-effective
cone,thenthereexistsa(good)coveringfamilyofcurve
(
C
t
)
suchthat
K
X
·
C
t
=0
.
(2)
If
K
X
ispseudo-effectiveandthereexistsagoodcoveringfamily
(
C
t
)
ofcurveswith
K
X
·
C
t
=0
,
then
κ
(
X
)

0
.
Inthelastsectionwewillprove(2)indimension4,andevenpartsofitinany
dimension.

§
3Zariskidecompositionandmovableintersections
Let
X
becompactKa¨hlerandlet
α

E

beinthe
interior
ofthepseudo–effective
cone.Inanalogywiththealgebraiccontextsuchaclass
α
iscalled“big”,anditcan
thenberepresentedbya
Ka¨hlercurrent
T
,i.e.aclosedpositive(1
,
1)-current
T
such
that
T
>
δω
forsomesmoothhermitianmetric
ω
andaconstant
δ

1.
3.1Theorem
(Demailly[Dem92],[Bou02b,3.1.24]
.
If
T
isaKa¨hlercurrent,thenone
canwrite
T
=lim
T
m
forasequenceofKa¨hlercurrents
T
m
whichhavelogarithmic
1poleswithcoefficientsin
m
Z
,i.e.therearemodifications
µ
m
:
X
m

X
suchthat
µ
⋆m
T
m
=[
E
m
]+
β
m
where
E
m
isaneffective
Q
-divisoron
X
m
withcoefficientsin
m
1
Z
(
the“fixedpart”
)
and
β
m
isaclosedsemi-positiveform
(
the“movablepart”
)
.
Proof
.Sincethisresulthasalreadybeenstudiedextensively,wejustrecallthemain
idea.Locallywecanwrite
T
=
i∂∂ϕ
forsomestrictlyplurisubharmonicpotential
ϕ
.
ByaBergmankerneltrickandtheOhsawa-Takegoshi
L
2
extensiontheorem,weget
localapproximations
1Xϕ
=lim
ϕ
m

m
(
z
)=log
|
g
ℓ,m
(
z
)
|
2
m2ℓwhere(
g
ℓ,m
)isaHilbertbasisofthespaceofholomorphicfunctionswhichare
L
2
with
respecttotheweight
e

2

.ThisHilbertbasisisalsoafamilyoflocalgeneratorsof
thegloballydefinedmultiplieridealsheaf
I
(
mT
)=
I
(

).Then
µ
m
:
X
m

X
is
obtainedbyblowing-upthisidealsheaf,sothat
µ
⋆m
I
(
mT
)=
O
(

mE
m
)
.
1Weshouldnoticethatbyapproximating
T

m
ω
insteadof
T
,wecanreplace
β
m
by
β
m
+
1
m
µ

ω
whichisabigclasson
X
m
;byplayingwiththemultiplicitiesofthe

§
3Zariskidecompositionandmovableintersections9

componentsoftheexceptionaldivisor,wecouldevenachievethat
β
m
isaKa¨hlerclass
on
X
m
,butthiswillnotbeneededhere.
Themorefamiliaralgebraicanaloguewouldbetotake
α
=
c
1
(
L
)withabigline
bundle
L
andtoblow-upthebaselocusof
|
mL
|
,
m

1,togeta
Q
-divisordecompo-
sition
⋆µ
m
L

E
m
+
D
m
,E
m
effective
,D
m
free
.
Suchablow-upisusuallyreferredtoasa“logresolution”ofthelinearsystem
|
mL
|
,
andwesaythat
E
m
+
D
m
isanapproximateZariskidecompositionof
L
.Wewillalso
usethisterminologyforKa¨hlercurrentswithlogarithmicpoles.
3.2Definition.
Wedefinethe
volume
,or
movableself-intersection
ofabigclass
α

E

tobe
Z
Vol(
α
)=sup
β
n
>
0
α∈TXewherethesupremumistakenoverallKa¨hlercurrents
T

α
withlogarithmicpoles,
and
µ

T
=[
E
]+
β
withrespecttosomemodification
µ
:
X
e

X
.
ByFujita[Fuj94]andDemailly-Ein-Lazarsfeld[DEL00],if
L
isabiglinebundle,
wehave
!n0nVol(
c
1
(
L
))=
m
l

im
+

D
m
=
m
l

im
+

n
h
(
X,mL
)
,
mandintheseterms,wegetthefollowingstatement.
3.3Proposition.
Let
L
beabiglinebundleontheprojectivemanifold
X
.Let
ǫ>
0
.
Thenthereexistsamodification
µ
:
X
ǫ

X
andadecomposition
µ

(
L
)=
E
+
β
with
E
aneffective
Q
-divisorand
β
abigandnef
Q
-divisorsuchthat
Vol(
L
)

ε
6
Vol(
β
)
6
Vol(
L
)
.

ItisveryusefultoobservethatthesupremuminDefinition3.2canactuallybe
computedbyacollectionofcurrentswhosesingularitiessatisfyafilteringproperty.
Namely,if
T
1
=
α
+
i∂∂ϕ
1
and
T
2
=
α
+
i∂∂ϕ
2
aretwoKa¨hlercurrentswithlogarithmic
polesintheclassof
α
,then
(3
.
4)
T
=
α
+
i∂∂ϕ,ϕ
=max(
ϕ
1

2
)
isagainaKa¨hlercurrentwithweakersingularitiesthan
T
1
and
T
2
.Onecoulddefine
aswell
(3
.
4

)
T
=
α
+
i∂∂ϕ,ϕ
=1log(
e
2

1
+
e
2

2
)
,
m2where
m
=lcm(
m
1
,m
2
)isthelowestcommonmultipleofthedenominatorsoccuring
in
T
1
,
T
2
.Now,takeasimultaneouslog-resolution
µ
m
:
X
m

X
forwhichthe
singularitiesof
T
1
and
T
2
areresolvedas
Q
-divisors
E
1
and
E
2
.Thenclearlythe
associateddivisorinthe
R
decomposition
µ
⋆m
T
=[
E
]+
β
isgivenby
E
=min(
E
1
,E
2
).
Bydoingso,thevolume
X
m
β
n
getsincreased,asweshallseeintheproofofTheorem
3.5below.

10Thepseudo-effectiveconeofcompactKa¨hlermanifolds

3.5Theorem
(Boucksom[Bou02b])
.
Let
X
beacompactKa¨hlermanifold.Wedenote
hereby
H
>
k,
0
k
(
X
)
theconeofcohomologyclassesoftype
(
k,k
)
whichhavenon-negative
intersectionwithallclosedsemi-positivesmoothformsofbidegree
(
n

k,n

k
)
.
(i)
Foreachinteger
k
=1
,
2
,...,n
,thereexistsacanonical“movableintersection
product”
k,kE
×···×
E

H
>
0
(
X
)
,
(
α
1
,...,α
k
)
7→h
α
1
·
α
2
···
α
k

1
·
α
k
i
suchthat
Vol(
α
)=
h
α
n
i
whenever
α
isabigclass.
(ii)
Theproductisincreasing,homogeneousofdegree
1
andsuperadditiveineachar-
gument,i.e.
′′′′′′
h
α
1
···
(
α
j
+
α
j
)
···
α
k
i
>
h
α
1
···
α
j
···
α
k
i
+
h
α
1
···
α
j
···
α
k
i
.
Itcoincideswiththeordinaryintersectionproductwhenthe
α
j

K
arenefclasses.
(iii)
ThemovableintersectionproductsatisfiestheTeissier-Hovanskiiinequalities
h
α
1
·
α
2
···
α
n
i
>
(
h
α
1
n
i
)
1
/n
...
(
h
α
nn
i
)
1
/n
(
with
h
α
jn
i
=Vol(
α
j
))
.
(iv)
For
k
=1
,theabove“product”reducestoa
(
nonlinear
)
projectionoperator
E

E
1

→h
α
i
ontoacertainconvexsubcone
E
1
of
E
suchthat
K

E
1

E
.Moreover,thereis
a“divisorialZariskidecomposition”
α
=
{
N
(
α
)
}
+
h
α
i
where
N
(
α
)
isauniquelydefinedeffectivedivisorwhichiscalledthe“negative
divisorialpart”of
α
.Themap
α
7→
N
(
α
)
ishomogeneousandsubadditive,and
N
(
α
)=0
ifandonlyif
α

E
1
.
(v)
Thecomponentsof
N
(
α
)
alwaysconsistofdivisorswhosecohomologyclassesare
linearlyindependent,especially
N
(
α
)
hasatmost
ρ
=rank
Z
NS(
X
)
components.
Proof
.Weessentiallyrepeattheargumentsdeveloppedin[Bou02b],withsomesimpli-
ficationsarisingfromthefactthat
X
issupposedtobeKa¨hlerfromthestart.
(i)Firstassumethatallclasses
α
j
arebig,i.e.
α
j

E

.Fixasmoothclosed(
n

k,n

k
)
semi-positive
form
u
on
X
.WeselectKa¨hlercurrents
T
j

α
j
withlogarithmicpoles,
andasimultaneouslog-resolution
µ
:
X
e

X
suchthat
µ

T
j
=[
E
j
]+
β
j
.
Weconsiderthedirectimagecurrent
µ

(
β
1

...

β
k
)(whichisaclosedpositivecurrent
ofbidegree(
k,k
)on
X
)andthecorrespondingintegrals

1

...

β
k

µ

u
>
0
.
Xe

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