Analytic techniques in algebraic geometry

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Niveau: Supérieur, Licence, Bac+2
Analytic techniques in algebraic geometry Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier Lectures given at the School held in Mahdia, Tunisia, July 14 – July 31, 2004 Analyse Complexe et Geometrie The purpose of this series of lectures is to explain some advanced techniques of Complex Analysis which can be applied to obtain fundamental results in algebraic geometry: vanishing of cohomology groups, embedding theorems, description of the geometric structure of projective algebraic varieties. Contents 0. Preliminary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Holomorphic Vector Bundles, Connections and Curvature . . . . . . . . . . . . . . . . . . . . 4 2. Bochner Technique and Vanishing Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 3. L2 Estimates and Existence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  • kahler

  • every compact

  • c∞ vector bundle

  • compact complex

  • kahler metric

  • complex measures

  • hermitian vector

  • closed positive

  • projective space

  • distribution ∑ ?j?ktjk


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Analytictechniquesinalgebraicgeometry

Jean-PierreDemailly
Universite´deGrenobleI,InstitutFourier

LecturesgivenattheSchoolheldinMahdia,Tunisia,July14–July31,2004
AnalyseComplexeetGe´ome´trie

Thepurposeofthisseriesoflecturesistoexplainsomeadvancedtechniquesof
ComplexAnalysiswhichcanbeappliedtoobtainfundamentalresultsinalgebraic
geometry:vanishingofcohomologygroups,embeddingtheorems,descriptionofthe
geometricstructureofprojectivealgebraicvarieties.

Contents
0.PreliminaryMaterial
........................................................
1
1.HolomorphicVectorBundles,ConnectionsandCurvature
....................
4
2.BochnerTechniqueandVanishingTheorems
.................................
8
23.
L
EstimatesandExistenceTheorems
......................................
14
4.MultiplierIdealSheaves
....................................................
19
5.NefandPseudoeffectiveCones
.............................................
28
6.NumericalCharacterizationoftheKa¨hlerCone
.............................
30
7.ConesofCurves
............................................................
38
8.DualityResults
............................................................
40
9.Approximationofpshfunctionsbylogarithmsofholomorphicfunctions
.....
42
10.ZariskiDecompositionandMovableIntersections
..........................
46
11.TheOrthogonalityEstimate
...............................................
52
12.ProofoftheMainDualityTheorem
.......................................
54
13.References
................................................................
56

0.Preliminarymaterial

Let
X
beacomplexmanifoldand
n
=dim
C
X
.Thebundleofdifferentialforms
oftypr(
p,q
)isdenotedby
Λ
p,q
T
X

.Weareespeciallyinterestedin
closedpositive
currents
oftype(
p,p
)
2T
=
i
p
T
JK
(
z
)
dz
J

dz
J
,dz
J
=
dz
j
1

...

dz
j
p
,dT
=0
.
X|
J
|
=
|
K
|
=
p
Recallthatacurrentisadifferentialformwithdistributioncoefficients,andthat
sucha(
p,p
)currentissaidtobepositive(inthe“mediumpositivity”sense)ifthe

2J.-P.Demailly,Analytictechniquesinalgebraicgeometry
distribution
λ
J
λ
K
T
JK
isapositivemeasureforallcomplexnumbers
λ
J
.The
Pcoefficients
T
JK
arethencomplexmeasures.Importantexamplesofclosedpositive
(
p,p
)-currentsarecurrentsofintegrationoveranalyticcyclesofcodimension
p
:
Z
=
c
j
Z
j
,
[
Z
]=
c
j
[
Z
j
]
XXwherethecurrent[
Z
j
]isdefinedbydualityas
Zh
[
Z
j
]
,u
i
=
u
|
Z
j
Zjforevery(
n

p,n

p
)testform
u
on
X
.Anotherimportantexampleofpositive
(1
,
1)-currentistheHessianform
T
=
i∂∂ϕ
ofaplurisubharmonicfunctiononan
openset


X
.A
Ka¨hlermetric
on
X
isapositivedefinitehermitian(1
,
1)-form
ω
(
z
)=

jk
(
z
)
dz
j

dz
k
suchthat

=0
,
X1

j,k

n
withsmoothcoefficients.Toeveryclosedreal(1
,
1)-form(orcurrent)
α
isassociated
itsDeRhamcohomologyclass
{
α
}∈
H
1
,
1
(
X,
R
)

H
2
(
X,
R
)
.
Wedenotehereby
H
k
(
X,
C
)(resp.
H
k
(
X,
R
))thecomplex(real)DeRhamcoho-
mologygroupofdegree
k
,andby
H
p,q
(
X,
C
)thesubspaceofclasseswhichcanbe
representedasclosed(
p,q
)-forms,
p
+
q
=
k
.
Wewillrelyonthenontrivialfactthatallcohomologygroupsinvolved(De
Rham,Dolbeault,
...
)canbedefinedeitherintermsofsmoothformsorintermsof
currents.Infact,ifweconsidertheassociatedcomplexesofsheaves,formsandcur-
rentsbothprovideacyclicresolutionsofthesamesheaf(locallyconstantfunctions,
resp.holomorphicsections),hencedefinethesamecohomologygroups.
Inthesequel,wearemostlyinterestedinthegeometryof
compactcomplex
manifolds
.Thecompactnessassumptionbringsmanyinterestingfeaturessuchas
finitessresultsforthecohomologyorthetopology,Stokestheorem,intersection
formulasofBezouttype,etc.A
projectivealgebraicmanifold
isaclosedsubmanifold
X
ofsomecomplexprojectivespace
P
N
=
P
C
N
definedbyafinitecollectionof
homogeneouspolynomialequations
P
j
(
z
0
,z
1
,...,z
N
)=0
,
1

j

k
(insuchawaythat
X
isnonsingular).AnimportanttheoremduetoChowstates
thateverycomplexanalyticsubmanifoldof
P
N
isinfactautomaticallyalgebraic,i.e.
definedasabovebyafinitecollectionofpolynomials.Wewillprovethisinsection4.
However,wewillsometimesneedtostudylocalsituations,andinthatcaseitis
alsousefultoconsiderthecaseof(pseudoconvex)opensetsin
C
n
.

(0.1)Definition.
a)
Ahermitianmanifoldisapair
(
X,ω
)
where
ω
isa
C

positivedefinite
(1
,
1)
-
formon
X
.

0.Preliminarymaterial3
b)
X
issaidtobeaKa¨hlermanifoldif
X
carriesatleastoneKa¨hlermetric
ω
.

Since
ω
isreal,theconditions

=0,
d

ω
=0,
d
′′
ω
=0areallequivalent.In
localcoordinatesweseethat
d

ω
=0ifandonlyif
∂ω
jk
=
∂ω
lk
,
1

j,k,l

n.
∂z
l
∂z
j
Asimplecomputationgives
nω=det(
ω
jk
)i
dz
j

dz
j
=2
n
det(
ω
jk
)
dx
1

dy
1
∧∧
dx
n

dy
n
,
^n
!
1

j

n
where
z
n
=
x
n
+i
y
n
.Thereforethe(
n,n
)-form
(0
.
2)
dV
=1
ω
n
!nnthen
X
ω
=
n
!Vol
ω
(
X
)
>
0.Thissimpleremarkalreadyimpliesthatcompact
ispos
R
itiveandcoincideswiththehermitianvolumeelementof
X
.If
X
iscompact,
Ka¨hlermanifoldsmustsatisfysomerestrictivetopologicalconditions:
(0.3)Consequence.
a)
If
(
X,ω
)
iscompactKa¨hlerandif
{
ω
}
denotesthecohomologyclassof
ω
in
H
2
(
X,
R
)
,then
{
ω
}
n
6
=0
.
b)
If
X
iscompactKa¨hler,then
H
2
k
(
X,
R
)
6
=0
for
0

k

n
.Infact,
{
ω
}
k
isa
nonzeroclassin
H
2
k
(
X,
R
)
.

(0.4)Example.
Thecomplexprojectivespace
P
n
isKa¨hler.AnaturalKa¨hlermetric
ω
FS
on
P
n
,calledthe
Fubini-Studymetric
,isdefinedby
ip

ω
FS
=
d

d
′′
log
|
ζ
0
|
2
+
|
ζ
1
|
2
+

+
|
ζ
n
|
2
π2where
ζ
0

1
,...,ζ
n
arecoordinatesof
C
n
+1
andwhere
p
:
C
n
+1
\{
0
}→
P
n
isthe
projection.Let
z
=(
ζ
1

0
,...,ζ
n

0
)benonhomogeneouscoordinateson
C
n

P
n
.
Thenacalculationshowsthat
ω
FS
=i
d

d
′′
log(1+
|
z
|
2
)

F
n
S
=1
.
Zπ2nPItisalsowell-knownfromtopologythat
{
ω
FS
}∈
H
2
(
P
n
,
Z
)isageneratorofthe
cohomologyalgebra
H

(
P
n
,
Z
).

(0.5)Example.
A
complextorus
isaquotient
X
=
C
n

byalattice
Γ
ofrank2
n
.
i
ω
jk
dz
j

dz
k
withconstantcoefficientsdefinesaKa¨hlermetricon
X
.
Th
P
en
X
isacompactcomplexmanifold.Anypositivedefinitehermitianform
ω
=
(0.6)Example.
Every(complex)submanifold
Y
ofaKa¨hlermanifold(
X,β
)isKa¨hler
withmetric
ω
=
β

Y
.Especially,allcomplexsubmanifoldsof
X

P
N
areKa¨hler

4J.-P.Demailly,Analytictechniquesinalgebraicgeometry
withKa¨hlermetric
ω
=
ω
FS

X
.Since
ω
FS
isin
H
2
(
P
,
Z
),therestriction
ω
isan
integralclassin
H
2
(
X,
Z
).Conversely,theKodairaembeddingtheorem[Kod54]
statesthateverycompactKa¨hlermanifold
X
possessingaKa¨hlermetric
ω
withan
integralcohomologyclass
{
ω
}∈
H
2
(
X,
Z
)canbeembeddedinprojectivespaceas
aprojectivealgebraicsubvariety.Wewillprovethisinsection4.

(0.7)Example.
Considerthecomplexsurface
X
=(
C
2
\{
0
}
)

where
Γ
=
{
λ
n
;
n

Z
}
,
λ<
1,actsasagroupofhomotheties.Since
C
2
\{
0
}
isdiffeomorphicto
R

+
×
S
3
,wehave
X

S
1
×
S
3
.Therefore
H
2
(
X,
R
)=0by
Ku¨nneth’sformula,andproperty0.3b)showsthat
X
isnotKa¨hler.Moregenerally,
onecantake
Γ
tobeaninfinitecyclicgroupgeneratedbyaholomorphiccontraction
of
C
2
,oftheform
z
1
7−→
λ
1
z
1
,
resp.
z
1
7−→
λz
1
,

pz
2
λ
2
z
2
z
2
λz
2
+
z
1
where
λ,λ
1

2
arecomplexnumberssuchthat0
<
|
λ
1
|≤|
λ
2
|
<
1,0
<
|
λ
|
<
1,and
p
apositiveinteger.ThesenonKa¨hlersurfacesarecalled
Hopfsurfaces
.

1.HermitianVectorBundles,ConnectionsandCurvature

Thegoalofthissectionistorecallthemostbasicdefinitionsofhemitiandifferential
geometryrelatedtotheconceptsofconnection,curvatureandfirstChernclassofa
linebundle.
Let
F
beacomplexvectorbundleofrank
r
overasmoothdifferentiablemani-
fold
M
.A
connection
D
on
F
isalineardifferentialoperatoroforder1
D
:
C

(
M,Λ
q
T
M⋆

F
)

C

(
M,Λ
q
+1
T
M⋆

F
)
suchthat
(1
.
1)
D
(
f

u
)=
df

u
+(

1)
deg
f
f

Du
forallforms
f

C

(
M,Λ
p
T
M⋆
),
u

C

(
X,Λ
q
T
M⋆

F
).Onanopenset


M
where
F
admitsatrivialization
θ
:
F
|



×
C
r
,aconnection
D
canbewritten
Du

θ
du
+
Γ

u
where
Γ

C

(


1
T
M⋆

Hom(
C
r
,
C
r
))isanarbitrarymatrixof1-formsand
d
actscomponentwise.Itistheneasytocheckthat
D
2
u

θ
(

+
Γ

Γ
)

u
on

.
Since
D
2
isagloballydefinedoperator,thereisaglobal2-form
(1
.
2)
Θ
(
D
)

C

(
M,Λ
2
T
M⋆

Hom(
F,F
))

1.HermitianVectorBundles,ConnectionsandCurvature5
suchthat
D
2
u
=
Θ
(
D
)

u
foreveryform
u
withvaluesin
F
.
Assumenowthat
F
isendowedwitha
C

hermitianmetricalongthefibersand
thattheisomorphism
F
|

×
C
r
isgivenbya
C

frame(
e
λ
).Wethenhavea
canonicalsesquilinearpairing
(1
.
3)
C

(
M,Λ
p
T
M⋆

F
)
×
C

(
M,Λ
q
T
M⋆

F
)
−→
C

(
M,Λ
p
+
q
T
M⋆

C
)
(
u,v
)
7−→{
u,v
}
givenby
{
u,v
}
=
XXX,λTheconnection
D
issaidtobe
hermitian
ifitsatisfiestheadditionalproperty
d
{
u,v
}
=
{
Du,v
}
+(

1)
deg
u
{
u,Dv
}
.
Assumingthat(
e
λ
)isorthonormal,oneeasilychecksthat
D
ishermitianifandonly
if
Γ

=

Γ
.Inthiscase
Θ
(
D
)

=

Θ
(
D
),thus
i
Θ
(
D
)

C

(
M,Λ
2
T
M⋆

Herm(
F,F
))
.

uλ∧vheλ,ei,u=uλ⊗eλ,v=v⊗e.(1.4)Specialcase.
Forabundle
F
ofrank1,theconnectionform
Γ
ofahermitian
connection
D
canbeseenasa1-formwithpurelyimaginarycoefficients
Γ
=i
A
(
A
real).Thenwehave
Θ
(
D
)=

=i
dA
.Inparticulari
Θ
(
F
)isaclosed2-form.The
firstChernclass
of
F
isdefinedtobethecohomologyclass
i2c
1
(
F
)
R
=
Θ
(
D
)

H
DR
(
M,
R
)
.
onπ2Thecohomologyclassisactuallyindependentoftheconnection,sinceanyother
connection
D
1
differsbyaglobal1-form,
D
1
u
=
Du
+
B

u
,sothat
Θ
(
D
1
)=
Θ
(
D
)+
dB
.Itiswell-knownthat
c
1
(
F
)
R
istheimagein
H
2
(
M,
R
)ofanintegral
class
c
1
(
F
)

H
2
(
M,
Z
);byusingtheexponentialexactsequence
0

Z
→E→E


0
,
c
1
(
F
)canbedefinedinCˇechcohomologytheoryastheimagebythecoboundary
map
H
1
(
M,
E

)

H
2
(
M,
Z
)ofthecocycle
{
g
jk
}∈
H
1
(
M,
E

)defining
F
;seee.g.
[GrH78]fordetails.

Wenowconcentrateourselvesonthecomplexanalyticcase.If
M
=
X
isa
complexmanifold
X
,everyconnection
D
onacomplex
C

vectorbundle
F
canbe
splittedinauniquewayasasumofa(1
,
0)andofa(0
,
1)-connection,
D
=
D

+
D
′′
.
Inalocaltrivialization
θ
givenbya
C

frame,onecanwrite
(1
.
5

)
D

u

θ
d

u
+
Γ


u,
(1
.
5
′′
)
D
′′
u

θ
d
′′
u
+
Γ
′′

u,

6J.-P.Demailly,Analytictechniquesinalgebraicgeometry
with
Γ
=
Γ

+
Γ
′′
.Theconnectionishermitianifandonlyif
Γ

=

(
Γ
′′
)

inanyor-
thonormalframe.Thusthereexistsauniquehermitianconnection
D
corresponding
toaprescribed(0
,
1)part
D
′′
.
Assumenowthatthebundle
F
itselfhasa
holomorphic
structure.Theunique
hermitianconnectionforwhich
D
′′
isthe
d
′′
operatordefinedin
§
1iscalledthe
Chernconnection
of
F
.Inalocalholomorphicframe(
e
λ
)of
E
|

,themetricisgiven
bythehermitianmatrix
H
=(
h
λ
),
h
λ
=
h
e
λ
,e

i
.Wehave
{
u,v
}
=
h
λ
u
λ

v

=
u


Hv,
X,λwhere
u

isthetransposedmatrixof
u
,andeasycomputationsyield
d
{
u,v
}
=(
du
)


Hv
+(

1)
deg
u
u


(
dH

v
+
Hdv
)
=
du
+
H

1
d

H

u


Hv
+(

1)
deg
u
u


(
dv
+
H

1
d

H

v
)
†usingthefactthat
dH
=
d

H
+
d

H
and
H
=
H
.ThereforetheChernconnection
D
coincideswiththehermitianconnectiondefinedby
1−Du

θ
du
+
Hd

H

u,
()6.1(D


θ
d

+
H

1
d

H


=
H

1
d

(
H

)
,D
′′
=
d
′′
.
Itisclearfromthisrelationsthat
D

2
=
D
′′
2
=0.Consequently
D
2
isgivenby
to
D
2
=
D

D
′′
+
D
′′
D

,andthecurvaturetensor
Θ
(
D
)isoftype(1
,
1).Since
d

d
′′
+
d
′′
d

=0,weget
(
D

D
′′
+
D
′′
D

)
u

H

1
d

H

d
′′
u
+
d
′′
(
H

1
d

H

u
)
θ=
d
′′
(
H

1
d

H
)

u.

(1.7)Proposition.
TheCherncurvaturetensor
Θ
(
F
):=
Θ
(
D
)
issuchthat
i
Θ
(
F
)

C

(
X,Λ
1
,
1
T
X⋆

Herm(
F,F
))
.
If
θ
:
E

×
C
r
isaholomorphictrivializationandif
H
isthehermitianmatrix
representingthemetricalongthefibersof
F

,then
1−i
Θ
(
F
)

θ
i
d
′′
(
Hd

H
)on

.

Let(
z
1
,...,z
n
)beholomorphiccoordinateson
X
andlet(
e
λ
)
1

λ

r
beanor-
thonormalframeof
F
.Writing
⋆i
Θ
(
F
)=
c
jkλ
dz
j

dz
k

e
λ

e

,
X1

j,k

n,
1

λ,

r
wecanidentifythecurvaturetensortoahermitianform
X(1
.
8)
Θ
e
(
F
)(
ξ

v
)=
c
jkλ
ξ
j
ξ
k
v
λ
v

1

j,k

n,
1

λ,

r

1.HermitianVectorBundles,ConnectionsandCurvature7

on
T
X

F
.Thisleadsinanaturalwaytopositivityconcepts,followingdefinitions
introducedbyKodaira[Kod53],Nakano[Nak55]andGriffiths[Gri69].

(1.9)Definition.
Thehermitianvectorbundle
F
issaidtobe
τ

∂/∂z
j

e
λ

T
X

F
.
a)
p
P
ositiveinthesenseofNakanoif
Θ
e
(
F
)(
τ
)
>
0
forallnonzerotensors
τ
=
b)
positiveinthesenseofGriffithsif
Θ
e
(
F
)(
ξ

v
)
>
0
forallnonzerodecomposable
tensors
ξ

v

T
X

F
;
Correspondingsemipositivityconceptsaredefinedbyrelaxingthestrictinequalities.

(1.10)Specialcaseofrank1bundles.
Assumethat
F
isalinebundle.Thehermitian
matrix
H
=(
h
11
)associatedtoatrivialization
θ
:
F

×
C
issimplyapositive
functionwhichwefindconvenienttodenoteby
e

ϕ
,
ϕ

C

(

,
R
).Inthiscasethe
curvatureform
Θ
(
F
)canbeidentifiedtothe(1
,
1)-form2
d

d
′′
ϕ
,and
i
Θ
(
F
)=i
d

d
′′
ϕ
=
dd
c
ϕ
ππ2isareal(1
,
1)-form.Hence
F
issemipositive(ineithertheNakanoorGriffithssense)
ifandonlyif
ϕ
ispsh,resp.positiveifandonlyif
ϕ
is
strictlypsh
.Inthissetting,
theLelong-Poincare´equationcanbegeneralizedasfollows:let
σ

H
0
(
X,F
)bea
nonzeroholomorphicsection.Then
(1
.
11)
dd
c
log
k
σ
k
=[
Z
σ
]

i
Θ
(
F
)
.
π2Formula(1.11)isimmediateifwewrite
k
σ
k
=
|
θ
(
σ
)
|
e

ϕ
andifweapply(1.20)to
theholomorphicfunction
f
=
θ
(
σ
).Asweshallseelater,itisveryimportantfor
theapplicationstoconsideralsosingularhermitianmetrics.

(1.12)Definition.
Asingular
(
hermitian
)
metriconalinebundle
F
isametricwhich
≃isgiveninanytrivialization
θ
:
F

−→

×
C
by
k
ξ
k
=
|
θ
(
ξ
)
|
e

ϕ
(
x
)
,x



F
x
where
ϕ

L
l1oc
(

)
isanarbitraryfunction,calledtheweightofthemetricwith
respecttothetrivialization
θ
.

If
θ

:
F


−→


×
C
isanothertrivialization,
ϕ

theassociatedweightand
g
∈O

(


)thetransitionfunction,then
θ

(
ξ
)=
g
(
x
)
θ
(
ξ
)for
ξ

F
x
,andso
ϕ

=
ϕ
+log
|
g
|
on


.Thecurvatureformof
F
isthengivenformallybythe
closed(1
,
1)-current
2i
π
Θ
(
F
)=
dd
c
ϕ
on

;ourassumption
ϕ

L
l1oc
(

)guarantees
that
Θ
(
F
)existsinthesenseofdistributiontheory.Asinthesmoothcase,
2i
π
Θ
(
F
)
isgloballydefinedon
X
andindependentofthechoiceoftrivializations,anditsDe
RhamcohomologyclassistheimageofthefirstChernclass
c
1
(
F
)

H
2
(
X,
Z
)in
H
2
DR
(
X,
R
).Beforegoingfurther,wediscusstwobasicexamples.

8J.-P.Demailly,Analytictechniquesinalgebraicgeometry
(1.13)Example.
Let
D
=
α
j
D
j
beadivisorwithcoefficients
α
j

Z
andlet
PF
=
O
(
D
)betheassociatedinvertiblesheafofmeromorphicfunctions
u
suchthat
div(
u
)+
D

0;thecorrespondinglinebundlecanbeequippedwiththesingular
set


X
then
θ
(
u
)=
ug

j
definesatrivializationof
O
(
D
)over

,thusour
metricdefinedby
k
u
k
=
|
Q
u
|
.If
g
j
isageneratoroftheidealof
D
j
onanopen
singularmetricisassociatedtotheweight
ϕ
=
α
j
log
|
g
j
|
.BytheLelong-Poincare´
Pequation,wefind

O
(
D
)=
dd
c
ϕ
=[
D
]
,
π2where[
D
]=
α
j
[
D
j
]denotesthecurrentofintegrationover
D
.

P(1.14)Example.
Assumethat
σ
1
,...,σ
N
arenonzeroholomorphicsectionsof
F
.
Thenwecandefineanatural(possiblysingular)hermitianmetricon
F

by
2Xk
ξ

k
2
=

ξ


j
(
x
)

for
ξ


F
x⋆
.
n≤j≤1Thedualmetricon
F
isgivenby
2k
ξ
k
2
=
|
θ
(
ξ
)
|
|
θ
(
σ
1
(
x
))
|
2
+
...
+
|
θ
(
σ
N
(
x
))
|
2
ϕ
(
x
)=log
1

j

N
|
θ
(
σ
j
(
x
))
|
21
/
2
.Inthiscase
ϕ
isapshfunction,thusi
Θ
(
F
)is
withrespec

tt
P
oanytrivialization

θ
.Theassociatedweightfunctionisthusgivenby
aclosedpositivecurrent.Letusdenoteby
Σ
thelinearsystemdefinedby
σ
1
,...,σ
N
1−andby
B
Σ
=
σ
j
(0)itsbaselocus.Wehaveameromorphicmap

Σ
:
X
r
B
Σ

P
N

1
,x
7→
(
σ
1
(
x
):
σ
2
(
x
):
...
:
σ
N
(
x
))
.
iThen
2
π
Θ
(
F
)isequaltothepull-backover
X
r
B
Σ
oftheFubini-Studymetric
ω
FS
=
2i
π
log(
|
z
1
|
2
+
...
+
|
z
N
|
2
)of
P
N

1
by
Φ
Σ
.

(1.15)Ampleandveryamplelinebundles.
Aholomorphiclinebundle
F
overa
compactcomplexmanifold
X
issaidtobe
a)
veryampleifthemap
Φ
|
F
|
:
X

P
N

1
associatedtothecompletelinearsystem
|
F
|
=
P
(
H
0
(
X,F
))
isaregularembedding
(
bythiswemeaninparticularthat
thebaselocusisempty,i.e.
B
|
F
|
=

)
.
b)
ampleifsomemultiple
mF
,
m>
0
,isveryample.

HereweuseanadditivenotationforPic(
X
)=
H
1
(
X,
O

),hencethesymbol
mF
denotesthelinebundle
F

m
.ByExample1.14,everyamplelinebundle
F
hasa
smoothhermitianmetricwithpositivedefinitecurvatureform;indeed,ifthelinear
system
|
mF
|
givesanembeddinginprojectivespace,thenwegetasmoothhermitian
metricon
F

m
,andthe
m
-throotyieldsametricon
F
suchthat
2i
π
Θ
(
F
)=
m
1
Φ
|
⋆mF
|
ω
FS
.Conversely,theKodairaembeddingtheorem[Kod54]tellsusthatevery
positivelinebundle
F
isample(see(4.14)forastraightforwardanalyticproofof
theKodairaembeddingtheorem).