Analytic techniques in algebraic geometry
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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
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
)=
iω
jk
(
z
)
dz
j
∧
dz
k
suchthat
dω
=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
dω
=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
,ζ
1
,...,ζ
n
arecoordinatesof
C
n
+1
andwhere
p
:
C
n
+1
\{
0
}→
P
n
isthe
projection.Let
z
=(
ζ
1
/ζ
0
,...,ζ
n
/ζ
0
)benonhomogeneouscoordinateson
C
n
⊂
P
n
.
Thenacalculationshowsthat
ω
FS
=i
d
′
d
′′
log(1+
|
z
|
2
)
,ω
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
/Γ
byalattice
Γ
ofrank2
n
.
i
ω
jk
dz
j
∧
dz
k
withconstantcoefficientsdefinesaKa¨hlermetricon
X
.
Th
P
en
X
