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Applications of the theory of L2 estimates and positive currents in algebraic geometry


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Applications of the theory of L2 estimates and positive currents in algebraic geometry Jean-Pierre Demailly Ecole d'ete de Mathematiques de Grenoble, June-July 2007 These notes are derived in part from the lectures “Multiplier ideal sheaves and analytic methods in algebraic geometry” given at the ICTP School held in Tri- este, Italy, April 24 – May 12, 2000 “Vanishing theorems and effective results in Algebraic Geometry”, as well as from a more detailed online book on Algebraic Geometry available at the author's home page Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Preliminary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Lelong Numbers and Intersection Theory . . . . . . . . . . . . . . . . . . . . . .

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  • been brought

  • vanishing theorems

  • l2 estimates

  • skoda's estimates

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  • condi- tion can

  • local holomorphic



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2Applications of the theory of L estimates
and positive currents in algebraic geometry
Jean-Pierre Demailly
´Ecole d’´et´e de Math´ematiques de Grenoble, June-July 2007
These notes are derived in part from the lectures “Multiplier ideal sheaves and
analytic methods in algebraic geometry” given at the ICTP School held in Tri-
este, Italy, April 24 – May 12, 2000 “Vanishing theorems and effective results in
Algebraic Geometry”, as well as from a more detailed online book on Algebraic
Geometry available at the author’s home page
0. Introduction ................................................................ 1
1. Preliminary Material ........................................................5
2. Lelong Numbers and Intersection Theory ...................................13
3. Holomorphic Vector Bundles, Connections and Curvature ...................22
4. K¨ahler identities and Hodge Theory ........................................27
5. Bochner Technique and Vanishing Theorems ................................37
26. L Estimates and Existence Theorems ......................................39
7. Numerically Effective and Pseudoeffective Line Bundles .....................51
8. A Simple Algebraic Approach to Fujita’s Conjecture ........................58
9. Holomorphic Morse Inequalities ............................................67
10. Effective Version of Matsusaka’s Big Theorem .............................69
11. Positivity Concepts for Vector Bundles ....................................74
212. Skoda’s L Estimates for Surjective Bundle Morphisms ....................81
213. The Ohsawa-Takegoshi L Extension Theorem .............................90
14. Invariance of Plurigenera of Varieties of General Type ....................106
15. Subadditivity of Multiplier Ideal Sheaves and Zariski Decomposition ......113
16. Hard Lefschetz Theorem with Multiplier Ideal Sheaves ....................118
17. Nef and Pseudoeffective Cones in K¨ahler Geometry .......................130
18. Numerical Characterization of the K¨ahler Cone ...........................133
19. Cones of Curves ......................................................... 142
20. Duality Results ..........................................................144
21. Approximation of psh functions by logarithms of holomorphic functions ...146
22. Zariski Decomposition and Movable Intersections .........................150
23. The Orthogonality Estimate ............................................. 156
24. Proof of the Main Duality Theorem ......................................158
References ...................................................................1592 J.-P. Demailly, Complex analytic techniques in algebraic geometry
Transcendental methods of algebraic geometry have been extensively studied since
a long time, starting with the work of Abel, Jacobi and Riemann in the nineteenth
century. More recently, in the period 1940-1970, the work of Hodge, Hirzebruch,
Kodaira,Atiyahrevealed deeper relationsbetween complexanalysis,topology,PDE
theory and algebraic geometry. In the last twenty years, gauge theory has proved to
be a very efficient tool for the study of many important questions: moduli spaces,
stable sheaves, non abelian Hodge theory, low dimensional topology...
Our main purpose here is to describe a few analytic tools which are useful to
study questions such as linear series and vanishing theorems for algebraic vector
bundles. One of the early successes of analytic methods in this context is Kodaira’s
use of the Bochner technique in relation with the theory of harmonic forms, during
the decade 1950-60. The idea is to represent cohomology classes by harmonic forms
and to prove vanishing theorems by means of suitable a priori curvature estimates.
TheprototypeofsuchresultsistheAkizuki-Kodaira-Nakanotheorem(1954):ifX is
a nonsingular projective algebraic variety and L is a holomorphic line bundle on X
pqwithpositivecurvature, thenH (X,Ω ⊗L)= 0 forp+q> dimX (throughout theX
p p ⋆ n ⋆paperwesetΩ =Λ T andK =Λ T ,n= dimX,viewingtheseobjectseitherXX X X
asholomorphicbundles oraslocallyfreeO -modules).Itisonlymuch laterthatanX
algebraic proof of this result has been proposed by Deligne-Illusie, via characteristic
p methods, in 1986.
A refinement of the Bochner technique used by Kodaira led, about ten years
2later, to fundamental L estimates due to H¨ormander [H¨or65], concerning solu-
tions of the Cauchy-Riemann operator. Not only vanishing theorems are proved,
but more precise information of a quantitative nature is obtained about solutions
2of∂-equations. The best way of expressing these L estimates is to use a geometric
settingfirst considered by Andreotti-Vesentini [AV65].More explicitly,suppose that
we have a holomorphic line bundle L equipped with a hermitian metric of weight
−2ϕe ,whereϕ isa (locallydefined) plurisubharmonic function; then explicitboundsR
2 2 −2ϕon the L norm |f| e of solutions is obtained. The result is still more usefulX
if the plurisubharmonic weight ϕ is allowed to have singularities. Following Nadel
[Nad89], we define the multiplier ideal sheaf I(ϕ) to be the sheaf of germs of holo-
2 −2ϕmorphicfunctionsf suchthat|f| e islocallysummable.ThenI(ϕ)isacoherent
qalgebraic sheaf overX andH (X,K ⊗L⊗I(ϕ)) = 0 for allq≥ 1 if the curvatureX
ofL is positive (as a current). This important result can be seen as a generalization
ofthe Kawamata-Viehweg vanishing theorem ([Kaw82],[Vie82]),which isone of the
cornerstones of higher dimensional algebraic geometry (especially of Mori’s minimal
model program). In the dictionary between analytic geometry and algebraic geome-
try, the idealI(ϕ) plays a very important role, since it directly converts an analytic
object into an algebraic one, and, simultaneously, takes care of singularities in a
very efficient way.
Another analytic tool used to deal with singularities is the theory of positive
currents introduced by Lelong [Lel57]. Currents can be seen as generalizations of
algebraic cycles, and many classical results of intersection theory still apply to cur-
rents. The concept of Lelong number of a current is the analytic analogue of the
concept of multiplicity of a germ of algebraic variety. Intersections of cycles corre-
spond to wedge products of currents (whenever these products are defined).
Besides the Kodaira-Nakano vanishing theorem, one of the most basic “effective0. Introduction 3
result” expected to hold in algebraic geometry is expressed in the following conjec-
ture of Fujita [Fuj87]: if L is an ample (i.e. positive) line bundle on a projective
n-dimensional algebraic varietyX, thenK +(n+1)L is generated by sections andX
K +(n+2)L is very ample. In the last decade, a lot of effort has been brought forX
the solution of this conjecture – and it seems indeed that a solution might finally
emerge in the first years or the third millenium – hopefully during this Summer
School! The first major results are the proof of the Fujita conjecture in the case of
long time), and the numerical criterion for the very ampleness of 2K +L given inX
[Dem93b], obtained by means of analytic techniques and Monge-Amp`ere equations
with isolated singularities. Alternative algebraic techniques were developed slightly
later by Koll´ar [Kol92],Ein-Lazarsfeld [EL93],Fujita [Fuj93], Siu [Siu95, 96],Kawa-
mata [Kaw97] and Helmke [Hel97]. We will explain here Siu’s method because it
is technically the simplest method; one of the results obtained by this method is3n+1the following effective result: 2K +mL is very ample for m ≥ 2 + . TheX n
basic idea is to apply the Kawamata-Viehweg vanishing theorem, and to combine
this with the Riemann-Roch formula in order to produce sections through a clever
inductionprocedure onthedimension ofthe baselociofthelinearsystemsinvolved.
Although Siu’s result is certainly not optimal, it is sufficient to obtain a nice
constructive proof of Matsusaka’s big theorem ([Siu93], [Dem96]). The result states
nthat there is an effective value m depending only on the intersection numbers L0
n−1andL ·K ,suchthatmLisveryampleform≥m .ThebasicideaistocombineX 0
results on thevery ampleness of 2K +mL togetherwiththe theoryof holomorphicX
′ ′ofmL−K form large. Again this step can be made algebraic (following sugges-X
tions by F. Catanese and R. Lazarsfeld), but the analytic formulation apparently
has a wider range of applicability.
2In the next sections, we pursue the study of L estimates, in relation with the
P 2thedivisionproblemf = g h canbesolvedholomorphicallywithverypreciseLj j
2 −pestimates, provided that the L norm of|f||g| is finite for some sufficiently large
exponent p (p > n = dimX is enough). Skoda’s estimates have a nice interpreta-
tion in terms of local algebra, and they lead to precise qualitative and quantitative
estimates in connection with the B´ezout problem. Another very important result
2is the L extension theorem by Ohsawa-Takegoshi [OT87, Ohs88], which has also
2been generalized later by Manivel [Man93]. The main statement is that every L
section f of a suitably positive line bundle defined on a subavariety Y ⊂ X can
2 ˜be extended to a L section f defined over the whole of X. The positivity condi-
tion can be understood in terms of the canonical sheaf and normal bundle to the
subvariety. The extension theorem turns outto have an incredible amount of impor-
tant consequences: among them, let us mention for instance Siu’s theorem [Siu74]
on the analyticity of Lelong numbers, the basic approximation theorem of closed
positive (1,1)-currents by divisors, the subadditivity propertyI(ϕ+ψ)⊂I(ϕ)I(ψ)
of multiplier ideals [DEL00], the restriction formula I(ϕ ) ⊂ I(ϕ) , .... A suit-|Y |Y
able combination of these results can be used to reprove Fujita’s result [Fuj94] on
approximate Zariski decomposition, as we show in section 15.
In the last section 16, we show how subadditivity can be used to derive an ap-
be approximated by a sequence of (almost) plurisubharmonic functions which are4 J.-P. Demailly, Complex analytic techniques in algebraic geometry
smooth outside an analytic set, and which define the same multiplier ideal sheaves.
From this, we derive a generalized version of the hard Lefschetz theorem for coho-
mology with values in a pseudo-effective line bundle; namely, the Lefschetz map is
surjective when the cohomology groups are twisted by the relevant multiplier ideal
for algebraic geometers. Although efficient algebraic techniques exist, our feeling is
that the analytic techniques are very flexible and offer a large variety of guidelines
for more algebraicquestions (including applicationstonumber theorywhich are not
discussed here). We made a special effort to use as little prerequisites and to be as
self-contained as possible; hence the rather long preliminary sections dealing with
basic facts of complex differential geometry. I am indebted to L. Ein, J. Koll´ar, R.
Lazarsfeld, Th. Peternell, M. Schneider and Y.T. Siu for many discussions on these
subjects over a period of time of two decades or more. These discussions certainly
had a great influence on my research work and therefore on the contents of the
present notes. We refer to Rob Lazarsfeld’s book [Laz04] for an extremely detailed
algebraic exposition of the subject of positivity in algebraic geometry.1. Preliminary Material 5
1.A. Dolbeault Cohomology and Sheaf Cohomology
p,q ⋆LetX be aC-analytic manifold of dimensionn. We denote byΛ T the bundle ofX
differentialformsofbidegree (p,q)onX,i.e.,differentialformswhichcanbewritten
as X
u = u dz ∧dz .I,J I J
Here (z ,...,z ) denote arbitrary local holomorphic coordinates, I = (i ,...,i ),1 n 1 p
J = (j ,...,j ) are multiindices (increasing sequences of integers in the range1 q
[1,...,n], of lengths|I|=p,|J|=q), and
dz :=dz ∧...∧dz , dz :=dz ∧...∧dz .I i i J j j1 p 1 q
p,q ∞Let E be the sheaf of germs of complex valued differential (p,q)-forms with C
′ ′′coefficients. Recall that the exterior derivative d splits as d=d +d where
X ∂uI,J′du = dz ∧dz ∧dz ,k I J
X ∂uI,J′′d u = dz ∧dz ∧dzk I J
areof type(p+1,q),(p,q+1)respectively.The well-knownDolbeault-Grothendieck
′′ ′′lemma asserts that any d -closed form of type (p,q) with q > 0 is locally d -exact
′′(this is the analogue for d of the usual Poincar´e lemma for d, see e.g. [H¨or66]). In
p,• ′′other words, the complex of sheaves (E ,d ) is exact in degree q > 0; in degree
p′′q = 0, Kerd is the sheaf Ω of germs of holomorphic forms of degree p on X.X
More generally, if F is a holomorphic vector bundle of rank r over X, there is
′′ ∞ p,q ⋆a natural d operator acting on the space C (X,Λ T ⊗F) of smooth (p,q)-XP
forms with values in F; if s = s e is a (p,q)-form expressed in terms of aλ λ1≤λ≤r P′′ ′′local holomorphic frame ofF, we simply define d s:= d s ⊗e , observing thatλ λ
the holomorphic transition matrices involved in changes of holomorphic frames do
′′not affect the computation of d . It is then clear that the Dolbeault-Grothendieck
lemma stillholds forF-valued forms. For every integerp = 0,1,...,n, the Dolbeault
p,qCohomology groups H (X,F) are defined to be the cohomology groups of the
complex of global (p,q) forms (graded by q):

p,q q ∞ p,• ⋆(1.1) H (X,F)=H C (X,Λ T ⊗F) .X
•isomorphism theorem): let (L ,d) be a resolution of a sheaf A by acyclic sheaves,
•i.e. a complex of sheaves (L ,δ) such that there is an exact sequence of sheaves
0 qj δ δ0 1 q q+10−→A −→L −→L −→···−→L −→L −→··· ,
s qand H (X,L )= 0 for allq≥ 0 and s≥ 1. Then there is a functorial isomorphism

q • q(1.2) H Γ(X,L ) −→H (X,A).6 J.-P. Demailly, Complex analytic techniques in algebraic geometry
p,q ∞We apply this to the following situation: let E(F) be the sheaf of germs of C
p,q ⋆ p,• ′′sections of Λ T ⊗F. Then (E(F) ,d ) is a resolution of the locally free O -XX
p p,qmodule Ω ⊗O(F) (Dolbeault-Grothendieck lemma), and the sheaves E(F) areX
∞acyclic as modules over the soft sheaf of ringsC . Hence by (1.2) we get
(1.3) Dolbeault Isomorphism Theorem (1953). For every holomorphic vector bundle
F on X, there is a canonical isomorphism
pp,q qH (X,F)≃H (X,Ω ⊗O(F)). X
If X is projective algebraic and F is an algebraic vector bundle, Serre’s GAGA
pqtheorem[Ser56]showsthatthealgebraicsheafcohomologygroupH (X,Ω ⊗O(F))X
computed with algebraic sections over Zariski open sets is actually isomorphic to
the analytic cohomology group. These results are the most basic tools to attack
algebraic problems via analytic methods. Another important tool is the theory of
in the decades 1940-1960.
1.B. Plurisubharmonic Functions
Plurisubharmonic functions havebeen introduced independently byLelongand Oka
in the study of holomorphic convexity. We refer to [Lel67, 69] for more details.
n(1.4) Definition. A functionu:Ω−→ [−∞,+∞[ defined on an open subsetΩ⊂C
is said to be plurisubharmonic (psh for short) if
a) u is upper semicontinuous ;
nb) for every complex line L⊂C , u is subharmonic on Ω∩L, that is, for all↾Ω∩L
na ∈ Ω and ξ ∈C with |ξ| < d(a,∁Ω), the function u satisfies the mean value
inequality Z 2π1 iθu(a)≤ u(a+e ξ)dθ.
2π 0
The set of psh functions on Ω is denoted by Psh(Ω).
We list below the most basic properties of psh functions. They all follow easily
from the definition.
(1.5) Basic properties.
a) Every function u ∈ Psh(Ω) is subharmonic, namely it satisfies the mean value
inequality on euclidean balls or spheres:
u(a)≤ u(z)dλ(z)
n 2nπ r /n! B(a,r)
1foreverya∈Ω andr<d(a,∁Ω).Eitheru≡−∞oru∈L oneveryconnectedloc
component of Ω.
b) For any decreasing sequence of psh functions u ∈ Psh(Ω), the limitu = limuk k
is psh on Ω.1. Preliminary Material 7
c) Let u ∈ Psh(Ω) be such that u ≡ −∞on every connected component of Ω. If
∞(ρ ) is a family of smoothing kernels, then u⋆ρ is C and psh onε ε

Ω = x∈Ω;d(x,∁Ω)>ε ,ε
the family (u⋆ρ ) is increasing in ε and lim u⋆ρ =u.ε ε→0 ε
pd) Let u ,...,u ∈ Psh(Ω) and χ : R −→ R be a convex function such that1 p
χ(t ,...,t ) is increasing in each t . Then χ(u ,...,u ) is psh on Ω. In partic-1 p j 1 p
u u1 pular u +···+u , max{u ,...,u }, log(e +···+e ) are psh on Ω. 1 p 1 p
2(1.6)Lemma. A functionu∈C (Ω,R) is psh onΩ if and only if the hermitian form
P 2Hu(a)(ξ)= ∂ u/∂z ∂z (a)ξ ξ is semipositive at every point a∈Ω.j k j k1≤j,k≤n
Proof. This is an easy consequence of the following standard formula
Z Z Z2π 11 2 dtiθu(a+e ξ)dθ−u(a)= Hu(a+ζξ)(ξ)dλ(ζ),
2π π t0 0 |ζ|<t
where dλ is the Lebesgue measure on C. Lemma 1.6 is a strong evidence that
plurisubharmonicity is the natural complex analogue of linear convexity.
For non smooth functions, a similar characterization of plurisubharmonicity can
be obtained by means of a regularization process.
(1.7) Theorem. If u∈ Psh(Ω), u ≡−∞on every connected component of Ω, then
nfor all ξ∈C
2X ∂ u ′Hu(ξ)= ξ ξ ∈D (Ω)j k∂z ∂zj k
′is a positive measure. Conversely, if v ∈ D (Ω) is such that Hv(ξ) is a positive
nmeasure for every ξ ∈ C , there exists a unique function u ∈ Psh(Ω) which is
locally integrable on Ω and such that v is the distribution associated to u.
that u is a function on a complex n-dimensional manifold X. If Φ : X → Y is a
2 ′ ′′ ⋆ ′ ′′holomorphic mapping and if v∈C (Y,R), we have dd (v◦Φ) =Φ dd v, hence

′H(v◦Φ)(a,ξ)=Hv Φ(a),Φ(a).ξ .
In particularHu, viewed as a hermitian form onT , does not depend on the choiceX
ofcoordinates(z ,...,z ).Therefore, thenotionofpsh function makessense onany1 n
complex manifold. More generally, we have
(1.8) Proposition. If Φ : X −→ Y is a holomorphic map and v ∈ Psh(Y), then
v◦Φ∈ Psh(X).
(1.9) Example. It is a standard fact that log|z| is psh (i.e. subharmonic) onC. Thus
0log|f|∈ Psh(X) for every holomorphic function f ∈H (X,O ). More generallyX

α α1 qlog |f | +···+|f | ∈ Psh(X)1 q8 J.-P. Demailly, Complex analytic techniques in algebraic geometry
0for every f ∈H (X,O ) and α ≥ 0 (apply Property 1.5d with u =α log|f |).j X j j j j
We will be especially interested in the singularities obtained at points of the zero
variety f =...=f = 0, when the α are rational numbers. 1 q j
(1.10) Definition. A psh function u∈ Psh(X) will be said to have analytic singular-
ities if u can be written locally as
α 2 2u = log |f | +···+|f | +v,1 N
whereα∈R ,v is a locally bounded function and thef are holomorphic functions.+ j
If X is algebraic, we say that u has algebraic singularities if u can be written as
above on sufficiently small Zariski open sets, with α∈Q and f algebraic.+ j
We then introduce the ideal J = J(u/α) of germs of holomorphic functions h
u/αsuch that|h|≤Ce for some constant C, i.e.

|h|≤C |f |+···+|f | .1 N
This is a globally defined ideal sheaf onX, locally equal to the integral closureI of
the ideal sheaf I = (f ,...,f ), thus J is coherent on X. If (g ,...,g ′) are local1 N 1 N
generators ofJ, we still have
α 2 2u= log |g | +···+|g ′| +O(1).1 N
If X is projective algebraic and u has analytic singularities with α ∈ Q , then+
u automatically has algebraic singularities. From an algebraic point of view, the
singularities of u are in 1:1 correspondence with the “algebraic data” (J,α). Later
on, we will see another important method for associating an ideal sheaf to a psh
(1.11) Exercise. Show that the above definition of the integral closure of an ideal
I is equivalent to the following more algebraic definition: I consists of all germs h
satisfying an integral equation
d d−1 kh +a h +...+a h+a = 0, a ∈I .1 d−1 d k
of the blow-up of X along the (non necessarily reduced) zero variety V(I).
1.C. Positive Currents
The reader can consult [Fed69] for a more thorough treatment of current theory.
Let us first recall a few basic definitions. A current of degree q on an oriented
differentiable manifold M is simply a differential q-form Θ with distribution co-
′qefficients. The space of currents of degree q over M will be denoted by D (M).
Alternatively, a current of degree q can be seen as an element Θ in the dual space′′ p pD (M) := D (M) of the space D (M) of smooth differential forms of degreep
p = dimM−q with compact support; the duality pairing is given by1. Preliminary Material 9
p(1.12) hΘ,αi= Θ∧α, α∈D (M).
S of M :
(1.13) h[S],αi= α, degα =p = dim S.R
Then [S] is a current with measure coefficients, and Stokes’ formula shows that
q−1d[S] = (−1) [∂S], in particular d[S] = 0 if S has no boundary. Because of this
′example, the integer p is said to be the dimension of Θ when Θ ∈ D (M). Thep
current Θ is said to be closed if dΘ = 0.
On a complex manifoldX, we have similar notions of bidegree and bidimension;
as in the real case, we denote by
′p,q ′D (X)=D (X), n = dimX,n−p,n−q
to[Lel57],acurrentΘ ofbidimension(p,p)issaidtobe(weakly) positiveifforevery
choice of smooth (1,0)-formsα ,...,α on X the distribution1 p
(1.14) Θ∧iα ∧α ∧...∧iα ∧α is a positive measure.1 1 p p
(1.15) Exercise. If Θ is positive, show that the coefficients Θ of Θ are complexI,J
measures, and that, up to constants, they are dominated by the trace measure
X X1 i ip −p ′ ′′ 2σ =Θ∧ β = 2 Θ , β = dd |z| = dz ∧dz ,Θ I,I j j
p! 2 2
which is a positive measure.P
Hint. Observe that Θ is invariant by unitary changes of coordinates and thatI,I
p,p ⋆the (p,p)-forms iα ∧α ∧...∧iα ∧α generate Λ T as aC-vector space. 1 1 p p nC
A current Θ = i Θ dz ∧dz of bidegree (1,1) is easily seen to bejk j k1≤j,k≤n P
positive if and only if the complex measure λ λ Θ is a positive measure forj k jk
nevery n-tuple (λ ,...,λ )∈C .1 n
(1.16) Example. If u is a (not identically−∞) psh function on X, we can associate
with u a (closed) positive current Θ = i∂∂u of bidegree (1,1). Conversely, every
closed positivecurrent of bidegree (1,1)can be written under this form on any open
2 1subsetΩ⊂X such thatH (Ω,R)=H (Ω,O)= 0, e.g. on small coordinate ballsDR
(exercise to the reader).
It is not difficult to show that a product Θ ∧...∧Θ of positive currents of1 q
bidegree (1,1) is positive whenever the product is well defined (this is certainly the
case if all Θ but one at most are smooth; much finer conditions will be discussedj
in Section 2).
We now discuss another very important example of closed positive current. In
fact,witheveryclosedanalyticsetA⊂X ofpuredimensionpisassociatedacurrent
of integration10 J.-P. Demailly, Complex analytic techniques in algebraic geometry
p,p(1.17) h[A],αi= α, α∈D (X),
obtained by integrating over the regular points ofA. In order to show that (1.17) is
acorrectdefinitionofacurrentonX,onemustshowthatA haslocallyfiniteareareg
in a neighborhood of A . This result, due to [Lel57] is shown as follows. Supposesing
that 0 is a singular point of A. By the local parametrization theorem for analytic
nsets, there is a linear change of coordinates onC such that all projections
π : (z ,...,z ) →(z ,...,z )I 1 n i i1 p
define a finite ramified covering of the intersection A∩Δ with a small polydisk
n pΔ in C onto a small polydisk Δ in C . Let n be the sheet number. Then theI I
p-dimensional area ofA∩Δ is bounded above by the sum of the areas of its projec-
tions counted with multiplicities, i.e.
Area(A∩Δ)≤ n Vol(Δ ).I I
The fact that [A] is positive is also easy. In fact
2iα ∧α ∧...∧iα ∧α =|det(α )| iw ∧w ∧...∧iw ∧w1 1 p p jk 1 1 p p
ifα = α dw intermsoflocalcoordinates(w ,...,w )onA .Thisshowsthatj jk k 1 p reg
allsuchformsare≥ 0inthecanonicalorientationdefinedbyiw ∧w ∧...∧iw ∧w .1 1 p p
More importantly,Lelong[Lel57]has shown that[A]isd-closed inX,even atpoints
of A . This last result can be seen today as a consequence of the Skoda-El Mirsing
extension theorem. For this we need the following definition: a complete pluripolar
set is a set E such that there is an open covering (Ω ) of X and psh functions uj j
−1on Ω with E ∩Ω = u (−∞). Any (closed) analytic set is of course completej j j
pluripolar (takeu as in Example 1.9).j
(1.18) Theorem (Skoda [Sko82], El Mir [EM84], Sibony [Sib85]). Let E be a closed
complete pluripolar set in X, and let Θ be a closed positive current on XrE such
that the coefficients Θ of Θ are measures with locally finite mass near E. ThenI,J
ethe trivial extension Θ obtained by extending the measures Θ by 0 on E is stillI,J
closed on X.
Lelong’s result d[A] = 0 is obtained by applying the Skoda-El Mir theorem to
Θ = [A ] on XrA .reg sing
Proof of Theorem 1.18. The statement is local on X, so we may work on a small
−1open set Ω such that E∩Ω =v (−∞), v ∈ Psh(Ω). Let χ :R→R be a convex
increasing function such that χ(t) = 0 for t ≤ −1 and χ(0) = 1. By shrinking Ω
−1and putting v = χ(k v⋆ρ ) with ε → 0 fast, we get a sequence of functionsk ε kk
∞v ∈ Psh(Ω)∩C (Ω) such that 0 ≤ v ≤ 1, v = 0 in a neighborhood of E∩Ωk k k
∞and limv (x) = 1 at every point of ΩrE. Let θ ∈ C ([0,1]) be a function suchk
that θ = 0 on [0,1/3],θ = 1 on [2/3,1] and 0≤θ≤ 1. Then θ◦v = 0 near E∩Ωk
˜and θ◦v → 1 on ΩrE. Therefore Θ = lim (θ◦v )Θ andk k→+∞ k
′ ′˜dΘ = lim Θ∧d(θ◦v )k