Multiplier ideal sheaves and analytic methods in

algebraic geometry

Jean-Pierre Demailly

Universit´e de Grenoble I, Institut Fourier

Lectures given at the ICTP School held in Trieste, Italy, April 24 – May 12, 2000

Vanishing theorems and eﬀective results in Algebraic Geometry

Contents

0. Introduction ................................................................ 1

1. Preliminary Material ........................................................4

2. Lelong Numbers and Intersection Theory ...................................12

3. Holomorphic Vector Bundles, Connections and Curvature ...................21

4. Bochner Technique and Vanishing Theorems ................................26

25. L Estimates and Existence Theorems ......................................31

6. Numerically Eﬀective and Pseudoeﬀective Line Bundles .....................40

7. A Simple Algebraic Approach to Fujita’s Conjecture ........................47

8. Holomorphic Morse Inequalities ............................................56

9. Eﬀective Version of Matsusaka’s Big Theorem ..............................58

10. Positivity Concepts for Vector Bundles ....................................63

211. Skoda’s L Estimates for Surjective Bundle Morphisms ....................71

212. The Ohsawa-Takegoshi L Extension Theorem .............................80

13. Invariance of Plurigenera of Varieties of General Type .....................96

14. Subadditivity of Multiplier Ideal Sheaves and Zariski Decomposition ......104

15. Hard Lefschetz Theorem with Multiplier Ideal Sheaves ....................109

References ...................................................................122

0.Introduction

Transcendental methodsofalgebraicgeometryhavebeenextensivelystudied sincea

verylongtime,startingwiththeworkofAbel,JacobiandRiemanninthenineteenth

century. More recently, in the period 1940-1970, the work of Hodge, Hirzebruch,

Kodaira, Atiyah revealed still deeper relations between complex analysis, topology,

PDE theory and algebraic geometry. In the last ten years, gauge theory has proved

to be a very eﬃcient toolfor 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’s2 J.-P. Demailly, Multiplier ideal sheaves and analytic methods

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 geometry, the ideal

I(ϕ) plays a very important role, since it directly converts an analytic object into

an algebraic one, and, simultaneously, takes care of the singularities in a very ef-

ﬁcient way. Another analytic tool used to deal with singularities is the theory of

positive currents introduced by Lelong [Lel57]. Currents can be seen as generaliza-

tions of algebraic cycles, and many classical results of intersection theory still apply

to currents. 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

correspond to wedge products of currents (whenever these products are deﬁned).

Besides the Kodaira-Nakano vanishing theorem, one of the most basic “eﬀective

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

surfacesbyReider[Rei88](thecaseofcurvesiseasyandhasbeenknownsinceavery

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 equations0. Introduction 3

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 is

3n+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

Morseinequalities([Dem85b]).TheMorseinequalitiesareusedtoconstructsections

′ ′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

Nullstellenstatzandwiththeextensionproblem.Skoda[Sko72b,Sko78]showedthatP

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 14.

In the last section 15, we show how subadditivity can be used to derive an ap-

proximationtheoremfor(almost)plurisubharmonicfunctions:anysuchfunctioncan

be approximated by a sequence of (almost) plurisubharmonic functions which are

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

sheaves.

Thesenotesareessentiallywrittenwiththeideaofservingasananalytictoolbox

for algebraic geometers. Although eﬃcient algebraic techniques exist, our feeling is

that the analytic techniques are very ﬂexible and oﬀer a large variety of guidelines4 J.-P. Demailly, Multiplier ideal sheaves and analytic methods

for more algebraic questions (including applications to number theory which 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 at least one decade. These discussions have

certainly had a great inﬂuence on my research work and therefore on the contents

of the present notes.

1.PreliminaryMaterial

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

|I|=p,|J|=q

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

∂zk|I|=p,|J|=q,1≤k≤n

X ∂uI,J′′d u = dz ∧dz ∧dzk I J

∂zk

|I|=p,|J|=q,1≤k≤n

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. [Ho¨r66]). 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):1. Preliminary Material 5

p,q q ∞ p,• ⋆(1.1) H (X,F)=H C (X,Λ T ⊗F) .X

Now,letusrecallthefollowingfundamentalresultfromsheaftheory(DeRham-Weil

•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).

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

plurisubharmonicfunctionsandpositivecurrentsoriginatedbyK.OkaandP.Lelong

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.6 J.-P. Demailly, Multiplier ideal sheaves and analytic methods

(1.5) Basic properties.

a) Every function u ∈ Psh(Ω) is subharmonic, namely it satisﬁes the mean value

inequality on euclidean balls or spheres:

Z

1

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.

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 j1≤j,k≤n k

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 k1≤j,k≤n

′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.

Inordertogetabettergeometricinsightofthisnotion,weassumemoregenerally

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, hence1. Preliminary Material 7

′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 q

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

2

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 2

′u= log |g | +···+|g | +O(1).1 N

2

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

function.

(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 k8 J.-P. Demailly, Multiplier ideal sheaves and analytic methods

Hint.Oneinclusionisclear.Toprovetheotherinclusion,considerthenormalization

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 by

Z

p(1.12) hΘ,αi= Θ∧α, α∈D (M).

M

Abasicexampleisthecurrentofintegration[S]overacompactorientedsubmanifold

S of M :

Z

(1.13) h[S],αi= α, degα =p = dim S.R

S

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

thespaceofcurrentsofbidegree(p,q)andbidimension(n−p,n−q)onX.According

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

1≤j≤n

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. n1 1 p p C

P

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 n1. Preliminary Material 9

(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 integration

Z

p,p(1.17) h[A],αi= α, α∈D (X),

Areg

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.

X

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

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.10 J.-P. Demailly, Multiplier ideal sheaves and analytic methods

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

k→+∞

′intheweaktopologyofcurrents.ItisthereforesuﬃcienttoverifythatΘ∧d (θ◦v )k

′′ ′ ′′˜ ˜ ˜convergesweaklyto0(notethatd Θ isconjugatetodΘ,thusd Θ willalsovanish).

′n−1,n−1 ′ ′n,n−1Assume ﬁrst thatΘ∈D (X). Then Θ∧d (θ◦v )∈D (Ω), and wek

have to show that

′ ′ ′ 1,0hΘ∧d(θ◦v ),αi =hΘ,θ (v )dv ∧αi −→ 0, ∀α∈D (Ω).k k k

k→+∞

1,0Asγ →Θh,iγ∧γiisanon-negativehermitianformonD (Ω),theCauchy-Schwarz

inequality yields