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Lectures by Jean-Pierre Demailly

Universit´e de Grenoble I, Institut Fourier

December 2009

Acompilationoflecturesgivenatvariousplaces(CIME1994,Trieste2000,Mahdia2004,

Grenoble 2007, Park City 2008, Beijing 2009 ...)

Contents

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

1. Preliminary material.............................................................................4

2. Lelong numbers and intersection theory.........................................................12

3. Hermitian vector bundles, connections and curvature............................................21

4. Bochner technique and vanishing theorems......................................................26

25. L estimates and existence theorems............................................................31

6. Numerically eﬀective and pseudo-eﬀective line bundles...........................................39

7. A simple algebraic approach to Fujita’s conjecture...............................................50

8. Holomorphic Morse inequalities.................................................................59

9. Eﬀective version of Matsusaka’s big theorem....................................................62

10. Positivity concepts for vector bundles..........................................................67

211. Skoda’s L estimates for surjective bundle morphisms..........................................74

212. The Ohsawa-Takegoshi L extension theorem..................................................84

13. Approximation of closed positive currents by analytic cycles....................................97

14. Subadditivity of multiplier ideals and Fujita’s approximate Zariski decomposition theorem.....117

15. Hard Lefschetz theorem with multiplier ideal sheaves..........................................122

16. Invariance of plurigenera of projective varieties................................................134

17. Numerical characterization of the Ka¨hler cone ................................................ 137

18. Structure of the pseudo-eﬀective cone and mobile intersection theory..........................147

19. Super-canonical metrics and abundance.......................................................162

20. Siu’s analytic approach and P˘aun’s non vanishing theorem....................................169

References....................................................................................... 172

§0. Introduction

The main purpose of these notes is to describe analytic techniques which are use-

ful to study questions such as linear series, multiplier ideals and vanishing theorems for

algebraicvectorbundles. One century aftertheground-breaking workofRiemannonge-

ometric aspects of function theory, the general progress achieved in diﬀerential geometry

and global analysis on manifolds resulted into major advances in the theory of algebraic

and analytic varieties of arbitrary dimension. One central unifying concept is the con-

cept of positivity, which can ve viewed either in algebraic terms (positivity of divisors

and algebraic cycles), or in more analytic terms (plurisubharmonicity, hermitian connec-

tions withpositive curvature). In this direction, one of the most basic result is Kodaira’s2 Jean-Pierre Demailly, Analytic methods in algebraic geometry

vanishing theorem for positive vector bundles (1953-54), which is a deep consequence of

the Bochner technique and of the theory of harmonic forms initiated by W.V.D. Hodge

during the 1940’s. This method quickly led Kodaira to the well-known embedding the-

orem for projective varieties, a far reaching extension of Riemann’s characterization of

abelian varieties. Further reﬁnements of the Bochner technique led ten years later to

2the theory of L estimates for the Cauchy-Riemann operator, (J.J. Kohn [Koh63, 64],

Andreotti-Vesentini [AV65], [Ho¨r65]). Not only vanishing theorems can be proved of re-

proved in that manner, but perhaps more importantly, extremely precise information of

a quantitative nature is obtained about solutions of ∂-equations, their zeroes, poles and

growth at inﬁnity.

What makes the theory extremely ﬂexible is the possibility to formulate existence

2theorems with a wide assortment of diﬀerent L norms, namely norms of the formR

2 −2ϕ|f| e where ϕ is a plurisubharmonic or strictly plurisubharmonic function on theX

given manifold or variety X. Here, the weight ϕ need not be smooth, and it is on

the contrary extremely important to allow weights which have logarithmic poles of theP

2form ϕ(z) = clog |g | , where c > 0 and (g ) is a collection of holomorphic func-j j

tions possessing a common zero zet Z ⊂ X. Following Nadel [Nad89], one deﬁnes the

multiplier ideal sheaf (ϕ) to be the sheaf of germs of holomorphic functions f such

2 −2ϕthat |f| e is locally summable. Then (ϕ) is a coherent algebraic sheaf over X and

qH (X,K ⊗L⊗ (ϕ)) = 0 for all q > 1 if the curvature of L is positive as a current.X

This important result can be seen as a generalization of the Kawamata-Viehweg vanish-

ing theorem ([Kaw82], [Vie82]), which is one of the cornerstones of higher dimensional

algebraic geometry, especially in relation with Mori’s minimal model program.

In the dictionary between analytic geometry and algebraic geometry, the ideal (ϕ)

playsa very important role, since it directly converts ananalyticobject into analgebraic

one, and, simultaneously, takes care of the 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 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 re-

sult” expected to hold in algebraic geometry is expressed in the following conjecture of

Fujita [Fuj87]: if L is an ample (i.e. positive) line bundle on a projective n-dimensional

algebraic variety X, then K +(n+1)L is generated by sections and K +(n+2)L isX X

very ample. In the last decade, a lot of eﬀort has been brought for 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 surfaces by Reider [Rei88] (the case of

curvesiseasyandhasbeenknownsinceaverylongtime),andthenumericalcriterionfor

the very ampleness of 2K +L given in [Dem93b], obtained by means of analytic tech-X

niques and Monge-Amp`ere equations with isolated singularities. Alternative algebraic

techniques were developed slightly later by Kolla´r [Kol92], Ein-Lazarsfeld [EL93], Fujita

[Fuj93], Siu [Siu95, 96], Kawamata [Kaw97] and Helmke [Hel97]. We will explain here

Siu’smethodbecauseitistechnicallythesimplestmethod; oneoftheresultsobtainedby3n+1

thismethodisthefollowingeﬀectiveresult: 2K +mL isveryampleform>2+ .X n

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ThebasicideaistoapplytheKawamata-Viehwegvanishingtheorem,andtocombinethis

with the Riemann-Roch formula in order to produce sections through a clever induction

procedure on the dimension of the base loci of the linear systems involved.

AlthoughSiu’sresultiscertainlynotoptimal,itissuﬃcienttoobtainaniceconstruc-

tive proof of Matsusaka’s big theorem ([Siu93], [Dem96]). The result states that there

n n−1is an eﬀective value m depending only on the intersection numbers L and L ·K ,0 X

such that mL is very ample for m > m . The basic idea is to combine results on the0

very ampleness of2K +mL together withthetheory of holomorphicMorse inequalitiesX

′ ′([Dem85b]). The Morse inequalities are used to construct sections of mL−K for mX

large. Again this step can be made algebraic (following suggestions by F. Catanese and

R.Lazarsfeld), buttheanalyticformulationapparentlyhasawiderrangeofapplicability.

2In the next sections, we pursue the study of L estimates, in relation with the Null-

stellenstatz and with the extension problem. Skoda [Sko72b, Sko78] showed that theP

2division problem f = g h can be solved holomorphically with very precise L esti-j j

2 −pmates, providedthattheL normof|f||g| isﬁniteforsomesuﬃciently largeexponent

p (p > n = dimX is enough). Skoda’s estimates have a nice interpretation in terms of

local algebra, and they lead to precise qualitative and quantitative estimates in con-

2nection with the B´ezout problem. Another very important result is the L extension

theorem by Ohsawa-Takegoshi [OT87, Ohs88], which has also been generalized later by

2Manivel [Man93]. The main statement is that every L section f of a suitably posi-

2 ˜tive line bundle deﬁned on a subavariety Y ⊂ X can be extended to a L section f

deﬁned over the whole of X. The positivity condition can be understood in terms of

the canonical sheaf and normal bundle to the subvariety. The extension theorem turns

out to have an incredible amount of important consequences: among them, let us men-

tion 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

property (ϕ +ψ) ⊂ (ϕ) (ψ) of multiplier ideals [DEL00], the restriction formula

(ϕ )⊂ (ϕ) , .... A suitable combination of these results yields another important|Y |Y

important result of Fujita [Fuj94] on approximate Zariski decomposition, as we show in

section 14.

In section 15, we show how subadditivity can be used to derive an “equisingular”

approximation theorem for (almost) plurisubharmonic functions: any such function can

beapproximatedbyasequenceof(almost)plurisubharmonicfunctionswhicharesmooth

outsideananalyticset,andwhichdeﬁnethesamemultiplieridealsheaves. Fromthis,we

deriveageneralizedversionofthehard Lefschetz theoremforcohomologywithvaluesina

pseudo-eﬀectivelinebundle;namely,theLefschetzmapissurjectivewhenthecohomology

groups are twisted by the relevant multiplier ideal sheaves.

Section 16 explains theproof ofSiu’s theorem on theinvariance of plurigenera, accor-

ding to a beautiful approach developped by Mihai P˘aun [Pau07]. The proofs consists of

an iterative process based on the Ohsawa-Takegoshi theorem, and a very clever limiting

argument for currents.

Sections 17 and 18 are devoted to the study of positive cones in Ka¨hler or projective

geometry. Recent “algebro-analytic” characterizations of the Ka¨hler cone ([DP04]) and

the pseudo-eﬀective cone of divisors ([BDPP04]) are explained in detail. This leads to

a discussion of the important concepts of volume and mobile intersections, following

S.Boucksom’s PhD work [Bou02]. As a consequence, we show that a projective algebraic

manifold has a pseudo-eﬀective canonical line bundle if and only if it is not uniruled.

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Section 19 presents further important ideas of H. Tsuji, later reﬁned by Berndtsson

andPa˘un, concerning theso-called“super-canonical metrics”,andtheirinterpretationin

termsoftheinvarianceofplurigeneraandoftheabundanceconjecture. Intheconcluding

section 20, we state Pa˘un’s version of the Shokurov-Hacon-McKernan-Siu non vanishing

theorem and give an account of the very recent approach of the proof of the ﬁniteness of

the canonical ring by Birkar-P˘aun [BiP09], based on the ideas of Hacon-McKernan and

Siu.

§1. Preliminary material

§1.A. Dolbeault cohomology and sheaf cohomology

p,q ⋆Let X be aC-analytic manifold of dimension n. We denote by Λ T the bundle ofX

diﬀerential forms of bidegree (p,q) on X, i.e., diﬀerential forms which can be written as

X

u= u dz ∧dz .I,J I J

|I|=p,|J|=q

Here (z , ...,z ) denote arbitrary local holomorphic coordinates onX,I = (i , ...,i ),1 n 1 p

J = (j , ...,j )aremultiindices(increasingsequencesofintegersintherange[1,...,n],1 q

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 be the sheaf of germs of complex valued diﬀerential (p,q)-forms with coef-

′ ′′ﬁcients. 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,16k6n

X ∂uI,J′′d u= dz ∧dz ∧dzk I J

∂zk

|I|=p,|J|=q,16k6n

are of type (p+1,q), (p,q + 1) respectively. The well-known Dolbeault-Grothendieck

′′ ′′lemma assertsthatanyd -closedformoftype(p,q)withq>0islocallyd -exact(thisis

′′the analogue ford of the usual Poincar´e lemma ford, see e.g. [H¨or66]). In other words,

p,• ′′ ′′the complex of sheaves ( ,d ) is exact in degree q> 0; in degree q =0, Kerd is the

p

sheaf Ω of germs of holomorphic forms of degree p on X.X

Moregenerally,ifF isaholomorphicvectorbundleofrankroverX,thereisanatural

′′ ∞ p,q ⋆d operator acting on the space (X,Λ T ⊗F) of smooth (p,q)-forms with valuesXP

inF; ifs= s e isa (p,q)-formexpressed in termsof a localholomorphicframeλ λ16λ6r P

′′ ′′of F, we simply deﬁne d s := d s ⊗e , observing that the holomorphic transitionλ λ

′′matrices involved in changes of holomorphic frames do not aﬀect the computation ofd .

ItisthenclearthattheDolbeault-GrothendiecklemmastillholdsforF-valuedforms. For

p,qevery integer p = 0,1,...,n, the Dolbeault Cohomology 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 (X,Λ T ⊗F) .X

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Now, let us recall the following fundamental result from sheaf theory (De Rham-Weil

•isomorphism theorem): let ( ,d) be a resolution of a sheaf by acyclic sheaves, i.e. a

•complex of sheaves ( ,δ) such that there is an exact sequence of sheaves

0 qj δ δ0 1 q q+10−→ −→ −→ −→···−→ −→ −→··· ,

s qand H (X, )=0 for allq>0 and s>1. Then there is a functorial isomorphism

q • q(1.2) H Γ(X, ) −→H (X, ).

p,q ∞Weapplythistothefollowingsituation: let (F) bethesheafofgermsof sections

pp,q ⋆ p,• ′′ofΛ T ⊗F. Then( (F) ,d )isaresolutionofthelocallyfree -moduleΩ ⊗ (F)XX X

p,q(Dolbeault-Grothendieck lemma), and the sheaves (F) are acyclic as modules over

∞the soft sheaf of rings . Hence by (1.2) we get

(1.3)DolbeaultIsomorphismTheorem(1953). Foreveryholomorphicvectorbundle

F on X, there is a canonical isomorphism

p,q q pH (X,F)≃H (X,Ω ⊗ (F)). X

If X is projective algebraic and F is an algebraic vector bundle, Serre’s GAGA the-

pqorem [Ser56] shows that the algebraic sheaf cohomology group H (X,Ω ⊗ (F)) com-X

putedwithalgebraicsectionsoverZariskiopensetsisactuallyisomorphictotheanalytic

cohomologygroup. Theseresultsarethemostbasictoolstoattackalgebraicproblemsvia

analytic methods. Another important tool is the theory of plurisubharmonic functions

and positive currents originated by K. Oka and P. Lelong in the decades 1940-1960.

§1.B. Plurisubharmonic functions

Plurisubharmonic functions have been introduced independently by Lelong and Oka

in the study of holomorphic convexity. We refer to [Lel67, 69] for more details.

n(1.4) Deﬁnition. A function u:Ω−→[−∞,+∞[ deﬁned on an open subset Ω⊂C is

said to be plurisubharmonic (psh for short) if

(a) u is upper semicontinuous ;

n(b) for every complex lineL⊂C , u is subharmonic on Ω∩L, that is, for alla∈Ω↾Ω∩L

nand ξ ∈C with |ξ|<d(a,∁Ω), the function u satisﬁes the mean value inequality

Z 2π1 iθu(a)6 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.

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

equality on euclidean balls or spheres:

Z

1

u(a)6 u(z)dλ(z)

n 2nπ r /n! B(a,r)

1for every a ∈ Ω and r < d(a,∁Ω). Either u ≡ −∞ or u ∈ L on every connectedloc

component of Ω.

(b) For any decreasing sequence of psh functions u ∈ Psh(Ω), the limit u = limu isk k

psh on Ω.

(c) Let u∈ Psh(Ω) be such that u ≡−∞on every connected component of Ω. If (ρ )ε

∞is a family of smoothing kernels, then u⋆ρ is and psh onε

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

p(d) 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 par-1 p j 1 p

u u1 pticular u +···+u , max{u , ...,u }, log(e +···+e ) are psh on Ω. 1 p 1 p

2(1.6) Lemma. A function u∈C (Ω,R) is psh on Ω if and only if the hermitian form

X

2Hu(a)(ξ)= ∂ u/∂z ∂z (a)ξ ξj k j k

16j,k6n

is semi-positive at every point a∈Ω.

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 onC. Lemma 1.6 is a strong evidence that plurisub-

harmonicity 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 for

nall ξ∈C

2X ∂ u ′Hu(ξ)= ξ ξ ∈ (Ω)j k∂z ∂zj k16j,k6n

′is a positive measure. Conversely, if v∈ (Ω) is such that Hv(ξ) is a positive measure

nfor 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.

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D§1. Preliminary material 7

Inordertogetabettergeometricinsightofthisnotion,weassumemoregenerallythat

u is a function on a complex n-dimensional manifold X. If Φ:X →Y is a holomorphic

2 ′ ′′ ⋆ ′ ′′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 particular Hu, viewed as a hermitian form on T , does not depend on the choiceX

of coordinates (z ,...,z ). Therefore, the notion of psh function makes sense on any1 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, ). More generallyX

α α1 qlog |f | +···+|f | ∈Psh(X)1 q

0for every f ∈ H (X, ) and α > 0 (apply Property 1.5d with u = α log|f |). Wej X j j j j

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 functionu∈Psh(X) will be said to have analytic singularities

if u can be written locally as

α 2 2u= log |f | +···+|f | +v,1 N

2

where α∈R , v is a locally bounded function and the f are holomorphic functions. If+ j

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 = (u/α) of germs of holomorphic functions h such

u/αthat |h|6Ce for some constant C, i.e.

|h|6C |f |+···+|f | .1 N

This is a globally deﬁned ideal sheaf on X, locally equal to the integral closure of

the ideal sheaf = (f , ...,f ), thus is coherent on X. If (g , ...,g ′) are local1 N 1 N

generators of , 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 auto-+

matically has algebraic singularities. From an algebraic point of view, the singularities

of u are in 1:1 correspondence with the “algebraic data” ( ,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 is

equivalent to the following more algebraic deﬁnition: consists of all germsh satisfying

an integral equation

d d−1 kh +a h +...+a h+a =0, a ∈ .1 d−1 d k

Hint. One inclusion is clear. To prove the other inclusion, consider the normalization of

the blow-up of X along the (non necessarily reduced) zero variety V( ).

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§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 coeﬃcients. The space

′qof currents of degree q over M will be denoted by (M). Alternatively, a current of

′′ pdegreeq canbeseenasanelement Θinthedualspace (M):= (M) ofthespacep

p(M) of smooth diﬀerential forms of degreep=dimM−q with compact support; the

duality pairing is given by

Z

p(1.12) hΘ,αi= Θ∧α, α∈ (M).

M

A basic example is the current of integration [S] over a compact oriented submanifold 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 d[S] =

q−1(−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 ∈ (M). The current Θ is said top

be closed if dΘ=0.

On a complex manifold X, we have similar notions of bidegree and bidimension; as

in the real case, we denote by

′p,q ′(X)= (X), n= dimX,n−p,n−q

the space of currents of bidegree (p,q) and bidimension (n−p,n−q) on X. According

to [Lel57], a current Θ of bidimension (p,p) is said to be (weakly) positive if for every

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

16j6n

which is a positive measure.P

Hint. Observe that Θ is invariant by unitary changes of coordinates and that theI,I

p,p ⋆(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 be positivejk j k16j,k6n P

if and only if the complex measure λ λ Θ is a positive measure for every n-tuplej k jk

n(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

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D§1. Preliminary material 9

positive current of bidegree (1,1) can be written under this form on any open subset

2 1Ω⊂X such that H (Ω,R)=H (Ω, )= 0, e.g. on small coordinate balls (exercise toDR

the reader).

It is not diﬃcult to show that a product Θ ∧...∧Θ of positive currents of bidegree1 q

(1,1) is positive whenever the product is well deﬁned (this is certainly the case if all Θj

but one at most are smooth; much ﬁner conditions will be discussed in Section 2).

We now discuss another very important example of closed positive current. In fact,

with every closed analytic set A ⊂ X of pure dimension p is associated a current of

integration

Z

p,p(1.17) h[A],αi= α, α∈ (X),

Areg

obtained by integrating over the regular points of A. In order to show that (1.17) is a

correct deﬁnition of a current onX, one must show thatA has locally ﬁnite area in areg

neighborhood ofA . This result, due to [Lel57] is shown as follows. Suppose that 0 issing

a singular point of A. By the local parametrization theorem for analytic sets, there is a

nlinear change of coordinates onC such that all projections

π :(z , ...,z ) →(z , ...,z )I 1 n i i1 p

ndeﬁne a ﬁnite ramiﬁed covering of the intersectionA∩Δ with a small polydisk Δ inC

ponto a small polydisk Δ in C . Let n be the sheet number. Then the p-dimensionalI I

area of A∩Δ is bounded above by the sum of the areas of its projections counted with

multiplicities, i.e. X

Area(A∩Δ)6 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

if α = α dw in terms of local coordinates (w , ...,w ) on A . This shows thatj jk k 1 p reg

all such forms are> 0 in the canonical orientation deﬁned by iw ∧w ∧...∧iw ∧w .1 1 p p

More importantly, Lelong [Lel57] has shown that [A] is d-closed in X, even at points of

A . This last result can be seen today as a consequence of the Skoda-El Mir extensionsing

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 u on Ω withj j j

−1E∩Ω =u (−∞). Any (closed) analytic set is of course complete pluripolar (take uj jj

as in Example 1.9).

(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. Then the trivialI,J

eextension Θ obtained by extending the measures Θ by 0 on E is still closed on X.I,J

Lelong’s result d[A] = 0 is obtained by applying the Skoda-El Mir theorem to Θ =

[A ] on XrA .reg sing

O

D10 Jean-Pierre Demailly, Analytic methods in algebraic geometry

Proof of Theorem 1.18. The statement is local on X, so we may work on a small open

−1set 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 6 −1 and χ(0) = 1. By shrinking Ω and putting

−1 ∞v =χ(k v⋆ρ )withε → 0fast, wegetasequenceoffunctionsv ∈Psh(Ω)∩ (Ω)k ε k kk

such that 0 6 v 6 1, v = 0 in a neighborhood of E∩Ω and limv (x) = 1 at everyk k k

∞point of ΩrE. Let θ ∈ ([0,1]) be a function such that θ = 0 on [0,1/3], θ = 1 on

[2/3,1] and 06θ61. Then θ◦v = 0 near E∩Ω and θ◦v → 1 on ΩrE. Thereforek k

˜Θ=lim (θ◦v )Θ andk→+∞ k

′ ′˜d Θ= lim Θ∧d (θ◦v )k

k→+∞

′in the weak topology of currents. It is therefore suﬃcient to verify that Θ∧d(θ◦v )k

′′ ′ ′′˜ ˜ ˜converges weakly to 0 (note thatd Θ is conjugate to d Θ, thus d Θ will also vanish).

′n−1,n−1 ′ ′n,n−1Assume ﬁrst that Θ∈ (X). Then Θ∧d (θ◦v )∈ (Ω), and we havek

to show that

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

k→+∞

1,0As γ →Θh,iγ∧γi is a non-negative hermitian form on (Ω), the Cauchy-Schwarz

inequality yields