Structure theorems for projective and K¨ahler varieties

Jean-Pierre Demailly

Universit´e de Grenoble I, Institut Fourier

Lectures given at the PCMI Graduate Summer School held at Park City in July 2008:

Analytic and Algebraic Geometry: Common Problems-Diﬀerent Methods

Contents

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

1. Preliminary material .....................................................................................2

2. Lelong numbers and intersection theory ..................................................................9

3. Holomorphic vector bundles, connections and curvature .................................................16

4. Bochner technique and vanishing theorems ..............................................................19

25.L Estimates and existence theorems ....................................................................23

6. Numerically eﬀective and pseudo-eﬀective line bundles .................................................. 30

7. Holomorphic Morse inequalities .........................................................................37

28. The Ohsawa-TakegoshiL extension theorem ........................................................... 39

9. Approximation of closed positive (1,1)-currents by divisors ..............................................49

10. Subadditivity of multiplier ideal sheaves and Zariski decomposition .....................................64

11. Hard Lefschetz theorem with multiplier ideal sheaves ...................................................68

12. Invariance of plurigenera of projective varieties .........................................................78

13. Positive cones in the (1,1) cohomology groups of compact K¨ahler manifolds .............................80

14. Numerical characterization of the K¨ahler cone ..........................................................83

15. Structure of the pseudo-eﬀective cone and mobile intersection theory ...................................89

16. Super-canonical metrics and abundance ...............................................................101

17. Siu’s analytic approach and Pa˘un’s non vanishing theorem ............................................107

Bibliography .............................................................................................108

0.Introduction

The main purpose of these notes is to describe analytic techniques which are useful to study questions such

as linear series, multiplier ideals and vanishing theorems for algebraic vector bundles. One century after the

ground-breaking work of Riemann on geometric 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 concept of positivity, which

can ve viewed either in algebraic terms (positivity of divisors and algebraic cycles), or in more analytic terms

(plurisubharmonicity, hermitian connections with positive curvature). In this direction, one of the most basic

result is Kodaira’s vanishing theorem for positive vector bundles (1953-54), which is a deep consequence of the

BochnertechniqueandofthetheoryofharmonicformsinitiatedbyW.V.D.Hodgeduringthe1940’s.Thismethod

quickly led Kodaira to the well-known embedding theorem for projective varieties, a far reaching extension of

Riemann’s characterization of abelian varieties. Further reﬁnements of the Bochner technique led ten years later

2to the 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 reproved 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 theorems with a wideR

2 2 −2ϕassortmentofdiﬀerentL norms,namelynormsofthe form |f| e whereϕisaplurisubharmonicorstrictly

X2 J.-P. Demailly, PCMI 2008, Structure theorems for projective and K¨ahler varieties

plurisubharmonicfunctiononthegivenmanifoldorvarietyX.Here,theweightϕneednotbesmooth,anditison

P 2the contrary extremely important to allow weights which have logarithmic poles of the formϕ(z)=clog |g | ,j

where c > 0 and (g ) is a collection of holomorphic functions possessing a common zero zet Z ⊂ X. Followingj

Nadel[Nad89],onedeﬁnesthemultiplier ideal sheafI(ϕ)tobethesheafofgermsofholomorphicfunctionsf such

2 −2ϕ qthat|f| e islocallysummable. ThenI(ϕ)is acoherentalgebraicsheafoverX andH (X,K ⊗L⊗I(ϕ)) = 0X

for all q> 1 if the curvature of L is positive as a current. This important result can be seen as a generalization

of the Kawamata-Viehweg vanishing 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 idealI(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 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).

2Another very important result is the L extension theorem by Ohsawa-Takegoshi [OT87, Ohs88] (see also

2Manivel [Man93]). The main statement is that every L section f of a suitably positive line bundle deﬁned on

2 ˜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 mention for instance

Siu’s theorem [Siu74] on the analyticity of Lelong numbers, Skoda’s division theorem for ideals of holomorphic

functions, a basic approximationtheoremof closedpositive (1,1)-currentsby divisors,the subadditivity property

I(ϕ +ψ) ⊂ I(ϕ)I(ψ) of multiplier ideals [DEL00], the restriction formula I(ϕ ) ⊂ I(ϕ) , ... . A suitable|Y |Y

combination of these results can be used to reproveFujita’s result [Fuj94] on approximate Zariski decomposition,

as detailed in section 10.

In section 11, we show how subadditivity can be used to derive an “equisingular” approximation theorem

for (almost) plurisubharmonic functions: any such function can 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 cohomology 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.

Section 12 explains the proof of Siu’s theorem on the invariance of plurigenera, according to a beautiful

approach developped by Mihai Pa˘un [Pau07]. The proofs consists of an iterative process based on the Ohsawa-

Takegoshi theorem, and a very clever limiting argument for currents.

Sections13to15aredevotedtothestudyofpositiveconesinK¨ahlerorprojectivegeometry.Recent“algebro-

analytic” characterizations of the K¨ahler 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.

Section 16 presents some important ideas of H. Tsuji, later reﬁned by Berndtsson and Pa˘un, concerning the

so-called “super-canonical metrics”, and their interpretation in terms of the invariance of plurigenera and of the

abundance conjecture. As the concluding section, 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.

I would like to thank the organizers of the Graduate Summer School on Analytic and Algebraic Geometry

held at the Park City Mathematical Institute in July 2008 for their invitation to give a series of lectures, and

thus for the opportunity of publishing these notes.1. Preliminary material 3

1.Preliminarymaterial

1.A. Dolbeault cohomology and sheaf cohomology

p,q ⋆LetX be aC-analytic manifold of dimensionn. We denote byΛ T the bundle of diﬀerential forms of bidegreeX

(p,q) onX, 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 )denotearbitrarylocalholomorphiccoordinates,I =(i ,...,i ),J =(j ,...,j )aremultiindices1 n 1 p 1 q

(increasing sequences of integers in the range [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,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 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

p,• ′′lemma ford, see e.g. [Ho¨r66]).Inother words,the complex ofsheaves(E ,d )is exactin degreeq> 0; indegree

p′′q = 0, Kerd is the sheafΩ 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 a natural d operator actingP

∞ p,q ⋆on the space C (X,Λ T ⊗F) of smooth (p,q)-forms with values in F; if s = s e is a (p,q)-formλ λX 16λ6rP

′′ ′′expressed in terms of a local holomorphic frame of F, 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 still holds for F-valued forms. For every integer

p,qp = 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 C (X,Λ T ⊗F) .X

Now, let us recall the following fundamental result from sheaf theory (De Rham-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 ∞ p,q ⋆We apply this to the following situation: let E(F) be the sheaf of germs of C sections of Λ T ⊗F. ThenX

pp,• ′′(E(F) ,d ) is a resolutionof the locally freeO -moduleΩ ⊗O(F) (Dolbeault-Grothendieck lemma), and theX X

p,q ∞sheavesE(F) are acyclic as modules over the soft sheaf of ringsC . Hence by (1.2) we get

(1.3) Dolbeault IsomorphismTheorem(1953). For every holomorphic vector bundleF onX, there is a canonical

isomorphism

p,q q pH (X,F)≃H (X,Ω ⊗O(F)). X

If X is projective algebraic and F is an algebraic vector bundle, Serre’s GAGA theorem [Ser56] shows that the

pqalgebraic sheaf cohomology groupH (X,Ω ⊗O(F)) computed with algebraic sections over Zariski open sets isX

actually isomorphic to the analytic cohomology group. These results are the most basic tools to attack algebraic4 J.-P. Demailly, PCMI 2008, Structure theorems for projective and Ka¨hler varieties

problems via 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.Afunctionu :Ω−→ [−∞,+∞[deﬁned on an open subsetΩ⊂C is said to be plurisubharmonic

(psh for short) if

(a) u is upper semicontinuous ;

n n(b) for every complex line L ⊂ C , u is subharmonic on Ω∩L, that is, for all a ∈ Ω and ξ ∈ C with↾Ω∩L

|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.

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

or spheres: Z

1

u(a)6 u(z)dλ(z)

n 2nπ r /n! B(a,r)

1for everya∈Ω and r<d(a,∁Ω). Either u≡−∞ or u∈L on every connected component of Ω.loc

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

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

∞kernels, then u⋆ρ is C 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 that χ(t ,...,t ) is increasing in each1 p 1 p

u u1 pt . Thenχ(u ,...,u ) is psh onΩ. In particular u +···+u , max{u ,...,u }, log(e +···+e ) are pshj 1 p 1 p 1 p

onΩ.

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

wheredλ is the Lebesgue measure onC. 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. Preliminary material 5

n(1.7) Theorem. If u∈ Psh(Ω), u ≡−∞on every connected component of Ω, then for all ξ∈C

X 2∂ u ′Hu(ξ)= ξ ξ ∈D (Ω)j k

∂z ∂zj k

16j,k6n

′ nis a positive measure. Conversely, if v∈D (Ω) is such that Hv(ξ) is a positive measure for every ξ ∈C , there

exists a unique functionu∈Psh(Ω) which is locally integrable onΩ and such thatv is the distribution associated

to u.

In order to get a better geometric insight of this notion, we assume more generally that u is a function on a

2complex n-dimensional manifold X. If Φ : X → Y is a 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 on T , does not depend on the choice of coordinates (z ,...,z ).X 1 n

Therefore, the notion of psh function makes sense on any 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 log|f|∈ Psh(X) for every

0holomorphic function f ∈H (X,O ). More generallyX

α α1 qlog |f | +···+|f | ∈Psh(X)1 q

0foreveryf ∈H (X,O )andα >0(applyProperty1.5dwithu =α log|f |).Wewillbeespeciallyinterestedj X j j j j

inthesingularitiesobtainedatpointsofthezerovarietyf =... =f = 0,whentheα arerationalnumbers. 1 q j

(1.10) Deﬁnition. A psh function u∈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 X is algebraic, we say+ j

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

u/αWe then introduce the ideal J =J(u/α) of germs of holomorphic functions h such that |h|6 Ce for some

constantC, i.e.

|h|6C |f |+···+|f | .1 N

ThisisagloballydeﬁnedidealsheafonX,locallyequaltotheintegralclosureI oftheidealsheafI =(f ,...,f ),1 N

′thusJ is coherent onX. If (g ,...,g ) are local generators ofJ, we still have1 N

α 2 2u = 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 algebraicpoint of view, the singularities ofu 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 idealI is equivalent to the following

more algebraic deﬁnition:I consists of all germs h satisfying an integral equation

d d−1 k

h +a h +...+a h+a = 0, a ∈I .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(I). 6 J.-P. Demailly, PCMI 2008, Structure theorems for projective and Ka¨hler varieties

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 degreeq on an oriented diﬀerentiable manifoldM is simply a diﬀerentialq-formΘ with

′qdistribution coeﬃcients. The space of currents of degree q over M will be denoted by D (M). Alternatively, a

′′ p pcurrent of degree q can be seen as an element Θ in the dual space D (M) := D (M) of the space D (M) ofp

smooth diﬀerential forms of degree p =dimM−q with compact support; the duality pairing is given by

Z

p(1.12) hΘ,αi = Θ∧α, α∈D (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

q−1Then [S] is a current with measure coeﬃcients, and Stokes’ formula shows thatd[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). The currentΘ is said to be closed if dΘ = 0.p

OnacomplexmanifoldX,wehavesimilarnotionsofbidegreeandbidimension;asinthe realcase,wedenote

by

′p,q ′

D (X)=D (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 onX the1 p

distribution

(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 complex measures, and that, up toI,J

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 the (p,p)-forms iα ∧α ∧I,I 1 1

p,p ⋆...∧iα ∧α generate Λ T as aC-vector space. p p nC

P

A current Θ = i Θ dz ∧dz of bidegree (1,1) is easily seen to be positive if and only if the complexjk j k16j,k6nP

nmeasure λ λ Θ is a positive measure for everyn-tuple (λ ,...,λ )∈C .j k jk 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 positive current of bidegree (1,1) can be written

2 1under this form on any open subsetΩ⊂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 productΘ ∧...∧Θ of positive currents of bidegree (1,1) is positive whenever1 q

the product is well deﬁned (this is certainly the case if allΘ but one at most are smooth; much ﬁner conditionsj

will be discussed in Section 2).

We nowdiscussanotherveryimportantexampleofclosedpositive current.Infact, witheveryclosedanalytic

set A⊂X of pure dimensionp is associated a current of integration

Z

p,p(1.17) h[A],αi = α, α∈D (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 on X, one must show that A has locally ﬁnite area in a neighborhood of A . This result, due toreg sing

[Lel57] is shown as follows. Suppose that 0 is a singular point of A. By the local parametrization theorem for

nanalytic sets, there is a linear change of coordinates onC such that all projections1. Preliminary material 7

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

ndeﬁne a ﬁnite ramiﬁed covering of the intersection A∩Δ with a small polydisk Δ in C onto a small polydisk

pΔ in C . Let n be the sheet number. Then the p-dimensional area of A∩Δ is bounded above by the sum ofI I

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 that all such forms are> 0 inj jk k 1 p reg

the canonical orientation deﬁned by iw ∧w ∧...∧iw ∧w . More importantly, Lelong [Lel57] has shown that1 1 p p

[A] is d-closed inX, even at points ofA . This last result can be seen today as a consequence of the Skoda-Elsing

Mir extension theorem. For this we need the following deﬁnition: a complete pluripolar set is a set E such that

−1there is an open covering (Ω ) ofX and psh functions u onΩ withE∩Ω =u (−∞). Any (closed) analyticj j j j j

set is of course complete pluripolar (take u 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 locallyI,J

eﬁnite mass nearE. Then the trivial extensionΘ obtained by extending the measuresΘ by 0 onE is still closedI,J

on X.

Lelong’s resultd[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 onX, so we may work on a small open setΩ such thatE∩Ω =

−1v (−∞), v ∈ Psh(Ω). Let χ : R → R be a convex increasing function such that χ(t) = 0 for t 6 −1 and

−1χ(0) = 1. By shrinking Ω and 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 06v 61,v =0 in a neighborhood ofE∩Ω and limv (x) = 1 at every pointk k k k

∞of ΩrE. Let θ ∈ C ([0,1]) be a function such that θ = 0 on [0,1/3], θ = 1 on [2/3,1] and 06 θ 6 1. Then

˜θ◦v = 0 near E∩Ω andθ◦v → 1 on ΩrE. Therefore Θ = lim (θ◦v )Θ andk k k→+∞ k

′ ′˜dΘ = lim Θ∧d(θ◦v )k

k→+∞

′in the weak topologyof currents.It is thereforesuﬃcient to verifythatΘ∧d (θ◦v ) convergesweaklyto 0 (notek

′′ ′ ′′˜ ˜ ˜that d Θ is conjugate to dΘ, thus d Θ will also vanish).

′n−1,n−1 ′ ′n,n−1Assume ﬁrst thatΘ∈D (X). ThenΘ∧d (θ◦v )∈D (Ω), and we have to show thatk

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

k→+∞

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

2 1,0