Regularization of currents and entropy

REGULARIZATION OF CURRENTS AND ENTROPY

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ABSTRACT. – LetT be a positive closed(p, p)-current on a compact Kähler manifold X. We prove the existence of smooth positive closed (p, p)-forms T_n⁺ and T_n⁻ such that T_n⁺ −T_n⁻→T weakly.

Moreover,Tn^±cXTwherecX>0is a constant independent ofT. We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed(1,1)-currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate for the topological entropy of meromorphic maps on compact Kähler manifolds.

2004 Elsevier SAS

RÉSUMÉ. – Soit T un (p, p)-courant positif fermé sur une variété kählérienne compacte X. Nous montrons l’existence de (p, p)-formes lisses, positives fermées T_n⁺ et T_n⁻ telles que T_n⁺−T_n⁻→T faiblement. De plus, on aT_n^±cXToùcX>0est une constante indépendante deT. Nous montrons aussi ce résultat pour les courants positifs pluriharmoniques. Nous étudions également le produit extérieur de(1,1)-courants positifs fermés à potentiel continu avec des courants pluriharmoniques positifs. Comme application, nous donnons une estimation de l’entropie topologique des applications méromorphes d’une variété kählérienne compacte.

2004 Elsevier SAS

1. Introduction

Let(X, ω)be a compact Kähler manifold of dimensionk. Demailly [7] has shown that for a positive closed(1,1)-currentTonX, there exist smooth positive closed(1,1)-formsT_n⁺which converge weakly (i.e. in the sense of currents) toT +cωwherec >0is a constant. Moreover, there is a constantc_X>0, independent ofT, such thatT_n⁺andcare bounded byc_XT. We refer to Demailly’s papers [6,7] for the basics on currents on complex manifolds. Recall that the mass of a positive(p, p)-current S is defined byS:=

XS∧ω^k−p. Our main result is the following theorem where positivity can be understood in the weak or strong sense.

THEOREM 1.1. – Let (X, ω)be a compact Kähler manifold of dimensionk. Then, for every positive closed(p, p)-currentT onX, there exist smooth positive closed(p, p)-formsT_n⁺ and T_n⁻ such thatT_n⁺−T_n⁻ converge weakly to the currentT. Moreover, Tn^±cXT where c_X>0is a constant independent ofT.

We deduce from this theorem the following corollary which is proved in [10] for projective manifolds.

COROLLARY 1.2. – Let(X, ω)be a compact Kähler manifold of dimensionk. Then, for every positive closed(p, p)-currentT onX, there exist smooth positive closed(p, p)-formsT_n⁺which converge weakly to a currentT withTT. Moreover,T_n⁺cXTand TcXT wherecX>0is a constant independent ofT.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

Let(X, ω)be another compact Kähler manifold of dimensionkkand letΠ :X→Xbe a surjective holomorphic map. We want to define the pull-back of the currentT by the mapΠ.

WhenΠis a finite map, this problem is studied in [19,12]. In general, the mapΠis a submersion only in the complement of an analytic subsetCofX. Letπdenote the restriction ofΠtoX\C.

Then,π^∗(T)is well defined and is a positive closed(p, p)-current onX\C. Let(T_n⁺)andcXbe as in Corollary 1.2. DefineSn:= Π^∗(T_n⁺). The(p, p)-formsSnare smooth and positive onX. Their classes inH^p,p(X,C)are bounded since(T_n⁺)is bounded. It follows that(Sn)is bounded. Taking a subsequence, we can assume thatS_n converge to a currentS. We also have Sπ^∗(T)on X\C. In particular, π^∗(T)has finite mass. Following Skoda [22], the trivial extensionπ^∗(T)ofπ^∗(T)onXis a positive closed current. So, we have the following corollary.

COROLLARY 1.3. – Let X,X,Π,π andT be as above. Then, the positive current π^∗(T) is well defined and closed. Moreover, there exists a constantc_Π>0independent ofT such that π^∗(T)cΠT. The mapT→π^∗(T)is l.s.c. in the sense that ifTn→T, then any cluster pointτof(π^∗(Tn))satisfiesτπ^∗(T).

In [19], Méo gave an example which shows that, in general, whenXandXare not compact, the currentπ^∗(T)onX\Cis not always of bounded mass nearC.

Consider a dominating meromorphic self-map f:X →X of X. Define fⁿ:=f ◦ · · · ◦f (n times) the nth iterate of f. We refer to the survey [20] for the theory of iteration of meromorphic maps. Let I_n be the indeterminacy set of fⁿ. ThenI_n is an analytic subset of codimension2ofX. Denote byΩf the set of pointsx∈X\I1such thatfⁿ(x)∈/I1for every n1. A subsetF⊂Ωf is called(n, )-separated, >0, if

0maxin−1dist

fⁱ(x), fⁱ(y)

forx, y∈F distinct.

The topological entropyh(f)(see [5]) is defined by h(f) := sup

>0

lim sup

n→∞

1

nlog max

#F, F(n, )-separated .

LetΓ_nbe the closure inXⁿof the set of points x, f(x), . . . , fⁿ⁻¹(x)

, x∈Ωf.

This is an analytic subset of dimensionkofXⁿ. LetΠi be the canonical projections ofXⁿon its factors. We consider onXⁿ the Kähler metricωn:=

Π^∗_i(ω). Define, following Gromov [17],

lov(f) := lim sup

n→∞

1 nlog

vol(Γ_n)

= lim sup

n→∞

1 nlog

Γn

ω_n^k

. (1)

Define also the dynamical degree of orderpoff by

dp:= lim sup

n→∞

X\In

f^n∗(ω^p)∧ω^k−p 1/n

. (2)

Using an inequality of Lelong [18], Gromov [17] proved thath(f)lov(f). Following Gromov and Yomdin [23,16,17], we have

h(f) = lov(f) = max

1pklogd_p

whenf is a holomorphic map. Using Corollary 1.2, we prove, in the same way as in [10], that the sequences in (1), (2) are convergent and that the dynamical degreesdp are bimeromorphic invariants off. More precisely, ifΠ :X→Xis a bimeromorphic map between compact Kähler manifolds, the dynamical degrees ofΠ⁻¹◦f◦Πare equal todp. Using Corollary 1.2, we also get the following result.

THEOREM 1.4. – Letfbe a dominating meromorphic self-map on a compact Kähler manifold X of dimensionk. Letd_pbe the dynamical degrees off. Then

h(f)lov(f) = max

1pklogdp.

This theorem gives a partial answer to a conjecture of Friedland [15] which says that h(f) = max_1pklogd_p. Theorem 1.4 is already proved in [10] for rational maps on projective manifolds. Corollary 1.2 permits to extend the proof to compact Kähler manifolds. One can also extend some results on meromorphic correspondences or transforms, which are proved in the projective case in [11] (see also [8]).

In the last two sections, we extend Theorem 1.1 to positive pluriharmonic currents and currents of class DSH. We also study the intersection of such currents with positive closed(1,1)-currents.

We thank the referee for his constructive observations that helped to improve the exposition.

2. A classical lemma

We give here a classical lemma that we use in Section 3. LetBdenote the unit ball inR^m. LetK(x, y)be a function with compact support inB×B, smooth inB×B\∆where∆is the diagonal ofB×B. Assume that, for every(x, y)

K(x, y)A|x−y|^2−m (3)

whereA >0is a constant andx= (x1, . . . , xm)are coordinates ofR^m. Observe that for everyy K(., y)

L^1+δA (4)

for someδ >0andA>0. Assume also that for everyx,y ∇K(x, y)A|x−y|^1−m. (5)

In this section, we identifyν, a current of degree0and of order0, with the current of degreem, νdy₁∧ · · · ∧dy_m. LetMdenote the set of Radon measures onR^m. We define a linear operator P onMby:

P µ(x) :=

y∈R^m

K(x, y) dµ(y).

Observe that the functionP µhas support inB. We have the following lemma.

LEMMA 2.1. – The operatorPmaps continuouslyMintoL^1+δ. It also maps continuouslyL^p intoL^q,L^∞intoC⁰andC⁰intoC¹, whereq=∞ifp⁻¹+ (1 +δ)⁻¹1andp⁻¹+ (1 +δ)⁻¹= 1 +q⁻¹otherwise.

All the assertions are easy to deduce from (3), (4) and (5) and the Hölder inequality.

3. Proof of Theorem 1.1

Let ∆ denote the diagonal of X ×X. We first give a weak regularization of the current of integration [∆]. LetX×X denote the blow-up of X×X along ∆. Following Blanchard [4], X×X is a Kähler manifold. Letπ:X×X →X×X be the canonical projection and

∆ := π⁻¹(∆). Then∆ is a smooth hypersurface in X×X. Ifγ is a closed strictly positive (k−1, k−1)-form onX×X, thenπ_∗(γ∧[∆]) is a non-zero positive closed(k, k)-current on X×X supported on∆. So, it is a multiple of[∆]. We chooseγso thatπ_∗(γ∧[∆]) = [∆]. We will use the following regularization of[∆].

Since[∆] is a positive closed(1,1)-current, there exist a quasi-p.s.h. functionϕand a smooth closed (1,1)-form Θ such that dd^cϕ= [∆] −Θ. Recall that d^c:= _2πⁱ (∂−∂). Demailly’s regularization theorem [7] implies the existence of smooth functionsϕnand of a smooth positive closed(1,1)-formΘonX×X such that

• dd^cϕ_n−Θ;

• ϕndecrease toϕ.

In this case, independently of Demailly’s theorem, we can construct the functions ϕn as follows. Observe thatϕ is smooth out of∆ andϕ⁻¹(−∞) =∆. Let χ:R∪ {−∞} →Rbe a smooth increasing convex function such that χ(x) = 0 on[−∞,−1],χ(x) =xon [1,+∞[

and0χ1. Defineχ_n(x) :=χ(x+n)−nandϕ_n:=χ_n◦ϕ. The functionsϕ_nare smooth decreasing toϕand we have

dd^cϕn= (χ_n◦ϕ) dϕ∧d^cϕ+ (χ_n◦ϕ) dd^cϕ (χ_n◦ϕ) dd^cϕ=−(χ_n◦ϕ)Θ−Θ (6)

where we choose the smooth positive closed formΘbig enough so thatΘ−Θis positive.

DefineΘ⁺_n := dd^cϕn+ ΘandΘ⁻_n := Θ−Θ thenΘ⁺_n −Θ⁻_n →[∆]. We have Θ^±_nc0

wherec0>0is a constant. The formsΘ^±_n are smooth. Define K_n^±:=γ∧Θ^±_n and K_n^±:=π_∗(K_n^±).

The (k, k)-forms K_n^± are positive closed with coefficients in L¹ and smooth out of ∆. We also have K_n⁺ −K_n⁻ →[∆] weakly and K_n^±c1, c1>0. This is what we call a weak regularization of[∆]. We will useK_n^±to regularize the currentT. The following lemma shows that the coefficients of K_n^± satisfy inequalities of type (3) and (5) for m= 2k. Then, the singularities ofK_n^±are the same as the singularities of the Bochner–Martinelli kernel.

LEMMA 3.1. – Let(x, y) = (x1, . . . , xk, y1, . . . , yk),|xi|<3,|yi|<3, be local holomorphic coordinates of a chart ofX×X such that∆ = (y= 0)in that chart. LetH_n^±be a coefficient of K_n^±in these coordinates. Then, there exists a constantA_n>0, depending onn, such that

H_n^±(x, y)A_n|y|^2−2k and |∇H_n^±|A_n|y|^1−2k for|xi|1,|yi|1andy= 0.

Proof. – By symmetry, it is sufficient to consider(x, y)in the open sectorS defined by the inequalities|xi|<3,|yi|<3,|yi|<3|y1|and prove the estimates in the sectorS defined by

|xi|<2,|yi|<2and|yi|<2|y1|(we can assume thaty1 is the largest coordinate of the point y= 0). LetSandS be the interiors ofπ⁻¹(S)and ofπ⁻¹(S)respectively. We consider the coordinate system(x, Y)ofSwith|x_i|<3,Y₁=y₁andY_i=y_i/y₁,|y₁|<3,|y_i|<3|y₁|for i= 2, . . . , k. We haveπ(x, Y) = (x, y)for(x, y)∈S. The equation of∆ inSisY₁= 0.

SinceK_n^±are smooth onS, they are finite sums of forms of type Φ(x, Y) =L(x, Y) dxI∧dxI∧dYJ∧dYJ

whereLis a smooth function,I,I,J,Jare subsequences of{1, . . . , k}and dxI= dxi1∧ · · · ∧dxim ifI={i1, . . . , im}.

Hence, inSthe formsK_n^±are finite sums of forms of typeπ_∗(Φ).

Observe thatπ_∗(Φ)is obtained fromΦ(x, Y)replacingY1byy1andYi byyi/y1. There are here at most2k−2factors of the formd(yi/y1) = dyi/y1−yidy1/y²₁or their conjugate. Hence, the coefficients ofπ_∗(Φ)onSare finite sums of

L(x, y1, y2/y1, . . . , yk/y1)P(y)y^−m₁ y⁻ⁿ₁

whereP is a homogeneous polynomial inyandy such thatdeg(P) + 2k−2m+n. Since SS,Lis bounded onSandL(x, y1, y2/y1, . . . , yk/y1)is bounded onS. The first estimate of the lemma follows.

For the second estimate, it is sufficient to observe that the coefficients in the gradient of L(x, y1, y2/y1, . . . , yk/y1)P(y)y^−m₁ y⁻ⁿ₁

are combinations of functions of the same type with homogeneous polynomials P such that deg(P) + 2k−1m+n. 2

Define

T_n^±(x) :=

y∈X

K_n^±(x, y)∧T(y).

(7)

Letπidenote the canonical projections ofX×X on its factors. We have T_n^±:= (π1)_∗

K_n^±∧π^∗₂(T) . (8)

Observe that π₂^∗(T) is well defined sinceπ2 is a submersion. The currents K_n^±∧π₂^∗(T) are positive closed and well defined on X ×X\∆. They are of finite mass since, for each n, Kn⁺(., y)L¹ is uniformly bounded. A priori, the mass depends on n. By Skoda’s extension theorem [22], their trivial extensions are positive and closed. It follows thatT_n^± are well defined and are positive closed currents on X. The use of Skoda’s theorem can be replaced by an argument similar to the one in the proof of the following lemma.

LEMMA 3.2. – The currents T_n⁺ −T_n⁻ converge weakly to T when n → ∞. Moreover, T_n^±cTwherec >0is a constant independent ofnandT.

Proof. – DefineΠ :=π2◦π. Observe thatΠis a submersion fromX×X ontoX andΠ_|^∆is a submersion from∆ ontoX. Indeed, consider chartsUV⊂X that we identify with open sets inC^k. Assume thatU is small enough and0∈U. We can, using the change of coordinates (z, w)→(z−w, w)onV×U, reduce to the product situationV ×U,UV ⊂C^k where∆ is identified to{0} ×U. The blow-up along{0} ×Uis still a product. SoΠ^∗of a current is just integration on fibers. We can use this local model for the assertions below.

The potential of∆is integrable with respect toΠ^∗(T)since its singularity is likelog dist(z,∆) and this function has bounded integral on fibers ofΠ. In particular,[∆] ∧Π^∗(T)is well defined (we use here thatTis closed) and is equal to(Π_|∆)^∗(T), and[∆] has no mass forΠ^∗(T)nor for K_n^±∧Π^∗(T). We then have

K_n^±∧π^∗₂(T) =π_∗ K_n^±∧Π^∗(T) (9)

since the formula is valid out of ∆ and there is no mass on ∆. The potentials of K_n⁺ are decreasing and the currentsK_n⁻are independent ofn, hence

K_n⁺∧Π^∗(T)−K_n⁻∧Π^∗(T)→γ∧[∆] ∧Π^∗(T) =γ∧(Π_|∆)^∗(T).

(10)

Sinceπ_|^∆ is a submersion onto∆, we have(Π_|^∆)^∗(T) = (π_|^∆)^∗(π_2|∆)^∗(T). Hence π_∗

γ∧(Π_|^∆)^∗(T)

= (π_2|∆)^∗(T).

This and (9), (10) imply that

K_n⁺∧π₂^∗(T)−K_n⁻∧π^∗₂(T)→(π_2|∆)^∗(T).

Taking the direct image underπ1givesT_n⁺−T_n⁻→T.

SinceΠis a submersion,Π^∗(T)c2Twherec2>0is independent ofT. Observe that since K_n^± are smooth we can computeK_n^±∧Π^∗(T) cohomologically. The cohomological classes ofK_n^±are bounded, hence there exists a constantc₃>0such that

K_n^±∧Π^∗(T)c3T.

It follows that

Tn^±=(π1)_∗π_∗ K_n^±∧Π^∗(T)cT wherec >0is independent ofnandT. 2

The proof of Theorem 1.1 is completed by the following three steps.

Step 1. We show first that we can choose in Theorem 1.1 forms T_n^± withL¹ coefficients.

DefineT_n^±as in (7) and (8). We use partitions of unity ofXand ofX×Xin order to reduce the problem to the case ofR^m. Following Lemmas 2.1 and 3.1, the formsT_n^±haveL¹coefficients.

Lemma 3.2 implies thatT_n⁺−T_n⁻→TandT_n^±cT. Of course, in general,T_n⁺−T_n⁻do not converge inL¹since the constantsAnin Lemma 3.1 depend onn.

Step 2. We can now assume that T is a form with L¹ coefficients. Define T_n^± as in (7) and (8). Lemmas 2.1 and 3.1 imply thatT_n^± are forms with coefficients inL^1+δ. We also have T_n⁺−T_n⁻→TandT_n^±cT. Hence, we can assume thatTis a form withL^1+δcoefficients.

We repeat this process N times withNδ⁻¹. Lemmas 2.1, 3.1 and 3.2 allow to reduce the problem to the case whereT is a form withL^∞coefficients. If we repeat this process two more times, we can assume thatTis aC¹form.

Step 3. Now assume thatTis of classC¹. We can also assume thatTis strictly positive. LetΩ be a smooth real closed(p, p)-form cohomologous toT. Using standard Hodge theory [6], there is a real(p−1, p−1)-formuof classC²such thatT= Ω + dd^cu. Let(un)be a sequence of real smooth(p−1, p−1)-forms such thatun→uinC² norm. The currentTn:= Ω + dd^cun

converges toTinC⁰norm. Moreover,T_nis positive fornbig enough sinceTis strictly positive.

This completes the proof of Theorem 1.1. 2

4. Pluriharmonic and DSH currents

In this section, we extend Theorem 1.1 to positive pluriharmonic currents, i.e. positive dd^c-closed currents, and currents of class DSH. We have the following result which is new even for bidegree(1,1)currents.

THEOREM 4.1. – LetTbe a positivedd^c-closed(p, p)-current on a compact Kähler manifold (X, ω). Then there exist smooth positive dd^c-closed forms T_n^± such that T_n⁺ −T_n⁻ →T. Moreover,T_n^±cXTwherecX>0is a constant independent ofT.

We deduce from this theorem the following corollary.

COROLLARY 4.2. – LetX,X,Π,πandCbe as in Corollary 1.3. IfTis as in Theorem 4.1, then the positivedd^c-closed currentπ^∗(T)is well defined. Moreover the operatorT→π^∗(T) is l.s.c. andπ^∗(T)cΠTwherecΠ>0is a constant independent ofT.

To prove the corollary, observe that by Theorem 4.1, the positive pluriharmonic currentπ^∗(T), which is well defined on X\C, has finite mass. Following Alessandrini and Bassanelli [1], π^∗(T)satisfiesdd^cπ^∗(T)0. Then, Stokes Theorem implies thatdd^cπ^∗(T) = 0.

Proof of Theorem 4.1. – We use the same idea as in Section 3. ClearlyT_n^± given by (7), (8) and (9) are pluriharmonic positive currents. We only need to check thatT_n⁺−T_n⁻→T. The rest of proof is the same as in Theorem 1.1.

Letϕandϕ_nbe quasi-p.s.h. functions as in Section 3. We want to prove the analog of (10):

(dd^cϕn+ Θ)∧Π^∗(T)→(Π_|∆)^∗(T).

(11)

The problem is local. Define S:= (Π_|^∆)^∗(T). We choose as in Lemmas 3.1 and 3.2 local holomorphic coordinates(x1, . . . , x2k)of an open setW⊂X×X,|xi|<1, so that inW

• ∆ = {x2k= 0}; henceψ:=ϕ−log|x2k|is smooth anddd^cψ=−Θ;

• Π(x1, . . . , x2k) = (x1, . . . , xk).

Defineτ(x1, . . . , x2k) := (x1, . . . , x_2k−1). SinceΠ = Π_|^∆◦τ, we haveΠ^∗(T) =τ^∗(S)inW. Observe that(dd^cϕ_n+ Θ)∧τ^∗(S)is supported in(ϕ <−n+ 2)and, by (6),

(dd^cϕ_n+ Θ)∧τ^∗(S)(1−χ_n◦ϕ)Θ∧τ^∗(S).

The definition of χn implies that the measures (1−χ_n ◦ϕ)Θ∧τ^∗(S) tend to 0. Hence, every limit value of(dd^cϕn+ Θ)∧τ^∗(S)is a positive currentR supported in∆. Following

Bassanelli [2], it is a current on∆ (this is true for every positive currentT supported in∆ such thatdd^cRis of order 0). Hence, in order to prove (11) we only have to check that

W

Ψ(x2k)(dd^cϕn+ Θ)∧τ^∗(Φ∧S)→

∆

Φ∧S

for every test(2k−p−1,2k−p−1)-formΦwith compact support in∆∩W and for every functionΨ(x_2k)supported in {|x_2k|<1}, such thatΨ(0) = 1. Observe that since τ^∗(Φ∧S) is proportional todx₁∧dx₁∧ · · · ∧dx_2k−1∧dx_2k−1only the component ofdd^cϕ_n+ Θ with respect todx_2k∧dx_2kis relevant. When(x₁, . . . , x_2k−1)is fixed, we have

x2k

Ψ(dd^c_x2kϕn+ Θ)→1

sincedd^c_x2kϕn+ Θconverges to the Dirac massδ0andΨ(0) = 1. The last integral is uniformly bounded with respect tonandx1, . . . , x_2k−1because by (6) one can prove that the masses of the measuresdd^c_x2kϕ_n+ Θon a compact set of{|x_2k|<1, x₁, . . . , x_2k−1fixed}are uniformly bounded. This implies the result. 2

Remark 4.3. – Theorem 4.1 implies that on an arbitrary compact Kähler manifold (X, ω) positive pluriharmonic currents T of bidegree (1,1) have finite energy. We then have T = Ω +∂S+∂S+i∂∂vwithΩsmooth closed,S,∂S,∂S inL² andv inL¹. The energy ofT is equal to

∂S∧∂S∧ω^k−2. The case of the projective space is treated in [14]. To extend the result to an arbitrary compact Kähler manifold, one has to use the approximation Theorem 4.1, to go from a priori estimates on smooth positive pluriharmonic forms to the estimates on positive pluriharmonic currents.

LetDSH^p(X)denote the space of(p, p)-currentsT=T1−T2whereTiare negative currents, such thatdd^cTi= Ω⁺_i −Ω⁻_i withΩ^±_i positive closed. Observe thatΩ⁺_i =Ω⁻_i . We define the DSH-norm ofT as

TDSH:= min

T1+T2+Ω⁺₁+Ω⁺₂, Ti, Ω^±_i as above . We say thatTn→TinDSH^p(X)ifTn→Tweakly and(TnDSH)is bounded.

The spacesDSH^p(X)are analogous to the space generated by quasi-p.s.h. functions. They are useful in order to study the regularity of Green currents in dynamics [11,12]. The proof of the following theorem, which gives the density of smooth forms inDSH^p(X), follows the lines of previous approximation results and is left to the reader. In this case, for the control of the mass ofT_n^±we need to estimate the mass ofdd^cϕn∧Π^∗(Ti). It is sufficient to estimate the mass of ϕnΠ^∗(dd^cTi)using the definition ofϕn.

THEOREM 4.4. – Let T be a current in DSH^p(X). Then there exist smooth real (p, p)- forms T_n such thatT_n→T. Moreover, T_nDSHc_XTDSH where c_X >0 is a constant independent ofT.

Remark 4.5. – We haveK_n⁺−K_n⁻−γ∧[∆] = γ∧dd^c(ϕn−ϕ)andϕn=ϕout of the set (ϕ <−n+ 2). Hencesupp(K_n⁺−K_n⁻)converge to∆, supp(K_n⁺−K_n⁻)converge to∆ and supp(T_n⁺−T_n⁻)converge tosupp(T).

We also have the following useful proposition.

PROPOSITION 4.6. – LetT be a continuous form andT_n^±be the forms defined in (7) and (8).

ThenT_n:=T_n⁺−T_n⁻ converge uniformly toT. WhenT is closed (resp.dd^c-closed) thenT_n^± are closed (resp.dd^c-closed).

Proof. – We can approximateTuniformly by smooth forms. We then assume thatTis smooth.

The form T_n−T is the push-forward of (K_n⁺−K_n⁻−γ∧[∆]) ∧Π^∗(T)by Π:=π₁◦π (see Lemma 3.1). The last current is equal toT_n:= dd^c(ϕ_n−ϕ)∧γ whereγ is a smooth form. Using a partition of unity, we reduce the problem to a local situation with the coordinates x= (x, x) = (x1, . . . , xk, xk+1, . . . , x2k), Π(x) =x, ∆ = (x 2k = 0) and γ of compact support as in the proof of Theorem 4.1. We have to check thatΠ_∗(T_n)(x) =

xT_n(x)converge uniformly to 0.

Observe that the last integral is taken in the neighbourhood(ϕ <−n+ 2)of(x2k= 0) and the formTn−dd^c_x(ϕn−ϕ)∧γis of order1/|x2k|since in the difference we get at most one derivative with respect tox2k. Hence, it is sufficient to estimate

x

dd^c_x(ϕn−ϕ)∧γ=

x

(ϕn−ϕ)∧dd^c_xγ.

It is clear that these forms converge uniformly to0. 2

5. Intersection of currents

We want to consider a class of positive pluriharmonic currents which are of interest in some problems of complex analysis and dynamics. Some of their properties are given in [21,1,2,13,9, 14]. Given a compact Kähler manifold(X, ω)of dimensionk, we want to define the intersection S∧T of a positive closed(1,1)-currentSwith a positive pluriharmonic currentT of bidegree (p, p),1pk−1. We have seen a special case of this situation in the last section.

We write S=α+ dd^cuwithα smooth and ua quasi-p.s.h. function. We say that uis a potential ofS.

THEOREM 5.1. – Assume thatuis continuous. Then S∧T is well defined and is a positive dd^c-closed current. MoreoverS∧T depends continuously onS andT in the following sense.

LetTn be positive pluriharmonic currents converging weakly toT. IfSn=α+ dd^cunwithun

continuous converging uniformly touthenS_n∧T_nconverges weakly toS∧T. In particular, it holds when theu_nare continuous and decrease tou.

We first prove the following proposition for smooth potentials. We will see later that it can be extended to continuous quasi-p.s.h. functionsv^± and thatdv^±∧R andd^cv^±∧Rare well defined in this case.

PROPOSITION 5.2. – Let v^± and v =v⁺−v⁻ be smooth real functions on X such that dd^cv^± = Θ^± −α where α is a smooth closed (1,1)-form and Θ^± are positive closed (1,1)-currents. Let R be a positive current inDSH^p(X)with dd^cR= Ω⁺−Ω⁻ whereΩ^± are positive closed currents. Then

dv∧d^cv∧R∧ω^k−p−1vL^∞

2 α∧R∧ω^k−p−1+ 3

v⁺L^∞+v⁻L^∞

Ω^±

.

In particular, ifRis positive pluriharmonic, we have

dv∧d^cv∧R∧ω^k−p−12vL^∞ [α]∧[R]∧[ω]^k−p−1.

Proof. – Observe that vL^∞ v⁺L^∞ +v⁻L^∞, Θ⁺=Θ⁻ and Ω⁺=Ω⁻. Hence

dv∧d^cv∧R∧ω^k−p−1

1

2

dd^cv²∧R∧ω^k−p−1 +

vdd^cv∧R∧ω^k−p−1

=1 2

v²∧dd^cR∧ω^k−p−1 +

v(Θ⁺−Θ⁻)∧R∧ω^k−p−1

1

2v²_L∞ (Ω⁺+ Ω⁻)∧ω^k−p−1+vL^∞ (Θ⁺+ Θ⁻)∧R∧ω^k−p−1

=v²L^∞Ω^±+ 2vL^∞ α∧R∧ω^k−p−1

+vL^∞ dd^c(v⁺+v⁻)∧R∧ω^k−p−1

vL^∞

v⁺L^∞+v⁻L^∞

Ω^±+ 2vL^∞ α∧R∧ω^k−p−1

+vL^∞ (v⁺+v⁻)∧(Ω⁺−Ω⁻)∧ω^k−p−1 vL^∞

2 α∧R∧ω^k−p−1+ 3

v⁺L^∞+v⁻L^∞

Ω^±

. 2

Proof of Theorem 5.1. – Observe that, whenun decreases to u, the Hartogs lemma implies thatu_nconverges uniformly tou. By Demailly’s regularization theorem [7], we can assume that u_n is smooth and uniformly convergent tou. So,S_n∧T_n is well defined. We will prove that S_n∧T_nconverges. This also implies that the limit depends only onSandT.

We first consider the case whereTn=T. We then have

S_n∧T=α∧T+ d(d^cu_n∧T)−d^c(du_n∧T)−dd^c(u_nT).

(12)

Proposition 5.2 applied tou_n−u_mand the Cauchy criterion imply thatdu_nandd^cu_nconverge inL²(T∧ω^k−p−1). HenceS_n∧Tconverges anddu∧T,d^cu∧T are well defined.

To complete the proof we write

S_n∧T_n−S∧T= dd^c(u_n−u)∧T_n+ dd^cu∧(T_n−T) +α∧(T_n−T).

The last term tends to zero. Proposition 5.2 implies that

d(u_n−u)∧d^c(u_n−u)∧T_n∧ω^k−p−1 has zero limit. An identity as in (12) and Cauchy–Schwarz’s inequality show that the first term tends to 0. For the second term, we use again (12). We only need estimates of the following type. Ifγis a test 1-form,Θis a smooth positive(k−p−1, k−p−1)-form andvis a smooth quasi-p.s.h. function withu−vL^∞anddd^cv−cω, then

du∧γ∧(T−T_n)∧Θ = dv∧γ∧(T−T_n)∧Θ

+ d(u−v)∧γ∧(T−Tn)∧Θ.

The first integral tends to zero. Cauchy–Schwarz’s inequality and Proposition 5.2 imply d(u−v)∧γ∧Tn∧Θ

²const d(u−v)∧d^c(u−v)∧Tn∧Θ constu−vL^∞T_n

and similarly forT. 2

In the same way, using the full strength of Proposition 5.2, one can prove the following theorem.

THEOREM 5.3. – Let T be a current inDSH^p(X)andS as in Theorem 5.1. ThenS∧T is well defined and belongs toDSH^p+1(X). MoreoverS∧T depends continuously on SandT. The topology on theT variable is the topology ofDSH^p(X).

It is enough to assumeTpositive and to modify (12) into

S_n∧T=α∧T+ d(d^cu_n∧T)−d^c(du_n∧T)−dd^c(u_nT) +u_ndd^cT.

Remarks 5.4. – If Si are positive closed (1,1)-currents with continuous potentials, then S1∧ · · · ∧Sm∧T is symmetric inSi since this is true whenSi andT are smooth. Letube a quasi-p.s.h. function on an open ball Ω⊂X. By the maximum regularization procedure as in [6], ifuis continuous we can extenduto a continuous quasi-p.s.h. function onX. Hence, dd^cu∧T is well defined onΩ.

WhenTis only a (positive pluriharmonic)(p, p)-current onΩ, we do not know how to define dd^cu∧T without additional hypothesis onu. Assume thatT,dT anddd^cT are of order 0 and uis locally integrable with respect to the coefficient measures ofT,dTanddd^cT. Then we can define

dd^cu∧T:= dd^c(uT) +udd^cT−d(ud^cT) + d^c(udT).

Ifunare p.s.h. and decrease touor ifunconverge uniformly tou, we have dd^cu_n∧T→dd^cu∧T.

WhenTis positive, we also have an inequality of Chern–Levine–Nirenberg type (see [6, p. 126]).

More precisely, ifK,Lare compact sets inΩwithLK, then dd^cu∧TLcK,L

uTK+udTK+udd^cTK

.

Note that positive harmonic currents associated to a lamination by Riemann surfaces satisfy the above hypothesis (see [3]).

IfT is of bidegree(1,1)we can extend Theorem 5.1 to currentsSwith bounded potential.

PROPOSITION 5.5. – LetT be a positive pluriharmonic current of bidegree(1,1)in(X, ω).

Ifuis a bounded quasi-p.s.h. function thendd^cu∧T is well defined. If(un)is a bounded se- quence of quasi-p.s.h. functions converging pointwise to u with dd^cun −cω, then dd^cun∧T→dd^cu∧T.

Proof. – We can assume thatu_nare smooth and positive. It is easy to check thatu²_nare quasi- p.s.h. and converge tou². It follows that∂u_n(resp.∂u_n) converge to∂u(resp.∂u) weakly in L²(X).

As in Theorem 5.1, we only need to show that∂un∧T (resp.∂un∧T) converges weakly.

Recall that we can writeT = Ω +∂S+∂S+i∂∂vwithΩsmooth closed,∂S,∂S inL²,vin L¹[14]. Ifγis a test1-form, we have

∂u_n∧T∧γ∧ω^k−1=− u_n∂T∧γ∧ω^k−1− u_nT∧∂γ∧ω^k−1

=− un∂∂S∧γ∧ω^k−1− unT∧∂γ∧ω^k−1. The second term tends to

uT ∧∂γ∧ω^k−1. The first term is equal to

− ∂un∧∂S∧γ∧ω^k−1+ un∂S∧∂γ∧ω^k−1

which converges to

− ∂u∧∂S∧γ∧ω^k−1+ u∂S∧∂γ∧ω^k−1 sinceu_n→uand∂u_n→∂uweakly inL².

The convergence of∂un∧T is proved in the same way. 2

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(Manuscrit reçu le 29 mars 2004 ; accepté, après révision, le 28 septembre 2004.)

Tien-Cuong DINH Université Paris-Sud, Mathématique - Bât. 425,

UMR 8628, 91405 Orsay, France

E-mail: [email protected]

Nessim SIBONY Université Paris-Sud, Mathématique - Bât. 425,

UMR 8628, 91405 Orsay, France E-mail: [email protected]