A Liouville theorem for plurisubharmonic currents

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

FREDJELKHADHRA, SOUADMIMOUNI

A Liouville theorem for plurisubharmonic currents

Tome XIX, n^o3-4 (2010), p. 651-674.

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A Liouville theorem for plurisubharmonic currents

Fredj Elkhadhra⁽¹⁾, Souad Mimouni⁽²⁾

ABSTRACT.— The goal of this paper is to extend the concepts of alge- braic and Liouville currents, previously defined for positive closed currents by M. Blel, S. Mimouni and G. Raby, to psh currents onC^{n. Thus, we}

study the growth of the projective mass of positive currents onC^nwhose

support is contained in a tubular neighborhood of an algebraic subvari- ety. We also give a sufficient condition guaranteeing that a negative psh current is Liouville. Moreover, we prove that every negative psh algebraic current is Liouville. For the particular case of closed currents, under ade- quate support conditions, we obtain a structure theorem.

R^´ESUM´E.— Le but de ces papiers est d’´etendre les concepts de courants alg´ebrique et Liouville pr´ec´edemment d´efinis pour les courants positifs ferm´es par M. Blel, S. Mimouni et G. Raby aux courants psh surC^{n. Nous}

´etudions alors la croissance de la masse projective des courants positifs d´efinis surCⁿ dont le support est contenu dans un voisinage tubulaire d’une sous-vari´et´e alg´ebrique. Ensuite, nous donnons une condition suff- isante, garantissant qu’un courant n´egatif et psh soit Liouville. De plus, on montre que tout courant n´egative psh et alg´ebrique est Liouville. Dans le cas particulier des courants ferm´es, et sous des conditions ad´equates sur le support, nous obtenons un th´eor`eme de structure.

(∗) Re¸cu le 19/06/2008, accept´e le 03/06/2010

(1) D´epartement de Math´ematique, Facult´e des sciences de Monastir, 5000 Monastir Tunisie.

[email protected]

(2) D´epartement de Math´ematique, Facult´e des sciences de Monastir, 5000 Monastir Tunisie.

[email protected]

1. Introduction

The class of positive, or negative plurisubharmonic (psh for short) cur- rents appears today as a tool for the study of analytic objects and as a natural extension of plurisubharmonic functions [Ga], [D-L], [D-E-E]... In [D-S], the authors used negative psh currents to describe polynomial hulls of compact sets inCⁿ. In particular, positive pluriharmonic currents have so many important application in non K¨ahler geometry [H-L],[A-B] and also for the study of laminations with singularities of a compact set inP² [F-S].

In the present work, we deal with thealgebraicand Liouvilleproperties of this class of currents related with certain support conditions. The following definition will be useful.

Definition 1.1. —LetT be a current of order zero and of bidimension (p, p) onCⁿ. One says thatT is algebraic if there exists a currentTof order zero onPⁿ such thatT=T onCⁿ andT= 0 on the hyperplane at infinity H_∞.

Letω_{F S} be the Fubini-Study K¨ahler form onPⁿ, its restriction toCⁿ is given by ω_{F S} =dd^clog(1 +|z|²) up to a constant. The topic of our paper is positive algebraic currents, i.e. currents T onCⁿ ⊂Pⁿ that have finite mass locally near the hyperplane at infinity. This is equivalently formulated by saying that the projective mass Tp.m =

CⁿT ∧ω_{F S}^p is finite, (p, p) denoting the bidimension ofT. Note thatT∧ω^p_{F S}is the trace measure ofT with respect to the Fubini-Study K¨ahler formωF S. On the other hand if we extend the concept of thedegree for positive currents, then it is clear that there is a one-to-one correspondence between the class of positive algebraic currents and those of finite projective mass or equivalently of finite degrees.

Thanks to the Demailly-Lelong-Jensen formula [De], a positive plurihar- monic currentT is algebraic if and only if the quantityνT(r) := ^σT({|z|<r})

τpp!r^2p

is bounded independently ofr, where σT :=T ∧β^p is the trace of T with respect to the flat metric β = dd^c|z|². Then, one recovers the definition given by [B-M-R] in case whenT is closed.

Example 1.2 of positive algebraic currents. —

1. Let L ={v ∈ psh(C²), v(z, w)log⁺||(z, w)||+O(1) at infinity}.

By [Le], for allv∈ Lthe currentdd^cv is algebraic. Conversely, all al- gebraic closed positive current of bidegree (1,1) onC²can be written cdd^cvwith c0 andv∈ L.

2. Letg be the H´enon map defined by g(z, w) = (z²+c+aw, z), with (a, c)∈C^∗×C. We denote byG⁺(z, w) = lim

n→∞(1/2ⁿ) log⁺||gⁿ(z, w)||

andG⁻(z, w) = lim

n→∞(1/2ⁿ) log⁺||g⁻ⁿ(z, w)||. By [B-S], we haveG⁺ ∈ L. It follows that the currentsT^±=dd^cG^±are algebraic onC². These currents, which are also called Green currents, play a central role in the theory of complexdynamics. Let be T⁻ the trivial extension of T⁻ onP² andϕa negative quasi-psh function onP². By [C-G], one hasϕ∈L¹(T⁻∧ω_{F S}). It follows that the currentϕ_|C2T⁻is negative and algebraic onC².

3. Let χ ∈ D([0,1]), ψ(z, w) = χ(|z|²) +iχ(|w|²), with (z, w) ∈ C², and T =i∂ψ∧∂ψ. Then, T is a positive pluriharmonic current on C². Moreover, it is not hard to see that T has a total finite pro- jective mass on C², therefore it is algebraic and the trivial exten- sionT is a positive pluriharmonic current on P² (see [D-E-E]). Let us note here that in [F-S], the authors give explicitly the current T and used it for the evaluation of the infimum of the energy (i.e.

inf{

P²T∧T, T 0 onP², dd^cT = 0,

P²T∧ω_{F S}= 1}).

By [B-M-R], a closed positive current of bidegree (1,1) and with tubular support (i.e. included in{|P|c^te}whereP is a non constant polynomial in C[z₁, ..., z_n]) is shown to be an algebraic current. In the first section of the present work, we will consider positive currents T whose support is contained in the tube {|P₁|+. . .+|P_s| c^te}. With adequate conditions on the polynomials P_j, j = 1, ..., s, we study the growth of the projective mass of T and the quantity ν_T according to whether T or −T is psh. In particular, we show that ifdd^cT = 0, thenT is algebraic. More precisely we prove :

Theorem 1.3. — LetP₁, ..., P_s,(s+k=n)be polynomials inC[z₁, ..., z_n] having the same degree δ. Suppose that the intersection of the zeros of their homogeneous parts of top degrees form an algebraic subset of codimensions inPⁿ⁻¹. LetT be a positive current with bidimension(p, p)onCⁿ such that pk. Assume thatdd^cT is negative and thatSuppT ⊂ {|P1|+...+|Ps|1}.

Then, there exists a constant c > 0 such that for all r 1 we have : νT(r)c. If furthermore T is also pluriharmonic, then T is an algebraic current.

In the special case where T is a negative plurisubharmonic function, Theorem 1.3 is classical without any assumption on the support. Notice that in [E-M], the authors establish Theorem 1.3 in the case when P1 =

z_k+1^δ , ..., P_s=z_n^δ (i.e. the support of T is contained in a strip). In the con- text of dynamics, this class of currents is interesting and many of them can be constructed as the invariant currents of certain polynomial endo- morphisms [Du],... Furthermore, S. Giret shows in his thesis [Gi] that the class of positive closed currents with support in a strip are well preserved by pulling-back by a blow up with smooth center.

On the other hand, it is important to point out that Theorem 1.3 deals with a much larger class of currents than the class of closed currents. In fact whenT isd−closed, Theorem 1.3 is an immediate consequence of Theorem 2.4 in [B-M-R], by almost the same proof as in corollary 2.5 of that paper.

Indeed, the condition that the algebraic hypersurfaces {Pi = 0} intersect properly at infinity is clearly preserved by taking the intersection with a general hyperplaneH ofCⁿ.

Denote byTp.m(r) =

{|z|r}T∧w^p_{F S}the projective mass ofT carried by {|z| r} and by N_T(r) = r

1 ν_T(t)/t dt the counting map associated to T. As indicated in the introduction if T is positive and pluriharmonic the quantity ν_T(r) coincides with Tp.m(r), hence a direct computation shows that T is algebraic if and only if ν_T(r) = O(1) or, equivalently, N_T(r) = O(log r) (this equivalence will be proved later). In the general situation, we obtain the following estimates:

Proposition 1.4. —Let T be a positive current of bidimension (p, p) onCⁿ.

1. In both cases whenT is psh ordd^cT is negative, we have the growth estimateνdd^cT(r) =O(νT(√

2r)).In particular, ifνT is bounded then dd^cT has finite total projective mass i.e. dd^cT is algebraic.

2. If T is psh then Tp.m(r) = O(νT(r)). In particular, when νT is bounded or equivalently, NT has logarithmic growth, then T is alge- braic. If dd^cT is negative, then there exists c, c > 0 such that for every r 2, we have : Tp.m(r) c+c(νT(r) +NT(√

2r)). In particular ifν_T is bounded, then the projective mass ofT carried by {|z|r} has at most logarithmic growth i.e.Tp.m(r) =O(log r).

Let T be a positive current on Cⁿ, we say that T is Liouville if for every holomorphic functionf onCⁿ, bounded on the support ofT one has:

T∧dd^c|f|²= 0. The previous definition coincides then with the definition given in [B-M-R] when T is closed. In the same paper, the authors prove that a closed positive algebraic current is a Liouville current. For negative psh currents we obtain our second main result:

Theorem 1.5. —Let T be a negative psh current of bidimension (p, p) on Cⁿ. Let u be a C² plurisubharmonic function on Cⁿ and bounded on the support of T. Assume that we have the following growth condition:

Tp.m(r) =O((log logr)^s)for somes0, then the currents T∧du∧d^cu andT∧dd^cuvanish, and thereforeT is Liouville. In particular, every alge- braic negative psh current is Liouville.

Notice that Theorem 1.5 asserts the following elementary statement:

there are no negative psh compactly supported currents of bi-dimension (p, p), ifp >0 (we will takeu=|z|² and remark thatT has a globally finite projective mass).

Another immediate consequence of Theorem 1.5 is the following: letP is a non constant polynomial in C[z₁, ..., z_n] and T be a negative psh current of bi-dimension (p, p) on Cⁿ with support contained in {|P| 1}. Then the current T ∧dP ∧dP vanishes when T is algebraic (Theorem 1.5 for u = |P|²). This allows us to prove a structure theorem: if F : Cⁿ → C^k is a equi-dimensional polynomial map and T is a closed positive algebraic current of bidegree (k, k) on Cⁿ supported on the inverse image byF of a compact subset of C^k, then T can be split into two currents, the first of which can be written as an average of integration currents on components of fibres ofF and the other is supported by an algebraic set containing the critical points ofF. More precisely, we prove the following theorem:

Theorem 1.6. —Let F= (P₁, ..., P_k) :Cⁿ→C^k be a polynomial map- ping such that for all t ∈C^k the codimension of the fiber F⁻¹(t) in Cⁿ is k. Let T be a positive closed algebraic current of bidegree(k, k)onCⁿ such that SuppT ⊂ {|P1|+...+|Pk|1}. Denote by V the space of connected components of different fibers P⁻¹(t) in Cⁿ, t ∈ C^k F(X), where X is the set of critical values of F, then there exists a unique positive measure µ onV and a positive closed algebraic currentR supported by an algebraic set containing the critical points ofFsuch thatT =

v∈V[P⁻¹(t)]vdµ(v)+R.

In the following result, we give conditions on T weaker than those on the support, guaranteeing that T is a Liouville current. Let π : Cⁿ → C be the orthogonal projection. By [B-E], the sliceT, π, zexists outside of a pluripolar set in C. Let us denote by β = dd^c|z|², β = dd^c|z |² and v_ε(z, z ) =|z|²+ε|z |²forε >0.

Theorem 1.7. —Let T be a closed positive current of bidegree (1,1) on Cⁿ. Assume that for all R > 0 there exists a function ε(R) such that

0 < ε(R) −→ 0 if R −→ ∞, with

{vε(R)<R}T ∧β ⁿ⁻¹ =o(^R_ε(R)ⁿ⁻¹) and a constant γ > 0 such that for almost all z, the mass ||T, π, z||({vε(R) <

R}) =O(R^n−2−γ).Then, for all positive psh function uonCⁿ and bounded on the support of T, we haveT ∧dd^cu= 0. In particular, T is a Liouville current.

Example 1.8. — InC², the class of closed positive currents (and in the same way negative psh) of bidegree (1,1) and having a support in a strip {|z |1}satisfies the hypothesis of theorem 1.7.

2. Preliminaries

Let Ω be an open set of Cⁿ. As usual Dp,q(Ω) denotes the space of smooth and compactly supported (p, q)−form on Ω. The dual D_p,q(Ω) is the space of currents of bidimension (p, q) or of bidegree (n−p, n−q). For T ∈ D_p,p(Ω), one says that T is positive if for all α1, ..., αp in D1,0(Ω), the distributionT∧iα1∧α1∧...∧iαp∧αp determines a positive measure on Ω. We say that T is plurisubharmonic (psh for short) if the current dd^cT is positive, and pluriharmonic ifdd^cT = 0. Let β = dd^c|z|², (where d=∂+∂, d^c= (i/2)(∂−∂)), be the K¨ahler form onCⁿ. LetT ∈Dp,p(Ω) be of order zero on Ω. Then

T =i^(n−p)2

|I|=|J|=n−p

TIJdzI∧d¯zJ,

withT_IJare complexmeasures. One defines the mass measure of the current T byT=

|I|=|J|=n−p|TIJ|, and the trace measure by:

σT = 1

4^pp!T∧β^p= (2^−p

|I|=n−p

TII)τn.

Let A be a closed subset in Ω and T a current of order zero on ΩA.

LetTbe the trivial extension ofT by zero acrossA. We say thatT exists if T has locally finite mass on Ω. In the remaining part of this paper, we denote by Tp.m(r) =

{|z|r}T ∧w^p_{F S} the projective mass of T carried by{|z|r} and Tp.m =

CⁿT ∧w^p_{F S} the total projective mass onCⁿ. Let T be a positive current of bidimension (p, p) such that the measure dd^cT∧β^p−1 is positive on Ω. Let ϕbe aC² function on Ω such that logϕ is plurisubharmonic on{z∈Ω; ϕ(z)>0}. Let

B(r) ={z∈Ω; ϕ(z)< r}, w=dd^cϕ and α=dd^clogϕ.

For 0< r₁ < r₂ such that SuppT ∩B(r₂) is relatively compact in Ω, one has the following Lelong-Jensen formula [De] which is our basic tool in this paper:

1 r₂^p

B(r2)

T∧w^p− 1 r^p₁

B(r1)

T∧w^p=

B(r1,r2)

T ∧α^p

+ _r2

r1

1 t^p − 1

r^p_{2 B(t)}dd^cT∧w^p−1 +

1 r^p₁ − 1

r^p₂

r1

0

dt

B(t)

dd^cT∧w^p−1.

Recall that a homogeneous polynomial of degreeδinnvariables depends on (δ+n−1)!/δ!(n−1)!) coefficients. Hence, a polynomial homogeneous map F = (F₁, ..., F_n) can be identified with an an element of C^N, where N =n(δ+n−1)!/δ!(n−1)!. Moreover, by [G-K-Z] p.427, there exists a unique polynomialRes(F₁, ..., F_n) in the coefficients ofF₁, ..., F_n, such that Res(F₁, ..., F_n) = 0 if and only if the map F in degenerate. With the last identification, the space of all homogeneous, non degenerate, polynomial maps of degreeδonCⁿ is an open subset ofC^N.

We thank Prof. H. El Mir and J.-P. Demailly for a number of remarks which contributed to improve this article. We also would like to thank the referee for valuable comments.

3. Proof of Theorem 1.3

The proof of Theorem 1.3 is divided into two steps : Proof. — Step 1: case when P₁=z^δ_k+1, ..., P_s=z^δ_n.

First let us suppose that p=k, otherwise T vanishes. In fact, the current T isC−flat, therefore ifπis the projection on C^k, the sliceT, π, zexists for almost allz ∈C^k and it is a positive current having a negativedd^c and a compact support in Cⁿ. By [D-E-E], we haveT, π, z= 0 for almost all z and alsoT = 0 by applying the slicing formula for the C−flat currents.

Let us now continue the proof for the interesting casep=k. For instance assume that T is smooth. Let χ ∈ D(R), χ(t) = 1 if |t| 1 and χ = 0 if |t| > 2. Let β = dd^c|z|², and for a = (a1, ..., ap) ∈ C^p, let us denote g(a) =

CⁿT∧χ(|z +a|²)β ^p. Then

2i ∂²g

∂a1∂a1

=

CⁿT∧ 2i∂²

∂a1∂a1

χ(|z +a|²)β ^p

=

CⁿT∧ 2i∂²

∂z1∂z1

χ(|z +a|²)β^p

=

CⁿT∧dd^c

χ(|z +a|²)i

2dz₂∧dz₂∧...∧ i

2dz_p∧dz_p

=

Cⁿχ(|z +a|²)dd^cT∧ i

2dz₂∧dz₂∧...∧ i

2dz_p∧dz_p Taking into account the fact thatdd^cT is negative, the last integral is nega- tive. Hence the functiona1→ −g(a1, a2, ..., ap) is negative and subharmonic onC, therefore it is constant with respect toa₁and is equal tog(0, a₂, ..., a_p).

By iteration, one shows that g is independent from the variablesa₁, ..., a_p. Then,g(a) =g(0) =

CⁿT∧χ(|z|²)β ^p. Thus, there exists a constantC >0 such that

|z|1,zT∧β^pC.let bej∈ {p+ 1, ..., n}, then

CⁿT∧χ²(|z|²)dd^c|z_j|²∧β^p−1 =

CⁿT∧dd^c(|z_j|²χ²(|z|²)β^p−1)

−

Cⁿ|zj|²T∧dd^c(χ²(|z|²))∧β ^p−1

−

CⁿT ∧dχ²(|z|²)∧d^c|zj|²∧β ^p−1

−

CⁿT ∧d|zj|²∧d^cχ²(|z|²)∧β ^p−1

= (1) + (2) + (3) + (3).

On the other hand, using Stokes’s theorem and the fact that|z_j|²χ²(|z|²)β^p−1 has a compact support relatively toT, we find

(1) =

CⁿT∧dd^c

|zj|²χ²(|z|²)β ^p−1

=

Cⁿdd^cT∧|zj|²χ²(|z|²)β ^p−10.

Hence, we get the inequality

CⁿT∧χ²(|z|²)dd^c|zj|²∧β ^p−1(2) + (3) + (3). (3.1)

The term numbered (2) satisfies:

(2) =−

Cⁿ|zj|²T∧dd^c(χ²(|z|²))∧β ^p−1C

1|z|2

T∧β ^pC1. The existence of constant C follows from the fact that |zj| is bounded on the support ofT, observing that |χ|,|χ|and |χ |are bounded. To obtain C1, we may slightly modifyχ by taking χ(t) = 1 if |t| 2 and χ = 0 if

|t|>3 and repeat the above argument. Letϕ∈D(R), 0ϕ1 andϕ= 1 on Suppχ. According to the Cauchy-Schwarz inequality, we have:

(3)

CⁿT∧2χ(|z|²)ϕ(|z|²)dχ(|z|²)∧d^c|zj|²∧β ^p−1 (1/ε)

CⁿT∧2ϕ²(|z|²)dχ(|z|²)∧d^cχ(|z|²)∧β^p−1 +ε

CⁿT∧2χ²(|z|²)d|z_j|²∧d^c|z_j|²∧β^p−1

(C2/ε)

1|z|2

T∧β ^p+ 4ε

CⁿT∧χ²(|z|²)dzj∧dzj∧β ^p−1. Choosingε= 1/8 and by (3.1), we get:

CⁿT∧χ²(|z|²)dd^c|zj|²∧β ^p−1C1+8C2+1 2

CⁿT∧χ²(|z|²)dd^c|zj|²∧β ^p−1. PutC3= 2(C1+ 8C2). Asdd^c|z |²=_n

j=p+1dd^c|zj|², we have:

CⁿT∧χ²(|z|²)dd^c|z |²∧β^p−1(n−p)C3. In order to show that the integral

CⁿT ∧χ²(|z|²)(dd^c|z |²)²∧β ^p−2 is finite, we use the last inequality and we rewrite the previous proof with β^p−1replaced bydd^c|z |²∧β^p−2. While proceeding by induction, we show that there exists a constantC4>0 such that for 1spwe have:

CⁿT∧χ²(|z|²)(dd^c|z |²)^s∧β^p−sC₄. It follows that there existsC5>0 such that:

|z|1,|z|1

T ∧β^p

CⁿT ∧χ²(|z|²)β^pC₅.

By the above induction argument, in order to prove the last inequality for T not smooth, we get the following

|z|1,z

T ∧β ^p C. In fact let T_ε

be a regularization of T and let g_ε be the function associated with T_ε. The sequenceT_ε converges weakly to T, and it is not hard to see that the sequenceg_ε(a) tends tog(a) (we may replaceχ(|z+a|²)β ^pbyχ(|z |)χ(|z+ a|²)β ^p to get a compactly supported test form, and we observe that the integral is unchanged sinceT has support in a strip). By the above argument we find thatg_εis constant with respect toa, thusg is constant as well and therefore the desired inequality follows. Forr >1, one can cover{z,|z|< r}

by at most ([r] + 1)^2p unit cubes, where [r] denotes the integer part of r.

Therefore

B(0,r)T ∧β^pC5([r] + 1)^2p.

Step 2: general case.The hypothesis implies that there exists an ho- mogeneous polynomial system (Qα1, ..., Qαp) ofC[z1, ..., zn] such that each polynomialQαj is of degreeδand so that the homogeneous parts of higher degrees of the polynomialsQα1, ..., Qαp, P1, ..., Ps vanish simultaneously at thesinglepoint 0. Therefore, the mapfα defined onCⁿ by

f_α(z) := (Q_α1(z), ..., Q_αp(z), P₁(z), ..., P_s(z)),

is proper and finite. The current (fα)_∗T is positive of bidimension (k, k) with negativedd^c onCⁿ. Moreover, Supp(fα)_∗T ⊂ {|zp+1|+...+|zn|1}.

Let|α|=|(α₁, ..., α_p)|=α₁+...+α_p. We claim that:There exists a different system of homogeneous polynomial (Q_α1, ..., Q_αk)_1|α|µ, that each of one satisfy the above condition of the map f_α and such that :

|z|^2pδ−2p(dd^c|z|²)^p

1|α|µ

dQα1∧dQ_α1∧...∧dQαp∧dQ_αp. In fact, in view of the characterization of the non degenerate polynomial homogeneous maps (see the end of section 2), one can makes an appropri- ate large choice of different system (Qα1, ..., Qαk)_1|α|µ (µ is big enough) so that the homogeneous part of degree δ of the map fα is non degener- ate and all the monomials of degree δ appear in the decomposition of the productdQα1∧dQ_α1∧...∧dQαp∧dQ_αp. More precisely, for a sufficiently large selection of different coefficients ofQα1, we can obtain the inequality:

α1dQ_α1 ∧dQ_α1 |z|^2δ−2dd^c|z|², and similarly for the other α_j. Let us now continue the proof of step 2. Letr1, we have :

B(r)

T∧ |z|^2pδ−2p(dd^c|z|²)^p

B(r)

T ∧

1|α|µ

dQα1∧dQ_α1∧...∧dQαp∧dQ_αp

B(r)

T ∧

1|α|µ

f_α^∗(dd^c|w|²)^p

=

1|α|µ

fα(B(r))

(fα)_∗T∧(dd^c|w|²)^p.

Let

|f_α|= s j=1

|P_j|+ p

l=1

|Q_αl|.

So, we have|fα(z)|cα(1+|z|²)^δfor suitable constantscα>0. This implies that B(r)⊂Kr=

1|α|µf_α⁻¹(B(c¹_αr^δ)), wherec¹_αare positive constants.

By replacing rwithc¹_αr^δ in the previous inequality, we obtain:

B(r)

T∧ |z|^2pδ−2p(dd^c|z|²)^p

Kr

T∧ |z|^2pδ−2p(dd^c|z|²)^p

1|α|µ

fα(Kr)

(fα)_∗T∧(dd^c|w|²)^p

1|α|µ

B(c¹_αr^δ)

(fα)_∗T∧(dd^c|w|²)^p. Let bej0∈N^∗such thatr/2^j0 1 and forj= 1, ..., j0, settingB(r/2^j, r/2^j−1)

=B(r/2^j−1)B(r/2^j). Then, (r/2^j)^2pδ−2p

B(r/2^j,r/2^j−1)

T∧(dd^c|z|²)^p

B(r/2^j,r/2^j−1)

T∧ |z|^2pδ−2p(dd^c|z|²)^p

1|α|µ

B(c¹_α(r/2^j−1)^δ)

(fα)_∗T∧(dd^c|w|²)^p (r/2^j−1)^2pδc

1|α|µ

(c¹_α)^2p.

The last inequality is a consequence ofstep 1(one can choosec¹_αbig enough so thatc¹_α

r/2^j0−1δ

1). We putc1=c

1|α|µ(c¹_α)^2p, therefore

B(r/2^j−1,r/2^j)T∧(dd^c|z|²)^p c₁(r/2^j)^2p−2pδ(r/2^j−1)^2pδ

= c12^2p−2pδr^2p(1/2^j−1)^2p.

AsB(r, r/2^j0) =∪^j_j=1⁰ B(r/2^j−1, r/2^j) andr/2^j0 1, it is easy to see that :

B(r)B(1)

T∧(dd^c|z|²)^p

B(r,r/2^j0)

T ∧(dd^c|z|²)^pc1r^2p. Now, we conclude the proof by an enough perturbation of the center so that we cover all the balls B(r). In particular, ifT is pluriharmonic, then according to the Lelong-Jensen formula, it is easy to see thatT has a finite total projective mass, therefore it is algebraic.

Remark 3.1. — In Theorem 1.3, the hypothesis thatdd^cT 0 is neces- sary as the following example shows: letD(0,1) be the unit disk inCand lethbe a positive subharmonic function on C. Pick f, g∈D(D(0,1))0 such thatg(z₂)dd^c|z₂|²−dd^cf(z₂), and

T =f(z2)dd^c|z1|²+g(z2)(h(z1) +|z1|²)dd^c|z2|².

ThenT is a positive psh current of bidegree (1,1) having a support in the strip{(z1, z2)∈C²,|z2|<1}, butνT(r) is not bounded.

As mentioned above, the logarithmic growth of the counting function NT(r) characterizes algebraic positive pluriharmonic currents on Cⁿ. The following result clarifies the relation between the growths of the projective mass of T on {|z| < r} and the quantity νT(r), when T is a positive or negative psh current.

Proposition 3.2. — Let T be a positive current of bidimension (p, p) onCⁿ.

1. In both cases whenT is psh ordd^cT is negative, we have the growth estimateν_dd^c_T(r) =O(ν_T(√

2r)).In particular, ifν_T is bounded then dd^cT has finite total projective mass i.e. dd^cT is algebraic.

2. If T is psh then Tp.m(r) = O(νT(r)). In particular, when ν_T is bounded or equivalently, N_T has logarithmic growth, then T is alge- braic. If dd^cT is negative, then there exists c, c > 0 such that for every r 2, we have : Tp.m(r) c+c(νT(r) +NT(√

2r)). In particular ifνT is bounded, then the projective mass ofT carried by {|z|r} has at most logarithmic growth, i.e.Tp.m(r) =O(log r).

Proof. — (1) Assume that dd^cT is negative and consider a function χ ∈ D(R) such that χ(t) = 1 if |t| 1, and χ = 0 if |t| > 2. Let be ν_dd^c_T(r) =_r2p−2¹

B(0,r)dd^cT∧β^p−1. Thanks to Stokes’ theorem we have:

νdd^cT(r) _r2p−2¹

B(0,√ 2r)

dd^cT ∧χ(|z|² r² )β^p−1

= 1

r^2p−2

B(0,√ 2r)

T∧dd^cχ(|z|²

r² )∧β^p−1

= 1

r^2p−2

B(0,√ 2r)

T∧χ(|z|² r² )β^p

r²

+ 1

r^2p−2

B(0,√ 2r)

T∧χ (|z|²

r² )d|z|²∧d^c|z|² r⁴ ∧β^p−1

= 1 r^2p

B(0,√ 2r)

T ∧χ(|z|² r² )β^p + 1

r^2p

B(0,√ 2r)

T∧χ (|z|²

r² )d|z|²∧d^c|z|²

r² ∧β^p−1.

As|χ|and|χ |are bounded, and d|z|²∧d^c|z|²|z|²dd^c|z|², then ν_dd^c_T(r)−cν_T(√

2r)−cν_T(√

2r)−c ν_T(√ 2r).

In the case of a psh current, we can reverse the above inequalities. Thus the desired estimate follows. In particular, if ν_T is bounded then the cur- rent dd^cT is algebraic (since it is closed). Observe that the fact thatν_T is bounded is equivalent toN_T having logarithmic growth. Indeed, sinceT is psh, thenν_T is increasing, so we haveN_T(r)ν_T(r) logr_r²

r ν_T(t)/t dt N_T(r²).

(2) Assume thatT is psh. By applying the Lelong-Jensen formula [De]

to the functionϕ(z) = 1 +|z|², one easily shows that the projective mass of T growth at most asνT (since the quantities involving dd^cT are positive).

Assume now that dd^cT is negative and choose c >0 such thatνdd^cT(r)

−cνT(√

2r), for allr >0. Let 2r1tr2, we denote byB(t) ={ϕ(z)<

t} andB(r₁, r₂) ={r₁ < ϕ(z)< r₂}. Using the negativity of the measure dd^cT∧β^p−1, a direct computation gives :

r2

r1

1 t^p − 1

r^p₂

dt

B(t)

dd^cT∧β^p−1 −c

_r2

2

(t−1) t^p

p−1

ν_T(√

2t−2)dt

−2c

^√_2r2−2

1

νT(t)/tdt=−2cNT(√

2r2−2).

1 r^p₁ − 1

r^p₂

r1

1

dt

B(t)

dd^cT∧β^p−1 c₁− c

r₁^p _r1

2

(t−1)^p−1ν_T(√

2t−2)dt c1−cN_T(√

2r1−2)c1−cN_T(√

2r2−2), where c₁ = ₂¹p

2 1 dt

B(t)dd^cT ∧β^p−1. Thus using the Lelong-Jensen for- mula, we deduce the following estimate:

B(r1,r2)

T∧(dd^clog(1 +|z|²))^p−c1+ν_T(√

r2−1) + 3cN_T(√

2r2−2).

Forr₁= 2 andr₂=r²+ 1, we haveTp.m(r)c₂+c₃(ν_T(r) +N_T(√ 2r)).

In particular ifν_T is bounded, it is clear thatN_T(r) =O(logr).Thus, the projective mass has at most logarithmic growth.

It is a classical fact that closed positive currents onPⁿ can be described as conical currents onCⁿ⁺¹. For negative psh currents, the following result is an immediate consequence of proposition 3.2:

Corollary 3.3. — Let T be an algebraic negative psh current of bidi- mension (p, p) on Cⁿ, then there exists a positive pluriharmonic current Θ of bidimension (p+ 1, p+ 1) and conical on Cⁿ⁺¹, which is the trivial extension of π^∗(T) across 0,π is canonic projectionπ:Cⁿ⁺¹{0} →Pⁿ. Proof. — By considering the current−T instead, we may assume that T is positive. SinceT is algebraic, by the Lelong-Jensen formulaνT must be bounded then according to proposition 3.2, the currentdd^cT is algebraic.

By [D-E-E], the residual current S = dd^cT −dd^cT is positive and closed, and supported by H_∞. This implies that the current dd^cT is negative on Pⁿ. Thanks to Stokes’ theorem, we have 0 =

Pⁿdd^cT∧ω_{F S}^p−10. Then the currentTis positive and pluriharmonic onPⁿ. Letπ:Cⁿ⁺¹{0} →Pⁿ be the canonical submersion. Then the currentπ^∗(T) is positive pluriharmonic of bidimension (p+1, p+1) onCⁿ⁺¹{0}, and by [D-E-E] it can be extended to a positive pluriharmonic current Θ onCⁿ⁺¹. It is clear that the current Θ is conical, i.e. h^∗_rΘ = Θ, for every r ∈ C^∗, with hr(z) = rz. It follows that νΘ(r)≡ is constant for allr >0. The Lelong-Jensen formula implies therefore Θ∧(dd^clog|z|)^p+1= 0 onCⁿ⁺¹{0}.

Remark 3.4. — LetT be a positive psh current of bidimension (p, p) on Cⁿ. Assume thatνT(r) is bounded. Then by proposition 3.2. the currentsT anddd^cT are algebraic. As a consequence, the trivial extension currentsT anddd^cT are of order zero onPⁿ. By [D-E-E], there exists a closed positive currentS supported byH_∞ such thatdd^cT=dd^cT −S. Then,T is adsh current (a currentTis saiddshifT =T₁−T₂anddd^cT_i= Ω⁺_i −Ω⁻_i ,i= 1,2.

with T_i negative and Ω^±_i are positive closed). The class of dsh currents recently introduced in complexdynamics turns out to be very useful. It is easy to see that the current π^∗(T) can be extended to a dsh and conical current onCⁿ⁺¹.