Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 287
Super-potentials allow to develop a theory of intersection of currents in higher
bidegree. Here, the fact that US has a value at every point (i.e. at every current
R ∈ Ck−p+1(Pk )) is crucial. Let S, S ′ be positive closed currents of bidegree (p, p)
and (p ′ , p ′ ) with p + p ′ ≤ k. We assume for simplicity that their masses are equal
to 1. We say that S and S ′ are wedgeable if US is finite at S ′ ∧ ω k−p−p′ +1
FS .This
property is symmetric on S and S ′ .If�S, �S ′ are more diffuse than S,S ′ and if S,S ′ are
wedgeable, then �S, �S ′ are wedgeable.
Let Φ be a real smooth form of bidegree (k − p − p ′ ,k − p − p ′ ). Write
ddcΦ = c(Ω + − Ω − ) with c ≥ 0andΩ ± positive closed of mass 1. If S and
S ′ are wedgeable, define the current S ∧ S ′ by
〈S ∧ S ′ ,Φ〉 := 〈S ′ ,ω p
FS ∧ Φ〉 + cUS(S ′ ∧ Ω + ) − cUS(S ′ ∧ Ω − ).
A simple computation shows that the definition coincides with the usual wedgeproduct
when S or S ′ is smooth. One can also prove that the previous definition does
not depend on the choice of c, Ω ± and is symmetric with respect to S,S ′ .IfS is of
bidegree (1,1), thenS,S ′ are wedgeable if and only if the quasi-potentials of S are
integrable with respect to the trace measure of S ′ . In this case, the above definition
coincides with the definition in Appendix A.3. We have the following general result.
Theorem A.48. Let Si be positive closed currents of bidegree (pi, pi) on P k with
1 ≤ i ≤ m and p1 +···+ pm ≤ k. Assume that for 1 ≤ i ≤ m−1,Si and Si+1 ∧...∧Sm
are wedgeable. Then, this condition is symmetric on S1,...,Sm. The wedge-product
S1 ∧ ... ∧ Sm is a positive closed current of mass �S1�...�Sm� supported on
supp(S1) ∩ ...∩ supp(Sm). It depends linearly on each variable and is symmet-
ric on the variables. If S (n)
i converge to Si in the Hartogs’ sense, then the S (n)
i are
wedgeable and S (n)
1
∧ ...∧ S(n)
m converge in the Hartogs’ sense to S1 ∧ ...∧ Sm.
We deduce from this result that wedge-products of PB currents are PB. One can
also prove that wedge-products of PC currents are PC. If Sn is defined by analytic
sets, they are wedgeable if the intersection of these analytic sets is of codimension
p1 + ···+ pm. In this case, the intersection in the sense of currents coincides with
the intersection of cycles, i.e. is equal to the current of integration on the intersection
of cycles where we count the multiplicities. We have the following criterion of
wedgeability which contains the case of cycles.
Proposition A.49. Let S,S ′ be positive closed currents on P k of bidegrees (p, p)
and (p ′ , p ′ ).LetW,W ′ be open sets such that S restricted to W and S ′ restricted to
W ′ are bounded forms. Assume that W ∪W ′ is (p + p ′ )-concave in the sense that
there is a positive closed smooth form of bidegree (k − p − p ′ + 1,k − p − p ′ + 1)
with compact support in W ∪W ′ . Then S and S ′ are wedgeable.
The following result can be deduced from Theorem A.48.
Corollary A.50. Let Si be positive closed (1,1)-currents on P k with 1 ≤ i ≤ p.
Assume that for 1 ≤ i ≤ p − 1, Si admits a quasi-potential which is integrable with