Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 277
Exercise A.25. With the notation of Exercise A.11, show that τ∗(ϕ) is p.s.h. if ϕ is
p.s.h.
Exercise A.26. Using that ν(S,a,r) is decreasing, show that if (Sn) is a sequence of
positive closed (p, p)-currents on X converging to a current S and (an) is a sequence
in X converging to a, then limsupν(Sn,an) ≤ ν(S,a).
Exercise A.27. Let S and S ′ be positive closed (1,1)-currents such that S ′ ≤ S.
Assume that the local potentials of S are bounded or continuous. Show that the local
potentials of S ′ are also bounded or continuous.
Exercise A.28. Let F be an L1 loc bounded family of p.s.h. functions on X.LetKbe a compact subset of X. Show that F is locally bounded from above and that there
is c > 0 such that �ddcu�K ≤ c for every u ∈ F . Prove that there is a constant ν > 0
such that ν(u,a) ≤ ν for u ∈ F and a ∈ K.
Exercise A.29. Let Yi, 1≤ i ≤ m, be analytic sets of pure codimension pi in P k .
Assume p1 + ···+ pm ≤ k. Show that the intersection of the Yi’s is a non-empty
analytic set of dimension ≥ k − p1 −···−pm.
A.3 Intersection, Pull-back and Slicing
We have seen that positive closed currents generalize differential forms and analytic
sets. However, it is not always possible to extend the calculus on forms or on
analytic sets to currents. We will give here some results which show how positive
closed currents are flexible and how they are rigid.
The theory of intersection is much more developed in bidegree (1,1) thanks
to the use of their potentials which are p.s.h. functions. The case of continuous
potentials was considered by Chern-Levine-Nirenberg [CLN]. Bedford-Taylor
[BD] developed a nice theory when the potentials are locally bounded. The case
of unbounded potentials was considered by Demailly [DE] and Fornæss-Sibony
[FS2, S2]. We have the following general definition.
Let S be a positive closed (p, p)-current on X with p ≤ k − 1. If ω is a fixed
Hermitian form on X as above, then S ∧ ω k−p is a positive measure which is called
the trace measure of S. In local coordinates, the coefficients of S are measures,
bounded by a constant times the trace measure. Now, if u is a p.s.h function on X,
locally integrable with respect to the trace measure of S, thenuS is a current on X
and we can define
dd c u ∧ S := dd c (uS).
Since u can be locally approximated by decreasing sequences of smooth p.s.h.
functions, it is easy to check that the previous intersection is a positive closed
(p + 1, p + 1)-current with support contained in supp(S). Whenu is pluriharmonic,
dd c u ∧ S vanishes identically. So, the intersection depends only on dd c u and on S.
If R is a positive closed (1,1)-current on X, one defines R ∧ S as above using local