Discrete Holomorphic Local Dynamical Systems

276 Tien-Cuong Dinh and Nessim Sibony
Theorem A.22. Let F be a family of p.s.h. functions on X which is bounded in
L1 loc (X). Let K be a compact subset of X. Then there are constants α > 0 and A > 0
such that
�e −αu �L1 (K) ≤ A
for every function u in F .
P.s.h. functions are in general unbounded. However, the last result shows that
such functions are nearly bounded. The above family F is uniformly bounded from
above on K. So, we also have the estimate
�e α|u| � L 1 (K) ≤ A
for u in F and for some (other) constants α,A. More precise estimates can be
obtained in terms of the maximal Lelong number of dd c u in a neighbourhood of K.
Define the Lelong number ν(u,a) of u at a as the Lelong number of dd c u at a.
The following result describes the relation with the singularity of p.s.h. functions
near a pole. We fix here a local coordinate system for X.
Proposition A.23. The Lelong number ν(u,a) is the supremum of the number ν
such that the inequality u(z) ≤ ν log�z − a� holds in a neighbourhood of a.
If S is a positive closed (p, p)-current, the Lelong number ν(S,a) can be computed
as the mass at a of the measure S ∧ (dd c log�z − a�) k−p . This property allows
to prove the following result, due to Demailly [DEM], which is useful in dynamics.
Proposition A.24. Let τ : (C k ,0) → (C k ,0) be a germ of an open holomorphic map
with τ(0) =0. Let d denote the multiplicity of τ at 0. Let S be a positive closed
(p, p)-current on a neighbourhood of 0. Then, the Lelong number of τ∗(S) at 0
satisfies the inequalities
ν(S,0) ≤ ν(τ∗(S),0) ≤ d k−p ν(S,0).
In particular, we have ν(τ∗(S),0)=0 if and only if ν(S,0)=0.
Assume now that X is a compact Kähler manifold and ω is a Kähler form on
X. IfS is a ddc-closed (p, p)-current, we can, using the ddc-lemma, define a linear
form on Hk−p,k−p (X,C) by [α] ↦→ 〈S,α〉. Therefore, the Poincaré duality implies
that S is canonically associated to a class [S] in H p,p (X,C). IfS is real then [S] is
in H p,p (X,R). IfS is positive, its mass 〈S,ω k−p 〉 depends only on the class [S].
So, the mass of positive ddc-closed currents can be computed cohomologically. In
Pk , the mass of ω p
is a probability measure. If H is a subspace of
FS is 1 since ωk FS
codimension p of P k , then the current associated to H is of mass 1 and it belongs to
the class [ω p
FS ].IfY is an analytic set of pure codimension p of Pk , the degree deg(Y)
of Y is by definition the number of points in its intersection with a generic projective
space of dimension p. One can check that the cohomology class of Y is deg(Y)[ω p
FS ].
The volume of Y, obtained using Wirtinger’s theorem A.2, is equal to 1 p! deg(Y ).