Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 271
There are three notions of positivity which coincide for the bidegrees (0,0),
(1,1), (k − 1,k − 1) and (k,k). Here, we only use two of them. They are dual to
each other. A (p, p)-form ϕ is (strongly) positive if at each point, it is equal to a
combination with positive coefficients of forms of type
( √ −1α1 ∧ α1) ∧ ...∧ ( √ −1αp ∧ α p),
where αi are (1,0)-forms. Any (p, p)-form can be written as a finite combination
of positive (p, p)-forms. For example, in local coordinates z, a(1,1)-form ω is
written as
ω =
k
∑
i, j=1
√
αij −1dzi ∧ dz j,
where αij are functions. This form is positive if and only if the matrix (αij) is
positive semi-definite at every point. In local coordinates z, the(1,1)-form dd c �z� 2
is positive. One can write dz1 ∧ dz2 as a combination of dz1 ∧ dz1, dz2 ∧ dz2,
d(z1 ± z2) ∧ d(z1 ± z2) and d(z1 ± √ −1z2) ∧ d(z1 ± √ −1z2). Hence, positive forms
generate the space of (p, p)-forms.
A (p, p)-current S is weakly positive if for every smooth positive (k − p,k − p)form
ϕ, S ∧ ϕ is a positive measure and is positive if S ∧ ϕ is a positive measure
for every smooth weakly positive (k − p,k − p)-form ϕ. Positivity implies weak
positivity. These properties are preserved under pull-back by holomorphic submersions
and push-forward by proper holomorphic maps. Positive and weakly
positive forms or currents are real. One can consider positive and weakly positive
(p, p)-forms as sections of some bundles of salient convex closed cones which are
contained in the real part of the vector bundle � p Ω 1,0 ⊗ � p Ω 0,1 .
The wedge-product of a positive current with a positive form is positive. The
wedge-product of a weakly positive current with a positive form is weakly positive.
Wedge-products of weakly positive forms or currents are not always weakly
positive. For real (p, p)-currents or forms S, S ′ , we will write S ≥ S ′ and S ′ ≤ S
if S − S ′ is positive. A current S is negative if −S is positive. A (p, p)-current or
form S is strictly positive if in local coordinates z, there is a constant ε > 0such
that S ≥ ε(dd c �z� 2 ) p . Equivalently, S is strictly positive if we have locally S ≥ εω p
with ε > 0.
Example A.12. Let Y be an analytic set of pure codimension p of X. Using the local
description of Y near a singularity in Theorem A.3 and Wirtinger’s theorem A.2,
one can prove that the 2(k − p)-dimensional volume of Y is locally finite in X.This
allows to define the following (p, p)-current [Y] by
�
〈[Y],ϕ〉 := ϕ
reg(Y )
for ϕ in D k−p,k−p (X), the space of smooth (k − p,k − p)-forms with compact
support in X. Lelong proved that this current is positive and closed [DEM, LE].