Discrete Holomorphic Local Dynamical Systems

280 Tien-Cuong Dinh and Nessim Sibony
current τ ∗ (S). The result still holds when X ′ is singular. In the case of bidegree
(1,1), we have the following result due to Méo [ME].
Proposition A.36. Let τ : X ′ → X be a holomorphic map between complex
manifolds. Assume that τ is dominant, that is, the image of τ contains an open
subset of X. Then the pull-back operator τ ∗ on smooth positive closed (1,1)-forms
can be extended in a canonical way to a continuous operator on positive closed
(1,1)-currents S on X.
Indeed, locally we can write S = dd c u with u p.s.h. The current τ ∗ (S) is then
defined by τ ∗ (S) := dd c (u ◦ τ). One can check that the definition does not depend
on the choice of u.
The remaining part of this paragraph deals with the slicing of currents. We only
consider a situation used in this course. Let π : X → V be a dominant holomorphic
map from X to a manifold V of dimension l and S a current on X. Slicing theory
allows to define the slice 〈S,π,θ〉 of some currents S on X by the fiber π −1 (θ).
Slicing theory generalizes the restriction of forms to fibers. One can also consider it
as a generalization of Sard’s and Fubini’s theorems for currents or as a special case
of intersection theory: the slice 〈S,π,θ〉 can be seen as the wedge-product of S with
the current of integration on π −1 (θ). We can consider the slicing of C-flat currents,
in particular, of (p, p)-currents such that S and dd c S are of order 0. The operation
preserves positivity and commutes with ∂, ∂. Ifϕ is a smooth form on X then
〈S ∧ ϕ,π,θ〉 = 〈S,π,θ〉∧ϕ. Here, we only consider positive closed (k − l,k − l)currents
S. In this case, the slices 〈S,π,θ〉 are positive measures on X with support
in π −1 (θ).
Let y denote the coordinates in a chart of V and λV := (dd c �y� 2 ) l the
Euclidean volume form associated to y. Letψ(y) be a positive smooth function
with compact support such that � ψλV = 1. Define ψε(y) := ε −2l ψ(ε −1 y) and
ψθ,ε(y) := ψε(y − θ). The measures ψθ,ελV approximate the Dirac mass at θ. For
every smooth test function Φ on X,wehave
〈S,π,θ〉(Φ)=lim
ε→0 〈S ∧ π ∗ (ψθ,ελV ),Φ〉
when 〈S,π,θ〉 exists. This property holds for all choice of ψ. Conversely, when the
previous limit exists and is independent of ψ, it defines the measure 〈S,π,θ〉 and
we say that 〈S,π,θ〉 is well-defined. The slice 〈S,π,θ〉 is well-defined for θ out of
a set of Lebesgue measure zero in V and the following formula holds for smooth
forms Ω of maximal degree with compact support in V:
�
〈S,π,θ〉(Φ)Ω(θ)=〈S ∧ π ∗ (Ω),Φ〉.
θ∈V
We recall the following result which was obtained in [DS7].
Theorem A.37. Let V be a complex manifold of dimension l and let π denote the
canonical projection from C k × V onto V. Let S be a positive closed current of
bidimension (l,l) on C k × V , supported on K × V for a given compact subset K