Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 281
of C k . Then the slice 〈S,π,θ〉 is well-defined for every θ in V and is a positive
measure whose mass is independent of θ. Moreover, if Φ is a p.s.h. function in a
neighbourhood of supp(S), then the function θ ↦→〈S,π,θ〉(Φ) is p.s.h.
The mass of 〈S,π,θ〉 is called the slice mass of S. The set of currents S as above
with bounded slice mass is compact for the weak topology on currents. In particular,
their masses are locally uniformly bounded on Ck ×V. In general, the slice 〈S,π,θ〉
does not depend continuously on S nor on θ. The last property in Theorem A.37
shows that the dependence on θ satisfies a semi-continuity property. More generally,
we have that (θ,S) ↦→〈S,π,θ〉(Φ) is upper semi-continuous for Φ p.s.h. We deduce
easily from the above definition that the slice mass of S depends continuously on S.
Exercise A.38. Let X,X ′ be complex manifolds. Let ν be a positive measure with
compact support on X such that p.s.h. functions on X are ν-integrable. If u is a p.s.h.
function on X × X ′ , show that x ′ ↦→ � u(x,x ′ )dν(x) is a p.s.h function on X ′ .Show
that if ν,ν ′ are positive measures on X,X ′ which are locally moderate, then ν ⊗ ν ′
is a locally moderate measure on X × X ′ .
Exercise A.39. Let S be a positive closed (1,1)-current on the unit ball of Ck .Let
π : � Ck → Ck be the blow-up of Ck at 0 and E the exceptional set. Show π∗ (S) is
equal to ν[E]+S ′ ,whereνistheLelong number of S at 0 and S ′ is a current without
mass on E.
A.4 Currents on Projective Spaces
In this paragraph, we will introduce quasi-potentials of currents, the spaces of d.s.h.
functions, DSH currents and the complex Sobolev space which are used as observables
in complex dynamics. We also introduce PB, PC currents and the notion of
super-potentials which are crucial in the calculus with currents in higher bidegree.
Recall that the Fubini-Study form ωFS on Pk satisfies �
Pk ωk FS = 1. If S is a
positive closed (p, p)-current, the mass of S is given by by �S� := 〈S,ω k−p
FS 〉.Since
H p,p (Pk ,R) is generated by ω p
p
FS , such a current S is cohomologous to cωFS where c
is the mass of S.So,S − cω p
FS is exact and the ddc-lemma, which also holds for currents,
implies that there exists a (p−1, p−1)-currentU, such that S = cω p
FS +ddcU. We call U a quasi-potential of S. We have in fact the following more precise result
[DS10].
Theorem A.40. Let S be a positive closed (p, p)-current of mass 1 in Pk . Then,
there is a negative form U such that ddcU = S − ω p
FS .Forr,swith 1 ≤ r < k/(k −1)
and 1 ≤ s < 2k/(2k − 1), we have
�U�Lr ≤ cr and �∇U�L s ≤ cs,
where cr,cs are constants independent of S. Moreover, U depends linearly and
continuously on S with respect to the weak topology on the currents S and the L r
topology on U.