Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 285
Let Ck−p+1(Pk ) denote the convex set of positive closed currents of bidegree
(k − p +1,k − p +1) and of mass 1, i.e. currents cohomologous to ω k−p+1
.LetSbe FS
a positive closed (p, p)-current on Pk . We assume for simplicity that S is of mass 1;
the general case can be deduced by linearity. The super-potential12 US of S is a function
on Ck−p+1(Pk ) with values in R ∪{−∞}.LetRbe a current in Ck−p+1(Pk ) and
UR a potential of R − ω k−p+1
FS . Subtracting from UR a constant times ω k−p
FS allows to
have 〈UR,ω p
FS 〉 = 0. We say that UR is a quasi-potential of mean 0 of R. Formally,
i.e. in the case where R and UR are smooth, the value of US at R is defined by
US(R) := 〈S,UR〉.
One easily check using Stokes’ formula that formally if US is a quasi-potential
of mean 0 of S, thenUS(R)=〈US,R〉. Therefore, the previous definition does not
depend on the choice of UR or US. By definition, we have US(ω k−p+1
FS )=0. Observe
also that when S is smooth, the above definition makes sense for every R and US is
a continuous affine function on Ck−p+1(Pk ). It is also clear that if US = US ′,then
S = S ′ . The following theorem allows to define US in the general case.
Theorem A.45. The above function US, which is defined on smooth forms R in
Ck−p+1(P k ), can be extended to an affine function on Ck−p+1(P k ) with values in
R ∪{−∞} by
US(R) := limsupUS(R ′ ),
where R ′ is smooth in Ck−p+1(P k ) and converges to R. We have US(R) =UR(S).
Moreover, there are smooth positive closed (p, p)-forms Sn of mass 1 and constants
cn converging to 0 such that USn + cn decrease to US. In particular, USn converge
pointwise to US.
For bidegree (1,1), there is a unique quasi-p.s.h. function uS such that dd c uS =
S − ωFS and 〈ω k FS ,uS〉 = 0. If δa denotes the Dirac mass at a, wehaveUS(δa) =
uS(a). Dirac masses are extremal elements in Ck(P k ). The super-potential US in this
case is just the affine extension of uS, that is, we have for any probability measure ν:
�
US(ν)=
�
US(δa)dν(a)=
uS(a)dν(a).
The function US extends the action 〈S,Φ〉 on smooth forms Φ to 〈S,U〉 where
U is a quasi-potential of a positive closed current. Super-potentials satisfy analogous
properties as quasi-p.s.h. functions do. They are upper semi-continuous and
bounded from above by a universal constant. Note that we consider here the weak
topology on Ck−p+1(P k ). We have the following version of the Hartogs’ lemma.
Proposition A.46. Let Sn be positive closed (p, p)-currents of mass 1 on Pk converging
to S. Then for every continuous function U on Ck−p+1 with US < U ,we
have USn < U for n large enough. In particular, limsupUSn ≤ US.
12 The super-potential we consider here corresponds to the super-potential of mean 0 in [DS10].
The other super-potentials differ from US by constants.