Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 153
is a proper map, and it realizes an isomorphism between Γf \ Z and U 0 Σ \ B, for
suitable analytic subsets Z ⊂ Γf and B ⊂ U 0 Σ , with B of codimension at least two.
The manifold UΣ × Y is Kähler. The Kähler form restricted to the graph of f
and pulled-back to its resolution gives a smooth, semipositive, closed (1,1)-form
ω on Γf . Fix a smaller (tubular) neighbourhood U ′ Σ of Σ, andsetU0 = UΣ \ U ′ Σ ,
U = π−1 (U0) ⊂ Γf . Up to restricting a little the initial UΣ , we may assume that the
ωn′ -volume of the shell U is finite (n ′ = n + 1 = dimΓf ). Our aim is to prove that
�
ω
Γf
n′
< +∞.
Indeed, this is the volume of the graph of f . Its finiteness, together with the analyticity
of the graph in U 0 Σ ×Y, imply that the closure of that graph in UΣ ×Y is still
an analytic subset of dimension n ′ , by Bishop’s extension theorem [Siu, Chi]. This
closure, then, is the graph of the desired meromorphic extension ¯f : UΣ ��� Y .
We shall apply Theorem 7.1. Hence, consider the space Pω(Γf ,U) of ω-plurisubharmonic
functions on Γf , nonpositive on U, and let us check conditions (A) and (B)
above, at the beginning of this Section.
Consider the open subset Ω ⊂ Γf where the functions of Pω(Γf ,U) are locally
uniformly bounded from above. It contains U, and it is a general fact that it is
locally pseudoconvex in Γf [Din, §3]. Therefore Ω ′ = Ω ∩ (Γf \ Z) is locally pseudoconvex
in Γf \ Z. Its isomorphic projection π(Ω ′ ) is therefore locally pseudoconvex
in U 0 Σ \ B. Classical characterizations of pseudoconvexity [Ran, II.5] show that
Ω0 = interior{π(Ω ′ )∪B} is locally pseudoconvex in U 0 Σ .FromΩ⊃U we also have
Ω0 ⊃ U0.
Ε
Ω 0 U 0 U 0
Σ
Take now in E the neighbourhood of infinity W0 = Ω0 ∪ (E \UΣ ). Because E \ Σ
is naturally isomorphic to E ∗ \ Σ ∗ , the isomorphism exchanging null sections and
sections at infinity, we can see W0 as an open subset of E ∗ ,sothatW = W0 ∪ Σ ∗ is a
neighbourhood of Σ ∗ , locally pseudoconvex in E ∗ . Because L is not pseudoeffective
by assumption, Lemma 7.4 says that W = E ∗ .Thatis,Ω0 = U 0 Σ .
This implies that the original Ω ⊂ Γf contains, at least, Γf \ Z. But, by the maximum
principle, a family of ω-plurisubharmonic functions locally bounded outside
an analytic subset is automatically bounded also on the same analytic subset. Therefore
Ω = Γf , and condition (A) of Theorem 7.1 is fulfilled.
W 0
Σ