Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 115
and this expression tends to 0 as k → +∞. Therefore f (b0)= f (a0), in contradiction
with (i) and (iii). ⊓⊔
It follows from this Lemma that UT = UT is Stein. ⊓⊔
Remark 2.7. We do not know if VT also is Stein, i.e. if VT = V T .
This Theorem allows to apply the results of Nishino and Yamaguchi discussed
above to holomorphic foliations on Stein manifolds. For instance: the set of
parabolic leaves of such a foliation is either full or complete pluripolar. A similar
point of view is pursued in [Suz].
3 The Unparametrized Hartogs Extension Lemma
In order to construct the leafwise universal covering of a foliation, we shall need an
extension lemma of Hartogs type. This is done in this Section.
Let X be a compact Kähler manifold. Denote by Ar, r ∈ (0,1), the semiclosed
annulus {r < |w| ≤1}, with boundary ∂Ar = {|w| = 1}. Given a holomorphic immersion
f : Ar → X
we shall say that f (Ar) extends to a disc if there exists a holomorphic map
g : D → X,
not necessarily immersive, such that f factorizes as g ◦ j for some embedding j :
Ar → D, sending ∂Ar to ∂D.Thatis, f itself does not need to extend to the full disc
{|w|≤1}, but it extends “after a reparametrization”, given by j.
Remark that if f is an embedding, and f (Ar) extends to a disc, then we can find
g as above which is moreover injective outside a finite subset. The image g(D) is a
(possibly singular) disc in X with boundary f (∂Ar). Such an extension g or g(D)
will be called simple extension of f or f (Ar). Note that such a g is uniquely defined
up to a Moëbius reparametrization of D.
Given a holomorphic immersion
f : D k × Ar → X
we shall say that f (D k × Ar) extends to a meromorphic family of discs if there
exists a meromorphic map
g : W ��� X
such that:
(i) W is a complex manifold of dimension k + 1 with boundary, equipped with a
holomorphic submersion W → D k all of whose fibers Wz, z ∈ D k , are isomorphic
to D;