Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 125
(iii) for every t ∈ T , the restriction of πT to Q −1
T (t)= �Lt coincides, after removal of
it
indeterminacies, with the holonomy covering �Lt → Lt → it(Lt) ⊂ X.
Proof. We firstly prove a similar statement for the regular foliation F 0 on X 0 .We
use Il’yashenko’s methodology [Il1]; an alternative but equivalent one can be found
in [Suz], we have already seen it at the beginning of the proof of Theorem 2.4.
In fact, in the case of a regular foliation the construction of V 0 T below is a rather
classical fact in foliation theory, which holds in the much more general context of
smooth foliations with real analytic holonomy.
Consider the space Ω F 0
T composed by continuous paths γ : [0,1] → X 0 tangent
to F 0 and such that γ(0) ∈ T , equipped with the uniform topology. On Ω F 0
T we put
the following equivalence relation: γ1 ∼ γ2 if γ1(0)=γ2(0), γ1(1)=γ2(1), andthe
loop γ1 ∗ γ −1
, has trivial holonomy.
Set
2 , obtained by juxtaposing γ1 and γ −1
2
V 0 T
F 0�
= ΩT ∼
with the quotient topology. Note that we have natural continuous maps
and
Q 0 T : V 0 T
→ T
π 0 T : V 0 T → X 0
defined respectively by [γ] ↦→ γ(0) ∈ T and [γ] ↦→ γ(1) ∈ X 0 . We also have a natural
section
q 0 T : T → V 0 T
which associates to t ∈ T the equivalence class of the constant path at t.Clearly,for
every t ∈ T the pointed fiber ((Q0 T )−1 (t),q 0 T (t)) is the same as ( � L0 t ,t), bythevery
definition of holonomy covering, and π0 T restricted to that fiber is the holonomy
covering map. Therefore, we just have to find a complex structure on V 0 T such that
all these maps become holomorphic.
We claim that V 0 T is a Hausdorff space. Indeed, if [γ1],[γ2] ∈ V 0 T are two nonseparated
points, then γ1(0)=γ2(0)=t, γ1(1)=γ2(1), and the loop γ1 ∗ γ −1
2 in the leaf
L0 t can be uniformly approximated by loops γ1,n ∗ γ −1
2,n in the leaves L0 tn (tn → t) with
trivial holonomy (so that [γ1,n]=[γ2,n] is a sequence of points of V 0 T converging to
both [γ1] and [γ2]). But this implies that also the loop γ1 ∗ γ −1
2 has trivial holonomy,
by the identity principle: if h ∈ Di f f (Dn−1 ,0) is the identity on a sequence of open
sets accumulating to 0, then h is the identity everywhere. Thus [γ1]=[γ2],andV0 T is
Hausdorff.
Now, note that π 0 T : VT → X 0 is a local homeomorphism. Hence we can pull back
to V 0 T the complex structure of X 0 , and in this way V 0 T becomes a complex manifold
of dimension n with all the desired properties. Remark that, at this point, π0 T has not
yet indeterminacy points, and so V 0 T is a so-called Riemann Domain over X 0 .
In order to pass from V 0 T to VT , we need to add to each fiber � L0 t of V 0 T the discrete
set �Lt \ � L 0 t .