Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 113
which contains Ω andsuchthatevery f ∈ OF (Ω) extends to some ˜f ∈ O(V T ).The
classical Cartan-Thullen-Oka theory [GuR] saysthatV T is a Stein manifold.
The projection Ω → T extends to a map
Q T : V T → T
thanks to OF (Ω) ↩→ O(V T ). Consider a fiber Q −1
T (t). It is not difficult to see that
the connected component of Q −1
T (t) which cuts Ω (⊂ V T ) is exactly the holonomy
covering �Lt of Lt, with basepoint t. The reason is the following one. Firstly,
if γ : [0,1] → Lt is a path contained in a leaf, with γ(0) =t, then any function
f ∈ OF (Ω) can be analytically prolonged along γ, by preserving the constancy
on the leaves. Secondly, if γ1 and γ2 are two such paths with the same endpoint
s ∈ Lt, then the germs at s obtained by the two continuations of f along γ1 and γ2
may be different. If the foliation has trivial holonomy along γ1 ∗ γ−1 2 , then the two
germs are certainly equal; conversely, if the holonomy is not trivial, then we can
find f such that the two final germs are different. This argument shows that �Lt is
naturally contained into Q −1
T (t). The fact that it is a connected component is just a
“maximality” argument (note that V T is foliated by the pull-back of F ,andfibers
of QT are closed subvarieties invariant by this foliation).
We denote by VT ⊂ V T (open subset) the union of these holonomy coverings, and
by QT the restriction of QT to VT .
Let us return to UT . We have a natural map (local biholomorphism)
FT : UT → VT
which acts as a covering between fibers (but not globally: see Examples 4.7 and 4.8
below). In particular, UT is a Riemann domain over the Stein manifold V T .
Lemma 2.5. UT is holomorphically separable.
Proof. Given p,q ∈ UT , p �= q, we want to construct f ∈ O(UT ) such that f (p) �=
f (q). The only nontrivial case (VT being holomorphically separable) is the case
where FT (p)=FT (q), in particular p and q belong to the same fiber �Lt.
We use the following procedure. Take a path γ in �Lt from p to q. It projects by
FT to a closed path γ0 in �Lt. Suppose that [γ0] �= 0inH1( �Lt,R). Thenwemayfind
a holomorphic 1-form ω ∈ Ω 1 ( �Lt) such that �
γ0
ω = 1. This 1-form can be holomorphically
extended from �Lt to VT ⊂ V T , because V T is Stein and �Lt is a closed
submanifold of it. Call �ω such an extension, and �ω = F∗ T ( �ω) its lift to UT .Onevery
(simply connected!) fiber �Lt of UT the 1-form �ω is exact, and can be integrated giv-
ing a holomorphic function ft(z)= � z
t �ω| �Lt
. We thus obtain a holomorphic function
f on UT , which separates p and q: f (p) − f (q)= � �
γ �ω = �ω γ0
= 1.
This procedure does not work if [γ0] =0: in that case, every ω ∈ Ω 1 ( �Lt) has
period equal to zero on γ0. But, in that case, we may find two 1-forms ω1,ω2 ∈
Ω 1 ( �Lt) such that the iterated integral of (ω1,ω2) along γ0 is not zero (this iterated
integral [Che] is just the integral along γ of φ1dφ2, whereφjis a primitive of ω j