Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 111
Proof. For every z ∈ D n we have a unique isomorphism
f (z,·) : P −1 (z) → C
such that, using the coordinates given by j,
f (z,0)=0 and f ′ (z,0)=1.
We want to prove that f is holomorphic in z.
Set Rε(z)= f (z,P−1 ε (z)) ⊂ C. By Koebe’s Theorem, the distorsion of f (z,·) on
compact subsets of D is uniformly bounded, and so D( 1 k ε) ⊂ f (z,D(ε)) ⊂ D(kε)
for every ε ∈ (0, 1
2 ) and for some constant k, independent on z. Therefore, for every
ε and z,
�
1
C \ D
k ε
�
⊂ Rε(z) ⊂ C \ D(kε).
In a similar way [Nis, II], Koebe’s Theorem gives also the continuity of the above
map f .
On the fibers of P0, which are all isomorphic to C∗ , we put the unique complete
hermitian metric of zero curvature and period (=length of closed simple geodesics)
equal to √ 2π. OnthefibersofPε, ε > 0, which are all hyperbolic, we put the
Poincaré metric multiplied by logε, whose (constant) curvature is therefore equal
to − 1
(logε) 2 . By a simple and explicit computation, the PoincarémetriconC\D(cε)
multiplied by logε converges uniformly to the flat metric of period √ 2π on C∗ ,as
ε → 0. Using this and the above bounds on Rε(z), we obtain that our fiberwise metric
on Uε Pε
→ Dn converges uniformly, as ε → 0, to our fiberwise metric on U0 → Dn (see
[Br4] for more explicit computations). Hence, from the plurisubharmonic variation
of the former we deduce the plurisubharmonic variation of the latter.
Our flat metric on P −1
idx∧d ¯x
0 (z) is the pull-back by f (z,·) of the metric
4|x| 2 on
R0(z)=C∗ . In the coordinates given by j, wehave
f (z,w)=w · e g(z,w) ,
with g holomorphic in w and g(z,0)=0foreveryz, by the choice of the normalization.
Hence, in these coordinates our metric takes the form
�
�
�
∂g
�1 + w
∂w (z,w)
�
�2
�
� · idw ∧ d ¯w
4|w| 2 .
Set F = log|1 + w ∂g
∂w |2 . We know, by the previous arguments, that F is plurisubhar-
monic. Moreover, ∂ 2 F
∂w∂ ¯w
≡ 0, by flatness of the metric. By semipositivity of the Levi
form we then obtain ∂ 2 F
∂w∂ ¯z k ≡ 0foreveryk. Hence the function ∂F
∂w
P0
is holomorphic,
that is the function ( ∂g
∂w + w ∂ 2g ∂w2 )(1 + w ∂g
∂w )−1 is holomorphic. Taking into account
that g(z,0) ≡ 0, we obtain from this that g also is fully holomorphic. Thus f is fully
holomorphic in the chart given by j, and hence everywhere. It follows that U is
isomorphic to a product. ⊓⊔