Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 157
Proof. By the discussion above (local triviality of UF along the leaves), the first
statement is equivalent to say that, if T ⊂ X 0 is a small transversal (say, isomorphic
to D n−1 ), then UT � T × C.
We use for this Theorem 2.3 of Section 2. We may assume that there exists an
embedding T × D j → UT sending fibers to fibers and T ×{0} to pT (T ).Thenweset
U ε T = UT \{j(T × D(ε))}.
In order to apply Theorem 2.3, we need to prove that the fiberwise Poincaré metric
on U ε T has a plurisubharmonic variation, for every ε > 0 small.
But this follows from Theorem 5.1 in exactly the same way as we did in
Proposition 6.2 of Section 6. We just replace, in that proof, the open subsets Ω j ⊂ US
(for S ⊂ T a generic disc) with
Then the fibration Ω ε j
Ω ε j = Ω j \{j(S × D(ε))}.
→ S is, for j large, a fibration by annuli, and its boundary in US
has two components: one is the Levi flat Mj, and the other one is the Levi-flat j(S ×
∂D(ε)). Then Theorem 2.1 of Section 2, or more simply the annular generalization
of Proposition 2.2, gives the desired plurisubharmonic variation on Ω ε j , and then on
U ε S by passing to the limit, and finally on U ε T by varying S.
Hence UT � T × C and UF is a locally trivial C-fibration over X 0 .
Let us now define explicitely the isomorphism between UF and the total space
EF of TF over X 0 .
Take p ∈ X 0 and let vp ∈ EF be a point over p. Thenvpisa tangent vector
to Lp at p, and it can be lifted to �Lp as a tangent vector �vp at p. Suppose �vp �= 0.
Then, because �Lp � C, �vp can be extended, in a uniquely defined way, to a complete
holomorphic and nowhere vanishing vector field �v on �Lp. If, instead, we have
�vp = 0, then we set �v ≡ 0. Take q ∈ �Lp equal to the image of p by the time-one flow
of �v. We have in this way defined a map (EF )p → �Lp, vp ↦→ q, which obviously
is an isomorphism, sending the origin of (EF )p to the basepoint of �Lp. Inother
words: because Lp is parabolic, we have a canonically defined isomorphism between
(TpLp,0) and (�Lp, p).
By varying p in X 0 , we thus have a bijective map
EF | X 0 → UF
sending the null section to pF (X 0 ), and we need to verify that this map is holomorphic.
This follows from the fact that UF (and EF also, of course) is a locally
trivial fibration. In terms of the previous construction, we take a local transversal
T ⊂ X 0 and a nowhere vanishing holomorphic section vp, p ∈ T ,ofEF over T .
The previous construction gives a vertical vector field �v on UT ,whichis,onevery
fiber, complete holomorphic and nowhere vanishing, and moreover it is holomorphic
along pT (T ) ⊂ UT . After a trivialization UT � T × C, sending pT (T ) to {w = 0},
this vertical vector field �v becomes something like F(z,w) ∂
∂w , with F nowhere