Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 107
[Dem]. When X is projective the above theorem follows also from results of [BMQ]
and [BDP], but with a totally different proof, untranslatable in our Kähler context.
Let us now describe in more detail the content of these notes.
In Section 2 we shall recall the results by Nishino and Yamaguchi on Stein fibrations
that we shall use later, and also some of Il’yashenko’s results. In Section
3 and 4 we construct the leafwise universal covering of (X,F ): we give an appropriate
definition of leaf Lp of F through a point p ∈ X \ Sing(F ) (this requires
some care, because some leaves are allowed to pass through some singular points),
and we show that the universal coverings �Lp can be glued together to get a complex
manifold. In Section 5 we prove that the complex manifold so constructed enjoys
some “holomorphic convexity” property. This is used in Section 6 and 8, together
with Nishino and Yamaguchi results, to prove (among other things) Theorem 1.1
above. The parabolic case requires also an extension theorem for certain meromorphic
maps into compact Kähler manifolds, which is proved in Section 7.
All this work has been developed in our previous papers [Br2,Br3,Br4,Br5] (with
few imprecisions which will be corrected here). Further results and application can
be found in [Br6]and[Br7].
2 Some Results on Stein Fibrations
2.1 Hyperbolic Fibrations
In a series of papers, Nishino [Nis] and then Yamaguchi [Ya1, Ya2, Ya3] studied
the following situation. It is given a Stein manifold U,ofdimensionn +1, equipped
with a holomorphic submersion P : U → D n with connected fibers. Each fiber P −1 (z)
is thus a smooth connected curve, and as such it has several potential theoretic invariants
(Green functions, Bergman Kernels, harmonic moduli...). One is interested
in knowing how these invariants vary with z, and then in using this knowledge to
obtain some information on the structure of U.
For our purposes, the last step in this program has been carried out by Kizuka
[Kiz], in the following form.
Theorem 2.1. [Ya1, Ya3, Kiz] If U is Stein, then the fiberwise Poincaré metricon
U P → D n has a plurisubharmonic variation.
This means the following. On each fiber P −1 (z), z ∈ D n , we put its Poincaré
metric, i.e. the (unique) complete hermitian metric of curvature −1 ifP −1 (z) is
uniformised by D, or the identically zero “metric” if P −1 (z) is uniformised by C
(U being Stein, there are no other possibilities). If v is a holomorphic nonvanishing
vector field, defined in some open subset V ⊂ U and tangent to the fibers of P, then
we can take the function on V
F = log�v�Poin