Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 145
is not a rational quasi-fibration, and all the local weights F introduced above are
plurisubharmonic, and not identically −∞.
Let us state two evident but important Corollaries.
Corollary 6.5. The canonical bundle KF of a hyperbolic foliation F is pseudoeffective.
Corollary 6.6. Given a hyperbolic foliation F , the subset
is complete pluripolar in X.
Sing(F ) ∪ Parab(F )
We think that the conclusion of this last Corollary could be strengthened. The
most optimistic conjecture is that Sing(F )∪Parab(F ) is even an analytic subset of
X. At the moment, however, we are very far from proving such a fact (except when
dimX = 2, where special techniques are available, see [MQ1] and[Br1]). Even
the closedness of Sing(F ) ∪ Parab(F ) seems an open problem! This is related to
the more general problem of the continuity of the leafwise Poincaré metric(which
would give, in particular, the closedness of its polar set). Let us prove a partial result
in this direction, following a rather standard hyperbolic argument [Ghy,Br2]. Recall
that a complex compact analytic space Z is hyperbolic if every holomorphic map of
C into Z is constant [Lan].
Theorem 6.7. Let F be a foliation by curves on a compact connected Kähler manifold
M. Suppose that:
(i) every leaf is hyperbolic;
(ii) Sing(F ) is hyperbolic.
Then the leafwise Poincaré metric is continuous.
Proof. Let us consider the function
F : U 0 → R, F(q)=log�v(q)�Poin
introduced just before Proposition 6.2. WehavetoprovethatF is continuous (the
continuity on the full U is then a consequence of Proposition 6.3). We have already
observed, during the proof of Proposition 6.2,thatF is upper semicontinuous, hence
let us consider its lower semicontinuity.
Take q∞ ∈ U 0 and take a sequence {qn}⊂U 0 converging to q∞. Foreveryn, let
ϕn : D → X be a holomorphic map into Lqn ⊂ X, sending 0 ∈ D to qn ∈ Lqn .For
every compact subset K ⊂ D, consider
IK = {�ϕ ′ n (t)� |t ∈ K,n ∈ N}⊂R
(the norm of ϕ ′ n is here computed with the Kähler metric on X).
Claim: IK is a bounded subset of R.