Discrete Holomorphic Local Dynamical Systems

154 Marco Brunella
Condition (B) is simpler [Din, §4]. We just have to exhibit a ω-plurisubharmonic
function on Γf which is nonpositive on U and exhaustive on Γf \U.OnU0 Σ we take
the function
ψ(z)=−logdist(z,Σ)
where dist(·,Σ) is the distance function from Σ, with respect to the Kähler metric
ω0 on UΣ . Classical estimates (Takeuchi) give dd c ψ ≥−C · ω0, for some positive
constant C. Thus
dd c (ψ ◦ π) ≥−C · π ∗ (ω0) ≥−C · ω
because ω ≥ π∗ (ω0). Hence 1 C (ψ ◦ π) is ω-plurisubharmonic on Γf . For a sufficiently
large C ′ > 0, 1
C (ψ ◦ π) −C′ is moreover negative on U, and it is exhaustive
on Γf \U. Thus condition (B) is fulfilled.
Finally we can apply Theorem 7.1, obtain the finiteness of the volume of the
graph of f , and conclude the proof of the theorem. ⊓⊔
Remark 7.5. We think that Theorem 7.3 should be generalizable to the following
“nonlinear” statement: if UΣ is any Kähler manifold and Σ ⊂ UΣ is a compact hypersurface
whose normal bundle is not pseudoeffective, then any meromorphic map
f : UΣ \ Σ ��� Y (Y Kähler compact) extends to ¯f : UΣ ��� Y. The difficulty is to
show that a locally pseudoconvex subset of U 0 Σ = UΣ \ Σ like Ω0 in the proof above
can be “lifted” in the total space of the normal bundle of Σ, preserving the local
pseudoconvexity.
8 Parabolic Foliations
We can now return to foliations.
As usual, let X be a compact connected Kähler manifold, dimX = n, andletF
be a foliation by curves on X different from a rational quasi-fibration. Let us start
with some general remarks, still following [Br5].
8.1 Global Tubes
The construction of holonomy tubes and covering tubes given in Section 4 can be
easily modified by replacing the transversal T ⊂ X 0 with the full X 0 . That is, all the
holonomy coverings �Lp and universal coverings �Lp, p ∈ X 0 , can be glued together,
without the restriction p ∈ T . The results are complex manifolds VF and UF ,of
dimension n + 1, equipped with submersions
sections
QF : VF → X 0 , PF : UF → X 0
qF : X 0 → VF , pF : X 0 → UF