Discrete Holomorphic Local Dynamical Systems

130 Marco Brunella
In fact, there is a common framework for the Stein case and the compact Kähler
case: the framework of holomorphically convex (not necessarily compact) Kähler
manifolds. Indeed, the only form of compactness that we need, in this Section and
also in the next one, is the following: for every compact K ⊂ X, there exists a (larger)
compact ˆK ⊂ X such that every holomorphic disc in X with boundary in K is fully
contained in ˆK. This property is obviously satisfied by any holomorphically convex
Kähler manifold, with ˆK equal to the usual holomorphically convex hull of K.
A more global point of view on holonomy tubes and covering tubes will be
developed in the last Section, on parabolic foliations.
4.4 Rational Quasi-Fibrations
We conclude this Section with a result which can be considered as an analog, in our
context, of the classical Reeb Stability Theorem for real codimension one foliations
[CLN].
Proposition 4.10. Let X be a compact connected Kähler manifold and let F be a
foliation by curves on X. Suppose that there exists a rational leaf Lp (i.e., �Lp = P).
Then all the leaves are rational. Moreover, there exists a compact connected Kähler
manifold Y , dimY = dimX − 1, a meromorphic map B : X ��� Y, and Zariski open
and dense subsets X0 ⊂ X,Y0 ⊂ Y, such that:
(i) B is holomorphic on X0 and B(X0)=Y0;
(ii) B : X0 → Y0 is a proper submersive map, all of whose fibers are smooth rational
curves, leaves of F .
Proof. It is sufficient to verify that all the leaves are rational; then the second part
follows by standard arguments of complex analytic geometry, see e.g. [CaP].
By connectivity, it is sufficient to prove that, given a covering tube UT ,ifsome
fiber is rational then all the fibers are rational. We can work, equivalently, with the
holonomy tube VT . Now, such a property was actually already verified in the proof
of Proposition 4.6, in the form of “nonexistence of vanishing cycles”. Indeed, the set
of rational fibers of VT is obviously open. To see that it is also closed, take a fiber �Lt
approximated by fibers �Ltn � P. Take an embedded cycle Γ ⊂ �Lt, approximated by
cycles Γn ⊂ �Ltn . Each Γn bounds in �Ltn two discs, one on each side. As in the proof
of Proposition 4.6, we obtain that Γ also bounds in �Lt two discs, one on each side.
Hence �Lt is rational. ⊓⊔
Such a foliation will be called rational quasi-fibration. A meromorphic map B
as in Proposition 4.10 is sometimes called almost holomorphic, because the image
of its indeterminacy set is a proper subset of Y , of positive codimension, contained
in Y \ Y0. IfdimX = 2thenB is necessarily holomorphic, and the foliation is a
rational fibration (with possibly some singular fibers). In higher dimensions one