Discrete Holomorphic Local Dynamical Systems

116 Marco Brunella
(ii) f factorizes as g ◦ j for some embedding j : D k × Ar → W, sending D k × ∂Ar to
∂W and {z}×Ar into Wz, foreveryz ∈ D k .
In particular, the restriction of g to the fiber Wz gives, after removal of indeterminacies,
a disc which extends f (z,Ar), and these discs depend on z in a meromorphic
way. The manifold W is differentiably a product of D k with D, but in general this
does not hold holomorphically. However, note that by definition W is around its
boundary ∂W isomorphic to a product D k × Ar.
We shall say that an immersion f : D k ×Ar → X is an almost embedding if there
exists a proper analytic subset I ⊂ D k such that the restriction of f to (D k \ I) × Ar
is an embedding. In particular, for every z ∈ D k \ I, f (z,Ar) is an embedded annulus
in X,and f (z,Ar), f (z ′ ,Ar) are disjoint if z,z ′ ∈ D k \ I are different.
The following result is a sort of “unparametrized” Hartogs extension lemma
[Siu, Iv1], in which the extension of maps is replaced by the extension of their images.
Its proof is inspired by [Iv1] and[Iv2]. The new difficulty is that we need to
construct not only a map but also the space where it is defined. The necessity of this
unparametrized Hartogs lemma for our future constructions, instead of the usual
parametrized one, has been observed in [ChI].
Theorem 3.1. Let X be a compact Kähler manifold and let f : D k × Ar → Xbe
an almost embedding. Suppose that there exists an open nonempty subset Ω ⊂ D k
such that f (z,Ar) extends to a disc for every z ∈ Ω. Then f(D k × Ar) extends to a
meromorphic family of discs.
Proof. Consider the subset
Z = { z ∈ D k \ I | f (z,Ar) extends to a disc }.
Our first aim is to give to Z a complex analytic structure with countable base. This is
a rather standard fact, see [Iv2] for related ideas and [CaP] for a larger perspective.
For every z ∈ Z, fix a simple extension
gz : D → X
of f (z,Ar). We firstly put on Z the following metrizable topology: we define the
distance between z1,z2 ∈ Z as the Hausdorff distance in X between the discs gz1 (D)
and gz2 (D). Note that this topology may be finer than the topology induced by the
inclusion Z ⊂ Dk :ifz1,z2 ∈ Z are close each other in Dk then gz1 (D), gz2 (D) are
close each other near their boundaries, but their interiors may be far each other
(think to blow-up).
Take z ∈ Z and take a Stein neighbourhood U ⊂ X of gz(D). Consider the subset
A ⊂ Dk \ I of those points z ′ such that the circle f (z ′ ,∂Ar) is the boundary of a
compact complex curve Cz ′ contained in U. Note that, by the maximum principle,
such a curve is Hausdorff-close to gz(D), ifz ′ is close to z. According to a theorem
of
�
Wermer or Harvey-Lawson [AWe, Ch.19], this condition is equivalent to say that
f (z ′ ,∂Ar) β = 0 for every holomorphic 1-form β on U (moment condition). These
integrals depend holomorphically on z ′ ,foreveryβ . We deduce (by noetherianity)
that A is an analytic subset of Dk \ I, on a neighbourhood of z. Foreveryz ′ ∈ A,