Discrete Holomorphic Local Dynamical Systems

140 Marco Brunella
boundaries Γt, t < t0. Given any sequence of holomorphic discs D θn
tn ⊂ Mtn , tn → t0,
we have proved that (up to subsequencing) D θn
tn
D∞ with ∂D∞ ⊂ Γt0
converges uniformly to some disc
. Given any point p ∈ Γt0 , we may choose the sequence Dθn
tn so
that ∂D∞ will contain p. It remains to check that all the discs so constructed glue
together in a real analytic way, giving Mt0 , and that this Mt0 glues to Mt, t < t0, ina
real analytic way, giving the Levi-flat extension over S × D(t0).
This can be seen using a Lemma from [BeG, §5]. It says that if D is an embedded
disc in a complex surface Y with boundary in a real analytic totally real surface
Γ ⊂ Y, and if the winding number (Maslov index) of Γ along ∂D is zero, then
D belongs to a unique embedded real analytic family of discs D ε , ε ∈ (−ε0,ε0),
D 0 = D, with boundaries in Γ (incidentally, in our real analytic context this can
be easily proved by the doubling argument used in Lemma 5.5, which reduces the
statement to the well known fact that a smooth rational curve of zero selfintersection
belongs to a unique local fibration by smooth rational curves). Moreover, if Γ is
moved in a real analytic way, then the family D ε also moves in a real analytic way.
For our discs D θ t ⊂ Mt, t < t0, the winding number of Γt along ∂D θ t is zero. By
continuity of this index, if D∞ is a limit disc then the winding number of Γt0 along
∂D∞ is also zero. Thus, D∞ belongs to a unique embedded real analytic family D ε ∞ ,
with ∂Dε ∞ ⊂ Γt0 . This family can be deformed, real analytically, to a family Dεt with
∂Dε t ⊂ Γt,foreverytclose to t0.Whent = tn, such a family Dε tn necessarily contains
D θn
tn , and thus coincides with Dθ tn for θ in a suitable interval around θn. Hence, for
every t < t0 the family D ε t coincides with D θ t ,forθ in a suitable interval.
In this way, for every limit disc D∞ we have constructed a piece
�
ε∈(−ε0,ε0)
of our limit Mt0 , this piece is real analytic and glues to Mt, t < t0, in a real analytic
way.
Because each p ∈ Γt0 belongs to some limit disc D∞, we have completed in this
way our construction of the Levi-flat hypersurface Mt0 , and the proof of Theorem 5.1
is finished.
6 Hyperbolic Foliations
We can now draw the first consequences of the convexity of covering tubes given
by Theorem 5.1, still following [Br2]and[Br3].
As in the previous Section, let X be a compact Kähler manifold of dimension
n, equipped with a foliation by curves F which is not a rational quasi-fibration.
Let T ⊂ X 0 be local transversal to F 0 . We firstly need to discuss the pertinence of
hypotheses (a) and (b) that we made at the beginning of Section 5.
Concerning (a), let us simply observe that Indet(ΠT) is an analytic subset of
codimension at least two in UT , and therefore its projection to T by PT is a countable
union of locally analytic subsets of positive codimension in T (a thin subset of T).
D ε ∞