Discrete Holomorphic Local Dynamical Systems

126 Marco Brunella
Take a vanishing end E ⊂ L 0 t ,oforderk, let f : D n−1 × Ar → X 0 be an almost
embedding adapted to E, andletg : W ��� X be a meromorphic family of discs
extending f , immersive outside F = Indet(g). Take also a parabolic end �E ⊂ � L0 t
projecting to E, with degree k. By an easy holonomic argument, the immersion
g| W\F : W \ F → X 0 can be lifted to V 0 T , as a proper embedding
�g : W \ F → V 0 T
which sends the central fiber W0 \ F0 to �E. Each fiber Wz \ Fz is sent by �g to a closed
subset of a fiber � L 0 t(z) , and each point of Fz corresponds to a parabolic end of � L 0 t(z)
projecting to a vanishing end of L 0 t(z) .
Now we can glue W to V 0 T using �g: this corresponds to compactify all parabolic
ends of fibers of V 0 T which project to vanishing ends and which are close to �E. By
doing this operation for every E and �E, we finally construct our manifold VT ,fibered
over T with fibers �Lt.ThemapπTextending (meromorphically) π0 T is then deduced
from the maps g above. ⊓⊔
The manifoldVT will be called holonomy tube over T . The meromorphic immersion
πT is, of course, very complicated: it contains all the dynamics of the foliation,
so that it is, generally speaking, very far from being, say, finite-to-one. Note, however,
that most fibers do not cut the indeterminacy set of πT ,sothatπT sends that
fibers to leaves of F 0 ; moreover, most leaves have trivial holonomy (it is a general
fact [CLN] that leaves with non trivial holonomy cut any transversal along a thin
subset), and so on most fibers πT is even an isomorphism between the fiber and the
corresponding leaf of F 0 . But be careful: a leaf may cut a transversal T infinitely
many times, and so VT will contain infinitely many fibers sent by πT to the same
leaf, as holonomy coverings (possibly trivial) with different basepoints.
4.3 Covering Tubes
The following proposition is similar, in spirit, to Proposition 4.5, but,asweshall
see, its proof is much more delicate. Here the Kähler assumption becomes really
indispensable, via the unparametrized Hartogs extension lemma. Without the Kähler
hypothesis it is easy to find counterexamples (say, for foliations on Hopf surfaces).
Proposition 4.6. There exists a complex manifold UT of dimension n, a holomorphic
submersion
PT : UT → T,
a holomorphic section
and a surjective holomorphic immersion
pT : T → UT ,
FT : UT → VT