Discrete Holomorphic Local Dynamical Systems

158 Marco Brunella
vanishing, F(z,·) holomorphic for every fixed z, andF(·,0) also holomorphic. The
completeness on fibers gives that F is in fact constant on every fiber, i.e. F = F(z),
and so F is in fact fully holomorphic. Thus �v is fully holomorphic on the tube. This
means precisely that the above map EF | X 0 → UF is holomorphic. ⊓⊔
Example 8.3. Consider a foliation F generated by a global holomorphic vector field
v ∈ Θ(X), vanishing precisely on Sing(F ). This means that TF is the trivial line
bundle, and EF = X × C. The compactness of X permits to define the flow of v
Φ : X × C → X
which sends {p}×C to the orbit of v through p, thatistoL0 p if p ∈ X 0 or to {p} if
p ∈ Sing(F ). Recalling that Lp = L0 p for a generic leaf, and observing that every L0 p
is obviously parabolic, we see that F is a parabolic foliation. It is also not difficult
to see that, in fact, Lp = L0 p for every leaf, i.e. there are no vanishing ends, and so
the map
ΠF : UF → X 0
is everywhere holomorphic, with values in X 0 .WehaveUF = X 0 ×C (by Theorem
8.2, which is however quite trivial in this special case), and the map ΠF : X 0 ×C →
X 0 can be identified with the restricted flow Φ : X 0 × C → X 0 .
Remark 8.4. It is a general fact [Br3] that vanishing ends of a foliation F produce
rational curves in X over which the canonical bundle KF has negative degree. In
particular, if KF is algebraically nef (i.e. KF ·C ≥ 0 for every compact curve C ⊂ X)
then F has no vanishing end.
8.3 Foliations by Rational Curves
We shall say that a foliation by curves F is a foliation by rational curves if for
every p ∈ X 0 there exists a rational curve Rp ⊂ X passing through p and tangent
to F . This class of foliations should not be confused with the smaller class of rational
quasi-fibrations: certainly a rational quasi-fibration is a foliation by rational
curves, but the converse is in general false, because the above rational curves Rp
can pass through Sing(F ) and so Lp (which is equal to Rp minus those points of
Rp ∩Sing(F ) not corresponding to vanishing ends) can be parabolic or even hyperbolic.
Thus the class of foliations by rational curves is transversal to our fundamental
trichotomy rational quasi-fibrations / parabolic foliations / hyperbolic foliations.
A typical example is the radial foliation in the projective space CPn , i.e. the foliation
generated (in an affine chart) by the radial vector field ∑z j ∂ : it is a foliation
∂zj
by rational curves, but it is parabolic. By applying a birational map of the projective
space, we can get also a hyperbolic foliation by rational curves.On the other
hand, it is a standard exercise in bimeromorphic geometry to see that any foliation
by rational curves can be transformed, by a bimeromorphic map, into a rational