Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 159
quasi-fibration. For instance, the radial foliation above can be transformed into a
rational quasi-fibration, and even into a P-bundle, by blowing-up the origin.
We have seen in Section 6 that the canonical bundle KF of a hyperbolic foliation
is always pseudoeffective. At the opposite side, for a rational quasi-fibration KF is
never pseudoeffective: its degree on a generic leaf (a smooth rational curve disjoint
from Sing(F )) is equal to −2, and this prevents pseudoeffectivity. For parabolic
foliations, the situation is mixed: the radial foliation in CP n has canonical bundle
equal to O(−1), which is not pseudoeffective; a foliation like in Example 8.3 has
trivial canonical bundle, which is pseudoeffective. One can also easily find examples
of parabolic foliations with ample canonical bundle, for instance most foliations
arising from complete polynomial vector fields in C n [Br4].
The following result, which combines Theorem 8.2 and Theorem 7.3,showsthat
most parabolic foliations have pseudoeffective canonical bundle.
Theorem 8.5. Let F be a parabolic foliation on a compact connected Kähler manifold
X. Suppose that its canonical bundle KF is not pseudoeffective. Then F is a
foliation by rational curves.
Proof. Consider the meromorphic map
ΠF : EF | X 0 � UF ��� X
given by Theorem 8.2. Because Sing(F )=X \ X 0 has codimension at least two,
such a map meromorphically extends [Siu] to the full space EF :
ΠF : EF ��� X.
The section at infinity of EF is the same as the null section of E∗ F , the total space
of KF .IfKF is not pseudoeffective, then by Theorem 7.3 ΠF extends to the full
EF = EF ∪{section at ∞}, as a meromorphic map
ΠF : EF ��� X.
By construction, ΠF sends the rational fibers of EF to rational curves in X tangent
to F , which is therefore a foliation by rational curves. ⊓⊔
Note that the converse to this theorem is not always true: for instance, a parabolic
foliation like in Example 8.3 has trivial (pseudoeffective) canonical bundle, yet it
can be a foliation by rational curves, for some special v. A parabolic foliation is a
foliation by rational curves if and only if the meromorphic map ΠF : EF ��� X
introduced in the proof above extends to the section at infinity, and this can possibly
occur even if KF is pseudoeffective, or even ample.
We have now completed our analysis of positivity properties of the canonical
bundle of a foliation, and their relation to uniformisation. We may resume the various
inclusions of the various classes of foliations in the diagram below.