Discrete Holomorphic Local Dynamical Systems

156 Marco Brunella
to the class of a path in D from q to p. In fact, the graphs pF (X 0 ) ⊂ UF and
qF (X 0 ) ⊂ VF are hypersurfaces everywhere transverse to � F and � F .
A moment of reflection shows also the following fact: the normal bundle of the
hypersurface pF (X 0 ) in UF (or qF (X 0 ) in VF ) is naturally isomorphic to TF | X 0,
the tangent bundle of the foliation restricted to X 0 . That is, the manifold UF (resp.
VF ) can be thought as an “integrated form” of the (total space of the) tangent bundle
of the foliation, in which tangent lines to the foliation are replaced by universal
coverings (resp. holonomy coverings) of the corresponding leaves. From this perspective,
which will be useful below, the map ΠF : UF ��� X is a sort of “skew
flow” associated to F , in which the “time” varies not in C but in the universal
covering of the leaf.
Let us conclude this discussion with a trivial but illustrative example.
Example 8.1. Suppose n = 1, i.e. X is a compact connected curve and F is the foliation
with only one leaf, X itself. The manifold VF is composed by equivalence
classes of paths in X, where two paths are equivalent if they have the same starting
point and the same ending point (here holonomy is trivial!). Clearly, VF is the product
X × X, QF is the projection to the first factor, qF is the diagonal embedding of
X into X × X, andπF is the projection to the second factor. Note that the normal
bundle of the diagonal Δ ⊂ X × X is naturally isomorphic to TX. The foliation � F is
the horizontal foliation, and note that its monodromy is trivial, corresponding to the
fact that the holonomy of the foliation is trivial. The manifold UF is the fiberwise
universal covering of VF , with basepoints on the diagonal. It is not the product of
X with the universal covering �X (unless X = P, of course). It is only a locally trivial
�X-bundle over X. The foliation � F has nontrivial monodromy: if γ : [0,1] → X is a
loop based at p, then the monodromy of � F along γ is the covering transformation
of the fiber over p (i.e. the universal covering of X with basepoint p) associated to γ.
The foliation � F can be described as the suspension of the natural representation
π1(X) → Aut(�X) [CLN].
8.2 Parabolic Foliations
After these preliminaries, let us concentrate on the class of parabolic foliations,i.e.
let us assume that all the leaves of F are uniformised by C. In this case, the Poincaré
metric on the leaves is identically zero, hence quite useless. But our convexity result
Theorem 5.1 still gives a precious information on covering tubes.
Theorem 8.2. Let X be a compact connected Kähler manifold and let F be a
parabolic foliation on X. Then the global covering tube UF is a locally trivial
C-fibration over X 0 , isomorphic to the total space of TF over X 0 ,byanisomorphism
sending pF (X 0 ) to the null section.