Discrete Holomorphic Local Dynamical Systems

106 Marco Brunella
By the classical Uniformisation Theorem, the universal covering of each leaf is
either the unit disc D (hyperbolic leaf) or the affine line C (parabolic leaf) or the
projective line P (rational leaf).
In these notes we shall assume that the ambient manifold X is a compact connected
Kähler manifold, and we will be concerned with the following problem: how
the universal covering �Lp of the leaf Lp through the point p depends on p ?Forinstance,
we may first of all ask about the structure of the subset of X formed by those
points through which the leaf is hyperbolic, resp. parabolic, resp. rational: is the set
of hyperbolic leaves open in X? Is the set of parabolic leaves analytic? But, even if
all the leaves are, say, hyperbolic, there are further basic questions: the uniformising
map of every leaf is almost unique (unique modulo automorphisms of the disc), and
after some normalization (to get uniqueness) we may ask about the way in which
the uniformising map of Lp depends on the point p. Equivalently, we may put on
every leaf its Poincaré metric, and we may ask about the way in which this leafwise
metric varies in the directions transverse to the foliation.
Our main result will be that these universal coverings of leaves can be glued
together in a vaguely “holomorphically convex” way. That is, the leafwise universal
covering of the foliated manifold (X,F ) can be defined and it has a sort of “holomorphically
convex” structure [Br2, Br3]. This was inspired by a seminal work of
Il’yashenko [Il1,Il2], who proved a similar result when X is a Stein manifold instead
of a compact Kähler one. Related ideas can also be found in Suzuki’s paper [Suz],
still in the Stein case. Another source of inspiration was Shafarevich conjecture on
the holomorphic convexity of universal coverings of projective (or compact Kähler)
manifolds [Nap].
This main result will allow us to apply results by Nishino [Nis] and Yamaguchi
[Ya1, Ya2, Ya3, Kiz] concerning the transverse variation of the leafwise Poincaré
metric and other analytic invariants. As a consequence of this, for instance, we shall
obtain that if the foliation has at least one hyperbolic leaf, then: (1) there are no
rational leaves; (2) parabolic leaves fill a subset of X which is complete pluripolar,
i.e. locally given by the poles of a plurisubharmonic function. In other words, the
set of hyperbolic leaves of F is either empty or potential-theoretically full in X.
These results are related also to positivity properties of the canonical bundle KF ,
along a tradition opened by Arakelov [Ara, BPV] in the case of algebraic fibrations
by curves and developed by Miyaoka [Miy, ShB] and then McQuillan and
Bogomolov [MQ1, MQ2, BMQ, Br1] in the case of foliations on projective manifolds.
From this point of view, our final result is the following ruledness criterion
for foliations:
Theorem 1.1. [Br3, Br5] Let X be a compact connected Kähler manifold and let
F be a foliation by curves on X. Suppose that the canonical bundle KF is not
pseudoeffective. Then through every point p ∈ X there exists a rational curve tangent
to F .
Recall that a line bundle on a compact connected manifold is pseudoeffective if it
admits a (singular) hermitian metric with positive curvature in the sense of currents