Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves
Marco Brunella
Abstract These lecture notes provide a full discussion of certain analytic aspects of
the uniformisation theory of foliations by curves on compact Kähler manifolds, with
emphasis on convexity properties and their consequences on positivity properties of
the corresponding canonical bundles.
1 Foliations by Curves and their Uniformisation
Let X be a complex manifold. A foliation by curves F on X is defined by a
holomorphic line bundle TF on X and a holomorphic linear morphism
τF : TF → TX
which is injective outside an analytic subset Sing(F ) ⊂ X of codimension at least
2, called the singular set of the foliation. Equivalently, we have an open covering
{Uj} of X and a collection of holomorphic vector fields v j ∈ Θ(Uj), with zero set
of codimension at least 2, such that
v j = g jkvk on Uj ∩Uk,
where g jk ∈ O ∗ (Uj ∩Uk) is a multiplicative cocycle defining the dual bundle T ∗ F =
KF , called the canonical bundle of F .
These vector fields can be locally integrated, and by the relations above these
local integral curves can be glued together (without respecting the time parametrization),
giving rise to the leaves of the foliation F .
M. Brunella
IMB - CNRS UMR 5584, 9 Avenue Savary, 21078 Dijon, France
e-mail: Marco.Brunella@u-bourgogne.fr
G. Gentili et al. (eds.), Holomorphic Dynamical Systems, 105
Lecture Notes in Mathematics 1998, DOI 10.1007/978-3-642-13171-4 3,
c○ Springer-Verlag Berlin Heidelberg 2010