  We propose and investigate two types, the latter with two variants, of
notions of partial hyperbolicity accounting for several classes of compact
complex manifolds behaving hyperbolically in certain directions, defined by a
vector subbundle of the holomorphic tangent bundle, but not necessarily in the
other directions. We study several classes of examples, prove implications
among these notions, give a sufficient criterion for the existence of an
Ahlfors current and a sufficient criterion for partial hyperbolicity in terms
of the signs of two curvature-like objects introduced recently by the
second-named author.
