  Given a positive singular hermitian metric of a pseudoeffective line bundle
on a complex Kaehler manifold, a singular foliation is constructed satisfying
certain analytic analogues of numerical conditions. This foliation refines
Tsuji's numerically trivial fibration and the Iitaka fibration. Using almost
positive singular hermitian metrics with analytic singularities on a
pseudo-effective line bundle, a foliation is constructed refining the nef
fibration. If the singularities of the foliation are isolated points, the
codimension of the leaves is an upper bound to the numerical dimension of the
line bundle, and the foliation can be interpreted as a geometric reason for the
deviation of nef and Kodaira-Iitaka dimension. Several surface examples are
studied in more details, $\mathbb{P}^2$ blown up in 9 points giving a counter
example to equality of numerical dimension and codimension of the leaves.
