  We develop an intersection theory for a singular hemitian line bundle with
positive curvature current on a smooth projective variety and irreducible
curves on the variety. And we prove the existence of a natural rational
fibration structure associated with the singular hermitian line bundle. Also
for any pseudoeffective line bundle on a smooth projective variety, we prove
the existence of a rational natural fibration structure associated with the
line bundle.
  We also characterize a numerically trivial singular hermitain line bundle on
a smooth projective variety.
