Numerical trivial fibrations
  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. 
Hajime Tsuji