Tsuji's Numerical Trivial Fibrations

The Reduction Map Theorem in H. Tsuji's work on numerical trivial fibrations is corrected and proven. To this purpose various definitions of Tsuji's new intersection numbers for pseudo-effective line bundles equipped with a positive singular hermitian metric are compared and their equivalence on sufficiently general smooth curves is shown. Numerically trivial varieties are characterized by a decomposition property of the curvature current. An important adjustment to the Reduction Map Theorem is to consider the fact that plurisubharmonic functions are singular on pluripolar sets.