Tsuji’s numerical trivial fibrations

This note grew out of an attempt to understand H. Tsuji’s work on numerical trivial foliations. In this paper, the Reduction Map Theorem [H. Tsuji, Numerically trivial fibrations, Preprint, 2000] 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. An important adjustment to the Reduction Map Theorem is to consider the fact that plurisubharmonic functions are singular on pluripolar sets. Then the author follows Tsuji’s argument for the proof of the Reduction Map Theorem. Another important result of the paper is the characterization of numerically trivial varieties by a decomposition property of the curvature current.