Pseudo-effective and numerically flat reflexive sheaves

Abstract

In this note, we discuss the concept of pseudoeffective vector bundle and also introduce pseudoeffective torsion-free sheaves over compact K\"ahler manifolds. We show that a pseudoeffective reflexive sheaf over a compact K\"ahler manifold with vanishing first Chern class is in fact a numerically flat vector bundle. A proof is obtained through a natural construction of positive currents representing the Segre classes of pseudoeffective vector bundles.