Pseudo-effective and numerically flat reflexive sheaves

TL;DR

This paper explores pseudoeffective and numerically flat reflexive sheaves on compact Kähler manifolds, establishing that pseudoeffective reflexive sheaves with zero first Chern class are actually numerically flat vector bundles, using positive currents and Segre classes.

Contribution

It introduces pseudoeffective torsion-free sheaves on compact Kähler manifolds and proves that such sheaves with vanishing first Chern class are numerically flat vector bundles, providing a new characterization.

Findings

Pseudoeffective reflexive sheaves with zero first Chern class are numerically flat.

Construction of positive currents representing Segre classes of pseudoeffective vector bundles.

Extension of the concept of pseudoeffectiveness to torsion-free sheaves on Kähler manifolds.

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.