Abundance for nef nowhere-torsion canonical bundles

Establish whether a Gorenstein projective variety with canonical singularities and a nef, nowhere-torsion canonical bundle necessarily has an ample canonical bundle, as predicted by the abundance conjecture.

Background

The paper studies nowhere-torsion line bundles as a condition sufficient for Bondal–Orlov reconstruction. It proves that a semi-ample nowhere-torsion line bundle on a proper variety is ample, and then relates this result to the abundance conjecture.

The unresolved issue is whether nefness can replace semi-ampleness for Gorenstein projective varieties with canonical singularities. A positive answer would imply ampleness of the canonical bundle under the stated nowhere-torsion hypothesis and would connect the paper’s reconstruction criterion with the expected abundance theory.