Semipositivity or semiampleness of the relative canonical bundle in smooth Kähler families

Establish that, for a smooth family of compact Kähler manifolds, if one fiber has a semipositive canonical line bundle, then the total-space canonical bundle is semipositive, or even semiample, in a neighborhood of every fiber.

Background

The paper proves local stability of canonical nefness for smooth fiberwise Kähler families and uses this result to establish local constancy of plurigenera when one fiber has nef canonical bundle. It then proposes a stronger statement concerning semipositivity and semiampleness of the canonical bundle on the total space.

The conjecture asks whether the fiberwise existence of a semipositive canonical metric propagates to a neighborhood of every fiber at the level of the total-space canonical bundle. This is stronger than the nefness results proved in the paper and is presented as an unresolved conjectural extension of the authors’ methods.