Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell

TL;DR

This paper explores the canonical bundle formula's role in Schnell's conjecture on algebraic fiber spaces, providing partial results under weakened assumptions and connecting non-vanishing and Campana--Peternell conjectures.

Contribution

It offers new partial results on Schnell's conjecture by weakening assumptions and applying the canonical bundle formula, advancing understanding of algebraic fiber spaces.

Findings

Partial verification of Schnell's conjecture under weakened assumptions.

Establishment of a link between non-vanishing and Campana--Peternell conjectures.

Proving Schnell's conjecture when the general fiber's canonical class is represented by a rigid current.

Abstract

We observe what the canonical bundle formula gives towards a conjecture of Schnell on algebraic fiber spaces, a question concerning the equivalence between the non-vanishing conjecture and the Campana--Peternell conjecture. As a result, we give a partial result on Schnell's conjecture under two independent assumptions. One weakens Schnell's assumption of the pseudo-effectivity of the canonical bundle of the base by adding some effective divisor supported on the ramification locus. The other is analogous to results on algebraic fiber spaces where the existence of good minimal models of a general fiber is assumed, but we use a priori a weaker assumption. More precisely, we prove Schnell's conjecture when the canonical class of the general fiber is represented by a rigid current.