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

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.