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

Abstract: We interpret a conjecture of Schnell on the equivalence of the non-vanishing and the Campana--Peternell conjectures, using the canonical bundle formula. As a result, we improve Schnell's assumption on pseudo-effectivity of the canonical bundle by adding extra effective divisors supported on the discriminant locus. We also give an inductive approach and an unconditional result for fourfolds, using rigid currents.