Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell
Hyunsuk Kim
TL;DR
This work analyzes Schnell's conjecture on algebraic fiber spaces by leveraging the canonical bundle formula to connect base positivity with the global Kodaira dimension. It establishes κ(X)=dim Y under two independent assumptions: (i) a small perturbation of the discriminant term yields pseudo-effectivity on the base, and (ii) rigidity of the general-fiber canonical class provides a metric-analytic constraint. The proofs combine Siu decomposition, Bergman-kernel metrics, and precise control of Lelong numbers to descend positivity from X to Y and show the base contribution K_Y+B_Y+M_Y is big. These results extend Schnell's approach by relaxing the base pseudo-effectivity condition and introducing the analytic notion of rigidity, with potential implications for the Campana--Peternell conjecture and related birational stability questions.
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.