Inductive approach to effective b-semiampleness

TL;DR

This paper reduces a conjecture about the effectiveness of the b-semiampleness of the moduli part in the canonical bundle formula to the case of one-dimensional bases and proves a linear equivalence result when the moduli part is numerically trivial.

Contribution

It simplifies the conjecture to lower-dimensional cases and establishes a linear equivalence for numerically trivial moduli parts based on topological invariants.

Findings

Reduction of the conjecture to one-dimensional base cases

Proof of linear equivalence for numerically trivial moduli parts

Dependence of the integer m on the Betti number of a canonical cover

Abstract

It has been conjectured by Prokhorov and Shokurov that the moduli part in the canonical bundle formula is effectively b-semiample. In this work we reduce this conjecture to the case where the base of the fibration has dimension one. Moreover, in the case where the moduli part M is numerically trivial, we prove the existence of an integer m, that depends only on the middle Betti number of a canonical covering of the fibre, such that mM is linearly equivalent to the trivial bundle.