Inductive Approach to Effective b-Semiampleness
Résumé
An lc-trivial fibration is the data of a pair (X, B) and a fibration such that (K X + B)| F is torsion where F is the general fiber. For such a fibration, we have an equality K X + B + 1/r(ϕ) = f * (K Z + B Z + M Z), where B Z is the discriminant of f, M Z is the moduli part, and ϕ is a rational function. It has been conjectured by Prokhorov and Shokurov that there exists an integer m = m(r, dim F) such that mM Z is base-point-free on some birational model Z of Z. In this work, we reduce this conjecture to the case where the base Z has dimension 1. Moreover, in the case, where M Z ≡ 0, we prove the existence of an integer m that depends only on the middle Betti number of a canonical covering of the fiber, such that mM Z ∼ 0.