Inductive Approach to Effective b-Semiampleness

An lc-trivial fibration is the data of a pair (X,B) and a fibration such that (KX+B)|F is torsion where F is the general fiber. For such a fibration, we have an equality KX+B+1/r(φ)=f*(KZ+BZ+MZ), where BZ is the discriminant of f, MZ 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 mMZ′ 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 MZ≡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 mMZ∼0.