Strong $(\delta ,n)$-Complements for Semi-Stable Morphisms

Abstract

We prove the boundedness of global strong $(\delta ,n)$-complements for generalized $ϵ$-log canonical pairs of Fano-type. We also prove some partial results towards boundedness of local strong $(\delta ,n)$-complements for semi-stable morphisms. As applications, we prove an effective generalized canonical bundle formula for generalized klt pairs and an effective generalized adjunction formula for exceptional generalized log canonical centers. Moreover, we prove that the existence of strong $(\delta ,n)$-complements implies a conjecture due to McKernan concerning the singularities of the base of a Mori fiber space.