Strong $(\delta,n)$-complements for semi-stable morphisms

TL;DR

This paper establishes boundedness results for strong complements in the context of semi-stable morphisms and generalized pairs, with applications to canonical bundle formulas and singularity conjectures.

Contribution

It proves boundedness of strong complements for generalized pairs of Fano type and partial results for semi-stable morphisms, advancing the understanding of their structure.

Findings

Boundedness of global strong complements for generalized pairs.

Effective formulas for canonical bundle and adjunction in generalized settings.

Implication of strong complements existence on a conjecture about Mori fiber spaces.

Abstract

We prove boundedness of global strong $(\delta,n)$-complements for generalized $\epsilon$-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 M$^{\rm c}$Kernan concerning the singularities of the base of a Mori fiber space.