Some remarks on the minimal model program for log canonical pairs

TL;DR

This paper investigates properties of the minimal model program for log canonical pairs, demonstrating that extremal Fano contractions preserve log canonical singularities and exploring related topics like canonical ring finite generation and moduli part semi-ampleness.

Contribution

It establishes that the target space of extremal Fano contractions from log canonical pairs remains log canonical and discusses the nefness and semi-ampleness of moduli parts in lc-trivial fibrations.

Findings

Target spaces of extremal Fano contractions are log canonical.

Nefness of moduli parts in lc-trivial fibrations is proven.

Observations on semi-ampleness and examples of non-Kähler manifolds are provided.

Abstract

We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact K\"ahler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-K\"ahler manifolds, flopping contractions, and so on, in order to clarify our results.