On log canonical rings

TL;DR

This paper explores the connections between key conjectures in minimal model theory, focusing on reducing the finite generation conjecture of log canonical rings for log canonical pairs to the case of purely log terminal pairs of log general type.

Contribution

It demonstrates that the finite generation conjecture for log canonical rings can be simplified by focusing on purely log terminal pairs, advancing understanding in minimal model theory.

Findings

Finite generation conjecture reduces to purely log terminal pairs

Connections established among conjectures in minimal model theory

Reduction simplifies the approach to proving finite generation

Abstract

We discuss the relationship among various conjectures in the minimal model theory including the finite generation conjecture of the log canonical rings and the abundance conjecture. In particular, we show that the finite generation conjecture of the log canonical rings for log canonical pairs can be reduced to that of the log canonical rings for purely log terminal pairs of log general type.