A valuative approach to the anticanonical minimal model program

TL;DR

This paper introduces a valuative approach to the anticanonical minimal model program, extending previous results to potentially klt triples and enabling the running of MMP in this broader context.

Contribution

It develops a framework using quasi-monomial valuations for potentially klt triples, extending Xu's results and facilitating MMP execution on these triples.

Findings

Log canonical threshold of potentially klt triples computed via quasi-monomial valuations

Framework allows MMP to be run on potentially klt triples and pairs

Extension of Xu's results to a broader class of triples

Abstract

In this paper, we show that the log canonical threshold of a potentially klt triple can be computed by a quasi-monomial valuation. The notion of potential triples provides a larger and more flexible framework to work with than that of generalized pairs. Our main result can be considered as an extension to the result of Xu on klt pairs. As an application of the main result, we show that we can run the MMP on any potentially klt triples and $-(K_X+\Delta)$-MMP on the potentially klt pairs.