Anticanonical divisor with good asymptotic base loci

TL;DR

This paper characterizes Fano type varieties using asymptotic base loci of certain divisors and establishes conditions for the existence of good minimal models in pairs with specific singularities.

Contribution

It provides a new characterization of Fano type varieties and extends results on the existence of good minimal models for potentially lc pairs.

Findings

Characterization of Fano type varieties via asymptotic base loci.

Conditions for the existence of good minimal models in potentially lc pairs.

An analogue of Birkar--Hu's result on minimal models.

Abstract

In this paper, we give a characterization of Fano type varieties in terms of the asymptotic base loci of $-(K_X+\Delta)$. We also show that for a potentially lc pair $(X,\Delta)$, if no plc centers are contained in the augmented base locus $\mathbf{B}_{+}(-(K_X+\Delta))$, then $(X,\Delta)$ has a good $-(K_X+\Delta)$-minimal model. This gives an analogous result of Birkar--Hu on the existence of good minimal models.