Anticanonical divisor with good asymptotic base loci

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.