Admissible extremal rays and the anticanonical minimal model program

Abstract

We develop an anticanonical minimal model program for projective $\mathbb{Q}$-factorial klt pairs $(X,Δ)$ such that $-(K_{X}+Δ)$ is pseudoeffective. We first prove the cone and contraction theorems for potentially klt triples and establish the existence of flips for such triples. Using these results, we introduce the notion of admissible extremal rays and show that $(K_{X}+Δ)$-positive admissible extremal rays are precisely the rays that are to be contracted by $-(K_{X}+Δ)$-MMP. The elementary MMP steps associated to such admissible rays preserve klt singularities. For potentially klt pairs $(X,Δ)$ with $\mathbb{Q}$-divisors $Δ$, we prove that every $(K_{X}+Δ)$-positive extremal ray is admissible and obtain a cone theorem for the $(K_{X}+Δ)$-positive part of the Mori cone. Assuming termination of the anticanonical minimal model program, we establish a dichotomy: potentially klt pairs admit anticanonical minimal models, whereas non-potentially klt pairs admit non-admissible models.