Minimal model program for normal pairs along log canonical locus

Abstract

Let $(X,Δ)$ be a normal pair with a projective morphism $X \to Z$ and let $A$ be a relatively ample $\mathbb{R}$-divisor on $X$. We prove the termination of some minimal model program on $(X,Δ+A)/Z$ and the abundance conjecture for its minimal model under assumptions that the non-nef locus of $K_{X}+Δ+A$ over $Z$ does not intersect the non-lc locus of $(X,Δ)$ and that the restriction of $K_{X}+Δ+A$ to the non-lc locus of $(X,Δ)$ is semi-ample over $Z$.