Minimal model program for normal pairs along log canonical locus

Abstract

Let $(X,\hat{I}”)$ 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 $(X,\hat{I}”+A)/Z$ and the abundance conjecture for its minimal model under assumptions that the non-nef locus of $K_X+\hat{I}”+A$ over $Z$ does not intersect the non-lc locus of $(X,\hat{I}”)$ and that the restriction of $K_X+\hat{I}”+A$ to the non-lc locus of $(X,\hat{I}”)$ is semi-ample over $Z$.

Final version, 79 pages. Expositions in the introduction were improved. Some properties on relative Nakayama--Zariski decomposition were added. Other minor changes were done. To appear in Forum Math. Sigma