Abstract Let (X, ) be a normal pair with a projective morphism X Z and let A be a relatively ample R -divisor on X. We prove the termination of some minimal model program on (X, +A) /Z and the abundance conjecture for its minimal model under assumptions that the non-nef locus of Kₗ+ +A over Z does not intersect the non-lc locus of (X, ) and that the restriction of Kₗ+ +A to the non-lc locus of (X, ) is semi-ample over Z.
Kenta Hashizume (Wed,) studied this question.