Under the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs (X/Z, ∆) such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most n. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.Date: 2019/08/25, version 0.56. 2010 Mathematics Subject Classification. 14E30.

We continue our study of the relation between log minimal models and various types of Zariski decompositions. Let (X, B) be a projective log canonical pair. We will show that (X, B) has a log minimal model if either K X + B birationally has a Nakayama-Zariski decomposition with nef positive part, or that K X + B is big and birationally it has a Fujita or CKM Zariski decomposition. Along the way we introduce polarized pairs (X, B + P ) where (X, B) is a usual projective pair and P is nef, and study the birational geometry of such pairs.

Abstract. The main goal of this paper is to construct an algebraic analogue of quasi-plurisubharmonic function (qpsh for short) from complex analysis and geometry. We define a notion of qpsh function on a valuation space associated to a quite general scheme. We then define the multiplier ideals of these functions and prove some basic results about them, such as subadditivity property, the approximation theorem. We also treat some applications in complex algebraic geometry.

We prove that the moduli b-divisor of an lc-trivial fibration from a log canonical pair is log abundant. This result follows from a theorem on the restriction of the moduli b-divisor, which is also obtained by E. Floris and V. Lazić [FL19]. We give an alternative proof in Section 4.3, based on a theory of lc-trivial morphisms. We also prove a theorem on extending a finite cover over a closed subvariety of the same degree, on a variety over an arbitrary field.

Abstract. Let (X, B) be a projective log canonical pair such that B is a Q-divisor, and that there is a surjective morphism f : X → Z onto a normal variety Z satisfying: K X + B ∼ Q f * M for some big Q-divisor M , and the augmented base locus B + (M ) does not contain the image of any log canonical centre of (X, B). We will show that (X, B) has a good log minimal model. An interesting special case is when f is the identity morphism.