Finiteness of log abundant log canonical pairs in log minimal model program with scaling

We study relations between the property of being log abundant for lc pairs and the termination of log MMP with scaling. We prove that any log MMP with scaling of an ample divisor starting with a projective dlt pair contains only finitely many log abundant dlt pairs.Comment: 42 pages. Following referee's comment, I removed Section 4 in the previous version and added a new section (Section 5 in the current version). Corollaries 3.12 and 3.13 were added. Exposition in the introduction was change

Minimal model theory for relatively trivial log canonical pairs

We study relative log canonical pairs with relatively trivial log canonical divisors. We fix such a pair $(X,\Delta)/Z$ and establish the minimal model theory for the pair $(X,\Delta)$ assuming the minimal model theory for all Kawamata log terminal pairs whose dimension is not greater than ${\rm dim}\,Z$. We also show the finite generation of log canonical rings for log canonical pairs of dimension five which are not of log general type.Comment: 38 pages, final version. The statement of Theorem 1.2 was replaced by that of Theorem 4.2, which was equivalent to Theorem 1.2. Accompanied by this, Lemma 4.1 and Theorem 4.2 was removed. Numbering of definitions and others in Section 2 and Section 4 was changed. Other minor changes. To appear in Ann. Inst. Fourier (Grenoble