Some remarks on the minimal model program for log canonical pairs

We prove that the target space of an extremal Fano contraction from a log canonical pair has only log canonical singularities. We also treat some related topics, for example, the finite generation of canonical rings for compact K\"ahler manifolds, and so on. The main ingredient of this paper is the nefness of the moduli parts of lc-trivial fibrations. We also give some observations on the semi-ampleness of the moduli parts of lc-trivial fibrations. For the reader's convenience, we discuss some examples of non-K\"ahler manifolds, flopping contractions, and so on, in order to clarify our results.Comment: 37 pages, v2: title changed, major revision, v3: update the reference list, v4: Section 6 on non-K\"ahler manifolds is new, v5: minor revision following referee's comment, v6: References were updated, final version, to appear in Kodaira Centennial issue of the Journal of Mathematical Sciences, the University of Toky