  We introduce the notion of basic slc-trivial fibrations. It is a
generalization of that of Ambro's lc-trivial fibrations. Then we study
fundamental properties of basic slc-trivial fibrations by using the theory of
variations of mixed Hodge structure on cohomology with compact support. More
precisely, we prove that the moduli part of a basic slc-trivial fibration is
b-strongly nef. Note that the notion of basic slc-trivial fibrations is closely
related to that of normal irreducible quasi-log canonical pairs. So the results
obtained in this paper will play an important role in the theory of quasi-log
schemes. Here we give a structure theorem for normal irreducible quasi-log
canonical pairs as an application of the main theorem. This result makes the
theory of quasi-log schemes more powerful and more flexible.
