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. (read more)