  In this paper, we investigate higher direct images of log canonical divisors.
After we reformulate Koll\'ar's torsion-free theorem, we treat the relationship
between higher direct images of log canonical divisors and the canonical
extensions of Hodge filtration of gradedly polarized variations of mixed Hodge
structures. As a corollary, we obtain a logarithmic version of
Fujita-Kawamata's semi-positivity theorem. By this semi-positivity theorem, we
generalize Kawamata's positivity theorem and apply it to the study of a log
canonical bundle formula. The final section is an appendix, which is a result
of Morihiko Saito.
