  We show that the dualizing sheaves of reduced simple normal crossings pairs
have a canonical weight filtration in a compatible way with the one on the
corresponding mixed Hodge modules by calculating the extension classes between
the dualizing sheaves of smooth varieties. Using the weight spectral sequence
of mixed Hodge modules, we then reduce the semi-positivity theorem for the
higher direct images of dualizing sheaves to the smooth case where the
assertion is well known. This may be used to simplify some constructions in a
recent paper of Y. Kawamata. We also give a simple proof of the semi-positivity
theorem for admissible variations of mixed Hodge structure in [FF] by using the
theories of Cattani, Kaplan, Schmid, Steenbrink, and Zucker. This generalizes
Kawamata's classical result in the pure case.
