  We study restriction of logarithmic Higgs bundles to the boundary divisor and
we construct the corresponding nearby-cycles functor in positive
characteristic. As applications we prove some strong semipositivity theorems
for analogs of complex polarized variations of Hodge structures and their
generalizations. This implies, e.g., semipositivity for the relative canonical
divisor of a semistable reduction in positive characteristic and it gives some
new strong results generalizing semipositivity even for complex varieties.
