  The goal of this paper is to give a new proof of a special case of the
Kodaira-Saito vanishing theorem for a variation of Hodge structure on the
complement of a divisor with normal crossings. The proof does not use the
theory of mixed Hodge modules, but instead reduces it to a more general
vanishing theorem for semistable nilpotent Higgs bundles, which is then proved
by using some facts about Higgs bundles in positive characteristic.
