  This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive
characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the
arguments here are in characteristic zero. The main result is a Kodaira
vanishing theorem for semistable parabolic Higgs bundles with trivial parabolic
Chern classes. This implies a general semipositivity theorem. This also implies
a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.
