On the decomposition theorem for intersection de Rham complexes

We establish a positive characteristic analogue of intersection cohomology for polarized variations of Hodge structure. This includes: a) the decomposition theorem for the intersection de Rham complex; b) the $E_1$-degeneration theorem for the intersection de Rham complex of a periodic de Rham bundle: c) the Kodaira vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle.