Intersection de Rham complexes in positive characteristic

Abstract

We establish a positive characteristic analogue of intersection cohomology theory for variations of Hodge structure. It includes: a) the de Rham-Higgs comparison 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–Saito vanishing theorem for the intersection cohomology groups of a periodic Higgs bundle. These results generalize the decomposition theorem of Deligne–Illusie [DI] and the de Rham–Higgs theorem of Ogus–Vologodsky [OV], the $E_1$-degeneration theorem of Deligne-Illusie [DI], Illusie [I90], Faltings [Fa] and the Kodaira-Saito vanishing theorem of Arapura [Ar]. As an application, we give an algebraic proof of the $E_1$-degeneration theorem due to Cattani-Kaplan–Schmid [CKS] and Kashiwara–Kawai [KK], and the vanishing theorem of Saito [Sa] for VHSs of geometric origin.