Nearby cycles and semipositivity in positive characteristic

Abstract

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.

A Lefschetz theorem for crystalline representations

“…In this section, we apply results due to Langer [Lan14,Lan19] in the vein of Langton to graded semistable logarithmic Higgs bundles.…”