Vanishing and Injectivity for Hodge Modules and R-Divisors

Abstract

We prove an injectivity and vanishing theorem for Hodge modules and $\mathbb{R}$-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in [Wu17]. In particular, the injectivity generalizes the fundamental injectivity of Esnault–Viehweg for normal crossing $\mathbb{Q}$-divisors, whereas the vanishing generalizes Kawamata–Viehweg vanishing for $\mathbb{Q}$-divisors. As a main application, we also deduce a Fujita-type freeness result for Hodge modules in the normal crossing case.