Abstract
We prove an injectivity and vanishing theorem for Hodge modules and 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 Q-divisors, whereas the vanishing generalizes Kawamata–Viehweg vanishing for Q-divisors. As a main application, we also deduce a Fujita-type freeness result for Hodge modules in the normal crossing case.
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