Vanishing and injectivity for R-Hodge modules and R-divisors

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We prove the injectivity and vanishing theorem for R-Hodge modules and R-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in Wu15. In particular, the injectivity generalizes the fundamental injectivity of Esnault-Viehweg for normal crossing Q-divisors, while the vanishing generalizes Kawamata-Viehweg vanishing for Q-divisors. As a main application, we also deduce a Fujita-type freeness result for R-Hodge modules in the normal crossing case.