  We prove a surjectivity theorem for the Deligne canonical extension of a
polarizable variation of Hodge structure with quasi-unipotent monodromy at
infinity along the lines of Esnault-Viehweg. We deduce from it several
injectivity theorems and vanishing theorems for pure Hodge modules. We also
give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded
piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules
of nearby cycles.
