Kodaira-type Vanishings via Non-abelian Hodge Theory

In this paper, we use non-abelian Hodge Theory to study Kodaira type vanishings and its generalizations. In particular, we generalize Saito vanishing using Mixed Twistor D-modules. We also generalize it to a Kawamata-Viehweg type vanishing using Q-divisors, and we also prove a relative version for a projective morphism.