An $$L^2$$ Dolbeault lemma and its applications to vanishing theorems

Abstract

In this paper, we will first build an $$L^2$$ Dolbeault lemma by analytic methods and Hormander $$L^2$$ estimates. Then as applications, we will prove some log Nadel type vanishing theorems on compact Kahler manifolds and some log Kawamata–Viehweg type vanishing theorems on projective manifolds. Some log Nakano–Demailly type vanishing theorems for vector bundles will be also discussed by the same methods; one of which generalizes the original Nakano vanishing theorem on compact Kahler manifolds.