An L2 Dolbeault lemma and its applications to vanishing theorems

In this paper, we will first build an L2\documentclass[12pt] Dolbeault lemma by analytic methods and Hormander L2\documentclass[12pt] 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.