Some Kollar-Enoki type injectivity and Nadel type vanishing theorems on compact Kahler manifolds

In this paper we will first show some Kollar-Enoki type injectivity theorems on compact Kahler manifolds, by using the Hodge theory, the Bochner- Kodaira-Nakano identity and the analytic method provided by O. Fujino and S. Matsumura in [15, 25, 36, 39]. We have some straightforward corollaries. In particular, we will show that our main injectivity theorem implies several Nadel type vanishing theorems on smooth projective manifolds. Second, by applying the transcendental method, especially the Demailly-Peternell-Schneider equisingular approximation theorem and the Hormander L2 estimates, we will prove some Nakano-Demailly type and Nadel type vanishing theorems for holomorphic vector bundles on compact Kahler manifolds, twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we will show that our first main vanishing theorem generalizes the classical Nakano-Demailly vanishing theorem while the second one contains the famous Nadel vanishing theorem as a special case.