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In this paper we will first show some Kollar-Enoki type injectivity theoremson 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. Inparticular, we will show that our main injectivity theorem implies severalNadel type vanishing theorems on smooth projective manifolds. Second, byapplying the transcendental method, especially the Demailly-Peternell-Schneiderequisingular approximation theorem and the Hormander L2 estimates, we willprove some Nakano-Demailly type and Nadel type vanishing theorems forholomorphic vector bundles on compact Kahler manifolds, twisted bypseudo-effective line bundles and multiplier ideal sheaves. As applications, wewill show that our first main vanishing theorem generalizes the classicalNakano-Demailly vanishing theorem while the second one contains the famousNadel vanishing theorem as a special case.

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