: In this talk, I will talk on the Nadel vanishing theorem for the multiplier ideal sheaf associated to a singular metric admitting an analytic Zariski decomposition (such as a metric with minimal singularities), under some weak assumption. For the proof, we generalize Koll ar's injectivity theorem to a line bundle with a singular metric. Further we establish the asymptotic vanishing theorem for high tensors powers of a line bundle.