Numerical dimension and a Kawamata-Viehweg-Nadel type vanishing theorem on compact K\"ahler manifolds

Let $X$ be a compact K"ahler manifold and let $(L,\phi )$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension of pseudo-effective line bundles with singular metrics, and then discuss the properties of this type numerical dimension. We finally prove a very general Kawamata-Viehweg-Nadel type vanishing theorem on an arbitrary compact K"ahler manifold.