  Let $X$ be a compact K\"ahler manifold and $(L,h)\rightarrow X$ be a
pseudoeffective line bundle, such that the curvature $i\Theta_{L,h}\geq 0$ in
the sense of currents. The main result of the present paper is that
$H^n(X,\mathcal{O}(\Omega^p_X\otimes L)\otimes \mathcal{I}(h))=0$ for $p\geq
n-nd(L,h)+1$. This is a generalization of Bogomolov's vanishing theorem.
