On a Bogomolov type vanishing theorem

Let $X$ be a compact Kähler manifold and $(L,h)\to X$ be a pseudoeffective line bundle, such that the curvature $i{\mathrm{\Theta }}_{L,h}\ge 0$ in the sense of currents. The main result of the present paper is that $H^n(X,\mathcal{O}({\mathrm{\Omega }}_X^p\otimes L)\otimes \mathcal{I}(h))=0$ for $p\ge n-nd(L,h)+1$. This is a generalization of Bogomolov’s vanishing theorem.