A weak Kawamata-Viehweg Vanishing Theorem

Using the techniques of J.-P. Demailly and Th. Peternell in math.AG/0208021 a weak version of Kawamata-Viehweg vanishing is proven for pseudo-effective line bundles on compact Kaehler manifolds. It is a weak version, since one has to adjust the line bundle $K_X+L$ with the upper regularized multiplier ideal sheaf $I_{+}(L)$ of $L$, and this is already not optimal in the nef case.