  We consider a complete nonsingular variety $X$ over $\bC$, having a normal
crossing divisor $D$ such that the associated logarithmic tangent bundle is
generated by its global sections. We show that $H^i\big(X, L^{-1} \otimes
\Omega_X^j(\log D)\big) = 0$ for any nef line bundle $L$ on $X$ and all $i < j
- c$, where $c$ is an explicit function of $(X,D,L)$. This implies e.g. the
vanishing of $H^i(X, L \otimes \Omega_X^j)$ for $L$ ample and $i > j$, and
gives back a vanishing theorem of Broer when $X$ is a flag variety.
