Vanishing theorems for Dolbeault cohomology of log homogeneous varieties

We consider a complete nonsingular variety $X$ over $\text{bC}$, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i\big(X, L^{-1} \otimes \Omega_X^j(\log D)\big) = 0$foranyneflinebundle$L$on$X$andall$i < j

• c$,where$c$isanexplicitfunctionof$(X,D,L)$. This implies e.g. the vanishing of $H^i(X,L\otimes {\mathrm{\Omega }}_X^j)$ for $L$ ample and $i>j$, and gives back a vanishing theorem of Broer when $X$ is a flag variety.