A vanishing theorem

Abstract: Let $E$ be a vector bundle and $L$ be a line bundle over a smooth projective variety $X$. In this article, we give a condition for the vanishing of Dolbeault cohomology groups of the form $H^{p,q}(X,\SSS^{\alpha}E\otimes \wedge^{\beta} E\otimes L)$ when $S^{\alpha+\beta}E \otimes L$ is ample. This condition is shown to be invariant under the interchange of $p$ and $q$. The optimality of this condition is discussed for some parameter values.