A vanishing theorem
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 $H^{p,q}(X,\SSS^αE\otimes \wedge^β E\otimes L)$ when $S^{α+β}E \otimes L$ 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.
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