A vanishing theorem

TL;DR

This paper establishes a vanishing theorem for certain Dolbeault cohomology groups of vector and line bundles over smooth projective varieties, providing conditions that are invariant under index interchange and discussing their optimality.

Contribution

It introduces a new vanishing criterion for Dolbeault cohomology groups involving symmetric and wedge powers of vector bundles and line bundles, with invariance properties.

Findings

Derived a vanishing condition for Dolbeault cohomology groups

Proved invariance of the condition under p and q interchange

Discussed the optimality of the vanishing condition for specific parameters

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.