Vanishing theorems for products of exterior and symmetric powers

TL;DR

This paper establishes vanishing theorems for cohomology groups of Schur functors applied to ample vector bundles over complex varieties, extending previous results and providing simpler proofs under weaker positivity assumptions.

Contribution

It offers a simplified proof of a vanishing theorem for hook partitions and extends vanishing results to broader positivity conditions and cohomology types.

Findings

Proved vanishing theorems for $H^{p,q}(X, S_I(E))$ with hook partitions.

Extended vanishing results to cases with ample $igwedge^m E$ and bounds on partition weight.

Showed vanishing conditions also hold when $p$ and $q$ are interchanged.

Abstract

For ample vector bundles $E$ over compact complex varieties $X$ and a Schur functor $S_I$ corresponding to an arbitrary partition $I$ of the integer $|I|$, one would like to know the optimal vanishing theorem for the cohomology groups $H^{p,q}(X, S_I(E))$, depending on the rank of $E$ and the dimension $n$ of $X$. Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved a vanishing theorem for the situation where the partition $I$ is a hook. Here we give a simpler proof of this theorem. We also treat the same problem under weaker positivity assumptions, in particular under the hypothesis of ample $\Lambda ^m E$ with $m\in \N^*$. In this case we also need some bound on the weight $|I|$ of the partition. Moreover, we prove that the same vanishing condition applies for $H^{q,p}(X, S_I(E))$, with $p,q$ interchanged.