The period-index problem for hyper-K\"ahler varieties via hyperholomorphic bundles

TL;DR

This paper establishes new bounds for the period-index problem in hyper-K"ahler varieties of $K3^{[n]}$-type, using hyperholomorphic bundles, and provides evidence supporting a conjecture relating dimension and Brauer classes.

Contribution

It introduces novel bounds for the period-index problem on hyper-K"ahler varieties of $K3^{[n]}$-type leveraging hyperholomorphic bundles, advancing understanding of their Brauer groups.

Findings

Dimension bounds for hyper-K"ahler varieties of $K3^{[n]}$-type.

Half-dimension bounds for most Brauer classes with higher Picard rank.

Support for Huybrechts' conjecture on Brauer classes and dimension.

Abstract

We prove new bounds for the period-index problem for hyper-K\"ahler varieties of $K3^{[n]}$-type using projectively hyperholomorphic bundles constructed by Markman. We show that $\mathrm{dim}(X)$ is a bound for any $X$ of $K3^{[n]}$-type. We also show that $\frac{1}{2}\mathrm{dim}(X)$ is a bound for most Brauer classes when the Picard rank of $X$ is at least two, providing evidence for a conjecture of Huybrechts.