The period-index problem for hyper-Kähler varieties via hyperholomorphic bundles

We prove new bounds for the period-index problem for hyper-Kähler varieties of $K{3}^{[n]}$-type using projectively hyperholomorphic bundles constructed by Markman. We show that $\mathrm{d}\mathrm{i}\mathrm{m}(X)$ is a bound for any $X$ of $K{3}^{[n]}$-type. We also show that $\frac{1}{2}\mathrm{d}\mathrm{i}\mathrm{m}(X)$ is a bound for most Brauer classes when the Picard rank of $X$ is at least two, providing evidence for a conjecture of Huybrechts.