New paper: The hyperkähler period-index conjecture is false

This post is about The hyperkähler period-index conjecture is false, joint with James Hotchkiss. If I were still posting fortnightly links, Alexander Perry: The period-index conjecture is false would have prominently featured, because it disproves (as the title suggests) the period-index conjecture, which is famously a theorem due to de Jong in dimension 2, with a beautiful proof. But Alex found counterexamples in every dimension at least 3, with some assistance from an LLM. The period-index conjecture To understand the statement of the period-index conjecture, let us consider a field $K$ of transcendence degree $d$ over an algebraically closed field $k$. We attach two integers to a Brauer class $\alpha\in\mathop{\rm Br}(K)$: the index $\mathop{\rm ind}(\alpha)$ (which measures the size of the unique division algebra in the Brauer class) the period $\mathop{\rm per}(\alpha)$ (which is the order of $\alpha$ in the torsion abelian group $\mathop{\rm Br}(K)$) The period-index conjecture,…