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arXiv · 2608.09436

The hyperk\"ahler period-index conjecture is false

Pieter Belmans·James Hotchkiss
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Abstract

Huybrechts conjectured that for every Brauer class $\alpha$ on a hyperk\"ahler variety $X$ we have that $\mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^{\dim(X)/2}$, strengthening the usual period-index conjecture. We show that this fails on certain hyperk\"ahler fourfolds, both in type $\mathrm{K}3^{[2]}$ and $\mathrm{Kum}^2$.

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Pieter Belmans, James Hotchkiss. 2026-08-10. The hyperk\"ahler period-index conjecture is false. https://arxiv.org/abs/2608.09436

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