The hyperkähler period-index conjecture is false

Abstract

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

Disclosure

“Hodge-theoretic index. The result for type Kum2 is given in Theorem 6, the results for type K3 [2] are given in Theorem 9 (for 2-torsion) and Theorem 12 (for 5-torsion). AI disclosure The starting points for this paper were two different LLM-assisted constructions of counterexamples, obtained independently by the authors. The paper is the synthesis of these construc- tions, with LLMs used to help with the copyediting. Acknowledgements We would like to thank Lie Fu and Alex Pe”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T05:07:30.042634Z

Structural counts