Huybrechts's hyperkähler period-index conjecture is false in dimension 4: on some K3^[2] and Kum2 fourfolds, period-2 classes have index divisible by 8, and on some K3^[2] fourfolds a period-5 class has index divisible by 125.
Exposant et indice d’algèbres simples centrales non ramifiées
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The hyperk\"ahler period-index conjecture is false
Huybrechts's hyperkähler period-index conjecture is false in dimension 4: on some K3^[2] and Kum2 fourfolds, period-2 classes have index divisible by 8, and on some K3^[2] fourfolds a period-5 class has index divisible by 125.