The period-index problem for hyperk\"ahler varieties: Lower and upper bounds

Abstract

It is expected that a stronger form of the period-index conjecture holds for hyperk\"ahler varieties. Following ideas of Hotchkiss, we provide further evidence for this expectation by proving a version in which the index is replaced by the Hodge-theoretic index. We also show that the hyperk\"ahler period-index conjecture is optimal. As an application, we prove that Mumford-Tate general hyperk\"ahler varieties cannot be covered by families of elliptic curves passing through a fixed point. By extending work of Hotchkiss, Maulik, Shen, Yin, and Zhang, we prove the hyperk\"ahler period-index conjecture for non-special coprime Brauer class on hyperk\"ahler varieties of K3^n-type without any restriction on the Picard number.