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

TL;DR

This paper investigates the period-index problem for hyperk"ahler varieties, providing evidence for a stronger conjecture, establishing optimal bounds, and proving the conjecture for certain classes of hyperk"ahler varieties.

Contribution

It introduces a Hodge-theoretic version of the period-index conjecture, proves its optimality, and confirms the conjecture for non-special coprime Brauer classes on K3^n-type hyperk"ahler varieties.

Findings

The hyperk"ahler period-index conjecture is optimal.

Mumford-Tate general hyperk"ahler varieties cannot be covered by elliptic curves.

The conjecture holds for non-special coprime Brauer classes on K3^n-type hyperk"ahler varieties.

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.