The period-index problem for hyperkähler varieties: Lower and upper bounds
Alessio Bottini, Daniel Huybrechts
TL;DR
The paper develops a comprehensive framework for the period–index problem on hyperkähler varieties, introducing Hodge-theoretic indices obtained by twisting the Mukai lattice and related structures. It proves sharp upper bounds ind(α) | per(α)^{dim(X)/2} in broad cases and establishes tight lower bounds in Mumford–Tate general settings, including non-special coprime Brauer classes on K3^{[n]}-type varieties, with special-prime-period cases addressed via framed lattice methods. The results culminate in strong statements for MT general varieties, Lagrangian fibrations, and Hilbert schemes of K3 surfaces, showing per(α)^n = ind(α) in several key scenarios, and provide obstructions and constructions (via δ, Δ and twisted K-theory) that illuminate the structure of the period–index problem in higher dimensions. The work also connects to geometric questions about covering families of curves, proving that very general hyperkähler varieties do not admit covering families of elliptic curves through a fixed point, linking arithmetic invariants to geometric spreading properties. Overall, the paper advances the Hyperkähler period–index conjecture, offering both optimality results and broad applicability to K3^{[n]}-type and MT-general hyperkähler varieties.
Abstract
It is expected that a stronger form of the period-index conjecture holds for hyperkähler 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 hyperkähler period-index conjecture is optimal. As an application, we prove that Mumford-Tate general hyperkähler 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 hyperkähler period-index conjecture for non-special coprime Brauer class on hyperkähler varieties of K3^n-type without any restriction on the Picard number.