The period-index problem for hyperk\"ahler manifolds

TL;DR

This paper conjectures a relationship between the index and period of unramified Brauer classes on projective hyperk"ahler manifolds, providing evidence by proving it for specific classes such as those with Lagrangian fibrations and Hilbert schemes of K3 surfaces.

Contribution

It introduces a new conjecture relating index and period of Brauer classes on hyperk"ahler manifolds and proves it for certain classes, advancing understanding in algebraic geometry.

Findings

Proved the conjecture for hyperk"ahler manifolds with Lagrangian fibrations.

Confirmed the conjecture for Hilbert schemes of K3 surfaces.

Provided evidence supporting the conjecture in specific geometric contexts.

Abstract

We conjecture that every unramified Brauer class $\alpha\in \text{Br}(X)$ on a projective hyperk\"ahler manifold $X$ satisfies $\text{ind}(\alpha)\mid\text{per}(\alpha)^{\dim(X)/2}$. We provide evidence for this conjecture by proving it for two large classes of projective hyperk\"ahler manifolds: For projective hyperk\"ahler manifolds admitting a Lagrangian fibration and for Hilbert schemes of K3 surfaces.