The period-index problem for hyperkähler manifolds

Abstract

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

37 pages. Minor corrections