The period-index problem for hyperkähler manifolds
Daniel Huybrechts
TL;DR
The paper conjectures that for projective hyperkähler X and Brauer class α, ind(α) divides per(α)^{dim(X)/2}, and provides substantial evidence by proving this across two major HK classes: those admitting a Lagrangian fibration and Hilbert schemes of K3 surfaces. It develops an abelian–fibration framework—twisting the dual fibration, constructing multi-sections, gluing line bundles on twists, and controlling obstruction classes via compactifications and rigidifications—then applies Fourier–Mukai theory to realize twisted sheaves and derive index bounds. Extending these methods to Jacobian fibrations, general projective varieties, and moduli spaces of sheaves on K3s, the work obtains uniform bounds ind(α) | per(α)^g or per(α)^n in several settings, including cases without coprimality assumptions. Collectively, the results advance understanding of Brauer groups on HK manifolds, provide concrete index bounds in higher dimensions, and illuminate how geometric structures like Lagrangian fibrations influence period–index phenomena and twisted-moduli constructions.
Abstract
We conjecture that every unramified Brauer class $α\in \text{Br}(X)$ on a projective hyperkähler manifold $X$ satisfies $\text{ind}(α)\mid\text{per}(α)^{\dim(X)/2}$. 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.