Hyperkähler manifold

Summary

A hyperkähler manifold is a Riemannian manifold that is Kähler with respect to three complex structures $I$, $J$, $K$ that multiply like the quaternion units. Its dimension is a multiple of $4$, and its holonomy group lies in the compact symplectic group $Sp(n)$. Hyperkähler manifolds are Ricci-flat, and they are one of the special holonomy cases in Berger's list. Examples include K3 surfaces with their Ricci-flat metrics and many moduli spaces from gauge theory.