Every hyperkähler manifold M has a 2-sphere of complex structures with respect to which the metric is Kähler. In particular, it is a hypercomplex manifold, meaning that there are three distinct complex structures, I, J, and K, which satisfy the quaternion relations Any linear combination with real numbers such that is also a complex structure on M. In particular, the tangent spaceTxM is a quaternionic vector space for each point x of M. Sp can be considered as the group of orthogonal transformations of which are linear with respect to I, J and K. From this it follows that the holonomy of the manifold is contained in Sp. Conversely, if the holonomy group of the Riemannian manifold M is contained in Sp, choose complex structures Ix, Jx and Kx on TxM which make TxM into a quaternionic vector space. Parallel transport of these complex structures gives the required quaternionic structure on M.
A hyperkähler manifold, considered as a complex manifold, is holomorphically symplectic. The converse is also true in the case of compact manifolds, due to Shing-Tung Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold, it is always equipped with a compatible hyperkähler metric. Such a metric is unique in a given Kähler class. Compact hyperkähler manifolds have been extensively studied using techniques from algebraic geometry, sometimes under a name holomorphically symplectic manifolds. The holonomy group of any Calabi-Yau metric on a simply connected compact holomorphically symplectic manifold with is exactly Sp; and if the simply connected Calabi-Yau manifold instead has, it is just the Riemannian product of lower-dimensional hyperkähler manifolds. This fact immediately follows from the Bochner formula for holomorphic forms on a Kähler manifold, together the Berger classification of holonomy groups; ironically, it is often attributed to Bogomolov, who incorrectly went on to claim in the same paper that compact hyperkähler manifolds actually do not exist!
Examples
Due to Kunihiko Kodaira's classification of complex surfaces, we know that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus. dimensions is a hyperkähler manifold, because SU is isomorphic to Sp As was discovered by Beauville, the Hilbert scheme of k points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension 4k. This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and generalized Kummer varieties. Non-compact, complete, hyperkähler 4-manifolds which are asymptotic to H/G, where H denotes the quaternions and G is a finite subgroup of Sp, are known as asymptotically locally Euclidean, or ALE, spaces. These spaces, and various generalizations involving different asymptotic behaviors, are studied in physics under the name gravitational instantons. The Gibbons–Hawking ansatz gives examples invariant under a circle action. Many examples of noncompact hyperkähler manifolds arise as moduli spaces of solutions to certain gauge theory equations which arise from the dimensional reduction of the anti-self dual Yang–Mills equations: instanton moduli spaces, monopole moduli spaces, spaces of solutions to Nigel Hitchin's self-duality equations on Riemann surfaces, space of solutions to Nahm equations. Another class of examples are the Nakajima quiver varieties, which are of great importance in representation theory.
Cohomology
show that the cohomology of any compact hyperkähler manifold embedds into the cohomology of a torus, in a way that preserves the Hodge structure.
