In differential geometry, a hyperkähler manifold is a Riemannian manifold <math>(M, g)</math> endowed with three integrable almost complex structures <math>I, J, K</math> that are Kähler with respect to the Riemannian metric <math>g</math> and satisfy the quaternionic relations <math>I^2=J^2=K^2=IJK=-1</math>. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.

Hyperkähler manifolds were first given this name by Eugenio Calabi in 1979.

Early history

Marcel Berger's 1955 paper on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1). Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan and Kraines who have independently proven that any such manifold admits a parallel 4-form <math>\Omega</math>. Bonan's later results include a Lefschetz-type result: wedging with this powers of this 4-form induces isomorphisms <math>

\Omega^{n-k}\wedge\bigwedge^{2k}T^*M=\bigwedge^{4n-2k}T^*M.</math>

Equivalent definition in terms of holonomy

Equivalently, a hyperkähler manifold is a Riemannian manifold <math>(M, g)</math> of dimension <math>4n</math> whose holonomy group is contained in the compact symplectic group .

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 <math>T^4</math>. (Every Calabi–Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because is isomorphic to .)

As was shown by Beauville, the Hilbert scheme of points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension . 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 , where denotes the quaternions and is a finite subgroup of , 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 embeds into the cohomology of a torus, in a way that preserves the Hodge structure.

Notes

See also

  • Quaternion-Kähler manifold
  • Hypercomplex manifold
  • Quaternionic manifold
  • Calabi–Yau manifold
  • Gravitational instanton
  • Hyperkähler quotient
  • Twistor theory

References

  • Kieran G. O’Grady, (2011) "Higher-dimensional analogues of K3 surfaces." MR2931873