hyperkähler manifold

Definition: Let M be a smooth manifoldMathworldPlanetmath, and I,J,K End(TM) endomorphismsPlanetmathPlanetmath of the tangent bundle satisfying the quaternionic relation


The manifold (M,I,J,K) is called hypercomplex if the almost complex structuresMathworldPlanetmath I, J, K are integrable. If, in addition, M is equipped with a Riemannian metric g which is Kähler with respect to I,J,K, the manifold (M,I,J,K,g) is called hyperkähler.

Since g is Kähler with respect to (I,J,K), we have


where denotes the Levi-Civita connectionMathworldPlanetmath. This means that the holonomy of lies inside the group Sp(n) of quaternionic-Hermitian endomorphisms. The converse is also true: a Riemannian manifold is hyperkähler if and only if its holonomy is contained in Sp(n). This definition is standard in differential geometry.

In physics literature, one sometimes assumes that the holonomy of a hyperkähler manifold is precisely Sp(n), and not its proper subgroupMathworldPlanetmath. In mathematics, such hyperkähler manifolds are called simple hyperkähler manifolds.

The following splitting theorem (due to F. Bogomolov) is implied by Berger’s classification of irreduciblePlanetmathPlanetmath holonomies.

Theorem: Any hyperkähler manifold has a finite covering which is a product of a hyperkähler torus and several simple hyperkähler manifolds.

Consider the Kähler forms ωI,ωJ,ωK on M:


An elementary linear-algebraic calculation implies that the 2-form ωJ+-1ωK is of Hodge type (2,0) on (M,I). This form is clearly closed and non-degenerate, hence it is a holomorphic symplectic formMathworldPlanetmath.

In algebraic geometryMathworldPlanetmathPlanetmath, the word “hyperkähler” is essentially synonymous with “holomorphically symplectic”, due to the following theorem, which is implied by Yau’s solution of Calabi conjecture (the famous Calabi-Yau theorem).

Theorem: Let (M,I) be a compactPlanetmathPlanetmath, Kähler, holomorphically symplectic manifold. Then there exists a unique hyperkähler metric on (M,I) with the same Kähler class.

Remark: The hyperkähler metric is unique, but there could be several hyperkähler structures compatible with a given hyperkähler metric on (M,I).


  • Bea Beauville, A. Varietes Kähleriennes dont la première classe de Chern est nulle. J. Diff. Geom. 18, pp. 755-782 (1983).
  • Bes Besse, A., Einstein Manifolds, Springer-Verlag, New York (1987)
  • Bo1 Bogomolov, F. On the decomposition of Kähler manifolds with trivial canonical class, Math. USSR-Sb. 22 (1974), 580-583.
  • Y Yau, S. T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I. Comm. on Pure and Appl. Math. 31, 339-411 (1978).
Title hyperkähler manifold
Canonical name HyperkahlerManifold
Date of creation 2013-03-22 15:50:16
Last modified on 2013-03-22 15:50:16
Owner tiphareth (13221)
Last modified by tiphareth (13221)
Numerical id 4
Author tiphareth (13221)
Entry type Definition
Classification msc 53C26
Synonym ”hyper-Kähler manifold”
Synonym ”hyper-Kählerian manifold”
Related topic Kahlermanifold
Related topic almostcomplexstructure
Related topic symplecticmanifold
Related topic quaternions
Related topic Quaternions
Related topic KahlerManifold
Related topic SymplecticManifold
Related topic AlmostComplexStructure
Defines ”hyperkähler manifold”
Defines ”hypercomplex manifold”