# hyperkähler manifold

Definition: Let M be a smooth manifold, and I,J,K $\in$ End(TM) endomorphisms of the tangent bundle satisfying the quaternionic relation

 $I^{2}=J^{2}=K^{2}=IJK=-Id_{TM}.$

The manifold (M,I,J,K) is called hypercomplex if the almost complex structures 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

 $\nabla I=\nabla J=\nabla K=0$

where $\nabla$ denotes the Levi-Civita connection. This means that the holonomy of $\nabla$ 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 subgroup. 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 irreducible 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 $\omega_{I},\omega_{J},\omega_{K}$ on M:

 $\omega_{I}(\cdot,\cdot):=g(\cdot,I\cdot),\ \ \omega_{J}(\cdot,\cdot):=g(\cdot,% J\cdot),\ \ \omega_{K}(\cdot,\cdot):=g(\cdot,K\cdot).$

An elementary linear-algebraic calculation implies that the 2-form $\omega_{J}+\sqrt{-1}\omega_{K}$ is of Hodge type (2,0) on (M,I). This form is clearly closed and non-degenerate, hence it is a holomorphic symplectic form.

In algebraic geometry, 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 compact, 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).

## References

• 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”