# hyperkähler manifold

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

$${I}^{2}={J}^{2}={K}^{2}=IJK=-I{d}_{TM}.$$ |

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” |