hyperkähler manifold

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

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

The manifold (M,I,J,K) is called hypercomplexif the almost complex structures I, J, Kare integrable. If, in addition, Mis equipped with a Riemannian metric g whichis Kähler with respect to I,J,K, themanifold (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 insidethe group Sp(n) of quaternionic-Hermitianendomorphisms. The converse is also true: aRiemannian manifold is hyperkähler if and onlyif its holonomy is contained in Sp(n).This definition is standard in differentialgeometry.

In physics literature, one sometimes assumesthat the holonomy of a hyperkähler manifold isprecisely Sp(n), and not its propersubgroup. In mathematics, such hyperkählermanifolds are called simple hyperkähler manifolds.

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

Theorem: Any hyperkählermanifold has a finite covering which is a productof a hyperkähler torus and several simplehyperkähler 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 impliesthat the 2-form ${\omega }_J+\sqrt{-1}{\omega }_K$ is of Hodge type (2,0)on (M,I). This form is clearly closed andnon-degenerate, hence it is a holomorphicsymplectic form.

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

Theorem: Let (M,I) be a compact, Kähler, holomorphicallysymplectic manifold. Then there exists a uniquehyperkähler metric on (M,I) with the same Kähler class.

Remark:The hyperkähler metric is unique, but there couldbe several hyperkähler structures compatible witha given hyperkähler metric on (M,I).