Hyperkähler manifold

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Contents

0.1 A definition
0.2 Quaternionic structure
0.3 Holomorphic symplectic form
0.4 Examples

1 Informal Reference

A definition

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) (here Sp(k) denotes a compact form of a symplectic group, identified with the group of quaternionic-linear unitary endomorphisms of an [n]-dimensional quaternionic Hermitian space). Hyperkähler manifolds are special classes of Kähler manifolds. They can be thought of as quaternionic analogues of Kähler manifolds. All hyperkähler manifolds are Ricci-flat and are thus Calabi-Yau manifolds (this can be easily seen by noting that Sp(k) is a subgroup of SU(2k)).

Hyperkähler manifolds were defined by E. Calabi in 1978.

Quaternionic structure

Every hyperkähler manifold M has a 2-sphere of almost complex structures with respect to which the metric is Kähler. These almost complex structures are integrable.

In particular, there are three distinct complex structures,

[I,J ] and [K]

which satisfy the quaternionic relation

[IJ = -JI = K].

Any linear combination

[aI + bJ + cK]

with

[a^2 + b^2 + c^2 = 1]

is also an integrable almost complex structure on M. In particular, T_xM is a quaternionic vector space for each point x of M. Sp(k) can be considered as the group of orthogonal transformations of [\mathbb^=\mathbb^] which are linear with respect to I, J and K. From this it follows that the holonomy of the manifold is contained in Sp(k). Conversely, if the holonomy group of the Riemannian manifold M is contained in Sp(k), choose complex structures I_x, J_x and K_x on T_xM which make T_xM into a quaternionic vector space. Parallel transport of these complex structures gives the required quaternionic structure on M.

Holomorphic symplectic form

A hyperkähler manifold (M,I,J,K), considered as a complex manifold (M,I), is holomorphically symplectic (equipped with a holomorphic, non-degenerate 2-form). The converse is also true, due to Yau's proof of the Calabi conjecture. Given a compact, Kähler, holomorphically symplectic manifold (M,I), it is always equipped with a compatible hyperkähler metric. Such a metric is unique in a given Kähler class. Compact hyperkähler manifolds have been extensively studied using techniques from algebraic geometry, sometimes under a name holomorphically symplectic manifolds. Due to Bogomolov's splitting theorem (1974), the holonomy group of a compact holomorphically symplectic manifold M is exactly Sp(k) if and only if M is simply connected and any pair of holomorphic symplectic forms on M are scalar multiples of each other.

Examples

Due to Kodaira's classification of complex surfaces, we know that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus [T^4].

Non-compact hyperkähler 4-manifolds which are asymptotic to H/G, where H denotes the quaternions and G is a finite subgroup of Sp(1), are known as Asymptotically locally Euclidean, or ALE, spaces. These space are studied in physics under the name gravitational instantons.

Every Calabi-Yau manifold in 4 real (2 complex) dimensions is a hyperkähler manifold, because SU(2) is isomorphic to Sp(1).

Informal Reference

Young-Hoon Kim, [Remarks on Hyperkäler Geometry]

See also: Calabi-Yau manifold

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