Hyperkähler manifold

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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.

1 Quaternionic structure
2 Holomorphic symplectic form
3 Examples
4 See also
5 External links

[edit] 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

I J = - J I = K

Any linear combination

a I + b J + c K

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 $\mathbb{R}^{4n}=\mathbb{H}^{n}$ 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.

[edit] 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 decomposition 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.

[edit] 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 $T 4$ .

A Hilbert scheme of points on a compact hyperkähler 4-manifold is again hyperkähler. This gives rise to two series of examples: Hilbert scheme of points on a K3 surface and generalized Kummer variety

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).