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Hyper-Kähler Manifold


A hyper-Kähler manifold can be defined as a Riemannian manifold of dimension 4n with three covariantly constant orthogonal automorphisms I, J, K of the tangent bundle which satisfy the quaternionic identities

 I^2=J^2=K^2=IJK=-1,
(1)

where -1 denotes the negative of the identity automorphism 1=id on the tangent bundle. The term hyper-Kähler is sometimes written without a hyphen (as hyperKähler) or without capitalization (as hyperkähler).

This definition is equivalent to several others commonly encountered in the literature; indeed, a manifold M^(4n) is said to be hyper-Kähler if and only if:

1. M is a holomorphically symplectic Kähler manifold with holonomy in Sp(n).

2. M is a holomorphically symplectic Kähler manifold which is Ricci-flat (i.e., which has zero scalar curvature.

The first of the above equivalences is referring to Berger's classification of the holonomy groups of Riemannian manifolds and implies that parallel transport preserves I, J, and K. Both this criterion and the criterion listed in the second of the above equivalences is used to differentiate hyperkähler manifolds from the similarly-named quaternionic-Kähler manifolds which have nonzero Ricci curvature and, in general, fail to be Kähler.

Hyper-kähler manifolds are necessarily Calabi-Yau manifolds and are Einstein manifolds with constant 0.

Generally, the automorphisms I,J,K:TM->TM are assumed to be integrable. The presence of these three complex structures induces three Kähler 2-forms omega_i, i=1,2,3, on M, namely

 omega_1(X,Y)=g(IX,Y),
(2)
 omega_2(X,Y)=g(JX,Y),
(3)

and

 omega_3(X,Y)=g(KX,Y)
(4)

for all X,Y in TM where, here, g is the Kähler/Riemannian metric on M. As the two equivalent definitions above indicate, hyperkähler manifolds are holomorphically symplectic, i.e., they have three holomorphic symplectic 2-forms induced by each of I, J, and K. For example, the 2-form omega_+ of the form

 omega_+(X,Y)=omega_2(X,Y)+iomega_3(X,Y)
(5)

is holomorphic and symplectic on (M,I) (where i denotes the standard imaginary unit). Calabi proved a partial converse which says that a compact holomorphically symplectic Kähler manifold admits a unique hyper-Kähler metric with respect to any of its Kähler forms.

All even-dimensional complex vector spaces and tori are hyper-Kähler. Further examples include the quaternions H=H^4, the cotangent bundle T^*CP^n of n-dimensional complex projective space, K3 surfaces, Hilbert schemes of points on compact hyper-kähler 4-manifolds, and generalized Kummer varieties, as well as various moduli spaces, spaces of solutions to Nahm's equations, and the Nakajima quiver varieties.


See also

Einstein Manifold, Flat Manifold, Kähler, Quaternion Kähler Manifold, Riemannian Manifold, Symplectic Manifold

This entry contributed by Christopher Stover

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References

Hitchin, N. "Hypkerkähler Manifolds." Séminaire N. Bourbaki 748, 137-166, 1991-1992.Verbitsky, M. "Hyperkähler Manifolds: Kähler Manifolds and Holonomy Groups." 2013a. http://verbit.ru/MATH/TALKS/SPB-2013/hk-1.pdf.Verbitsky, M. "Hyperkähler Manifolds: Calabi-Yau Theorem, Bochner Vanishing and Hodge Theory." 2013b. http://verbit.ru/MATH/TALKS/SPB-2013/hk-2.pdf.Verbitsky, M. "Hyperkähler Manifolds: Cohomology in Mathematics and Physics." 2013c. http://www.pdmi.ras.ru/EIMI/2013/Cohomology/verbitsky3.pdf.

Cite this as:

Stover, Christopher. "Hyper-Kähler Manifold." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Hyper-KaehlerManifold.html

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