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{{technical|date=November 2013}}
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{{unreferenced|date=November 2013}}
In [[differential geometry]], a '''hyperkähler manifold''' is a [[Riemannian manifold]] of dimension 4''k'' and [[holonomy group]] contained in [[Symplectic group|Sp(''k'')]] (here Sp(''k'') denotes a compact form of a [[symplectic group]], identified
with the group of quaternionic-linear unitary endomorphisms
of an <math>n</math>-dimensional quaternionic Hermitian space). Hyperkähler manifolds are special classes of [[Kähler manifold]]s. They can be thought of as [[quaternion]]ic analogues of Kähler manifolds. All hyperkähler manifolds are [[Ricci-flat]] and are thus [[Calabi–Yau]] manifolds (this can be easily seen by noting that [[Symplectic group|Sp(''k'')]] is a [[subgroup]] of [[Special unitary group|SU(2''k'')]]).
 
Hyperkähler manifolds were defined by [[Eugenio Calabi|E. Calabi]] in 1978.
 
==Quaternionic structure==
Every hyperkähler manifold ''M'' has a [[sphere|2-sphere]] of complex structures (i.e.  [[almost complex structure|integrable almost complex structures]]) with respect to which the [[metric tensor|metric]] is Kähler.
 
In particular, it is an [[almost quaternionic manifold]], meaning that there are three distinct complex structures, ''I'', ''J,'' and ''K,'' which satisfy the [[quaternions|quaternion relations]]
 
:<math>I^2 = J^2 = K^2 = IJK = -1.\,</math>
 
Any linear combination
 
:<math>aI + bJ + cK \, </math>
 
with <math>a, b, c</math> real numbers such that
 
:<math>a^2 + b^2 + c^2 = 1 \, </math>
 
is also a complex structure on ''M''. In particular, the tangent space ''T''<sub>''x''</sub>''M'' is a quaternionic vector space for each point ''x'' of ''M''. Sp(''k'') can be considered as the group of orthogonal transformations of <math>\mathbb{R}^{4k}=\mathbb{H}^k</math> 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''<sub>''x''</sub>, ''J<sub>x</sub>'' and ''K''<sub>''x''</sub> on ''T''<sub>''x''</sub>''M'' which make ''T''<sub>''x''</sub>''M'' into a quaternionic vector space. [[Parallel transport]] of these complex structures gives the required quaternionic structure on&nbsp;''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 in the case of compact manifolds, due to [[Shing-Tung Yau|Yau]]'s proof of the [[Eugenio Calabi|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 [[Fedor Bogomolov|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.
 
==Examples==
Due to [[Kunihiko Kodaira|Kodaira]]'s classification of complex surfaces, we know
that any [[compact space|compact]] hyperkähler 4-manifold is either a [[K3 surface]] or a compact torus <math>T^4</math>. 
(Every [[Calabi–Yau]] manifold in 4 (real) dimensions is a hyperkähler manifold, because [[Special unitary group|SU(2)]] is isomorphic to [[Symplectic group|Sp(1)]].)
 
A [[Hilbert scheme]] of points on a compact hyperkähler 4-manifold is again hyperkähler.  This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and [[Hilbert scheme|generalized Kummer varieties]].
 
Non-compact, complete, hyperkähler 4-manifolds which are asymptotic to '''H'''/''G'', where '''H''' denotes the [[quaternion]]s and ''G'' is a finite [[subgroup]] of Sp(1), are known as [[asymptotically locally Euclidean]], or ALE, spaces. These spaces, and various generalizations involving different asymptotic behaviors, are studied in [[physics]] under the name [[gravitational instanton]]s. The [[Gibbons–Hawking ansatz]] gives examples invariant under a circle action.
 
Many examples of noncompact hyperkähler manifolds arise as moduli spaces of solutions to certain gauge theory equations which arise from the dimensional reduction of the anti-self dual Yang–Mills equations:
instanton moduli spaces, monopole moduli spaces, spaces of solutions to Hitchin's self-duality equations on [[Riemann surface]]s, space of solutions to Nahm's equations.  
Another class of examples are the [[Nakajima quiver variety|Nakajima quiver varieties]], which are of great importance in representation theory.
 
==See also==
*[[Quaternionic-Kähler manifold]]
*[[Hypercomplex manifold]]
*[[Calabi–Yau manifold]]
 
==External links==
* Maciej Dunajski and Lionel J. Mason(2000)"[ftp://ftp.esi.ac.at/pub/Preprints/esi821.pdf Hyper Kahler Hierarchies and their Twistor Theory]"
 
{{DEFAULTSORT:Hyperkahler manifold}}
[[Category:Structures on manifolds]]
[[Category:Complex manifolds]]
[[Category:Riemannian manifolds]]
[[Category:Algebraic geometry]]

Latest revision as of 08:11, 18 March 2014

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If you are purchasing a game to one's child, appear for person that allows several individuals carry out together. Gaming can be a singular activity. Nonetheless, it's important to reinforcement your youngster to be societal, and multiplayer conflict of clans trucos events can do that. They allow siblings and also buddies to all take a moment and laugh and strive together.

Pay attention to a game's evaluation when purchasing a great gift. This evaluation will allow you to discover what age level clash of clans hack tool is most suitable for and will let you know when the sport is violent. It can help you figure out whether it is advisable to buy the sport.

A great deal as now, there exists minimum social options / abilities with this game i.e. there is not any chat, battling to team track linked with friends, etc but perhaps we could expect distinct to improve soon being Boom Beach continues to be in their Beta Mode.

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