Indifferential geometry, ahyperkähler manifold is aRiemannian manifold${\displaystyle (M,g)}$ endowed with threeintegrable almost complex structures${\displaystyle I,J,K}$ that areKähler with respect to theRiemannian metric${\displaystyle g}$ and satisfy thequaternionic relations${\displaystyle I^2=J^2=K^2=IJK=-1}$. In particular, it is ahypercomplex manifold. All hyperkähler manifolds areRicci-flat and are thusCalabi–Yau manifolds.^[a]

Hyperkähler manifolds were first given this name byEugenio Calabi in 1979.^[1]

Marcel Berger's 1955 paper^[2] on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1). Interesting results were proved in the mid-1960s in pioneering work byEdmond Bonan^[3] and Kraines^[4] who have independently proven that any such manifold admits a parallel 4-form${\displaystyle \mathrm{\Omega }}$. Bonan's later results^[5] include a Lefschetz-type result: wedging with this powers of this 4-form induces isomorphisms${\displaystyle {\mathrm{\Omega }}^{n-k}\wedge \stackrel{2k}{\bigwedge }{T}^{\ast }M=\stackrel{4n-2k}{\bigwedge }{T}^{\ast }M.}$

Equivalently, a hyperkähler manifold is a Riemannian manifold${\displaystyle (M,g)}$ of dimension${\displaystyle 4n}$ whoseholonomy group is contained in thecompact symplectic groupSp(n).^[1]

Indeed, if${\displaystyle (M,g,I,J,K)}$ is a hyperkähler manifold, then the tangent spaceT_xM is aquaternionic vector space for each pointx ofM, i.e. it is isomorphic to${\displaystyle {\mathbb{H}}^n}$ for some integer${\displaystyle n}$, where${\displaystyle \mathbb{H}}$ is the algebra ofquaternions. Thecompact symplectic groupSp(n) can be considered as the group of orthogonal transformations of${\displaystyle {\mathbb{H}}^n}$ which are linear with respect toI,J andK. From this, it follows that theholonomy group of the Riemannian manifold${\displaystyle (M,g)}$ is contained inSp(n). Conversely, if the holonomy group of a Riemannian manifold${\displaystyle (M,g)}$ of dimension${\displaystyle 4n}$ is contained inSp(n), choose complex structuresI_x,J_x andK_x onT_xM which makeT_xM into a quaternionic vector space.Parallel transport of these complex structures gives the required complex structures${\displaystyle I,J,K}$ onM making${\displaystyle (M,g,I,J,K)}$ into a hyperkähler manifold.

Every hyperkähler manifold${\displaystyle (M,g,I,J,K)}$ has a2-sphere ofcomplex structures with respect to which themetric${\displaystyle g}$ isKähler. Indeed, for any real numbers${\displaystyle a,b,c}$ such that

${\displaystyle a^2+b^2+c^2=1}$

the linear combination

${\displaystyle aI+bJ+cK}$

is acomplex structures that is Kähler with respect to${\displaystyle g}$. If${\displaystyle {\omega }_I,{\omega }_J,{\omega }_K}$ denotes theKähler forms of${\displaystyle (g,I),(g,J),(g,K)}$, respectively, then the Kähler form of${\displaystyle aI+bJ+cK}$ is

${\displaystyle a{\omega }_I+b{\omega }_J+c{\omega }_K.}$

A hyperkähler manifold${\displaystyle (M,g,I,J,K)}$, considered as a complex manifold${\displaystyle (M,I)}$, is holomorphically symplectic (equipped with a holomorphic, non-degenerate, closed 2-form). More precisely, if${\displaystyle {\omega }_I,{\omega }_J,{\omega }_K}$ denotes theKähler forms of${\displaystyle (g,I),(g,J),(g,K)}$, respectively, then

${\displaystyle \mathrm{\Omega }:={\omega }_J+i{\omega }_K}$

is holomorphic symplectic with respect to${\displaystyle I}$.

Conversely,Shing-Tung Yau's proof of theCalabi conjecture implies that acompact,Kähler, holomorphically symplectic manifold${\displaystyle (M,I,\mathrm{\Omega })}$ is always equipped with a compatible hyperkähler metric.^[6] Such a metric is unique in a given Kähler class. Compact hyperkähler manifolds have been extensively studied using techniques fromalgebraic geometry, sometimes under the nameholomorphically symplectic manifolds. The holonomy group of any Calabi–Yau metric on a simply connected compact holomorphically symplectic manifold of complex dimension${\displaystyle 2n}$ with${\displaystyle H^{2,0}(M)=1}$ is exactlySp(n); and if the simply connected Calabi–Yau manifold instead has${\displaystyle H^{2,0}(M)\ge 2}$, it is just theRiemannian product of lower-dimensional hyperkähler manifolds. This fact immediately follows from the Bochner formula for holomorphic forms on a Kähler manifold, together theBerger classification of holonomy groups; ironically, it is often attributed to Bogomolov, who incorrectly went on to claim in the same paper that compact hyperkähler manifolds actually do not exist!

For any integer${\displaystyle n\ge 1}$, the space${\displaystyle {\mathbb{H}}^n}$ of${\displaystyle n}$-tuples ofquaternions endowed with theflat Euclidean metric is a hyperkähler manifold. The first non-trivial example discovered is theEguchi–Hanson metric on the cotangent bundle${\displaystyle T^{\ast }{S}^2}$ of thetwo-sphere. It was also independently discovered byEugenio Calabi, who showed the more general statement that cotangent bundle${\displaystyle T^{\ast }{\mathbb{C}\mathbb{P}}^n}$ of anycomplex projective space has acomplete hyperkähler metric.^[1] More generally, Birte Feix and Dmitry Kaledin showed that the cotangent bundle of anyKähler manifold has a hyperkähler structure on aneighbourhood of itszero section, although it is generally incomplete.^[7][8]

Due toKunihiko Kodaira's classification of complex surfaces, we know that anycompact hyperkähler 4-manifold is either aK3 surface or a compacttorus${\displaystyle T^4}$. (EveryCalabi–Yau manifold in 4 (real) dimensions is a hyperkähler manifold, becauseSU(2) is isomorphic toSp(1).)

As was shown by Beauville, theHilbert scheme ofk points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension4k.^[6] This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface andgeneralized Kummer varieties.

Non-compact, complete, hyperkähler 4-manifolds which are asymptotic toH/G, whereH denotes thequaternions andG is a finitesubgroup ofSp(1), are known asasymptotically locally Euclidean, or ALE, spaces. These spaces, and various generalizations involving different asymptotic behaviors, are studied inphysics under the namegravitational instantons. TheGibbons–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 dualYang–Mills equations: instanton moduli spaces,^[9]monopole moduli spaces,^[10] spaces of solutions toNigel Hitchin'sself-duality equations onRiemann surfaces,^[11] space of solutions toNahm equations. Another class of examples are theNakajimaquiver varieties,^[12] which are of great importance in representation theory.

Kurnosov, Soldatenkov & Verbitsky (2019) show that the cohomology of any compact hyperkähler manifold embeds into the cohomology of a torus, in a way that preserves theHodge structure.

• Quaternion-Kähler manifold
• Hypercomplex manifold
• Quaternionic manifold
• Calabi–Yau manifold
• Gravitational instanton
• Hyperkähler quotient
• Twistor theory

• Dunajski, Maciej; Mason, Lionel J. (2000), "Hyper-Kähler hierarchies and their twistor theory",Communications in Mathematical Physics,213 (3):641–672,arXiv:math/0001008,Bibcode:2000CMaPh.213..641D,doi:10.1007/PL00005532,MR 1785432,S2CID 17884816
• Kieran G. O’Grady, (2011) "Higher-dimensional analogues of K3 surfaces."MR2931873
• Hitchin, Nigel (1991–1992),"Hyperkähler manifolds",Séminaire N. Bourbaki,34 (Talk no. 748):137–166,MR 1206066
• Kurnosov, Nikon; Soldatenkov, Andrey; Verbitsky, Misha (2019), "Kuga-Satake construction and cohomology of hyperkähler manifolds",Advances in Mathematics,351:275–295,arXiv:1703.07477,doi:10.1016/j.aim.2019.04.060,MR 3952121,S2CID 119124485

• Structures on manifolds
• Complex manifolds
• Riemannian manifolds
• Differential geometry
• Quaternions

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