Hyperkähler manifold

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 a $k$ -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 the special unitary group $SU(2 k)$ ).

Hyperkähler manifolds were defined by Eugenio Calabi in 1978.

Quaternionic structure

Every hyperkähler manifold $M$ has a 2-sphere of complex structures (i.e. integrable almost complex structures) with respect to which the metric is Kähler.

In particular, it is a hypercomplex manifold, meaning that there are three distinct complex structures, $I$ , $J$ , and $K$ , which satisfy the quaternion relations

I^{2}=J^{2}=K^{2}=IJK=-1.\,

Any linear combination

aI+bJ+cK\,

with $a,b,c$ real numbers such that

a^{2}+b^{2}+c^{2}=1\,

is also a complex structure on $M$ . In particular, the tangent space $T x M$ is a quaternionic vector space for each point $x$ of $M$ . $Sp(k)$ can be considered as the group of orthogonal transformations of $\mathbb {R} ^{4k}=\mathbb {H} ^{k}$ 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 $T x M$ which make $T x M$ 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 in the case of compact manifolds, due to Shing-Tung 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. The holonomy group of any Calabi–Yau metric on a simply connected compact holomorphically symplectic manifold with $H^{2,0}(M)=1$ is exactly $Sp(k)$ ; and if the simply connected Calabi–Yau manifold instead has $H^{2,0}(M)\geq 2$ , it is just the Riemannian product of lower-dimensional hyperkähler manifolds. This fact immediately follows from the Bochner formula for holomorphic forms on a Kähler manifold, together the Berger 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!

Examples

Due to Kunihiko Kodaira's classification of complex surfaces, we know that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus $T^{4}$ . (Every Calabi–Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because $SU(2)$ is isomorphic to $Sp(1)$ .)

As was discovered by Beauville, the Hilbert scheme of $k$ points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension $4k$ . This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and generalized Kummer varieties.

Non-compact, complete, 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 spaces, and various generalizations involving different asymptotic behaviors, are studied in physics under the name gravitational instantons. 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 Nigel Hitchin's self-duality equations on Riemann surfaces, space of solutions to Nahm equations. Another class of examples are the Nakajima quiver varieties, which are of great importance in representation theory.

Cohomology

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 the Hodge structure.

External links

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

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