Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold <math>(M, g)</math> endowed with three integrable almost complex structures <math>I, J, K</math> that are Kähler with respect to the Riemannian metric <math>g</math> and satisfy the quaternionic relations <math>I^2=J^2=K^2=IJK=-1</math>. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.Template:Efn

Hyperkähler manifolds were first given this name by Eugenio Calabi in 1979.<ref name="calabi">Template:Cite journal</ref>

Early historyEdit

Marcel Berger's 1955 paper<ref>Template:Cite journal</ref> 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 by Edmond Bonan<ref>Template:Cite journal</ref> and Kraines<ref>Template:Cite journal</ref> who have independently proven that any such manifold admits a parallel 4-form <math>\Omega</math>. Bonan's later results<ref>Template:Cite journal</ref> include a Lefschetz-type result: wedging with this powers of this 4-form induces isomorphisms <math>

\Omega^{n-k}\wedge\bigwedge^{2k}T^*M=\bigwedge^{4n-2k}T^*M.</math>

Equivalent definition in terms of holonomyEdit

Equivalently, a hyperkähler manifold is a Riemannian manifold <math>(M, g)</math> of dimension <math>4n</math> whose holonomy group is contained in the [[Symplectic_group#Sp(n)|compact symplectic group Template:Math]].<ref name="calabi" />

Indeed, if <math>(M, g, I, J, K)</math> is a hyperkähler manifold, then the tangent space Template:Math is a quaternionic vector space for each point Template:Math of Template:Math, i.e. it is isomorphic to <math>\mathbb{H}^n</math> for some integer <math>n</math>, where <math>\mathbb{H}</math> is the algebra of quaternions. The [[Symplectic_group#Sp(n)|compact symplectic group Template:Math]] can be considered as the group of orthogonal transformations of <math>\mathbb{H}^n</math> which are linear with respect to Template:Math, Template:Math and Template:Math. From this, it follows that the holonomy group of the Riemannian manifold <math>(M, g)</math> is contained in Template:Math. Conversely, if the holonomy group of a Riemannian manifold <math>(M, g)</math> of dimension <math>4n</math> is contained in Template:Math, choose complex structures Template:Math, Template:Math and Template:Math on Template:Math which make Template:Math into a quaternionic vector space. Parallel transport of these complex structures gives the required complex structures <math>I, J, K</math> on Template:Math making <math>(M, g, I, J, K)</math> into a hyperkähler manifold.

Two-sphere of complex structuresEdit

Every hyperkähler manifold <math>(M, g, I, J, K)</math> has a 2-sphere of complex structures with respect to which the metric <math>g</math> is Kähler. Indeed, for any real numbers <math>a, b, c</math> such that

the linear combination

is a complex structures that is Kähler with respect to <math>g</math>. If <math>\omega_I, \omega_J, \omega_K</math> denotes the Kähler forms of <math>(g, I), (g, J), (g, K)</math>, respectively, then the Kähler form of <math>aI + bJ + cK</math> is

<math>a \omega_I + b \omega_J + c \omega_K.</math>

Holomorphic symplectic formEdit

A hyperkähler manifold <math>(M, g, I, J, K)</math>, considered as a complex manifold <math>(M, I)</math>, is holomorphically symplectic (equipped with a holomorphic, non-degenerate, closed 2-form). More precisely, if <math>\omega_I, \omega_J, \omega_K</math> denotes the Kähler forms of <math>(g, I), (g, J), (g, K)</math>, respectively, then

<math>\Omega := \omega_J + i\omega_K</math>

is holomorphic symplectic with respect to <math>I</math>.

Conversely, Shing-Tung Yau's proof of the Calabi conjecture implies that a compact, Kähler, holomorphically symplectic manifold <math>(M,I,\Omega)</math> is always equipped with a compatible hyperkähler metric.<ref name=beauville /> Such a metric is unique in a given Kähler class. Compact hyperkähler manifolds have been extensively studied using techniques from algebraic geometry, sometimes under the name holomorphically symplectic manifolds. The holonomy group of any Calabi–Yau metric on a simply connected compact holomorphically symplectic manifold of complex dimension <math>2n</math> with <math>H^{2,0}(M)=1</math> is exactly Template:Math; and if the simply connected Calabi–Yau manifold instead has <math>H^{2,0}(M)\geq 2</math>, 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!

ExamplesEdit

For any integer <math>n \ge 1</math>, the space <math>\mathbb{H}^n</math> of <math>n</math>-tuples of quaternions endowed with the flat Euclidean metric is a hyperkähler manifold. The first non-trivial example discovered is the Eguchi–Hanson metric on the cotangent bundle <math>T^*S^2</math> of the two-sphere. It was also independently discovered by Eugenio Calabi, who showed the more general statement that cotangent bundle <math>T^*\mathbb{CP}^n</math> of any complex projective space has a complete hyperkähler metric.<ref name="calabi" /> More generally, Birte Feix and Dmitry Kaledin showed that the cotangent bundle of any Kähler manifold has a hyperkähler structure on a neighbourhood of its zero section, although it is generally incomplete.<ref>Feix, B. Hyperkähler metrics on cotangent bundles. J. Reine Angew. Math. 532 (2001), 33–46.</ref><ref>Kaledin, D. A canonical hyperkähler metric on the total space of a cotangent bundle. Quaternionic structures in mathematics and physics (Rome, 1999), 195–230, Univ. Studi Roma "La Sapienza", Rome, 1999.</ref>

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 <math>T^4</math>. (Every Calabi–Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because Template:Math is isomorphic to Template:Math.)

As was discovered by Beauville,<ref name="beauville">Beauville, A. Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differential Geom. 18 (1983), no. 4, 755–782 (1984).</ref> the Hilbert scheme of Template:Math points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension Template:Math. 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 Template:Math, where Template:Math denotes the quaternions and Template:Math is a finite subgroup of Template:Math, 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,<ref>Maciocia, A. Metrics on the moduli spaces of instantons over Euclidean 4-space. Comm. Math. Phys. 135 (1991), no. 3, 467–482.</ref> monopole moduli spaces,<ref>Atiyah, M.; Hitchin, N. The geometry and dynamics of magnetic monopoles. M. B. Porter Lectures. Princeton University Press, Princeton, NJ, 1988.</ref> spaces of solutions to Nigel Hitchin's self-duality equations on Riemann surfaces,<ref>Hitchin, N. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.</ref> space of solutions to Nahm equations. Another class of examples are the Nakajima quiver varieties,<ref>Nakajima, H. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), no. 2, 365–416.</ref> which are of great importance in representation theory.

CohomologyEdit

Template:Harvtxt show that the cohomology of any compact hyperkähler manifold embeds into the cohomology of a torus, in a way that preserves the Hodge structure.