Quaternion-Kähler manifold

In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1). Although this definition includes hyperkähler manifolds, these are often excluded from the definition of a quaternion-Kähler manifold by imposing the condition that the scalar curvature is nonzero, or that the holonomy group is equal to Sp(n)·Sp(1). The definition introduced by Edmond Bonan^[1] in 1965, uses a 3-dimensional subbundle H of End(TM) of endomorphisms of the tangent bundle to a Riemannian M, that in 1976 Stefano Marchiafava and Giuliano Romani called Il fibrato di Bonan .^[2] For M to be quaternion-Kähler, H should be preserved by the Levi-Civita connection and pointwise isomorphic to the imaginary quaternions which act on TM preserving the metric. Simultaneously, in 1965, Edmond Bonan and Vivian Yoh Kraines^[3] constructed the parallel 4-form. It was not until 1982 that Edmond Bonan proved an outstanding result : the analogue of hard Lefschetz theorem ^[4] for compact Sp(n)·Sp(1)-manifold.

Ricci curvature

Quaternion-Kähler manifolds appear in Berger's list of Riemannian holonomies as the only manifolds of special holonomy with non-zero Ricci curvature. In fact, these manifolds are Einstein. If an Einstein constant of a quaternion-Kähler manifold is zero, it is hyperkähler. This case is often excluded from the definition. That is, quaternion-Kähler is defined as one with holonomy reduced to Sp(n)·Sp(1) and with non-zero Ricci curvature (which is constant).

Quaternion-Kähler manifolds divide naturally into those with positive and negative Ricci curvature.

Examples

There are no known examples of compact quaternion-Kähler manifolds which are not locally symmetric or hyperkähler. Symmetric quaternion-Kähler manifolds are also known as Wolf spaces. For any simple Lie group G, there is a unique Wolf space G/K obtained as a quotient of G by a subgroup

K=K_{0}\cdot \operatorname {SU}(2).

Here, SU(2) is the subgroup associated with the highest root of G, and K₀ is its centralizer in G. The Wolf spaces with positive Ricci curvature are compact and simply connected.

If G is Sp(n+1), the corresponding Wolf space is the quaternionic projective space

{\mathbb {H}}\operatorname {P}^{n}.

It can be identified with a space of quaternionic lines in Hⁿ⁺¹.

It is conjectured that all quaternion-Kähler manifolds with positive Ricci curvature are symmetric.

Twistor spaces

Questions about quaternion-Kähler manifolds of positive Ricci curvature can be translated into the language of algebraic geometry using the methods of twistor theory (this approach is due to Penrose and Salamon). Let M be a quaternionic-Kähler manifold, and H the corresponding subbundle of End(TM), pointwise isomorphic to the imaginary quaternions. Consider the corresponding S²-bundle S of all h in H satisfying h² = −1. The points of S are identified with the complex structures on its base. Using this, it is can be shown that the total space Z of S is equipped with an almost complex structure.

Salamon proved that this almost complex structure is integrable, hence Z is a complex manifold. When the Ricci curvature of M is positive, Z is a projective Fano manifold, equipped with a holomorphic contact structure.

The converse is also true: a projective Fano manifold which admits a holomorphic contact structure is always a twistor space, hence quaternion-Kähler geometry with positive Ricci curvature is essentially equivalent to the geometry of holomorphic contact Fano manifolds.

References

↑ Bonan, Edmond (1965), "Structure presque quaternale sur une variété différentiable", C. R. Acad. Sci. Paris, 261: 5445–5448 .
↑ S.Marchiafava; G.Romani (1976), "Sui fibrati con struttura quaternionale generalizzata", Annali di Matematica pura ed applicata, 107: 131–157, doi:10.1007/bf02416470
↑ Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc, 71,3, 1: 526–527 .
↑
E. Bonan, (1982), "Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique", C. R. Acad. Sci. Paris, série I, 295: 115–118 .
- $\ \ h<n,\ \Lambda ^{{2n+2h}}=\Omega ^{h}\wedge \Lambda ^{{2n-2h}}$
- $\ \ p\leq 2n,\ \phi _{{p}}=\sum _{{0}}^{{[p/4]}}\Omega ^{{h}}\wedge \mu _{{p-4h}},\ \ \Omega \wedge *\mu _{{p-4h}}=0$
- $\ \ p<n-1,\ b_{{2p}}\leq b_{{2p+4}}$
- $\ \ p\leq n-1,\ b_{{2p-1}}\leq b_{{2p+1}}$
- E. Bonan, (1983), "Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique", C. R. Acad. Sci. Paris, série I, 296: 601–602 .
- T.Nagano; M.Takeuchi (1983), "Signature of quaternionic Kaehler manifolds", Proc.Japan Acad., 59: 384–386, doi:10.3792/pjaa.59.384 .
- Swann, Andrew.F. (1990), HyperKähler and Quaternionic Kähler Geometry (PDF) .
- Edmond Bonan, Isomorphismes sur une variété presque hermitienne quaternionique, Proc. of the Meeting on Quaternionique Structures in Math.and Physics SISSA , Trieste, (1994), 1-6.

Besse, Arthur Lancelot, Einstein Manifolds, Springer-Verlag, New York (1987)
Salamon, Simon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143-171.
Dominic Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000.

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