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

1. Bonan, Edmond (1965), "Structure presque quaternale sur une variété différentiable", C. R. Acad. Sci. Paris, 261: 5445–5448.
2. S.Marchiafava; G.Romani (1976), "Sui fibrati con struttura quaternionale generalizzata", Annali di Matematica pura ed applicata, 107: 131–157, doi:10.1007/bf02416470
3. Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc, 71,3, 1: 526–527.
4. 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.

• S.Marchiafava; G.Romani (1976), "Sui fibrati con struttura quaternionale generalizzata", Annali di Matematica pura ed applicata, 107: 131–157, doi:10.1007/bf02416470.
• 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.