Hypercomplex manifold

In differential geometry, a hypercomplex manifold is a manifold with the tangent bundleequipped with an action by the algebra of quaternionsin such a way that the quaternions $I,\; J,\; K$define integrable almost complex structures.

Examples

Every hyperkähler manifold is also hypercomplex.The converse is not true. The Hopf surface: (with $\{Bbb\; Z\}$ actingas a multiplication by a quaternion $q$, $|q|>1$) ishypercomplex, but not Kähler,hence not hyperkähler either.To see that the Hopf surface is not Kähler,notice that it is diffeomorphic to a product$S^1\; imes\; S^3,$ hence its odd cohomologygroup is odd-dimensional. By Hodge decomposition,odd cohomology of a compact Kähler manifoldare always even-dimensional.

In 1988, left-invarianthypercomplex structures on some compact Lie groupswere constructed by the physicistsPh. Spindel, A. Sevrin, W. Troost, A. Van Proeyen. In 1992, D. Joyce rediscovered this construction, and gave a complete classification of left-invariant hypercomplex structures on compact Lie groups. Here is the complete list.

:$T^4,\; SU(2l+1),\; T^1\; imes\; SU(2l),\; T^l\; imes\; SO(2l+1),$ :$T^\{2l\}\; imes\; SO(4l),\; T^l\; imes\; Sp(l),\; T^2\; imes\; E\_6,$ :$T^7\; imes\; E^7,\; T^8\; imes\; E^8,\; T^4\; imes\; F\_4,\; T^2\; imes\; G\_2$

where $T^i$ denotes an $i$-dimensional compact torus.

It is remarkable that any compact Lie group becomeshypercomplex after it is multiplied by a sufficientlybig torus.

Basic properties

Hypercomplex manifolds as such were introduced byCharles Boyer in 1988. He also proved that in real dimension 4, the only compact hypercomplexmanifolds are the complex torus $T^4$, the Hopf surface and the K3 surface.

Much earlier (in 1955) M. Obata studied affine connectionsassociated with quaternionic structures. His constructioncan be applied in hypercomplex geometry, giving what is calledthe Obata connection. Obata connection is a connectionpreserving the quaterionic action which is torsion-free.Obata proved that such a connection exists and is unique.

Twistor spaces

There is a 2-dimensional sphere of quaternions$Lin\{Bbb\; H\}$ satisfying $L^2=-1$.Each of these quaternions gives a complexstructure on a hypercomplex manifold "M". Thisdefines an almost complex structure on the manifold$M\; imes\; S^2$, which is fibered over$\{Bbb\; C\}P^1=S^2$ with fibers identified with $(M,\; L)$. This complex structure is integrable, as followsfrom Obata theorem. This complex manifoldis called the twistor space of $M$.If "M" is $\{Bbb\; H\}$, then its twistor spaceis isomorphic to .

References

[1] Boyer, Charles P."A note on hyper-Hermitian four-manifolds",Proc. Amer. Math. Soc. 102 (1988), no. 1, 157--164.

[2] Joyce, Dominic, "Compact hypercomplex and quaternionic manifolds", J. Differential Geom. 35(1992) no. 3, 743-761

[3] Obata, M., "Affine connections on manifolds with almost complex, quaternionic or Hermitian structure", Jap. J. Math., 26 (1955), 43-79.

[4] Ph. Spindel, A. Sevrin, W. Troost, A. Van Proeyen"Extended supersymmetric $sigma$-models on group manifolds", Nucl. Phys. B308 (1988) 662-698.