Hypercomplex manifold

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

1 Examples
2 Basic properties
3 Twistor spaces
4 Reference

[edit] Examples

Every hyperkähler manifold is also hypercomplex. The converse is not true. The Hopf surface

$\bigg({\Bbb H}\backslash 0\bigg)/{\Bbb Z}$

(with ${\Bbb Z}$ acting as a multiplication by a quaternion $q$ , $| q | > 1$ ) is hypercomplex, but not Kähler, hence not hyperkähler either. To see that the Hopf surface is not Kähler, notice that it is diffeomorphic to a product $S^1\times S^3,$ hence its odd cohomology group is odd-dimensional. By Hodge decomposition, odd cohomology of a compact Kähler manifold are always even-dimensional.

In 1988, left-invariant hypercomplex structures on some compact Lie groups were constructed by the physicists Ph. 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.

$T^4, SU(2l+1), T^1 \times SU(2l), T^l \times SO(2l+1),$

$T^{2l}\times SO(4l), T^l \times Sp(l), T^2 \times E_6,$

$T^7\times E^7, T^8\times E^8, T^4\times F_4, T^2\times G_2$

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

It is remarkable that any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus.

[edit] Basic properties

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

Much earlier (in 1955) M. Obata studied affine connections associated with quaternionic structures. His construction can be applied in hypercomplex geometry, giving what is called the Obata connection. Obata connection is a connection preserving the quaterionic action which is torsion-free. Obata proved that such a connection exists and is unique.

[edit] Twistor spaces

There is a 2-dimensional sphere of quaternions $L\in{\Bbb H}$ satisfying $L 2 = - 1$ . Each of these quaternions gives a complex structure on a hypercomplex manifold M. This defines an almost complex structure on the manifold $M\times S^2$ , which is fibered over ${\Bbb C}P^1=S^2$ with fibers identified with $(M, L)$ . This complex structure is integrable, as follows from Obata theorem. This complex manifold is called the twistor space of $M$ . If M is ${\Bbb H}$ , then its twistor space is isomorphic to ${\Bbb C}P^3\backslash {\Bbb C}P^1$ .