Hypercomplex manifold explained

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.

If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex.^[1]

Examples

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

( { H}\backslash 0 ) /{ Z} (with { 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 productS ^{1 x} S ^{3,} hence its odd cohomologygroup is odd-dimensional. By Hodge decomposition,odd cohomology of a compact Kähler manifoldare always even-dimensional. In fact Hidekiyo Wakakuwa proved^[2] that on a compact hyperkähler manifold b _{2p+1} \equiv 0 mod 4 . Misha Verbitsky has shown that any compacthypercomplex manifold admitting a Kähler structure is also hyperkähler.

In 1988, left-invariant hypercomplex structures on some compact Lie groupswere constructed by the physicists Philippe Spindel, Alexander Sevrin, Walter Troost, and Antoine Van Proeyen. In 1992, Dominic 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( 2 l +1 ), T ^{1} x SU( 2 l), T ^{l} x SO( 2 l +1 ), T ^{2l} x SO( 4 l), T ^{l} x Sp(l), T ^{2} x E _{6,} T ^{7 x} E ^{7,} T ^{8 x} E ^{8,} T ^{4 x} F _{4,} T ^{2 x} 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 studied by Charles 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) Morio Obata studied affine connection associated with almost hypercomplex structures (under the former terminology of Charles Ehresmann^[3] of almost quaternionic structures). His construction leads to what Edmond Bonan called the Obata connection^[4] which is torsion free, if and only if, "two" of the almost complex structures

I, J, K are integrable and in this case the manifold is hypercomplex.

Twistor spaces

There is a 2-dimensional sphere of quaternions

L\in{ H} satisfying L ^{2=-1} .Each of these quaternions gives a complexstructure on a hypercomplex manifold M. Thisdefines an almost complex structure on the manifoldM x S ^{2} , which is fibered over{ C}P ^{1=S} ^{2} with fibers identified with (M, L) . This complex structure is integrable, as followsfrom Obata's theorem (this was first explicitly proved by Dmitry Kaledin^[5]). This complex manifoldis called the twistor space of M .If M is { H} , then its twistor spaceis isomorphic to { C}P ^{3\backslash} { C}P ^{1} .

See also

• Quaternionic manifold
• Hyperkähler manifold

References

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Notes and References

1. Book: 0804.2814. Manev. Mancho. Contemporary Aspects of Complex Analysis, Differential Geometry and Mathematical Physics. S. Dimiev and K. Sekigawa. World Sci. Publ.. Hackensack, NJ . 2005. 174–186. Sekigawa. Kouei. 10.1142/9789812701763_0016. Some Four-Dimensional Almost Hypercomplex Pseudo-Hermitian Manifolds. 2005 . 978-981-256-390-3.
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5. Kaledin . Dmitry . Dmitry Kaledin . alg-geom/9612016 . Integrability of the twistor space for a hypercomplex manifold . 1996.