Indifferential geometry, ahypercomplex manifold is amanifold with thetangent bundleequipped with anaction bythe algebra of quaternionsin such a way that the quaternions${\displaystyle I,J,K}$define integrablealmost complex structures.

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

Everyhyperkähler manifold is also hypercomplex.The converse is not true. TheHopf surface

${\displaystyle (\mathbb{H}\mathrm{\setminus }0)/\mathbb{Z}}$

(with${\displaystyle \mathbb{Z}}$ actingas a multiplication by a quaternion${\displaystyle q}$,${\displaystyle |q|>1}$) ishypercomplex, but notKähler,hence nothyperkähler either.To see that the Hopf surface is not Kähler,notice that it is diffeomorphic to a product${\displaystyle S^1\times S^3,}$ hence its odd cohomologygroup is odd-dimensional. ByHodge decomposition,odd cohomology of a compactKähler manifoldare always even-dimensional. In fact Hidekiyo Wakakuwa proved^[2] that on a compacthyperkähler manifold${\displaystyle b_{2p+1}\equiv 0mod4}$.Misha Verbitsky has shown that any compacthypercomplex manifold admitting a Kähler structure is also hyperkähler.^[3]

In 1988, left-invariant hypercomplex structures on some compactLie 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.

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

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

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

where${\displaystyle T^i}$ denotes an${\displaystyle i}$-dimensional compact torus.

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

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${\displaystyle T^4}$, theHopf surface and theK3 surface.

Much earlier (in 1955) Morio Obata studiedaffine connection associated withalmost hypercomplex structures (under the former terminology ofCharles Ehresmann^[4] ofalmost quaternionic structures). His construction leads to whatEdmond Bonan called theObata connection^[5][6] which istorsion free, if and only if, "two" of the almost complex structures${\displaystyle I,J,K}$ are integrable and in this case the manifold is hypercomplex.

There is a 2-dimensional sphere of quaternions${\displaystyle L\in \mathbb{H}}$ satisfying${\displaystyle L^2=-1}$.Each of these quaternions gives a complexstructure on a hypercomplex manifoldM. Thisdefines an almost complex structure on the manifold${\displaystyle M\times S^2}$, which is fibered over${\displaystyle \mathbb{C}{P}^1=S^2}$ with fibers identified with${\displaystyle (M,L)}$. This complex structure is integrable, as followsfrom Obata's theorem (this was first explicitly proved byDmitry Kaledin^[7]). This complex manifoldis called thetwistor space of${\displaystyle M}$.IfM is${\displaystyle \mathbb{H}}$, then its twistor spaceis isomorphic to${\displaystyle \mathbb{C}{P}^3\mathrm{\setminus }\mathbb{C}{P}^1}$.

• Quaternionic manifold
• Hyperkähler manifold

• Complex manifolds
• Structures on manifolds

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