hypercomplex manifold

Every hyperkähler manifold is also hypercomplex.

The converse is not true. The Hopf surface

:$\backslash bigg(\{\backslash mathbb\; H\}\backslash backslash\; 0\backslash bigg)/\{\backslash mathbb\; Z\}$

(with $\{\backslash mathbb\; Z\}$ 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

$S^1\backslash times\; S^3,$ hence its odd cohomology

group is odd-dimensional. By Hodge decomposition,

odd cohomology of a compact Kähler manifold

are always even-dimensional. In fact Hidekiyo Wakakuwa proved

{{citation |first =Hidekiyo| last = Wakakuwa | title =On Riemannian manifolds with homogeneous holonomy group Sp(n)| volume =10| issue = 3 | journal = Tôhoku Mathematical Journal|year =1958| pages =274–303 | doi=10.2748/tmj/1178244665| doi-access =free}}.

that on a compact hyperkähler manifold $\backslash \; b\_\{2p+1\}\backslash equiv\; 0\; \backslash \; mod\; \backslash \; 4$.

Misha Verbitsky has shown that any compact

hypercomplex manifold admitting a Kähler structure is also hyperkähler.{{citation |first = Misha| last = Verbitsky| authorlink= Misha Verbitsky | title =Hypercomplex structures on Kaehler manifolds | volume =15 | issue = 6| journal = GAFA|year =2005| pages =1275–1283 | doi=10.1007/s00039-005-0537-4| arxiv =math/0406390}}

In 1988, left-invariant hypercomplex structures on some compact Lie groups

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

:$T^\{2l\}\backslash times\; SO(4l),\; T^l\; \backslash times\; Sp(l),\; T^2\; \backslash times\; E\_6,$

:$$

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.

Hypercomplex manifolds as such were studied 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) Morio Obata studied affine connection associated with almost hypercomplex structures (under the former terminology of Charles Ehresmann{{citation | first =Charles| last =Ehresmann| authorlink=Edmond Bonan | title =Sur la théorie des espaces fibrés| journal = Coll. Top. Alg., Paris | year =1947 }}. of almost quaternionic structures). His construction leads to what Edmond Bonan called the Obata connection{{citation | first =Edmond| last =Bonan | authorlink=Edmond Bonan | title =Tenseur de structure d'une variété presque quaternionienne | journal = C. R. Acad. Sci. Paris | volume =259 | year =1964 | pages = 45–48}}{{citation | first =Edmond| last =Bonan |authorlink=Edmond Bonan|title =Sur les G-structures de type quaternionien| journal = Cahiers de Topologie et Géométrie Différentielle Catégoriques| volume =9 | year =1967| issue =4 | pages = 389–463| url = http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1967__9_4/CTGDC_1967__9_4_389_0/CTGDC_1967__9_4_389_0.pdf }}. 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.

There is a 2-dimensional sphere of quaternions

$L\backslash in\{\backslash mathbb\; H\}$ 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

$M\backslash times\; S^2$, which is fibered over

$\{\backslash mathbb\; C\}P^1=S^2$ with fibers identified with $(M,\; L)$.

This complex structure is integrable, as follows

from Obata's theorem (this was first explicitly proved by

Dmitry Kaledin{{cite arXiv |last=Kaledin

|first=Dmitry |author-link=Dmitry Kaledin |eprint=alg-geom/9612016

|title=Integrability of the twistor space for a hypercomplex manifold |date=1996}}). This complex manifold

is called the twistor space of $M$.

If M is $\{\backslash mathbb\; H\}$, then its twistor space

is isomorphic to $\{\backslash mathbb\; C\}P^3\backslash backslash\; \{\backslash mathbb\; C\}P^1$.

• Quaternionic manifold
• Hyperkähler manifold

{{reflist}}

{{refbegin}}

• {{citation | first =Charles P.| last =Boyer | title =A note on hyper-Hermitian four-manifolds| journal =Proceedings of the American Mathematical Society | volume =102 |issue=1| year =1988| pages =157–164| doi = 10.1090/s0002-9939-1988-0915736-8 | doi-access =free}}.
• {{citation | first =Dominic| last = Joyce |author1-link=Dominic Joyce|title =Compact hypercomplex and quaternionic manifolds| journal =Journal of Differential Geometry | volume =35 |issue=3| year =1992| pages =743–761 | doi=10.4310/jdg/1214448266| doi-access =free}}.
• {{citation | first = Morio| last =Obata | title = Affine connections on manifolds with almost complex, quaternionic or Hermitian structure| journal =Japanese Journal of Mathematics | volume =26| year =1955| pages =43–79}}.
• {{citation|title=Extended supersymmetric $\backslash sigma$-models on group manifolds|last1=Spindel|first1=Ph.|last2=Sevrin|first2=A. |last3=Troost|first3=W.|last4=Van Proeyen|first4=A.|journal=Nuclear Physics |volume= B308 |year=1988|pages=662–698}}.

{{refend}}

Category:Complex manifolds

Category:Structures on manifolds