In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle
equipped with an action

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 by the algebra of quaternions
in such a way that the quaternions
define integrable almost complex structures

Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...

.

Examples

Every hyperkähler manifold

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...

 is also hypercomplex.
The converse is not true. The Hopf surface
(with acting
as a multiplication by a quaternion , ) is
hypercomplex, but not Kähler

Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

,
hence not hyperkähler

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...

 either.
To see that the Hopf surface is not Kähler,
notice that it is diffeomorphic to a product
hence its odd cohomology
group is odd-dimensional. By Hodge decomposition,
odd cohomology of a compact Kähler manifold

Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

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

Dominic Joyce

Dominic D. Joyce is a British mathematician, currently a professor at the University of Oxford and a fellow of Lincoln College since 1995. His undergraduate and doctoral studies were at Merton College. He undertook a DPhil in geometry under the supervision of Simon Donaldson, completed in 1992...

rediscovered this construction, and
gave a complete classification of
left-invariant hypercomplex structures on compact Lie groups.
Here is the complete list.


where denotes an -dimensional compact torus.

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

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
, the Hopf surface and
the K3 surface

K3 surface

In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....

.

Much earlier (in 1955) M. Obata studied affine connection

Affine connection

In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...

s
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.

Twistor spaces

There is a 2-dimensional sphere of quaternions
satisfying .
Each of these quaternions gives a complex
structure on a hypercomplex manifold M. This
defines an almost complex structure on the manifold
, which is fibered over
with fibers identified with .
This complex structure is integrable, as follows
from Obata theorem. This complex manifold
is called the twistor space of .
If M is , then its twistor space
is isomorphic to .