# Hopf manifold

In complex geometry, a **Hopf manifold** (Hopf 1948) is obtained
as a quotient of the complex vector space
(with zero deleted)
by a free action of the group of
integers, with the generator
of acting by holomorphic contractions. Here, a *holomorphic contraction*
is a map
such that a sufficiently big iteration
puts any given compact subset
onto an arbitrarily small neighbourhood of 0.

Two dimensional Hopf manifolds are called Hopf surfaces.

## Examples

In a typical situation, is generated
by a linear contraction, usually a diagonal matrix
, with
a complex number, . Such manifold
is called *a classical Hopf manifold*.

## Properties

A Hopf manifold is diffeomorphic to . For , it is non-Kähler. In fact, it is not even symplectic because the second cohomology group is zero.

## Hypercomplex structure

Even-dimensional Hopf manifolds admit hypercomplex structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.

## References

- Hopf, Heinz (1948), "Zur Topologie der komplexen Mannigfaltigkeiten",
*Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948*, Interscience Publishers, Inc., New York, pp. 167–185, MR 0023054 - Ornea, L. (2001), "H/h110270", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4