Hopf manifold


In complex geometry, a Hopf manifold is obtained
as a quotient of the complex vector space

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
maps any given compact subset of
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.