In complex geometry, a Hopf manifold 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 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.