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 \gamma of \Gamma acting by holomorphic contractions. Here, a holomorphic contraction is a map such that a sufficiently big iteration ;\gamma^N 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, \Gamma is generated by a linear contraction, usually a diagonal matrix q\cdot Id, with a complex number, 0<|q|<1. Such manifold is called a classical Hopf manifold.

Properties

A Hopf manifold is diffeomorphic to. For n\geq 2, 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.