Hopf manifold

In complex geometry, a Hopf manifold  is obtained as a quotient of the complex vector space (with zero deleted) $$({\mathbb C}^n\backslash 0)$$ by a free action of the group $$\Gamma \cong {\mathbb Z}$$ of integers, with the generator $$\gamma$$ of $$\Gamma$$ acting by holomorphic contractions. Here, a holomorphic contraction is a map $$\gamma:\; {\mathbb C}^n \to {\mathbb C}^n$$ such that a sufficiently big iteration $$\;\gamma^N$$ maps any given compact subset of $${\mathbb C}^n$$ 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 $$q\in {\mathbb C}$$ a complex number, $$0<|q|<1$$. Such manifold is called a classical Hopf manifold.

Properties
A Hopf manifold $$H:=({\mathbb C}^n\backslash 0)/{\mathbb Z}$$ is diffeomorphic to $$S^{2n-1}\times S^1$$. 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.