# Hopf manifold

In complex geometry, a Hopf manifold (Hopf 1948) 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 $\gamma :\;{\mathbb {C} }^{n}\mapsto {\mathbb {C} }^{n}$ 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.