Hopf manifold

In complex geometry, a Hopf manifold {{harv|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 $\gamma:\; {\mathbb C}^n \to {\mathbb C}^n$

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.

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.

Even-dimensional Hopf manifolds admit

The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.

{{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 | publisher=Interscience Publishers, Inc., New York |mr=0023054 | year=1948 | chapter=Zur Topologie der komplexen Mannigfaltigkeiten | pages=167–185}}
{{SpringerEOM|title=Hopf manifold|first=Liviu |last=Ornea}}