Hopf manifold

In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) $({\mathbb{ℂ}}^n\mathrm{\setminus }0)$ by a free action of the group $\mathrm{\Gamma }\cong \mathbb{\mathbb{Z}}$ of integers, with the generator $\gamma$ of $\mathrm{\Gamma }$ 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, $\mathrm{\Gamma }$ 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 $H:=({\mathbb{ℂ}}^n\mathrm{\setminus }0)/\mathbb{\mathbb{Z}}$ is diffeomorphic to $S^{2n-1}\times S^1$. For $n\ge 2$, 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.

References

• Hopf, Heinz (1948), "Zur Topologie der komplexen Mannigfaltigkeiten", Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, pp. 167–185 
• Hazewinkel, Michiel, ed. (2001), "Hopf manifold", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page