Hopf manifold

In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) ${\displaystyle ({\mathbb {C}}^{n}\backslash 0)}$ by a free action of the group ${\displaystyle \Gamma \cong {\mathbb {Z}}}$ of integers, with the generator $\gamma$ of $\Gamma$ acting by holomorphic contractions. Here, a holomorphic contraction is a map ${\displaystyle \gamma :\;{\mathbb {C}}^{n}\mapsto {\mathbb {C}}^{n}}$ such that a sufficiently big iteration ${\displaystyle \;\gamma ^{N}}$ puts any given compact subset ${\Bbb C}^n$ 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 ${\displaystyle q\cdot Id}$, with ${\displaystyle q\in {\mathbb {C}}}$ a complex number, ${\displaystyle 0<|q|<1}$. Such manifold is called a classical Hopf manifold.

Properties

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

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, MR 0023054
• Ornea, L. (2001), "H/h110270", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4