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 ${\displaystyle \gamma }$ of ${\displaystyle \Gamma }$ acting by holomorphic contractions. Here, a holomorphic contraction is a map ${\displaystyle \gamma :\;{\mathbb {C} }^{n}\to {\mathbb {C} }^{n))$ such that a sufficiently big iteration ${\displaystyle \;\gamma ^{N))$ maps any given compact subset of ${\displaystyle {\mathbb {C} }^{n))$ onto an arbitrarily small neighbourhood of 0.

Two-dimensional Hopf manifolds are called Hopf surfaces.

Examples

In a typical situation, ${\displaystyle \Gamma }$ 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 ${\displaystyle n\geq 2}$, 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.