Hopf manifold explained

In complex geometry, a Hopf manifold is obtainedas a quotient of the complex vector space(with zero deleted)

({ C} ^{n\backslash} 0 ) by a free action of the group \Gamma \cong { Z} ofintegers, with the generator \gamma of \Gamma acting by holomorphic contractions. Here, a holomorphic contractionis a map \gamma: { C} ^{n} \to { C} ^{n} such that a sufficiently big iteration \gamma ^{N} maps any given compact subset of { C} ^{n} onto an arbitrarily small neighbourhood of 0.

Two-dimensional Hopf manifolds are called Hopf surfaces.

Examples

In a typical situation,

\Gamma is generatedby a linear contraction, usually a diagonal matrix q ⋅ Id , with q\in { C} a complex number, 0 <|q|< 1 . Such manifoldis called a classical Hopf manifold.

Properties

A Hopf manifold

H: = ({ C} ^{n\backslash} 0 )/{ Z} is diffeomorphic to S ^{2n-1} x S ^{1} .For n\geq 2 , it is non-Kähler. In fact, it is not evensymplectic because the second cohomology group is zero.

Hypercomplex structure

Even-dimensional Hopf manifolds admithypercomplex structure.The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.