In complex geometry

Complex geometry

In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....

, a Hopf manifold is obtained
as a quotient of the complex vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

(with zero deleted)
by a free action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of the group of
integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, with the generator
of acting by holomorphic contractions

Contraction mapping

In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...

. Here, a holomorphic contraction
is a map
such that a sufficiently big iteration
puts any given compact subset
onto an arbitrarily small neighbourhood of 0.

Two dimensional Hopf manifolds are called Hopf surface

Hopf surface

In complex geometry, a Hopf surface is a compact complex surface obtainedas a quotient of the complex vector space C2 \ 0...

s.

Examples

In a typical situation, is generated
by a linear contraction, usually a diagonal matrix
, with
a complex number, . Such manifold
is called a classical Hopf manifold.

Properties

A Hopf manifold
is diffeomorphic to .
It is non-Kähler

Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

. Indeed,
the first cohomology group of H
is odd-dimensional. By Hodge decomposition,
odd cohomology of a compact Kähler manifold

Kähler manifold

In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

are always even-dimensional.

Hypercomplex structure

Even-dimensional Hopf manifolds admit
hypercomplex structure

Hypercomplex manifold

In differential geometry, a hypercomplex manifold is a manifold with the tangent bundleequipped with an action by the algebra of quaternionsin such a way that the quaternions I, J, Kdefine integrable almost complex structures.- Examples :...

.
The Hopf surface is the only compact hypercomplex manifold

Hypercomplex manifold

In differential geometry, a hypercomplex manifold is a manifold with the tangent bundleequipped with an action by the algebra of quaternionsin such a way that the quaternions I, J, Kdefine integrable almost complex structures.- Examples :...

 of quaternionic dimension 1 which is not hyperkähler

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...

.