View source for Calabi–Eckmann manifold

In [[Complex geometry|complex geometry]], a part of mathematics, a '''Calabi–Eckmann manifold''' (or, often, '''Calabi–Eckmann space'''), named after [[Biography:Eugenio Calabi|Eugenio Calabi]] and [[Biography:Beno Eckmann|Beno Eckmann]], is a [[Complex manifold|complex]], [[Homogeneous space|homogeneous]], non-[[Kähler manifold]], homeomorphic to a product of two odd-dimensional spheres of dimension ≥&nbsp;3.
The Calabi–Eckmann manifold is constructed as follows.  Consider the space <math>{\mathbb C}^n\backslash \{0\} \times {\mathbb C}^m\backslash \{0\}</math>, where <math>m,n>1</math>, equipped with an action of the group <math>{\mathbb C}</math>:

: <math>t\in {\mathbb C}, \ (x,y)\in {\mathbb C}^n\backslash \{0\} \times {\mathbb C}^m\backslash \{0\} \mid  t(x,y)= (e^tx, e^{\alpha t}y)</math>

where <math>\alpha\in {\mathbb C}\backslash {\mathbb R}</math> is a fixed [[Complex number|complex number]].  It is easy to check that this action is free and proper, and the corresponding orbit space ''M'' is homeomorphic to <math>S^{2n-1}\times S^{2m-1}</math>. Since ''M'' is a quotient space of a holomorphic action, it is also a complex manifold.  It is obviously homogeneous, with a transitive holomorphic action of <math>\operatorname{GL}(n,{\mathbb C}) \times \operatorname{GL}(m, {\mathbb C})</math>

A Calabi–Eckmann manifold ''M'' is non-Kähler, because <math>H^2(M)=0</math>. It is the simplest example of a non-Kähler
manifold which is simply connected (in dimension 2, all simply connected compact complex manifolds are Kähler).

The natural projection

: <math>{\mathbb C}^n\backslash \{0\} \times {\mathbb C}^m\backslash \{0\}\mapsto {\mathbb C}P^{n-1}\times {\mathbb C}P^{m-1}</math>

induces a holomorphic map from the corresponding Calabi–Eckmann manifold ''M'' to <math>{\mathbb C}P^{n-1}\times {\mathbb C}P^{m-1}</math>.  The fiber of this map is an elliptic curve ''T'', obtained as a quotient of <math>\mathbb C</math> by the lattice <math>{\mathbb Z} + \alpha\cdot {\mathbb Z}</math>. This makes ''M'' into a [[Principal bundle|principal]] ''T''-bundle.

Calabi and Eckmann discovered these manifolds in 1953.<ref>{{citation|first1=Eugenio|last1= Calabi|authorlink1=Eugenio Calabi|
first2=Benno|last2= Eckmann|title= A class of compact complex manifolds which are not algebraic|journal=[[Annals of Mathematics]]|volume= 58|pages= 494–500|year=1953}}</ref>

==Notes==

{{reflist}}

{{DEFAULTSORT:Calabi-Eckmann Manifold}}
[[Category:Complex manifolds]]

{{Sourceattribution|Calabi–Eckmann manifold}}