Calabi–Eckmann manifold

In complex geometry, a part of mathematics, a Calabi–Eckmann manifold (or, often, Calabi–Eckmann space), named after Eugenio Calabi and Beno Eckmann, is a complex, homogeneous, non-Kähler manifold, homeomorphic to a product of two odd-dimensional spheres of dimension ≥ 3.

The Calabi–Eckmann manifold is constructed as follows. Consider the space {\mathbb C}^n\backslash {0} \times {\mathbb C}^m\backslash {0}, where m,n1, equipped with an action of the group {\mathbb C}:

: 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)

where \alpha\in {\mathbb C}\backslash {\mathbb R} is a fixed complex number. It is easy to check that this action is free and proper, and the corresponding orbit space M is homeomorphic to S^{2n-1}\times S^{2m-1}. 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 \operatorname{GL}(n,{\mathbb C}) \times \operatorname{GL}(m, {\mathbb C})

A Calabi–Eckmann manifold M is non-Kähler, because H^2(M)=0. 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

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

induces a holomorphic map from the corresponding Calabi–Eckmann manifold M to {\mathbb C}P^{n-1}\times {\mathbb C}P^{m-1}. The fiber of this map is an elliptic curve T, obtained as a quotient of \mathbb C by the lattice {\mathbb Z} + \alpha\cdot {\mathbb Z}. This makes M into a principal T-bundle.

Calabi and Eckmann discovered these manifolds in 1953.{{citation|first1=Eugenio|last1= Calabi|authorlink1=Eugenio Calabi| first2=Benno|last2= Eckmann|authorlink2=Beno Eckmann|title= A class of compact complex manifolds which are not algebraic|journal= Annals of Mathematics|volume= 58|pages= 494–500|year=1953}}

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