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 $$m,n>1$$, 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.