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 ${\Bbb C}^n\backslash 0 \times {\Bbb C}^m\backslash 0$, m,n > 1, equipped with an action of a group ${\Bbb C}$:

$t\in {\Bbb C}, \ (x,y)\in {\Bbb C}^n\backslash 0 \times {\Bbb C}^m\backslash 0\ \ |\ \ t(x,y)= (e^tx, e^{\alpha t}y)$

where $\alpha\in {\Bbb C}\backslash {\Bbb R}$ 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 S²ⁿ⁻¹ × 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 $GL(n,{\Bbb C}) \times GL(m, {\Bbb C})$

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

${\Bbb C}^n\backslash 0 \times {\Bbb C}^m\backslash 0\mapsto {\Bbb C}P^{n-1}\times {\Bbb C}P^{m-1}$

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

Calabi and Eckmann discovered these manifolds in 1953.^[1]

Notes

1. E. Calabi and B. Eckmann: A class of compact complex manifolds which are not algebraic. Annals of Mathematics, 58, 494–500 (1953)