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 ${ℂ}^n\mathrm{\setminus }\{0\}\times {ℂ}^m\mathrm{\setminus }\{0\}$, where $m,n>1$, equipped with an action of the group $ℂ$:

$t\in ℂ,(x,y)\in {ℂ}^n\mathrm{\setminus }\{0\}\times {ℂ}^m\mathrm{\setminus }\{0\}\mid t(x,y)=(e^{t}x,e^{\alpha t}y)$

where $\alpha \in ℂ\mathrm{\setminus }ℝ$ 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 $\mathrm{GL}(n,ℂ)\times \mathrm{GL}(m,ℂ)$

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

${ℂ}^n\mathrm{\setminus }\{0\}\times {ℂ}^m\mathrm{\setminus }\{0\}\mapsto ℂP^{n-1}\times ℂP^{m-1}$

induces a holomorphic map from the corresponding Calabi–Eckmann manifold M to $ℂP^{n-1}\times ℂP^{m-1}$. The fiber of this map is an elliptic curve T, obtained as a quotient of $ℂ$ 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.