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

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

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 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 $GL(n,{\mathbb {C} })\times GL(m,{\mathbb {C} })$

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.