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

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

where ${\displaystyle \alpha \in {\mathbb {C} }\backslash {\mathbb {R} }}$ 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 ${\displaystyle S^{2n-1}\times S^{2m-1}}$. 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 ${\displaystyle \operatorname {GL} (n,{\mathbb {C} })\times \operatorname {GL} (m,{\mathbb {C} })}$

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

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

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

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