Calabi–Eckmann manifold explained

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\backslash\{0\} x {C}^m\backslash\{0\}

, where

m,n>1

, equipped with an action of the group

{C}

t\in{C}, (x,y)\in{C}^n\backslash\{0\} x {C}^m\backslash\{0\}\midt(x,y)=(e^tx,e^\alphay)

where

\alpha\in{C}\backslash{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 x 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,{C}) x \operatorname{GL}(m,{C})

A Calabi–Eckmann manifold M is non-Kähler, because

H^2(M)=0

. It is the simplest example of a non-Kählermanifold which is simply connected (in dimension 2, all simply connected compact complex manifolds are Kähler).

The natural projection

{C}^n\backslash\{0\} x {C}^m\backslash\{0\}\mapsto{C}P^n-1 x {C}P^m-1

induces a holomorphic map from the corresponding Calabi–Eckmann manifold M to

{C}P^n-1 x {C}P^m-1

. The fiber of this map is an elliptic curve T, obtained as a quotient of

by the lattice

{Z}+\alpha ⋅ {Z}

. This makes M into a principal T-bundle.

Calabi and Eckmann discovered these manifolds in 1953.