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 $^n\backslash \ \times ^m\backslash \$, where $m,n>1$, equipped with an action of the group $$:

: $t\in , \ (x,y)\in ^n\backslash \ \times ^m\backslash \ \mid t(x,y)= (e^tx, e^y)$

where $\alpha\in \backslash$ 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 $S^\times S^$. 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 $\operatorname(n,) \times \operatorname(m, )$

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

: $^n\backslash \ \times ^m\backslash \\mapsto P^\times P^$

induces a holomorphic map from the corresponding Calabi–Eckmann manifold ''M'' to $P^\times P^$. The fiber of this map is an elliptic curve ''T'', obtained as a quotient of $\mathbb C$ by the lattice $+ \alpha\cdot$. This makes ''M'' into a principal ''T''-bundle.

Calabi and Eckmann discovered these manifolds in 1953.

Notes

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Complex manifolds