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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 \( {\Bbb C}^n\backslash 0 \times {\Bbb C}^m\backslash 0, m,n > 1 \), equipped with an action of a group \( {\Bbb C}: \)
\( t\in {\Bbb C}, \ (x,y)\in {\Bbb C}^n\backslash 0 \times {\Bbb C}^m\backslash 0\ \ |\ \ t(x,y)= (e^tx, e^{\alpha t}y) \)
where \( \alpha\in {\Bbb C}\backslash {\Bbb R} \) 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 × S2m−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 \( GL(n,{\Bbb C}) \times GL(m, {\Bbb C}) \)
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
\( {\Bbb C}^n\backslash 0 \times {\Bbb C}^m\backslash 0\mapsto {\Bbb C}P^{n-1}\times {\Bbb C}P^{m-1} \)
induces a holomorphic map from the corresponding Calabi–Eckmann manifold M to \( {\Bbb C}P^{n-1}\times {\Bbb C}P^{m-1} \). The fiber of this map is an elliptic curve T, obtained as a quotient of \( \Bbb C \) by the lattice \( {\Bbb Z} + \alpha\cdot {\Bbb Z} \). This makes M into a principal T-bundle.
Calabi and Eckmann discovered these manifolds in 1953.[1]
Notes
E. Calabi and B. Eckmann: A class of compact complex manifolds which are not algebraic. Annals of Mathematics, 58, 494–500 (1953)
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