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

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

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.[1]

Notes

Calabi, Eugenio; Eckmann, Benno (1953), "A class of compact complex manifolds which are not algebraic", Annals of Mathematics, 58: 494–500

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[1] Calabi, Eugenio; Eckmann, Benno (1953), "A class of compact complex manifolds which are not algebraic", Annals of Mathematics, 58: 494–500