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Calabi–Eckmann manifold

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

t ∈ C ,   ( x , y ) ∈ C n ∖ { 0 } × C m ∖ { 0 } ∣ t ( x , y ) = ( e t x , e α t y ) {\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 α ∈ C ∖ R {\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 S 2 n − 1 × S 2 m − 1 {\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 GL ⁡ ( n , C ) × GL ⁡ ( m , C ) {\displaystyle \operatorname {GL} (n,{\mathbb {C} })\times \operatorname {GL} (m,{\mathbb {C} })}

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

C n ∖ { 0 } × C m ∖ { 0 } ↦ C P n − 1 × C P m − 1 {\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 C P n − 1 × C P m − 1 {\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 C {\displaystyle \mathbb {C} } by the lattice Z + α ⋅ Z {\displaystyle {\mathbb {Z} }+\alpha \cdot {\mathbb {Z} }} . This makes M into a principal T-bundle.

Calabi and Eckmann discovered these manifolds in 1953.

Notes

  1. 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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