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
, m,n > 1, equipped with an action of a group
C
:
t
∈
C
,
(
x
,
y
)
∈
C
n
∖
0
×
C
m
∖
0
|
t
(
x
,
y
)
=
(
e
t
x
,
e
α
t
y
)
where
α
∈
C
∖
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
G
L
(
n
,
C
)
×
G
L
(
m
,
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
C
n
∖
0
×
C
m
∖
0
↦
C
P
n
−
1
×
C
P
m
−
1
induces a holomorphic map from the corresponding Calabi–Eckmann manifold M to
C
P
n
−
1
×
C
P
m
−
1
. The fiber of this map is an elliptic curve T, obtained as a quotient of
C
by the lattice
Z
+
α
⋅
Z
. This makes M into a principal T-bundle.
Calabi and Eckmann discovered these manifolds in 1953.
