Contemporary Mathematics
Volume 337, 2003
On Some Complex Manifolds with Torus Symmetry
Kinetsu Abe, Dimitar Grantcharov, and Gueo Grantcharov
1.
Introduction
One of the first examples of complex non-Kahler manifolds are the products S
2n+l
x
S 2m+1 of odd-dimensional spheres equipped with the Calabi-Eckmann complex
structure. A standard approach to this construction is to consider S 2n+1
X
S 2m+1
as the total space of a principal torus bundle over cpn
X
cpm, Wang
[12]
general-
ized the construction and characterized all compact complex homogeneous spaces
as the total spaces of torus bundles over Kahlerian C-spaces (or generalized flag
manifolds). Other generalizations of the Calabi-Eckmann construction were given
by Abe
[1, 2]
and Blair-Ludden-Yano [5]. They used the tools of contact geometry
to construct the so called bicontact manifolds in the latter two papers.
In the present note we consider the total spaces of principal torus bundles over
Kahler manifolds. We start with the observation that if the fiber is even-dimensional
and the characteristic classes are of type (1,1), then the total space admits a complex
structure - a fact which is implicit in many papers. The aim of the paper is to
consider two partial cases of this construction in more details. The construction of
the complex structures makes use of connection 1-forms, a situation which resembles
some properties of the contact geometry.
After the preliminaries which contain the general results, in Section 3 we consider a
generalization of
the bicontact structures. Analogously to the results of [2], we show
that such structures satisfy a local Darboux Lemma and their complex subspaces
admit fibrations. Another case of the torus construction is considered in Section
4- the examples of manifolds constructed by Lopez di Medrano and Verjovsky [9]
and Meersseman
[10].
From [11], it is known that under an additional restriction
the manifolds arise from the torus bundle construction above. Our observation is
that they also appear as the zero set of a well-known moment map in symplectic
geometry. We hope that this will be useful in the further investigation of their
1991 Mathematics Subject Classification. Primary 32L05, Secondary 53C15, 53D20.
Key words and phrases. toric bundles, moment map.
The first author is partially supported by NSF grant CCR-0226504.
The third author is supported by NSF grant DMS-0209306 and EDGE.
©
2003 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/337/06047
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