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