ON SOME COMPLEX MANIFOLDS WITH TORUS SYMMETRY

3

PROOF. We first note that by taking a sub bundle we may assume that the

fiber is of dimension two. Then we see, using the spectral sequence arguments from

Lemma 5.1 (see section 5), that the Betti numbers satisfy

b2 (M)

=

b2 (X)-

2 for

a general base X. Moreover, any class in

H

2

(

M, JR.)

is a pull-back of a class in

H

2

(X,JR). Therefore, there is no class wE H

2

(M,IR) with wn

=f.

0. This implies

that M does not admit Kahler structure. D

Next we consider the Riemannian metrics on

M.

The following result was first

proved by Berard-Bergery

[6]:

THEOREM 2.2. The total space of a torus bundle over a manifold with a positive

Ricci curvature admits a metric with positive Ricci curvature provided that it has a

finite fundamental group.

3. Contact-type Structures and Torus Actions

Here we consider a special type of torus actions which generalizes the notion of

bicontact structure of

[2, 5].

Let X

1

,

X2, ... , X2k be commuting non-vanishing vector fields on a Hermitian man-

ifold

(M,

J,

g),

preserving the Hermitian structure and satisfying X 2i = J X 2i-l for

i

=

1, ... ,k. Consider the forms ui defined via ui(Xj)

=

6f,and each ui vanishes

on the orthogonal complement of span{X1, X2, ... , X2k} which is also a complex

subspace.

DEFINITION 3.1. We say that the action of a 2k-torus on (M, J, g) is holomor-

phic totally non-degenerate if there are forms ui as above that satisfy the following

conditions:

i) At least one of the forms ui has exterior derivative a (1,1} form (hence all have).

ii) There are integers ni not all 0 with n

1

+

n

2

+ .. +

n 2k = dimcM - k such that

the induced 1-forms satisfy (dul)n,+l

=

0, (du2)n2+l

=

0, ... , (du2k)n

2k+l

=

0, and

that u1

1\

(dul)n,

1\

u2

1\

(du2)n

2

••

1\

u2k

1\

(du2k)n

2

k

=f.

0 everywhere.

One can easily check that u2i

=

Ju2i-l and that the orbits are holomorphic sub-

manifolds. When

k

=

1, we have the bicontact manifolds of

[2, 5].

As in the bicontact case, we have:

THEOREM 3.1. (Darboux Lemma) There are local coordinates xi on M such that

ui

=

dx?- ~x;j-

1

dx?. In case when the base space X is a manifold, it is locally

a product

of Kahler manifolds.

PROOF. Since the proof is identical to the one in

[5],

we only sketch the ideas.

Define the subspaces Ti of the tangent space TpM as

Ti

=

na#iKer(ua) na#i Ker(dua)

E

TM.

Then Li Ti

=

TM and dimTilp

=

ni. We consider Ti's as subbundles and we easily

see that they are involutive and that the forms ui are contact on the corresponding