2 KINETSU ABE, DIMITAR GRANTCHAROV, AND GUEO GRANTCHAROV
properties. Finally, in Section 5 we present an explicit example of the total space
M of a bundle over the complex surface X
=
CP2 #2CP2
.
M was also considered in
[11],
where it was implicitly identified with a S 1-quotient of #5S3 x S
4
.
We identify
M,
using an old result by Goldsein and Linninger [8]. In particular, we obtain an
explicit example of a complex non-Kahler six-manifold with vanishing first Chern
class. The geometry of such manifolds became of interest recently because of its
relation to string theory
[3,
4].
2. Preliminaries
In this section we make some general observations about the even-dimensional torus
bundles over complex manifolds. The next proposition is probably well-known, but
for the sake of completeness we provide the proof. We assume that the Chern class
of the bundle is identified with a k-tuple of classes in H
2
(X,
JR)
and that a class is
of type (1,1) if it admits a representative 2-form which is of type (1,1) with respect
to the complex structure on
X.
PROPOSITION
2.1.
Suppose that M is the total space of an even-dimensional prin-
cipal torus bundle over a complex manifold X with characteristic classes of type
{1,1}. Then M admits a complex structure. Moreover, any free holomorphic action
of a compact complex torus with orbits which are complex submanifolds arises in
this form.
PROOF.
(Sketch) Since we have a complex structure on the base and connection
forms with curvature of type (1,1), we can construct an almost complex structure
on the total space in the standard way:
i) On the horizontal spaces 1t we have the horizontal lift of the base complex
structure.
ii)
The vertical spaces V are tangent spaces to even-dimensional tori hence carry a
natural complex structure.
Finally to check integrability we use the horizontal lifts of base (1,0) vector fields
Xh, yh and vertical (1,0) vector fields V, W. Then one can check that: [V, W]
E
V(l,O),
[Vh, Xh]
=
0 and [Xh, Yh]
=
w(X, Y)
=
0, where w is the curvature form
(vertical valued after some identifications). It is of type (1,1) and vanishes on any
two (1,0) vectors.
For the converse statement we use a general result about locally free group actions
that provide the structure of a principal torus bundle of
M.
Since the actions are
holomorphic, the quotients are equipped with a complex structure. Finally, the
characteristic classes of any holomorphic bundle are of type ( 1, 1). 0
Another feature of the above spaces is that they are in general non-Kahler. More
precisely we have:
THEOREM
2.1.
Suppose that the base of the fibration M
--+
X is Kahler and the
characteristic classes of the fibration are linearly independent. Then the complex
structure on M is non-Kiihler.
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