ON SOME COMPLEX MANIFOLDS WITH TORUS SYMMETRY
7
Let's fix N from Lemma 5.1. As a corollary of the lemma, the Betti numbers of N
are bo(N) = b2(N) = b4(N) = b5(N) = 1 and b3(N) = 4, so b2(N) = b2(X) - 2
as claimed in Theorem 2.1. We need the following result about the classification of
simply connected 6-manifolds with a free S 1-action:
THEOREM 5.1. [8] Let M be a compact smooth simply-connected 6-manifold with
torsion-free cohomology and b2(M)
=
1. Then M is diffeomorphic to one of the
following:
i)S2 x
S
4
#2S3
x
S
3
if M is spin and
ii) S 2xS4 #2S3 x S
3
if M is not spin. Here S 2xS4 denotes the non-trivial S
4
-
bundle over
S
2
.
From Lemma 5.1 and the Theorem 5.1, we can identify N as soon as we know if it
is spin or not. Thus, we obtain our main result:
THEOREM 5.2. The spaces S
2
x
S
4
#2S3
x
S
3
and S 2xS4 #2S3
x
S
3
admit complex
structures and Ricci positive metrics. The moduli space of the complex structures
on each of them is disconnected. Moreover, the first space admits also a complex
structure with vanishing first Chern class.
PROOF. In order to see when
N
is spin, we use the standard splitting
T(l,o)
(N)
=
1r*T(l,O)
(X) EB C2
,
where
C2
is the trivial tangent bundle to the fiber. Then
c
1
(N)
=
1r*(c1
(X)),
and this class is proportional to the generator of
H
2
(N,Z).
In our case, c1
(X)
= 3w1
+
w2
+
w3
.
Changing the basis of
H
2
(X, Z),
we obtain
both spin and non-spin manifolds
N.
More precisely,
w;_
:=
kw
1
+
w3,
w~
:=
w2
,
and
w~
=
(k-
1)w1
+
w3
form a basis of
H
2
(X, Z)
for every
k
E
Z.
Then we can
choose.:=
1r*(w~)
as a generator of
H
2
(N,
Z) and easily calculate:
1r*(c1
(X))
= 7r*(3w1
+
w2
+
w3)
= 7r*((4-
k)w;_
+
w~
+
(k-
3)w~)
=
(k-
3).
Therefore, N is spin manifold for an odd k and non-spin for an even k. We can
now identify N using Theorem 5.1. In particular, we also obtain that for different
k's we have complex structures in different components of the moduli space, since
they have different first Chern classes. The positivity of the Ricci tensor follows
from Theorem 2.2.
0
REMARK 5.1. The existence of complex structures on the spaces in Theorem
5.2
is
also obtained in
[11].
However, the existence of different components of the moduli
space as well as the existence of a complex structure with vanishing first Chern
class has not been known. We also note that the previously known explicit examples
of manifolds with a positive Ricci curvature on connected sums are the manifolds
#kSm
X
sm.
References
[1] K.Abe,
A generalisation of Hopf fibration,
Tohoku Math.
J.
30 (1978), 177-210.
[2] K.Abe,
On a class of Hermitian manifolds,
Invent. Math. 51 (1979), 103-121.
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