6
KINETSU ABE, DIMITAR GRANTCHAROV, AND GUEO GRANTCHAROV
5. Examples in Dimension Six
In this section X=
X(~)=
CIP2#2CIP2.
We fix the basis {w1,w2,w3} of
H
2(X,Z)
dual to {O(l),E1,E2}, where E1 and
E2
are the exceptional divisors of
X.
The
structure of
H* (X,
Z) is determined by the relations
w~
=
-w~
=
-w~
and
w1 /\w2
=
w2 1\ W3
= w1
1\ W3
= 0. In example 5.3 in
[11],
Verjovsky and Meerssemman show
that the toric manifold X is a base of a specific T 2-fiber bundle N arising from
their construction. We start with a more general case:
LEMMA
5.1.
Let N with
1r1(N)
=
0
be the total space of a principal T
2
-bundle
over X such that the Chern classes of the bundle are part of a set of generators of
H
2
(X,Z). Then H
2
(N,Z)
=
Z and the cohomology ring of N has no torsion.
PROOF. There exist connection forms
(lh,
02) on N with curvatures
(d01,
d02).
Since the curvatures represent the first Chern classes of the bundle, the condition in
the lemma implies that we may assume
d01
= w1,
d02
= w2 . Then 01IT2, 02IT2, 01/\
02IT;
generate the Z-module
H* (T;,
Z) when restricted to any fiber
r;,
~
E
X~
This
and the standard Lerray's theorem imply that the E 2-term of the Lerray spectral
sequence for the bundle
is:
z
= 01/\02
0
z;j
0
z
z~
= o1,o2 0
z~®z;j
0
z~
z
0
'Z,;j
=
W1,W2,W3
0
Z =
w~
Here
e1, ... , ek
denotes the Z-module generated by e1
, ... , ek.
For
d2
we have:
d
E O,l
E2,0
Ll Ll 2 : 2
--+
2 ,
Ul,
U2,
--+
WI, W2
so it is an injection in this term. Similarly,
d2:
Eg'
2
--+
E~' 1 ; 01
1\02--+ w11\ 02
+
w2 1\01
is an injection. Consequently,
d2 :
E~' 2 --+
Ei·\
01
1\02 1\ w--+
WI
1\021\ w
+
W2 1\01
1\ w
for any
w
E H
2
(X, lR). In particular, it is a surjection in this term. Finally
d
.
E2,1
E4,0.
Ll
A A
2 · 2 --+ ,
Ui
1\
Wj
--+
Wi
1\
Wj
is surjective. Hence the
E
3
-term is:
0 0
z
0
z
0 0
z4
0 0
z
0
z
0 0
In particular the spectral sequence collapses at the E
3
-term and we complete the
proof.
D
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