ON SOME COMPLEX MANIFOLDS WITH TORUS SYMMETRY 5
It is known that for any LM-V manifold there exists a natural fibration
G
defined
by commuting holomorphic vector fields. In the case that the given LM-V manifold
satisfies condition (K), Meersseman and Verjovsky proved the following:
THEOREM
4.1.
[11] Let
(A1, ... ,An)
form an admissible configuration satisfying
condition (K). Then
i) The leaves of the fibration G are compact complex tori and the quotient space is
a toric projective variety with only orbifold singularities X (
~).
ii) The projection
M---+
X(~)
is a holomorphic Seifert bundle.
Moreover, one can see that
X (
~)
is a manifold if the configuration A with condition
(K) satisfies an additional assumption that the basis of
(1)
can be completed to a
basis of
zn.
In [11] the manifold M has also been identified as a presymplectic (or
Poisson) reduction of
en.
Now we want to describe the fibration
M
---+
X(~)
in
terms of "genuine" moment map of a torus action on the complex projective space
cpn.
This will also provide another proof of the above theorem for the smooth
case and will lead to additional insight into the properties of M.
Consider the standard action of the real torus
rn
= { (
e27riXl' e27riX
2
' .•• ,
e
27riXn) I
X;
E
JR}
on
cpn-
1'
which in homogeneous coordinates has the form:
[
l
[
27riX1
27riXn
l
z1, ... Zn---+ e Z1, ... ,e Zn.
The moment map of this action with respect to the Fubini-Study form is:
A more general form of the above action is the weighted T
2m-action
on
cpn-
1
induced by
(A1, ... ,An):
[z
Z
l
---+
[e27rial
,x
z
e27ri(h ,x
z
e27rial
,x
z
e27rif31
,x
z
l
1, ···
n
1, 1, ... ,
n, n'
where x
=
(x
1 , ... ,
X2m) E
JR2m,
Aj
=
O:j
+
i/3j-
·,·is the the standard bilinear
form of
JR2m.
The moment map for this action with respect to the Fubini-Study
form is:
tLA([z])
=
(I:f=1o:;lz;l
2
/I:f=11z;l
2,
I:f=1/3;lz;l
2
/I:f=11z;l
2).
Then the zero level set of
{LA
is exactly
Thus we have proved:
THEOREM
4.2.
The LM- V manifold M is diffeomorphic to Z
=
p,;\
1
(0).
It is known that
Z---+
X(~)
is a principal torus bundle in the smooth case and a
Seifert fibration in the orbifold case [7]. From Theorem
2.1
we obtain a complex
structure on Z.
A corollary for the Riemannian geometry of such bundles is:
COROLLARY
4.1.
The LM- V manifolds satisfying condition (K) with base Fano
manifold admit a metric with positive Ricci curvature.
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