4
KINETSU ABE, DIMITAR GRANTCHAROV, AND GUEO GRANTCHAROV
(transversal at each point) integral submanifolds. Applying the standard Darboux
lemma completes the proof. D
In analogy to the bicontact case, we have the following Theorem:
THEOREM 3.2. Every complex subvariety of M is fibred by [-dimensional l 1
subvarietes of the fibres of
1r :
M
-
X .
PROOF. Again we only sketch the proof which is analogous to the one given in
[2].
If N is a subvariety with the (necessary dense) set ofregular points R(N), then
we have to prove that TxR(N)
nTx(1r-
1
(1r(x)) =f.
0. This is clear if dimcN
~
n-
k,
where n
=
dimcM. If dimcN ::::; n- k and Nc is not contained in a fiber, then there
is an open set of points where Nc is transversal to the fibers. From Definition 3.1 (ii)
follows that
d(
u
1
+ ... +
u
2k)n-k
=f.
0 when evaluated on complex planes transversal
to the vertical subspace. Hence
JN
d(u
1
+ ... +
u
2k)dimcN
=f.
0. On the other hand,
it vanishes by Stokes' theorem. D
4. LM-V Manifolds and Moment Maps
Here we consider the manifolds constructed by Lopez de Medrano and Verjovsky
[9],
and generalized by Meersseman
[10].
We give a geometric description for the
case when they are torus bundles over complex manifolds. We also identify them
with the zero sets of appropriate moment maps.
We start with the description of the LM-V manifolds. Fix integers
m
and
n
with
n
2m. Take
(A1, ... ,An)
to be a set of
n
vectors in
em
satisfying the following
conditions:
i) The Siegel condition: 0 E
CH(A1, ... ,An),
where CH denotes the convex hull.
ii) The weak hyperbolicity condition: 0
~
CH(Aip ... ,Ai
2
m)
for any 2m-tuple
1::::;
i1 ::::; ... ::::; i2m ::::;
n.
Such ann-tuple of vectors is said to form an "admissible configuration". Then using
a remark by Haefliger about the leaf space of a transversally holomorphic foliation,
Lopez de Medrano and Verjorvsky
[9]
and Meersseman
[9]
constructed a complex
structure on the following manifold:
Clearly M is not a complex submanifold of
cpn.
The manifold has many interesting
properties - see
[9, 10].
Here we consider a partial case by imposing further the
following restriction on
(A1, ... ,An)
DEFINITION 4.1. An admissible configuration fulfills condition (K) if for the real
space of solutions of the system:
(1)
there exists a basis with integer coordinates.
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