Date: Sun, 21 Jan 2001 16:15:40 -0500
From: Kiumars Kaveh <kaveh@math.toronto.edu>
Subject: Two questions from Toronto!

This is Kiumars from Toronto. I have two questions regarding
topology of complex Lie group actions.
I will appreciate any comment by somebody in the topology mailing
list.

Please respond to me to kaveh@math.toronto.edu
Here are the questions:

1. As you know, if E is a complex vector bundle of rank k over a
compact complex manifold M, Chern classes of E can be thought of
as Poincare dual to loci of points where generic sections of E become
dependent. More precisely:
Let s_1, ..., s_k be "generic" sections of E (i.e. if for each i, s_i+1
intersects the subspace of E spanned by s_1,...,s_i transversally)
and let D_i be the set of points in M where s_1,..., s_i become dependent.
then r-th Chern class of E is Poincare dual to D_k-r+1.

Now suppose a complex Lie group G is acting on a
complex manifold M, I am interested in the case where G is reductive
and M is a homogenuous space of G. Now can one get Chern classes
of M, as Poincare dual of loci of points where some generating vector
fields of the action become dependent?

2. As before let a complex Lie group G acts on a complex manifold M. I
am interested in the case where M is a homogenous space
of G. I say a submanifold N of M is "invariantly parallelizable" if there
are "generating vector fields" v_1,...,v_n ,n=dim(N), such that they
span TN at each point x of N.
Equivalently: if \h = Lie algebra generated by v_i's then one can see that
N is invariant under the \h-action and N is a homogenuous space of H =
exp(\h) say N = H/K. The vectors v_i span a complement \m of \k in \h
and  for all x\in H, \Ad_x(\k) \cap \m = 0.

Question: When one can expect to have a stratification of M into
invariantly parallelizable submanifolds?
My guess is that for solvable G one may say something.

regards,
Kiumars Kaveh
kaveh@math.toronto.edu
*******************************************************************************
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Kiumars Kaveh
Department of Mathematics
University of Toronto, Toronto, ON  M5S 3G3
tel: (416)978-3646(Work) (416)925-6758(Home)