406 research outputs found
Remarks on contact structures and vector fields on isolated complete intersection singularities
Let be an isolated complete intersection complex singularity ( can
also be smooth at 0). Let be its link, its canonical contact
structure and \D_X the complex vector bundle associated to . We prove
that the bundle \D_X is trivial if and only if the Milnor number of
satisfies modulo . This follows from a
general theorem stating that the complex orthogonal complement of a vector
field in with an isolated singularity at 0 is trivial iff the GSV-index of
is a multiple of . We have also an application to foliation theory:
a holomorphic foliation in a ball \B_r around the origin in \C^3,
with an isolated singularity at 0, admits a normal section (away
from 0) iff its multiplicity (or local index) is even, and this happens iff its
normal bundle in \B_r \setminus \{0\} is topologically trivial
Indices of Vector Fields on Singular Varieties: An Overview
Indices of vector fields on (complex analytic) singular varieties have been
considered by various authors from several different viewpoints. All these
indices coincide with the classical local index of Poincar\'e-Hopf when the
ambient variety is a smooth manifold. One has the Schwartz index, the local
Euler obstruction, the GSV-index, the virtual index and the homological index
(and probably other indices too). In this article we give a brief overview of
all these indices and the relations amongst them. We also indicate their
relations with the various generalisations to complex analytic singular
varieties of the Chern classes of complex manifolds
Codimension one foliations with Bott-Morse singularities I
We study codimension one (transversally oriented) foliations \fa on
oriented closed manifolds having non-empty compact singular set
\sing(\fa) which is locally defined by Bott-Morse functions. We prove that if
the transverse type of \fa at each singular point is a center and \fa has a
compact leaf with finite fundamental group or a component of \sing(\fa) has
codimension and finite fundamental group, then all leaves of \fa are
compact and diffeomorphic, \sing(\fa) consists of two connected components,
and there is a Bott-Morse function such that f\colon M
\setminus \sing(\fa) \to (0,1) is a fiber bundle defining \fa and
\sing(\fa) = f^{-1}(\{0,1\}). This yields to a topological description of the
type of leaves that appear in these foliations, and also the type of manifolds
admiting such foliations. These results unify, and generalize, well known
results for cohomogeneity one isometric actions and a theorem of Reeb for
foliations with Morse singularities of center type. In this case each leaf of
\fa is a sphere fiber bundle over each component of \sing(\fa).Comment: 3 figure
Morse Theory and the topology of holomorphic foliations near an isolated singularity
Let be the germ at of a
holomorphic foliation of dimension , , with an isolated
singularity at . We study its geometry and topology using ideas
that originate in the work of Thom concerning Morse theory for foliated
manifolds. Given and a real analytic function on
with a Morse critical point of index 0 at , we look
at the corresponding polar variety . These are the points
of contact of the two foliations, where is tangent to the fibres
of . This is analogous to the usual theory of polar varieties in algebraic
geometry, where holomorphic functions are studied by looking at the
intersection of their fibers with those of a linear form. Here we replace the
linear form by a real quadratic map, the Morse function .
We then study by looking at the intersection of its leaves with
the level sets of , and the way how these intersections change as the sphere
gets smaller.Comment: Accepted for publication by the Journal of Topolog
The degeneration of the boundary of the Milnor fibre to the link of complex and real non-isolated singularities
We study the boundary of the Milnor fibre of real analytic singularities f:
(\bR^m,0) \to (\bR^k,0), , with an isolated critical value and the
Thom -property. We define the vanishing zone for and we give necessary
and sufficient conditions for it to be a fibre bundle over the link of the
singular set of . In the case of singularities of the type \fgbar:
(\bC^n,0) \to (\bC,0) with an isoalted critical value, holomorphic, we
further describe the degeneration of the boundary of the Milnor fibre to the
link of \fgbar. As a milestone, we also construct a L\^e's polyhedron for
real analytic singularities of the type \fgbar: (\bC^2,0) \to (\bC,0) such
that either or depends only on one variable
On Discrete Subgroups of automorphism of
We study the geometry and dynamics of discrete subgroups of
\PSL(3,\mathbb{C}) with an open invariant set \Omega \subset \PC^2 where
the action is properly discontinuous and the quotient contains
a connected component whicis compact. We call such groups {\it
quasi-cocompact}. In this case is a compact complex projective
orbifold and is a {\it divisible set}. Our first theorem refines
classical work by Kobayashi-Ochiai and others about complex surfaces with a
projective structure: We prove that every such group is either virtually affine
or complex hyperbolic. We then classify the divisible sets that appear in this
way, the corresponding quasi-cocompact groups and the orbifolds
. We also prove that excluding a few exceptional cases, the
Kulkarni region of discontinuity coincides with the equicontinuity region and
is the largest open invariant set where the action is properly discontinuous
On the L\^e-Milnor fibration for real analytic maps
In this paper, we study the topology of real analytic map-germs with isolated
critical value , with . We
compare the topology of with the topology of the compositions , where are the
projections , for . As a main result, we give necessary and sufficient
conditions for to have a L\^e-Milnor fibration in the tube
Milnor fibrations of meromorphic functions
In analogy with the holomorphic case, we compare the topology of
Milnor fibrations associated to a meromorphic germ f/g : the local Milnor
fibrations given on Milnor tubes over punctured discs around the critical
values of f/g, and the Milnor fibration on a sphere.Comment: 20 pages, 5 figure
Milnor numbers and Euler obstruction
We determine the relation between the local Euler obstruction of a
holomorphic function and different generalizations of the Milnor number for
functions on singular spaces.Comment: 9 page
Proportionality of Indices of 1-Forms on Singular Varieties
This article is about 1-forms on complex analytic varieties and it is
particularly relevant when the variety has non-isolated singularities. We first
show how the radial extension technique of M.-H. Schwartz can be adapted to
1-forms, allowing us to define the Schwartz index of 1-forms with isolated
singularities on singular varieties. Then we see how MacPherson's local Euler
obstruction, adapted to 1-forms in general, relates to the Schwartz index, thus
obtaining a proportionality theorem for these indices analogous to the one for
vector fields. We also extend the definition of the GSV-index to 1-forms with
isolated singularities on (local) complete intersections with non-isolated
singularities that satisfy the Thom -condition, thus extending to this
setting the index introduced by Ebeling and Gusein-Zade. When the form is the
differential of a holomorphic function , this index measures the number of
critical points of a generic perturbation of on a local Milnor fiber. We
prove the corresponding proportionality theorem for this index.
These constructions can be made global on compact varieties and provide an
alternative way for studying the various characteristic classes of singular
varities and the relations amongst them.Comment: 14 page
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