Remarks on contact structures and vector fields on isolated complete intersection singularities

Let $(X,0)$ be an isolated complete intersection complex singularity ($X$ can also be smooth at 0). Let $K$ be its link, $\cal X$ its canonical contact structure and \D_X the complex vector bundle associated to $\cal X$. We prove that the bundle \D_X is trivial if and only if the Milnor number of $X$ satisfies $\mu(X,0) \equiv (-1)^{n-1}$ modulo $(n-1)!$. This follows from a general theorem stating that the complex orthogonal complement of a vector field in $X$ with an isolated singularity at 0 is trivial iff the GSV-index of $v$ is a multiple of $(n-1)!$. We have also an application to foliation theory: a holomorphic foliation $\cal F$ in a ball \B_r around the origin in \C^3, with an isolated singularity at 0, admits a $C^\infty$ 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 $M$ 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 $\ge 3$ 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 $f:M \to [0,1]$ 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 $\mathcal{F}$ be the germ at $\mathbf{0} \in \mathbb{C}^n$ of a holomorphic foliation of dimension $d$, $1 \leq d < n$, with an isolated singularity at $\mathbf{0}$. We study its geometry and topology using ideas that originate in the work of Thom concerning Morse theory for foliated manifolds. Given $\mathcal{F}$ and a real analytic function $g$ on $\mathbb{C}^n$ with a Morse critical point of index 0 at $\mathbf{0}$, we look at the corresponding polar variety $M= M(\mathcal{F},g)$. These are the points of contact of the two foliations, where $\mathcal{F}$ is tangent to the fibres of $g$. 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 $g$. We then study $\mathcal{F}$ by looking at the intersection of its leaves with the level sets of $g$, 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), $m\geq k$, with an isolated critical value and the Thom $a_f$-property. We define the vanishing zone for $f$ and we give necessary and sufficient conditions for it to be a fibre bundle over the link of the singular set of $f^{-1}(0)$. In the case of singularities of the type \fgbar: (\bC^n,0) \to (\bC,0) with an isoalted critical value, $f, g$ 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 $f$ or $g$ depends only on one variable

On Discrete Subgroups of automorphism of PC2P^2_C

We study the geometry and dynamics of discrete subgroups $\Gamma$ of \PSL(3,\mathbb{C}) with an open invariant set \Omega \subset \PC^2 where the action is properly discontinuous and the quotient $\Omega/\Gamma$ contains a connected component whicis compact. We call such groups {\it quasi-cocompact}. In this case $\Omega/\Gamma$ is a compact complex projective orbifold and $\Omega$ 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 $\Omega/\Gamma$. 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 $f: (\mathbb{R}^m,0) \to (\mathbb{R}^n,0)$, with $1 <n <m$. We compare the topology of $f$ with the topology of the compositions $\pi_i^* \circ f$, where $\pi_i^*: \mathbb{R}^n \to \mathbb{R}^{n-1}$ are the projections $(t_1, \dots, t_n) \mapsto (t_1, \dots, t_{i-1}, t_{i+1}, \dots, t_n)$, for $i=1, \dots, n$. As a main result, we give necessary and sufficient conditions for $f$ 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 $Eu_f$ of a holomorphic function $f$ 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 $a_f$-condition, thus extending to this setting the index introduced by Ebeling and Gusein-Zade. When the form is the differential of a holomorphic function $h$, this index measures the number of critical points of a generic perturbation of $h$ 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