arXiv subjects

Jose Seade

Publications and source records attributed to Jose Seade.

14 recordsLinked to original sources

arXiv2026

Generic vector fields on isolated complex hypersurface germs

We study holomorphic vector fields on isolated hypersurface singularities and derive global obstructions to the existence of holomorphic vector fields on compact singular varieties. For a hypersurface germ $(V,0)$ with an isolated singularity, we characterize the generic elements in the space of holomorphic vector fields with isolated singularity in terms of the GSV-index. Letting $\tau(V,0)$ denote the Tjurina-Greuel number, we prove that the minimal possible index is bounded below by $1+(-1)^{\dim(V)}\tau(V,0)$. We further prove that equality holds if the vector field admits an extension to $\mathbb{C}^{n+1}$ with a nondegenerate singularity at $\underline{0}$ and, in the case $n$ is odd, that such extensions, when they exist, form an open dense subset of the set of vector fields with an isolated singularity at $\underline{0}$. This yields a description of the generic vector fields on weighted homogeneous hypersurface germs. As a consequence, we obtain a characterization of weighted homogeneous hypersurface germs. Also, as applications to singular hypersurfaces in complex manifolds, we derive constraints on compact singular varieties admitting holomorphic vector fields. In particular, we show that an irreducible compact singular complex curve carrying a nontrivial holomorphic vector field with zeros is rational and has at most two singular points. We further prove that, for singular surfaces in K\"{a}hler 3-folds satisfying suitable positivity assumptions on the adjoint line bundle, the geometric genus is greater of equal than the irregularity.

math.AG↗

arXiv2021

Comparison of Limit Sets for the Action of Kleinian Groups in $\mathbb{C}P^n$

We compare different notions of limit sets for the action of Kleinian groups on the $n-$dimensional projective space via the irreducible representation $\varrho:PSL(2,\mathbb{C})\to PSL(n+1,\mathbb{C}).$ In particular, we prove that if the Kleinian group is convex-cocompact, the Myrberg and the Kulkarni limit coincide.

math.DS↗

arXiv2019

On the Chern classes of singular complete intersections

We consider two classical extensions for singular varieties of the usual Chern classes of complex manifolds, namely the total Schwartz-MacPherson and Fulton-Johnson classes, $c^{SM}(X)$ and $c^{FJ}(X)$. Their difference (up to sign) is the total Milnor class ${\mathcal M}(X)$, a generalization of the Milnor number for varieties with arbitrary singular set. We get first Verdier-Riemann-Roch type formulae for the total classes $c^{SM}(X)$ and $c^{FJ}(X)$, and use these to prove a surprisingly simple formula for the total Milnor class when $X$ is defined by a finite number of local complete intersection $X_1,\cdot \ldots \cdot,X_r$ in a complex manifold, satisfying certain transversality conditions. As applications we obtain a Parusi\'{n}ski-Pragacz type formula and an Aluffi type formula for the Milnor class, and a description of the Milnor classes of $X$ in terms of the global L\^e classes of the $X_i$.

math.AG↗

arXiv2018

On Milnor's fibration theorem and its offspring after 50 years

Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is two-fold. On the one hand, it should serve as an introduction to the topic for non-experts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading.

math.AG↗

arXiv2018

Discrete parabolic groups in ${\rm PSL}(3, \Bbb{C})$

We study and classify the purely parabolic discrete subgroups of $PSL(3,\Bbb{C})$. This includes all discrete subgroups of the Heisenberg group ${\rm Heis}(3,\Bbb{C})$. While for $PSL(2,\Bbb{C})$ every purely parabolic subgroup is Abelian and acts on $\Bbb{P}^1_\Bbb{C}$ with limit set a single point, the case of $PSL(3,\Bbb{C})$ is far more subtle and intriguing. We show that there are five families of purely parabolic discrete groups in $PSL(3,\Bbb{C})$, and some of these actually split into subfamilies. We classify all these by means of their limit set and the control group. We use first the Lie-Kolchin Theorem and Borel's fixed point theorem to show that all purely parabolic discrete groups in $PSL(3,\Bbb{C})$ are virtually triangularizable. Then we prove that purely parabolic groups in $PSL(3,\Bbb{C})$ are virtually solvable and polycyclic, hence finitely presented. We then prove a slight generalization of the Lie-Kolchin Theorem for these groups: they are either virtually unipotent or else Abelian of rank 2 and of a very special type. All the virtually unipotent ones turn out to be conjugate to subgroups of the Heisenberg group ${\rm Heis}(3,\Bbb{C})$. We classify these using the obstructor dimension introduced by Bestvina, Kapovich and Kleiner. We find that their Kulkarni limit set is either a projective line, a cone of lines with base a circle or else the whole $\Bbb{P}^2_\Bbb{C}$. We determine the relation with the Conze-Guivarc'h limit set of the action on the dual projective space $\check{\Bbb{P}}^2_\Bbb{C}$ and we show that in all cases the Kulkarni region of discontinuity is the largest open set where the group acts properly discontinuously.

math.DS↗

arXiv2011

Codimension one foliations with Bott-Morse singularities II

We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among other things, a good topological understanding of the leaves one may have around a center-type component of the singular set, and also of the topology of its basin. The stability theorem further allows the description of the topology of the boundary of the basin and how the topology of the leaves changes when passing from inside to outside the basin. This is described via fiberwise Milnor-Wallace surgery. A key-point for this is to show that if the boundary of the basin of a center is non-empty, then it contains a saddle; in this case we say that the center and the saddle are paired. We then describe the possible pairings one may have in dimension three and use a construction motivated by the classical saddle-node bifurcation, that we call foliated surgery, that allows the reduction of certain pairings of singularities of a foliation. This is used together with our previous work on the topic to prove an extension for 3-manifolds of Reeb's sphere recognition theorem.

math.GT↗

arXiv2009

A finiteness theorem for dual graphs of surface singularities

Consider a fixed connected, finite graph $Γ$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically Gorenstein surface singularity having $Γ$ as the dual graph of the minimal resolution, the weights $p_i$ of the vertices being the arithmetic genera of the corresponding irreducible components. As a consequence we get that if $Γ$ is not a cycle, then there is a finite number of possibilities of self-intersection numbers which one can attach to the vertices which are of valency $\geq 2$, such that one gets the dual graph of the minimal resolution of a numerically Gorenstein surface singularity. Moreover, we describe precisely the situations when there exists an infinite number of possibilities for the self-intersections of the component corresponding to some fixed vertex of $Γ$.

math.CV↗

arXiv2009

Refinements of Milnor's Fibration Theorem for Complex Singularities

Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_ε$ be a ball in $U$ of sufficiently small radius $ε>0$, centred at $\underline{0}\in\mathbb{C}^n$. We show that $f$ has an associated canonical pencil of real analytic hypersurfaces $X_θ$, with axis $V$, which leads to a fibration $Φ$ of the whole space $(X \cap \mathbb{B}_ε) \setminus V$ over $\mathbb{S}^1 $. Its restriction to $(X \cap \mathbb{S}_ε) \setminus V$ is the usual Milnor fibration $ϕ= \frac{f}{|f|}$, while its restriction to the Milnor tube $f^{-1}(\partial \D_η) \cap \mathbb{B}_ε$ is the Milnor-Lê fibration of $f$. Each element of the pencil $X_θ$ meets transversally the boundary sphere $\mathbb{S}_ε= \partial \B_ε$, and the intersection is the union of the link of $f$ and two homeomorphic fibers of $ϕ$ over antipodal points in the circle. Furthermore, the space ${\tilde X}$ obtained by the real blow up of the ideal $(Re(f), Im(f))$ is a fibre bundle over $\mathbb{R} \mathbb{P}^1$ with the $X_θ$ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.

math.AG↗

arXiv2008

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.

math.AG↗

arXiv2006

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 $μ(X,0) \equiv (-1)^{n-1}$ 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.

math.AG↗

arXiv2006

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 $\sing(\fa)$.

math.DG↗

arXiv2006

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 Poincaré-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.

math.AG↗

arXiv2004

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.

math.CV↗

arXiv2003

Global Euler obstruction and polar invariants

For an affine complex algebraic singular space Y, we define a global Euler obstruction Eu(Y) which extends the Euler-Poincare characteristic of a nonsingular Y. Using Lefschetz pencils, we express Eu(Y) as alternating sum of global polar invariants.

math.AG↗