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José Seade

Publications and source records attributed to José Seade.

16 recordsLinked to original sources
arXiv2026

On the topology of complex map-germs and a general Lê-Greuel formula

Consider a singular holomorphic map-germ $f: (X,\underline{0}) \to (\mathbb C,0)$ where $X$ is a singular complex analytic variety in $\mathbb C^N$, and another holomorphic map-germ $g: (X,\underline{0}) \to (\mathbb C,0)$ which is "sufficiently good" relatively to $f$. We use stratified Morse theory to determine up to homeomorphism, the topology of the Milnor fiber $F_f$ out from the slice $F_{g,f}$ and the Morse data of a Morsification of the restriction of $g$ to $F_f$. This generalizes classical results for the case where $X$ is non-singular, and it provides a general formula comparing the Euler characteristics of $F_f$ and $F_{g,f}$. Restricting to the case where the singularity of $X$ at $\underline{0}$ is isolated, the formula for the difference of the Euler characteristics becomes algebraic and easily computable, generalizing in two directions the classical Lê-Greuel formula for the Milnor number of isolated complete intersection germs (ICIS): Firstly, $X$ can have an isolated singularity, and secondly $f$ can have arbitrary critical set. This unifies several known formulae in this vein: i) Lê-Greuel for ICIS of arbitrary codimension; ii) the formula relating the Milnor number of a curve with that of a function on it, and an extension of it for surfaces; iii) the formula for determinantal singularities; iv) and the one for the image Milnor number. All of these are special cases of our general formula.

math.AG↗
arXiv2026

Transverse Analytic Envelopes and Semiglobal Dynamics

We associate with every singular orbit of a diagonal holomorphic action by a complex vector group a canonical transverse analytic envelope, defined by the holomorphic first integrals on a local transversal. This envelope is a normal affine toric germ. It provides a holomorphic separation of the transverse leaves even when their local leaf space fails the $T_1$ separation axiom. We determine exactly which part of the residual linear action is recovered by this envelope. Residual actions with a fixed labelled envelope form Grassmannian families, and the envelope determines the residual action precisely when the integral resonance relations span the full complex relation space. For one-dimensional residual actions, this criterion is reflected in Baum--Bott residues and, in transverse dimension two, in the Camacho--Sad indices. Along positive-dimensional singular orbits, the envelopes carry canonical transport and a flat connection, which may be viewed as a singular counterpart of transverse holonomy in the regular setting. The connection is logarithmic when the support weights are independent, and its residues and holonomy encode semiglobal information not contained in the pointwise envelope. At full resonance rank, the labelled envelope together with the logarithmic residues reconstructs the ambient weight configuration up to linear equivalence. Thus, the transverse analytic envelope and its canonical flat connection provide a framework for measuring exactly the information lost in passing from the transverse dynamics to holomorphic first integrals, and for determining when this information suffices to reconstruct the original linear action.

math.CV↗
arXiv2026

Asymptotic flag geometry of complex Kleinian groups in $\mathbb{P}^2_\mathbb{C}$

We organize several natural notions of limit set for discrete subgroups of $\mathrm{PSL}(3,\mathbb{C})$ around a common asymptotic structure encoded by pseudo--projective degeneration and by the full or partial flag data carried by divergent sequences. In the $θ$-divergent case, the two projections of the full-flag limit set recover the attracting and repelling projective boundary sets, while the Myrberg limit set is the union of the limiting projective lines. Thus the equicontinuity region is the complement of a canonical line configuration; under the usual three-line general-position hypothesis, this same configuration is the Kulkarni limit set and the equicontinuity and Kulkarni ordinary regions coincide.

math.DS↗
arXiv2026

Holomorphic Linear $\C^k$-Actions, Trace Foliations, and Higher-Rank Poincaré Dynamics

We study the orbit decomposition on $\C^n$ generated by diagonal holomorphic $\C^k$-actions in the higher-rank setting of the classical Poincaré--Siegel dichotomy for linear vector fields. The coordinate stratification determines the dimensions and isotropy groups of the leaves and, under a maximal-rank condition, gives a precise description of the orbit structure on every coordinate stratum. For configurations in the Poincaré domain, a separating real direction provides a global conical model of the punctured orbit foliation by its traces on Euclidean spheres. We construct the corresponding radially reparametrized action on a sphere, describe its leaves as homogeneous spaces, and prove that orbit closures are constrained by coordinate supports. In particular, a limit point cannot acquire a new nonzero coordinate, although nonclosed trace leaves may also accumulate within a fixed support stratum. We show that the geometry of the weight configuration determines the complex dimensions of the leaves, whereas the arithmetic of their effective isotropy groups determines their diffeomorphism types. This gives rise to a threshold phenomenon across the coordinate stratification: the diffeomorphism type of the leaves is rigid in the low- and high-dimensional regimes, but becomes arithmetically unstable in the intermediate range $k<|I|<2k$. Finally, we show that these singular foliations admit canonical local transverse holomorphic structures in the spirit of Haefliger's transverse geometry for regular foliations. These structures determine intrinsic transverse pseudogroups, yielding a well-defined local transverse holomorphic geometry for the orbit foliation.

math.DS↗
arXiv2026

Non-degenerate mixed maps and contact structures

We study the geometry and topology of real analytic maps $\mathbb{C}^n \to \mathbb{C}^k$, where $n > k$, regarded as mixed maps, defined below. Firstly, we give two natural families of mixed isolated complete intersection singularities, called mixed ICIS, which are interesting on their own. We consider the notion of (partial) non-degeneracy for mixed maps; we prove that these define mixed ICIS and that, under suitable conditions, admit a local Milnor fibration. Then, building on previous constructions due to Oka, we obtain natural contact structures and adapted open books on a particular class of mixed links. Finally, we look at mixed links that are diffeomorphic to holomorphic ones, and we address the problem of comparing different contact structures.

math.AG↗
arXiv2026

On the connectedness of the singular set of holomorphic foliations

Let $\mathcal{F}$ be a singular holomorphic foliation of dimension $k>1$ on a projective $n$-manifold $X$. Assume that the determinant of the normal sheaf of $\mathcal{F}$ is ample (as is always the case when $X=\mathbb{P}^{n}$), and that the singular set $Sing(\mathcal{F})$ has dimension $\leq k-1$. We show that the union of those irreducible components of $Sing(\mathcal{F})$ of dimension exactly $k-1$ is necessarily connected. Consequently, we obtain a Bott-type topological obstruction to the integrability of singular holomorphic distributions, echoing Bott's vanishing theorem, and we answer a question of Cerveau for codimension-one foliations on $\mathbb{P}^{3}$.

math.AG↗
arXiv2023

Elementary groups in $\PSL(3,\C)$

In this paper, we give a classification of the subgroups of $\textrm{PSL}(3, \mathbb{C})$ that act on $\mathbb{P}_{\mathbb{C}}^2$ in such a way that their Kulkarni limit set has finitely many lines in general position lines. These are the elementary groups.

math.GR↗
arXiv2019

A note on 3-manifolds and complex surface singularities

This article is motivated by the original Casson invariant regarded as an integral lifting of the Rochlin invariant. We aim to defining an integral lifting of the Adams e-invariant of stably framed 3-manifolds, perhaps endowed with some additional structure. We succeed in doing so for manifolds which are links of normal complex Gorenstein smoothable singularities. These manifolds are naturally equipped with a canonical $\SU$-frame. To start we notice that the set of homotopy classes of $\SU$-frames on the stable tangent bundle of every closed oriented 3-manifold is canonically a $\mathbb Z$-torsor. Then we define the $\widehat E$-invariant for the manifolds in question, an integer that modulo 24 is the Adams e-invariant. The $\widehat E$-invariant for the canonical frame equals the Milnor number plus 1, so this brings a new viewpoint on the Milnor number of the smoothable Gorenstein surface singularities.

math.AG↗
arXiv2016

On the Lê-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 $π_i^* \circ f$, where $π_i^*: \mathbb{R}^n \to \mathbb{R}^{n-1}$ 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 Lê-Milnor fibration in the tube.

math.DG↗
arXiv2012

On Discrete Subgroups of automorphism of $P^2_C$

We study the geometry and dynamics of discrete subgroups $Γ$ of $\PSL(3,\mathbb{C})$ with an open invariant set $Ω\subset \PC^2$ where the action is properly discontinuous and the quotient $Ω/Γ$ contains a connected component whicis compact. We call such groups {\it quasi-cocompact}. In this case $Ω/Γ$ is a compact complex projective orbifold and $Ω$ 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 $Ω/Γ$. 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.

math.DS↗
arXiv2012

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 Lê'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.

math.CV↗
arXiv2011

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.

math.CV↗
arXiv2011

Milnor Fibrations and the Thom Property for maps $f \bar g$

We prove that every map-germ ${f \bar g}: (\C^n,\0) {\to}(\C,0)$ with an isolated critical value at 0 has the Thom $a_{f \bar g}$-property. This extends Hironaka's theorem for holomorphic mappings to the case of map-germs $f \bar g$ and it implies that every such map-germ has a Milnor-Lê fibration defined on a Milnor tube. One thus has a locally trivial fibration $ϕ: \mathbb S_\e \setminus K \to \mathbb S^1$ for every sufficiently small sphere around $\0$, where $K$ is the link of $f \bar g$ and in a neighbourhood of $K$ the projection map $ϕ$ is given by $f \bar g / | f \bar g|$.

math.AG↗
arXiv2008

On the Equicontinuity Region of Discrete Subgroups of PU(1,n)

Let $ G $ be a discrete subgroup of PU(1,n). Then $ G $ acts on $\mathbb {P}^n_\mathbb C$ preserving the unit ball $\mathbb {H}^n_\mathbb {C}$, where it acts by isometries with respect to the Bergman metric. In this work we determine the equicontinuty region $Eq(G)$ of $G$ in $\mathbb P^n_{\mathbb C}$: It is the complement of the union of all complex projective hyperplanes in $\mathbb {P}^n_{\mathbb C}$ which are tangent to $\partial \mathbb {H}^n_\mathbb {C}$ at points in the Chen-Greenberg limit set $Λ_{CG}(G )$, a closed $G$-invariant subset of $\partial \mathbb {H}^n_\mathbb {C}$, which is minimal for non-elementary groups. We also prove that the action on $Eq(G)$ is discontinuous.

math.DS↗
arXiv2005

Fibered Multilinks and singularities $f \bar g$

In this article we extend Milnor's fibration theorem for complex singularities to the case of singularities $f \bar g:(X,P) \to (C,0))$ defined on a complex analytic singularity germ $(X,P)$, with $f, g$ holomorphic and $f \bar g$ having an isolated critical value at $0 \in C$. This can also be regarded as a result for meromorphic germs. Then we strenghten this fibration theorem when $X$ has complex dimension 2, obtaining a fibration theorem for multilinks that extends previous work by Pichon. We prove that the multilink $L_{f \bar g}$ in $L_X$ (the link of $X$), is fibred iff the map $f \bar g$ has an isolated critical value at $0 \in C$, and in this case the map $\frac{f \bar g}{|f \bar g|}$ defined on $L_X \setminus L_{f \bar g}$ is a multilink fibration.We also give a combinatorial criterium, easy to verify, to decide when is $L_{f \bar g}$ a fibred multilink. We finally prove a realization theorem for fibred multilinks.

math.AG↗