arXiv subjects

Aurélio Menegon

Publications and source records attributed to Aurélio Menegon.

6 recordsLinked to original sources

arXiv2026

Normalized Milnor Fibrations for Real Analytic Maps

Milnor's fibration theorem and its generalizations play a central role in the study of singularities of complex and real analytic maps. In the complex analytic case, the Milnor fibration on the sphere is always given by the normalized map $f/|f|$. In contrast, for real analytic maps the existence of such a normalized Milnor fibration generally fails, even when a Milnor--Le fibration exists on a tube. For locally surjective real analytic maps $f:(R^n,0)->(R^k,0)$ with isolated critical value, the existence of a Milnor--Le fibration on a tube is guaranteed under a transversality condition. However, the associated fibration on the sphere need not be given by the normalized map $f/||f||$, unless an additional regularity condition (d-regularity) is imposed. In this paper we show that this apparent obstruction is not intrinsic. More precisely, we prove that for any such map satisfying the transversality property, there exists a homeomorphism $h:(R^k,0)->(R^k,0)$ of the target space such that the composition $h^{-1}f$ becomes d-regular. As a consequence, the normalized map $(h^{-1}f)/||h^{-1}f||$ defines a smooth locally trivial fibration on the sphere, which is equivalent to both the Milnor--Le fibration on the tube and the Milnor fibration on the sphere. Our result reveals a closer topological parallel between real and complex analytic singularities than previously recognized, without changing the topological type of the singularity.

math.CV↗

arXiv2025

Perplex Analysis and Geometry of Singularities

We develop a real-analytic framework, called perplex analysis, in which the complex, split-complex, and dual numbers arise as members of a single four-parameter family of two-dimensional commutative real algebras. Within this unified setting we define differentiability through a generalized Cauchy-Riemann structure, extending several features of complex geometry to a broader real-analytic context. Two main results illustrate the analytic and geometric scope of the theory: a Lojasiewicz gradient inequality for perplex-analytic functions, providing quantitative control of critical behavior; and a Milnor-Le type fibration theorem for nondegenerate algebras, describing the local topology of singularities. The framework reveals a continuous transition between complex and hyperbolic geometries, with the dual boundary exhibiting new infinitesimal phenomena linked to zero divisors. These results connect generalized complex geometry, hypercomplex analysis, and singularity theory within a single analytic formalism.

math.CV↗

arXiv2025

Embedded topological triviality of separable families of singularities

Understanding how singularities behave under small perturbations is a central theme in singularity theory. In this paper we establish sufficient conditions for families of analytic function-germs on a germ of a complex analytic space to admit an embedded topological trivialization. Our results extend previous work of the third author and collaborators, moving from abstract triviality to the embedded setting. As an application, we obtain new instances of topological stability, including a broad class of $μ$-constant deformations. These findings provide a new insight into the long-standing $μ$-constant conjecture, one of the major open problems in the field.

math.AG↗

arXiv2022

On deformations of isolated singularity functions

We study multi-parameters deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic spaces. We prove that if such a deformation has no coalescing of singular points, then it has constant topological type. This extends some classical results due to Lê \& Ramanujam (1976) and Parusiński (1999), as well as a recent result due to Jesus-Almeida and the first author. It also provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in $\C^3$ to have constant topological type. On the other hand, for complex isolated singularity families defined on an isolated determinantal singularity, we prove that $μ$-constancy implies constant topological type.

math.CV↗

arXiv2021

Equivalence of Milnor and Milnor-Lê fibrations for real analytic maps

In [22] Milnor proved that a real analytic map $f\colon (R^n,0) \to (R^p,0)$, where $n \geq p$, with an isolated critical point at the origin has a fibration on the tube $f|\colon B_ε^n \cap f^{-1}(S_δ^{p-1}) \to S_δ^{p-1}$. Constructing a vector field such that, (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he "inflates" the tube to the sphere, to get a fibration $φ\colon S_ε^{n-1} \setminus f^{-1}(0) \to S^{p-1}$, but the projection is not necessarily given by $f/ \|f\|$ as in the complex case. In the case $f$ has isolated critical value, in [9] it was proved that if the fibres inside a small tube are transverse to the sphere $S_ε$, then it has a fibration on the tube. Also in [9], the concept of $d$-regularity was defined, it turns out that $f$ is $d$-regular if and only if the map $f/|f\|\colon S_ε^{n-1} \setminus f^{-1}(0) \to S^{p-1}$ is a fibre bundle equivalent to the one on the tube. In this article, we prove the corresponding facts in a more general setting: if a locally surjective map $f$ has a linear discriminant $Δ$ and a fibration on the tube $f|\colon B_ε^n \cap f^{-1}(S_δ^{p-1} \setminus Δ) \to S_δ^{p-1} \setminus Δ$, then $f$ is $d$-regular if and only if the map $f/ \|f\|\colon S_ε^{n-1} \setminus f^{-1}(Δ) \to S^{p-1} \setminus \mathcal{A}$ (with $\mathcal{A}$ the radial projection of $Δ$ on $S^{p-1}$) is a fibre bundle equivalent to the one on the tube. We do this by constructing a vector field $\tilde{w}$ which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that $f/ \|f\|$ is constant on the integral curves of $\tilde{w}$.

math.AG↗

arXiv2018

Milnor-Lê type fibrations for subanalytic maps

We prove a Milnor-Lê type fibration theorem for a subanalytic map $f: X \to Y$ between subanalytic sets $X \subset \mathbb{R}^m$ and $Y \subset \mathbb{R}^n$. Moreover, if $f$ extends to an analytic map $\mathbb{R}^m \to \mathbb{R}^n$, we define the singular set and the discriminant set of $f$, in a stratified sense. Then we give Milnor-Lê type fibration theorems for $f$ outside its discriminant, as well as over it. We give examples of each situation approached in this paper.

math.AG↗