Iberoamerican Webminar of Young Researchers in
Singularity Theory and related topics

It is classically known that there are germs of codimension one foliations in ambient dimension three that do not have an invariant surface, the property they share is that they are dicritical foliations. Thus, the question of how much transcendental can be the leaves of the foliation arises in these cases. Brunella’s Alternative in a local version conjectures that each leaf must contain at least an analytic germ of curve passing through the origin. In this talk we introduce some of the ingredients that allow us to deal with this problem and we also present the family of foliations for which we want to give an answer to this question.

Given any smooth surface in $\mathbb{R}^3$ (or a complex surface in $\mathbb{C}^3$) we associate, to any point on $M$, any integer $k>1$ and any plane, a holomorphic map-germ of the form $F^k(x,y)=(x,y^k, f(x,y))$. This kind of map-germs are called $k$-folding map-germs. In these two talks we describe the interplay between the topology of $k$-folding map-germs and the extrinsic differential geometry of $M$.

On the second session, we introduce the topological classification of $k$-folding map germs exhibited by generic surfaces, and show how their occurrence relates to old and new robust features of $M$.

This is a joint work with Guillermo Peñafort Sanchís (U. Valencia).

In this talk, we will present a link criterion for normal embedding of definable sets in o-minimal structures. Namely, we prove that given a definable germ $(X, 0)\subset (\mathbb{R}^n,0)$ with $(X\setminus\{0\},0)$ connected and a continuous definable function $\rho: (X,0) \to \mathbb{R}_{\geq 0}$ such that $\rho(x) \sim \|x\|$, then $(X,0)$ is Lipschitz normally embedded (LNE) if and only if $(X,0)$ is link Lipschitz normally embedded (LLNE) with respect to $\rho$. This is a generalization of Mendes--Sampaio's result for the subanalytic case.

In the works [1] and [2] we develope a new metric algebraic topology, called the Moderately Discontinuous Homology and Homotopy in the context of subanalytic germs in R^n (with a supplementary metric structure) that satisfies the analogues of the usual theorems in Algebraic Topology: long exact sequences, relatv case, Mayer Vietoris, Seifert van Kampen for special coverings...

This theory captures bilipschitz information, or in other words, quasi isometric invariants. The typical examples to which it applies are subalaytic germs with the inner metric (the length metric induced by the euclidean metric) and with the outer metric (the restriction of the euclidean metric).

A subanalytic germ is topologically a cone over its link and the moderately discontinuous theory captures the different speeds, with respect to the distance to the origin, in which the topology of the lik collapses towards the origin.

In this talk, I will present the most important concepts in the theory and some results or applications that we got until the present.

References:
[1] (with J. Fernández de Bobadilla, S. Heinze, E. Sampaio) Moderately discontinuous homology. To appear in Comm. Pure App. Math.. Available in arXiv: 1910.12552 ou sur https://arxiv.org/pdf/1910.12552.pdf
[2] (with J. Fernández de Bobadilla, S. Heinze) Moderately discontinuous homotopy. Submitted. Available in ArXiv:2007.01538 or in https://arxiv.org/pdf/2007.01538.pdf

This talk concerns Gabrielov’s rank Theorem, a fundamental result in local complex and real-analytic geometry, proved in the 1970’s. Contrasting with the algebraic case, it is not in general true that the analytic rank of an analytic map (that is, the dimension of the analytic-Zariski closure of its image) is equal to the generic rank of the map (that is, the generic dimension of its image). This phenomenon is involved in several pathological examples in local real-analytic geometry. Gabrielov’s rank Theorem provides a formal condition for the equality to hold.

Despite its importance, the original proof is considered very difficult. There is no alternative proof in the literature, besides a work from Tougeron, which is itself considered very difficult. I will present a new work in collaboration with Andr ́e Belotto da Silva and Guillaume Rond, where we provide a complete proof of Gabrielov’s rank Theorem, for which we develop formal-geometric techniques, inspired by ideas from Gabrielov and Tougeron, which clarify the proof.

I will start with some fundamental examples of the phenomenon at hand, and expose the main ingredients of the strategy of this difficult proof.

Given a totally nonholonomic distribution of rank two $\Delta$ on a three-dimensional manifold $M$, it is natural to investigate the size of the set of points $\mathcal{X}^x$ that can be reached by singular horizontal paths starting from a same point $x \in M$. In this setting, the Sard conjecture states that $\mathcal{X}^x$ should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.

I will present a reformulation of the conjecture in terms of the behavior of a singular foliation. By exploring this geometrical framework, in a recent work in collaboration with A. Figalli, L. Rifford and A. Parusinski, we show that the strong version of the conjecture holds for three dimensional analytic varieties, that is, the set $\mathcal{X}^x$ is a countable union of semi-analytic curves. Next, by studying the regularity of the solutions of the set $\mathcal{X}^x$, we show that sub-Riemannian geodesics are all $C^1$. Our methods rely on resolution of singularities of surfaces, vector-fields and metrics; regularity analysis of Poincaré transition maps; and on a symplectic argument, concerning a transversal metric of an isotropic singular foliation.

The Hodge locus was introduced by Grothendieck in 1966 to study the Hodge conjecture in families (aka Variational Hodge conjecture). Even for surfaces, where the Hodge conjecture is known to hold, its components are far from being well-understood. In this special case the Hodge locus coincides with the classical Noether-Lefschetz locus, that was studied at the end of the 80's and beginning of the 90's by Green, Voisin, Harris among others. Their main tool was the use of infinitesimal variations of Hodge structures (IVHS).

In this talk I will survey recent results about the computation of periods of algebraic cycles in hypersurfaces, their relation with IVHS and some applications to the study of the Hodge locus and Variational Hodge conjecture.

Hodge ideals were recently introduced by Popa and Mustata to study, mainly with birational techniques, the Hodge filtration on the sheaf of meromorphic functions on a variety along a divisor. They measure the difference between its Hodge and pole order filtrations, the latter always containing the former. They are also a generalization of multiplier ideals, since the zeroth Hodge ideal is the adjoint ideal of the divisor.

Up to now, most results dealt with isolated singularities; in this talk I will comment on a joint work with Ch. Sevenheck and L. Narváez Macarro, where we study and compute Hodge ideals for certain free divisors, by means of Hodge modules and D-module theory.

Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi\colon(\mathbb{C}^n,0)\to(\mathbb{C},0)$ and $f\colon(\mathbb{C}^n,0)\to\mathbb{C}$ such that the Bruce-Roberts number $\mu_{BR}(f,X)$ is finite. In this work we prove that $$\mu_{BR}(f,X)=\mu(f)+\mu(X\cap f^{-1}(0),0)+\mu(X,0)-\tau(X,0),$$ where $\mu$ and $\tau$ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. We also prove that the logarithmic characteristic variety $LC(X,0)$ is Cohen-Macaulay, both results generalize [1].

This is a joint work with J. J. Nuno-Ballesteros (Universitat de Valencia, SPAIN), B. Orefice-Okamoto (UFSCar, BRAZIL) and J.N. Tomazella, (UFSCar, BRAZIL).

References:
[1] J. J. Nu ̃no Ballesteros, B. Or ́efice-Okamoto, J. N. Tomazella, The Bruce-Roberts number of a function on a weighted homogeneous hypersurface, Q. J. Math.64(2013), no. 1, 269-280

We investigate properties of the contact exponent (in the sense of Hironaka [3]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic.
We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [1].

This is a joint work with Arkadiusz Płoski (see [2]).

References:
[1] García Barroso, E. and A. Płoski. An approach to plane algebroid branches. Rev. Mat. Complut., 28 (1) (2015), 227-252.
[2] García Barroso, E. and A. Płoski. Contact exponent and Milnor number of plane curve singularities. Analytic and Algebraic Geometry, 3. T. Krasinski, Stanisław Spodzieja (Eds.) Lodz University Press (2019), 93-109. http://dx.doi.org/10.18778/8142-814-9.08
[3] Hironaka, H. Introduction to the theory of infinitely near singular points, Memorias del Instituto Jorge Juan 28, Madrid 1974.

We will talk about Lê’s vanishing polyhedra for analytic maps and we will use them to prove a join theorem for mixed functions. This provides a tool for understanding the topology of some families of real analytic singularities, which is still somehow an unexplored area in Singularity Theory.

This is a joint work with José Luis Cisneros-Molina.

In this talk I will present an accessible view on applications of the Alexander Module of the complement of a plane curve complement on its topology. In particular, I will describe different strategies to use these modules to solve two problems: the Zariski pair problem and the quasi-projectivity Serre's problem. The first one arises when trying to determine whether or not two plane curves that have the same degree, same irreducible components, and same topological type of singularities, might have non-homeomorphic complements. The second one aims to deciding whether or not a given finitely presented group is the fundamental group of a quasi-projective variety (such as the complement of a plane curve).

To illustrate the Zariski-pair type of problems Rybnikov's famous example will be discussed, where two line arrangements with the same combinatorics are shown to have non-homeomorphic complements. As an example of Serre's problem, a characterization of the quasi-projective even Artin groups will be presented.

We consider a real bivariate polynomial function vanishing at the origin and exhibiting a strict local minimum at this point. We work in a neighbourhood of the origin in which the non-zero level curves of this function are smooth Jordan curves. Whenever the origin is a Morse critical point, the sufficiently small levels become boundaries of convex disks. Otherwise, these level curves may fail to be convex.

The aim of this talk is two-fold. Firstly, to study a combinatorial object measuring this non-convexity; it is a planar rooted tree. And secondly, we want to characterise all possible topological types of these objects. To this end, we construct a family of polynomial functions with non-Morse strict local minima realising a large class of such trees.

This is a joint work with Jorge Martín-Morales, Wim Veys & Juan Viu-Sos.

The study of zeta functions of hypersurfaces, allows one to determine some invariants of the singularity defining the hypersurface. A common strategy is to use a classical embedded resolution of the singularity, which gives a list of possible 'poles' from which some invariants can be read of. The list is usually very large and a major and difficult problem (closely connected with the Monodromy Conjecture) is determining the true poles. In this work we propose to use a partial resolution of singularities to deal with this problem. We use an embedded Q-resolution, where the final ambient space may contain quotient singularities. This machinery allows us to give some explicit formulas for motivic and topological zeta functions in terms of Q-resolutions, generalizing in particular some results of Veys for curves and providing in general a reduced list of candidate poles.