(Lecture notes), preprint .
Motivic integration was introduced by Kontsevich to show that birationally equivalent Calabi-Yau manifolds have the same Hodge numbers. To do so, he constructed a certain motivic measure on the arc space of a complex variety, taking values in a completion of the Grothendieck ring of algebraic varieties. Later, Denef and Loeser, together with the works of Looijenga and Batyrev, developed in a series of articles a more complete theory of the subject, with applications in the study of varieties and singularities. In particular, they developed a motivic zeta function, generalizing the usual ($p$-adic) Igusa zeta function and Denef-Loeser topological zeta function.
These notes are a basic introduction to geometric motivic integration, the precedent $p$-adic ideas associated with it, and the theory of the above zeta functions related to them. We focus in practical computations and ideas, providing examples and a recent formula obtained by means of partial resolutions.
(Submitted), arXiv:1912.01751 .
Effective periods are defined by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of $\mathbb{Q}$-rational functions over $\mathbb{Q}$-semi-algebraic domains in $\mathbb{R}^d$. The Kontsevich-Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stoke's formula.
In this paper, we discuss about possible geometric interpretations of this conjecture, viewed as a generalization of the Hilbert's third problem for compact semi-algebraic sets as well as for rational polyhedron equipped with piece-wise algebraic forms. Based on partial known results for analogous Hilbert's third problems, we study obstructions of possible geometric schemas to prove this conjecture.
Advances in Mathematics 370 (2020), journal . arXiv:1911.03354 .
We study motivic zeta functions for $\mathbb{Q}$-divisors in a $\mathbb{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the action of a nonabelian group, from which we compute some invariants of motivic nature after constructing a $\mathbb{Q}$-resolution.
Experimental Mathematics 29 (2020), no. 1, 28–35, journal . arXiv:1704.04152 .
We prove that the fundamental group of the complement of a real complexified line arrangement is not determined by its intersection lattice, providing a counter-example for a problem of Falk and Randell. We also deduce that the torsion of the lower central series quotients is not combinatorially determined, which gives a negative answer to a question of Suciu.
Mathematische Annalen 374 (2019), no. 1-2, 1–35, journal . arXiv:1702.00922 , including two appendices with detailled pictures of Zariski pairs.
A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This invariant is based on the weight counting over the points of the arrangement dual configuration, located in particular chambers of the real projective plane $\mathbb{RP}^2$, dealing only with geometrical properties.
Using this dual point of view, we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in $\mathbb{CP}^2$ (i.e. Zariski pairs), which are distinguished by this invariant. In particular, we obtain new Zariski pairs of 13, 15 and 17 lines defined over $\mathbb{Q}$ and containing only double and triple points. For each one of them, we can derive degenerations, containing points of multiplicity 2, 3 and 5, which are also Zariski pairs.
We explicitly compute the moduli space of the combinatorics of one of these examples, and prove that it has exactly two connected components. We also obtain three geometric characterizations of these components: the existence of two smooth conics, one tangent to six lines and the other containing six triple points, as well as the collinearity of three specific triple points.
Accepted for publication in International Journal of Number Theory. arXiv:1509.01097 , including an appendix with pseudo-codes of the main procedures.
The ${\overline{\mathbb Q}}$-algebra of periods was introduced by Kontsevich and Zagier as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of ${\mathbb Q}$-rational functions over ${\mathbb Q}$-semi-algebraic domains in ${\mathbb R}^d$. The Kontsevich-Zagier period conjecture affirms that any two different integral expressions of a given period are related by a finite sequence of transformations only using three rules respecting the rationality of the functions and domains: additions of integrals by integrands or domains, change of variables and Stoke's formula.
In this paper, we prove that every non-zero real period can be represented as the volume of a compact ${\overline{\mathbb Q}}\cap{\mathbb R}$-semi-algebraic set, obtained from any integral representation by an effective algorithm satisfying the rules allowed by the Kontsevich-Zagier period conjecture.
(Submitted), arXiv:1412.0137 .
Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$–derivations. First, we give a dynamical interpretation of $\mathcal{D}(\mathcal{A})$ as the set of polynomial vector fields which posses $\mathcal{A}$ as invariant set. We characterize polynomial vector fields having an infinite number of invariant lines. Then we prove that the minimal degree of polynomial vector fields fixing only a finite set of lines in $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.
Proc. XIII Intern. Conf. on Maths. and its Appl., (40), 61-66 (2016), journal .