Tautological systems and free divisors

We introduce tautological system defined by prehomogenous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor satisfying a certain finiteness condition, we show that these systems underly mixed Hodge modules. A dimensional reduction is considered and gives rise to one-dimensional differential systems generalizing the quantum differential equation of projective spaces

La Geometría Algebraica: punto de encuentro de las Matemáticas

Libro completo en: http://www.rasc.es/assets/rasdc---memorias-vol.-6-(1998-2001).pd

On the modules of m-integrable derivations in non-zero characteristic

Let k be a commutative ring and A a commutative k-algebra. Given a positive integer m, or m = ∞, we say that a k-linear derivation δ of A is m-integrable if it extends up to a Hasse–Schmidt derivation D = (Id, D1 = δ, D2, . . . , Dm) of A over k of length m. This condition is automatically satisfied for any m under one of the following orthogonal hypotheses: (1) k contains the rational numbers and A is arbitrary, since we can take Di = δ i i! ; (2) k is arbitrary and A is a smooth k-algebra. The set of m-integrable derivations of A over k is an A-module which will be denoted by Iderk(A; m). In this paper we prove that, if A is a finitely presented k-algebra and m is a positive integer, then a k-linear derivation δ of A is m-integrable if and only if the induced derivation δp : Ap → Ap is m-integrable for each prime ideal p ⊂ A. In particular, for any locally finitely presented morphism of schemes f : X → S and any positive integer m, the S-derivations of X which are locally mintegrable form a quasi-coherent submodule Ider S(OX; m) ⊂ Der S(OX) such that, for any affine open sets U = Spec A ⊂ X and V = Spec k ⊂ S, with f(U) ⊂ V , we have Γ(U,Ider S(OX; m)) = Iderk(A; m) and Ider S(OX; m)p = IderOS,f(p) (OX,p; m) for each p ∈ X. We also give, for each positive integer m, an algorithm to decide whether all derivations are m-integrable or not.Ministerio de Educación y CienciaFondo Europeo de Desarrollo Regiona

The local duality theorem in D-module theory

These notes are devoted to the Local Duality Theorem for D-modules, which asserts that the topological Grothendieck-Verdier duality exchanges the de Rham complex and the solution complex of holonomic modules over a complex analytic manifold. We give Mebkhout’s original proof and the relationship with Kashiwara-Kawai’s proof. In that way we are able to precise the commutativity of some diagrams appearing in the last one.Ce cours est consacré au théorème de du alité locale pour les D-modules, qui affirme que la dualité topologique de Grothendieck-Verdier échange le complexe de de Rham et le complexe des solutions des modules holonomes sur une variété analytique complexe. On donne la preuve originale de Mebkhout en faisant le rapport avec la preuve de Kashiwara-Kawai. Ceci nous permet de préciser la commutativité de certains diagrammes dans cette dernière.Dirección General de Enseñanza SuperiorFondo Europeo de Desarrollo Regiona

Higher derivations of modules and the Hasse-Schmidt module

In this paper we revisit Ribenboim's notion of higher derivations of modules and relate it to the recent work of De Fernex and Docampo on the sheaf of differentials of the arc space. In particular, we derive their formula for the K\"ahler differentials of the Hasse-Schmidt algebra as a consequence of the fact that the Hasse-Schmidt algebra functors commute.Comment: 13 page

En recuerdo de Alexander Grothendieck: prólogo para una lectura de su vida y obra

En este artículo repasamos algunas de las claves de las contribuciones matemáticas de Alexander Grothendieck y del contexto en el que se gestaron, y nos asomamos así a la vida y obra de una de las figuras más influyentes de las Matemáticas contemporáneas

Continuous division of linear differential operators and faithful flatness of D∞X over DX

In these notes we prove the faithful flatness of the sheaf of infinite order linear differential operators over the sheaf of finite order linear differential operators on a complex analytic manifold. We give the Mebkhout-Narv´aez’s proof based on the continuity of the division of finite order differential operators with respect to a natural topology. We reproduce the proof of the continuity theorem given by Hauser-Narváez, which is simpler than the original proof.Dans ce cours on démontre la fidèle platitude du faisceau d’opérateurs différentiels linéaires d’ordre infini sur le faisceau d’opérateurs différentiels linéaires d’ordre fini d’une variété analytique complexe lisse. La preuve que nous donnons est celle de Mebkhout-Narváez, qui utilise la continuité de la division d’opérateurs différentiels d’ordre fini par rapport à une topologie naturelle. Nous réproduisons la preuve de Hauser-Narváez du théorème de continuité, qui est plus simple que la preuve originale.Ministerio de Ciencia y TecnologíaFondo Europeo de Desarrollo Regiona

Primer Encuentro Conjunto RSME-UMA, Buenos Aires, 11 a 15 de diciembre de 2017

En la semana del lunes 11 al viernes 15 de diciembre de 2017 se llevó a cabo en Buenos Aires el Primer Encuentro Conjunto entre la Unión Matemática Argentina (UMA) y la RSME. Se celebraba el centenario de la primera visita del Profesor Don Julio Rey Pastor a la Argentina, el punto de partida del estrecho vínculo que desde entonces ha existido entre las comunidades matemáticas de ambos países. El encuentro, que convocó a unos 900 participantes, se organizó alrededor de tres ejes principales: Actividades Científicas, Educación, y Divulgación. Las dos primeras se desarrollaron en el predio de la Facultad de Ciencias Exactas y Naturales de la Universidad de Buenos Aires, mientras que las actividades de divulgación, dirigidas a un pú- blico general, se llevaron a cabo en el amplio Centro Cultural de la Ciencia. Pueden consultarse los detalles, incluida la composición de los Comités Organizador y Científico, en la web del congreso: http://uma2017.dm.uba.ar

On the reduced Bernstein-Sato polynomial of Thom-Sebastiani singularities

Given two holomorphic functions $f$ and $g$ defined in two respective germs of complex analytic manifolds $(X,x)$ and $(Y,y)$, we know thanks to M. Saito that, as long as one of them is Euler homogeneous, the reduced (or microlocal) Bernstein-Sato polynomial of the Thom-Sebastiani sum $f+g$ can be expressed in terms of those of $f$ and $g$. In this note we give a purely algebraic proof of a similar relation between the whole functional equations that can be applied to any setting (not necessarily analytic) in which Bernstein-Sato polynomials can be defined.Comment: 8 pages, final versio