Let $X$ be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of $𝒟$-modules on $X$ induces a fully faithful functor on a subcategory of germs of formal holonomic $𝒟$-modules. Further, given a germ $ℳ$ of holonomic $𝒟$-module, we obtain some results linking the subanalytic sheaf of tempered solutions of $ℳ$ and the classical formal and analytic invariants of $ℳ$.

We confirm a conjecture of Bernstein–Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.

The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot. However in full generality it is proven only for zeta functions associated to polynomials in two variables.In this article we work with zeta functions associated to an ideal. First we work in arbitrary dimension and obtain a formula (like the one of A’Campo) to compute the “Verdier monodromy” eigenvalues associated to an...

Nous donnons une condition topologique assurant l’existence d’un pôle pour le prolongement méromorphe de la distribution $f^{\lambda }$ dans le cadre analytique réel.

L’objet de cet article est de démontrer un théorème «  à la Thom-Sebastiani  » pour les développements asymptotiques des intégrales-fibres des fonctions du type $f\oplus g:(x,y)\to f\left(x\right)+g\left(y\right)$ sur $({ℂ}^p\times {ℂ}^q,(0,0))$ en terme des développements asymptotiques des intégrales-fibres associées aux germes holomorphes $f:({ℂ}^p,0)\to (ℂ,0)$ et $g:({ℂ}^q,0)\to (ℂ,0)$. Ceci se ramène à calculer les développements asymptotiques d’une convolution $\Phi *\Psi$ à partir des développements asymptotiques de $\Phi$ et $\Psi$ modulo les termes non singuliers.Pour obtenir un résultat précis donnant la non nullité des termes...

We consider tracefree meromorphic rank 2 connections over compact Riemann surfaces of arbitrary genus. By deforming the curve, the position of the poles and the connection, we construct the global universal isomonodromic deformation of such a connection. Our construction, which is specific to the tracefree rank 2 case, does not need any Stokes analysis for irregular singularities. It is thereby more elementary than the construction in arbitrary rank due to B. Malgrange and I. Krichever and it includes...

$\mu$-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo $\pm id$. Second, marked singularities are defined and global moduli spaces for right equivalence...