# D-modules and Singularities

in honor of Michel Granger
Angers, May 2-3, 2016

## Abstracts

Daniel Barlet : A note on some fiber-integrals

We remark that the study of a fiber-integral of the type

either in the local case where ρ ≡ 1 around 0 is $\C^\infty$ and compactly supported near the origin which is a singular point of {f = 0} in $\C^n+1$ , or in a global setting where f : X → D is a proper holomorphic function on a complex manifold X, smooth outside {f = 0} with ρ ≡ 1 near {f = 0}, for given holomorphic (n+1)−forms ω and ω' , that a better control on the asymptotic expansion of F when $s \mapsto 0$, is obtained by using the Bernstein polynomial of the “frescos” associated to f and ω and to f and ω' (a fresco is a “small” Brieskorn module corresponding to the differential equation deduced from the Gauss-Manin system of f at 0) than to use the Bernstein polynomial of the full Gauss-Manin system of f at the origin. We illustrate this in the local case in some rather simple (non quasi-homogeneous) polynomials, where the Bernstein polynomial of such a fresco is explicitly evaluate.

André Galligo : Trente Cinq ans après : Déformations Equisingulières des Germes de Courbes Gauches réduites

Monique Lejeune-Jalabert : La solution algébrique au problème de Nash pour les surfaces normales de deFernex-Docampo.

Je présenterai les grandes lignes de leur preuve algébrique de la surjectivité de l'application de Nash pour les surfaces en caractéristique zéro, à partir de l'étude du comportement de certains diviseurs canoniques.

Philippe Maisonobe : Filtration Relative, l’Idéal de Bernstein et ses pentes

Soit f_i : X → C, pour i entier compris entre 1 et p, des fonctions analytiques définies au voisinage d’un compact K d’une variété analytique complexe X. Notons F le produit des f_i et posons si \phi : X → C désigne une fonction C^1 à support compact dans K :

$$I_{\phi} (s_1, \ldots,s_p) = \int_X \mid f_1 (x) \mid^{s_1}\ldots \mid f_p (x) \mid^{s_p} \phi (x) \; dx \wedge d\overline{x} \; .$$

Tout comme dans le cas p = 1, en utilisant le théorème de résolution des singularités d’H. Hironaka, on peut montrer que I(s_1;...; s_p) qui est une fonction définie à priori pour Re s_i > 0 se prolonge en fonction méromorphe avec des pôles situés sur des hyperplans de C^p. F. Loeser étudie ces intégrales dans [?] et appelle pente de (f_1;...; f_p) les directions de leurs hyperplans polaires. Dans certains cas géométriques, il majore cet ensemble de pentes par un ensemble de formes linéaires liées à la géométrie du discriminant du morphisme (f_1;...; f_p) : X → C^p.

$$b (s_1, \ldots ,s_p) m f_1^{s_1} \ldots f_p^{s_p} \in {\cal D}_X[s_1, \ldots ,s_p]\, m f_1^{s_1+1} \ldots f_p^{s_p+1} \; .$$

Considérons D_X l’anneau des opérateurs différentiels et D_X[s_1;...; s_p] = C_X[s_1;...; s_p]\otimes_ C D_X. Soit m une section d’un D_X-Module holonome, notons B(m; x_0; f_1;...; f_p) l’idéal de C[s_1;...; s_p] des polynômes b vérifiant au voisinage de x_0 :

$$\prod_{H\in {\cal H} } \prod_{i\in I_{\cal H} } (H(s_1, \ldots, s_p) + \alpha _{H,i}) \in {\cal B}(m,x_0, f_1, \ldots ,f_p) \; ,$$

Ces polynômes sont appelés polynômes de Bernstein de (m; f_1;...; f_p) au voisinage de x_0. Suivant J. Bernstein, ils permettent de construire un prolongement des intégrales I(s_1;...; s_p). C. Sabbah montre l’existence pour tout x_0\in X d’un ensemble fini H de formes linéaires à coefficients premiers entre eux dans N telles que : où \alpha_{H,i} sont des nombres complexes. L’objet de cet exposé est notamment de montrer l’existence d’un ensemble H minimal. De plus, lorsque m est une section d’un module holome régulier, nous expliciterons cet ensemble géométriquement à partir de la variété caractéristique du système différentiel engendré par m.

Zoghman Mebkhout : Sur le Théorème de la Monodromie p-adique en dimensions supérieures.

Nous présentons dans cet exposé le Théorème de la Monodromie p-adique en un point générique d’une hypersurface d’une variété algébrique lisse sur un corps de caractéristique p>0. Pour cela nous définissons le fonceur $\Psi_t$ à l’aide de la théorie des Modules Spéciaux.

David Mond : Homology groups of the multiple point spaces in the disentangement of a map-germ

This is joint work with Isaac Bird. We calculate the ranks of the homology groups of the multiple point spaces of a stable perturbation of a corank 2 map from 3-space to 4-space. Unlike the case for corank 1 maps, these spaces are not Milnor fibres of complete intersections, and we do not have equations for them. We use a variety of techniques, principally the image-computing spectral sequence and a theorem of Theo de Jong on the virtual number of D_∞ points. This is part of a wider project to understand the identifications giving rise to the vanishing homology of images and discriminants.

Luis Narváez-Macarro : On the right D-module structure on the top differential forms through Hasse-Schmidt derivations

In characteristic zero and under smoothness conditions, it is very easy to describe the D-module structure on the module of top differential forms: it is enough to start with the action of derivations by means of the opposite of the Lie derivative, and check that is is compatible with Lie brackets and Leibniz rule, and it is linear over the ring of functions. This is due to the fact that, in that case, the ring D of differential operators is the enveloping algebra of the Lie-Rinehart algebra of derivations. If we are working in nonzero characteristic, things are much more subtle. If we want to proceed in a coordinate-free way, one needs to use some finiteness hypotheses, the relative duality formalism and co-stratifications. However, by using coordinates. it is easy to exhibit the right D-module module even in cases without any finiteness condition. In this talk I will explain how the notion of Hasse-Schmidt derivation can help us in understanding, by pure algebraic methods. the above right D-module structure intrinsically.

Adam Parusinski : Arc-wise equisingularity and Whitney's fibering conjecture.

Varchenko showed that if a complex or real analytic set is Zariski equisingular along an affine subspace T then it is locally topologically trivial along T. Using Whitney interpolation we construct explicitly a trivialization of a Zariski equisingular set that is moreover analytic on real analytic arcs and (complex resp. real) analytic with respect to the parameter space T. Then, given an algebraic set or a germ of an analytic set, we construct its stratification that fibers this set, locally along each stratum, into analytic submanifolds with the strong continuity of tangent spaces, analogous to Verdier's condition (w). This shows Whitney's fibering conjecture. Our construction is algorithmic, involves only linear changes of coordinates and computation of subsequent discriminants. This is a joint work with Laurentiu Paunescu.

Mathias Schulze : Duality on value semigroups

The semigroup of values is a classical combinatorial invariant associated to a curve singularity. It is defined by taking all regular elements of the corresponding ring to the integral closure and taking a multivaluation. For a curve singularity with $r$ branches the semigroup of values is a submonoid of $\NN^r$. In the irreducible case it is a numerical semigroup, otherwise it is not even finitely generated. Due to Lejeune-Jalabert and Zariski, this value semigroup determines the topological type of plane complex curves. As observed by Kunz in the irreducible case, the Gorenstein property of a curve singularity is equivalent to a symmetry of gaps and non-gaps in the (numerical) value semigroup. Delgado generalized this result to the reducible case introducing a non-obvious notion of symmetry of a semigroup. A canonical (fractional) ideal on a curve singularity defines a duality on fractional ideals. On the other hand taking multivaluations as above associates to any fractional ideal a value semigroup ideal. Such value semigroup ideals satisfy certain natural axioms defining the class of so-called good semigroup ideals. Barucci, D'Anna and Fröberg gave an example of a good semigroup that does not come from a ring. Extending Delgado's symmetry result, D'Anna described the value semigroup ideals of canonical ideals. In the Gorenstein case, Delphine Pol described the value semigroup ideal of duals. Unifying the work of D'Anna and Pol we establish a purely combinatorial duality on good semigroup ideals that mirrors the duality on fractional ideals. The talk is based on joint work with Philipp Korell and Laura Tozzo.