Abstract. -Let f, g : U → A 1 be two regular functions from the smooth affine complex variety U to the affine line. The associated exponential Gauß-Manin systems on the affine line are defined to be the cohomology sheaves of the direct image of the exponential differential system O U e g with respect to f . We prove that its holomorphic solutions admit representations in terms of period integrals over topological chains with possibly closed support and with rapid decay condition.
Résumé (Représentations intégrales des solutions des systèmes de Gauß-Manin exponentiels)Soient f, g : U → A 1 deux fonctions régulières sur une variété affine lisse U à valeurs dans la droite affine. On leurs associe des systèmes de Gauß-Manin sur la droite affine définis comme étant les faisceaux de cohomology de l'image directe par f du système différentiel exponentiel O U e g . Nous prouvons que leurs solutions holomorphes admettent des représentations sous forme d'intégrales de périodes sur des chaînes topologiques à support éventuellement fermé avec une condition de décroissance rapide.Texte reçu le 5 novembre 2007 et le 7 mars
Abstract. -In D-modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf O by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex f + (Oe g
Isolated hypersurfacesingularities come equipped with a Milnor lattice, a Z-lattice of finite rank, and a set of distinguished Z-bases of this lattice. Usually these bases are constructed from one morsification and all possible choices of distinguished systems of paths. But what does one obtain if one considers all possible morsifications and one fixed distinguished system of paths? Looijenga asked this question 1974 for the simple singularities. He and Deligne found that one obtains a bijection between Stokes regions in a universal unfolding and the set of distinguished bases modulo signs. This allows to see the base space of the universal unfolding as an atlas of Stokes data. Here we reprove their result and extend it to the simple elliptic singularities. We use more conceptual arguments, moduli spaces of marked singularities (i.e. Teichmüller spaces for singularities), extensions of them to F-manifolds, and the actions of symmetries of singularities on the Milnor lattices and these moduli spaces. We use and extend results of Jaworski on the Lyashko-Looijenga maps for the simple elliptic singularities. The sections 2 and 3 give a survey on singularities and the associated objects which allows to read the paper independently of other sources.
Let f and g be two regular functions on U smooth affine variety. Let M be a regular holonomic D U -module. We are interested in the irregularity of the complex f + (Me g ). More precisely, we relate the irregularity number at c of the systems) with the characteristic cycles of the systems• Let U be a smooth affine variety over C and g : U → C be a regular function on U . We denote by O U the sheaf of regular functions on U and by D U the sheaf of algebraic differential operators on U .Let In this paper, we consider two regular functions f, g : U → C. We are interested in the irregularity of the cohomology modules of the direct image by f of a D U -module, Me g , where M is regular and holonomic.
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