Hypergeometric periods for a tame polynomial

We analyse the Gauss-Manin system of differential equations---and its Fourier transform---attached to regular functions satisfying a tameness assupmption on a smooth affine variety over C (e.g. tame polynomials on C^{n+1}). We give a solution to the Birkhoff problem and prove Hodge-type results analogous to those existing for germs of isolated hypersurface singularities.Comment: AMS-LaTeX with amsart.sty. Uses XY-pic package. 43 page

Fourier-Laplace transform of irreducible regular differential systems on the Riemann sphere

We show that the Fourier-Laplace transform of an irreducible regular differential system on the Riemann sphere underlies, when one only considers the part at finite distance, a polarizable regular twistor $\mathcal{D}$-module. The associated holomorphic bundle out of the origin is therefore equipped with a natural harmonic metric with a tame behaviour near the origin.Comment: 13+11 pages, AMSlatex. Original version followed by an Erratum added may 2007 after publicatio

Wild twistor D-modules

We propose a definition of (polarized) wild twistor D-modules, generalizing to objects with irregular singularities that of (polarized) regular twistor D-modules. We give a precise analysis in dimension one.Comment: 49 pages, revised versio

Differential systems of pure Gaussian type

We give the transformation rule for the Stokes data of the Laplace transform of a differential system of pure Gaussian type.Comment: 31 pages. V2: final version to appear in Izv. Mat

On a twisted de Rham complex

We show that, given a projective regular function f on a smooth quasi-projective variety over C, the corresponding cohomology groups of the algebraic de Rham complex with twisted differential d-df and of the complex of algebraic forms with differential df have the same dimension (a result announced by Barannikov and Kontsevitch). We generalize the result to de Rham complexes with coefficients in a mixed Hodge Module.Comment: AMS-LaTeX with amsart.sty. 15 page

Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at infinity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor product with a certain sheaf of formal microlocal differential operators instead of the formal completion.Comment: 23 pages, this improves and generalizes some results in arXiv:1012.3818; v2, abstract is modified, and Cor. 1 and 3.4-5 are added; v3, Remark 3.6 adde

Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II)

We give an explicit description of the canonical Frobenius structure attached (by the results of the first part of this article) to the polynomial f(u_0,...,u_n)=w_0u_0+...+w_nu_n restricted to the torus u_0^{w_0}...u_n^{w_n}=1, for any family of positive integers w_0,...,w_n such that gcd(w_0,...,w_n)=1.Comment: 22 pages, 3 figures, LaTeX + smf classes available at http://smf.emath.fr/Publications/Formats/index.html Typos correcte

Semicontinuity of the spectrum at infinity

We prove that, for an analytic family of ``weakly tame'' regular functions on an affine manifold, the spectrum at infinity of each function of the family is semicontinuous in the sense of Varchenko.Comment: AMS-LaTeX with amsart.sty. 10 page

The local Laplace transform of an elementary irregular meromorphic connection

We give a definition of the topological local Laplace transformation for a Stokes-filtered local system on the complex affine line and we compute in a topological way the Stokes data of the Laplace transform of a differential system of elementary type.Comment: 56 pages, 21 figures. V2: Final version to appear in Rend. Sem. Mat. Univ. Padov