Associated to analytic Hamiltonian vector fields in ${ℂ}^4$ having an equilibrium point satisfying a non semisimple $1:-1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

Notre étude porte sur une catégorie de structures de Poisson singulières holomorphes au voisinage de $0\in {ℂ}^n$ et admettant une forme normale formelle polynomiale i.e. un nombre fini d’invariants formels. Les séries normalisantes sont divergentes en général. On montre l’existence de transformations normalisantes holomorphes sur des domaines sectoriels de la forme $a\<\arg x^R\<b$, où $x^R$ est un monôme associé au problème. Il suit une classification analytique.

En choisissant des “caractères” et des “logarithmes”, méromorphes sur $ℂ$, construits à l’aide de la fonction Gamma d’Euler, et en utilisant des séries de factorielles convergentes, nous sommes en mesure, dans une première partie, de donner une “forme normale” pour les solutions d’un système aux différences singulier régulier. Nous pouvons alors définir une matrice de connexion d’un tel système. Nous étudions ensuite, suivant une idée de G.D. Birkhoff, le lien de celles-ci avec le problème de la classification...

Etant donné un système différentiel linéaire de niveau unique quelconque, nous explicitons des formules donnant les multiplicateurs de Stokes en fonction de constantes de connexion dans le plan de Borel, généralisant ainsi les formules obtenues dans l’article Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems (M. Loday-Richaud, P. Remy). Pour ce faire, nous nous ramenons à un système de niveaux $\le 1$ par la méthode classique de réduction du rang ; puis, nous...

This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry.

This paper is divided in two parts. In the first part we study a convergent $q$-analog of the divergent Euler series, with $q\in (0,1)$, and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$-difference equation. In the second part, we work under the assumption $q\in (1,+\infty )$. In this case, at least four different $q$-Borel sums of a divergent power series solution of an irregular singular...

The Euler-MacLaurin summation formula compares the sum of a function over the lattice points of an interval with its corresponding integral, plus a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series,...

This article continues earlier work of the author on non-linear systems of ordinary differential equations, published in Asymptotic Analysis 15 (1997), MR no. 98g:34015b. There, a completely formal theory was presented, while here we are concerned with a semi-formal approach: Solutions of non-linear systems of ordinary meromorphic differential equations are represented as, in general divergent, power series in several free parameters. The coefficients, aside from an exponential polynomial, a general...

We study germs of singular holomorphic vector fields at the origin of ${ℂ}^n$ of which the linear part is $1$-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....

In this work we compute the Stokes matrices of the ordinary differential equation satisfied by the hypergeometric integrals associated to an arrangement of hyperplanes in generic position. This generalizes the computation done by J.-P. Ramis for confluent hypergeometric functions, which correspond to the arrangement of two points on the line. The proof is based on an explicit description of a base of canonical solutions as integrals on the cones of the arrangement, and combinatorial relations between...

This article is concerned with moduli spaces of connections on bundles on Riemann surfaces, where the structure group of the bundle may vary in different regions of the surface. Here we will describe such moduli spaces as complex symplectic manifolds, generalising the complex character varieties of Riemann surfaces.