Publications

2003

• Groupe fondamental premier à p, nombre de Milnor des singularités isolées, motifs de dimension inférieure ou égale à 1
  
  • Orgogozo Fabrice
  , 2003. Dans le premier chapitre, on démontre divers résultats sur le plus grand quotient du groupe fondamental étale premier aux caractéristiques, parmi lesquels la formule de Künneth et l'invariance par changement de corps séparablement clos pour les schémas de type fini sur un corps. Ces énoncés sont déduits de faits généraux sur les images directes de champs, une fois spécialisés au cas des torseurs sous un groupe constant fini d'ordre inversible sur la base. Des résultats analogues pour le groupe fondamental modéré sont également discutés. Au deuxième chapitre, on déduit de la formule du conducteur, conjecturée par S. Bloch, celle de P. Deligne exprimant, dans le cas d'une singularité isolée, la dimension totale des cycles évanescents en fonction du nombre de Milnor. En particulier, la formule de Deligne est établie en dimension relative un. Dans le troisième chapitre, on compare les 1-isomotifs de P. Deligne sur un corps avec la théorie de V. Voevodsky en dimension inférieure à 1.
• Singularités lagrangiennes
  
  • Sevenheck Christian
  , 2003. This thesis develops a deformation theory for lagrangian singularities. We define four each lagrangian singularity a complex of modules with a non-linear differential whose first cohomology is isomorphic to the space of infinitesimal deformations of the singularity. The second cohmology contains informations on the obstruction theory. This complex is related to the theory of differential modules. We show that under a geometric condition, its cohomology forms constructible sheaves. We describe a method using computer algebra to determine this cohomology for quasi-homogeneous surfaces.
• Altérations et groupe fondamental premier à p
  
  • Orgogozo Fabrice
  Bulletin de la société mathématique de France, Société Mathématique de France, 2003, 131 (1), pp.123--147. Nous démontrons divers résultats sur le plus grand quotient du groupe fondamental étale premier aux caractéristiques, parmi lesquels la formule de Künneth et l'invariance par changement de corps séparablement clos pour les schémas de type fini sur un corps. Ces énoncés sont déduits de faits généraux sur les images directes de champs, une fois spécialisés au cas des torseurs sous un groupe constant fini d'ordre inversible sur la base. Des résultats analogues pour le groupe fondamental modéré sont également discutés.
• Gauss-Manin systems, Brieskorn lattices and Frobenius structures (I)
  
  • Douai Antoine
  • Sabbah Claude
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2003, 53 (4), pp.1055-1116. We associate to any convenient nondegenerate Laurent polynomial on the complex torus (C^*)^n a canonical Frobenius-Saito structure on the base space of its universal unfolding. According to the method of K. Saito (primitive forms) and of M. Saito (good basis of the Gauss-Manin system), the main problem, which is solved in this article, is the analysis of the Gauss-Manin system of the Laurent polynomial (or its universal unfolding) and of the corresponding Hodge theory.
• Monodromy invariants in symplectic topology
  
  • Auroux Denis
  , 2003. This text is a set of lecture notes for a series of four talks given at I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick overview of symplectic topology and its main tools: symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants and Floer homology. The second and third lectures focus on symplectic Lefschetz pencils: existence (following Donaldson), monodromy, and applications to symplectic topology, in particular the connection to Gromov-Witten invariants of symplectic 4-manifolds (following Smith) and to Fukaya categories (following Seidel). In the last lecture, we offer an alternative description of symplectic 4-manifolds by viewing them as branched covers of the complex projective plane; the corresponding monodromy invariants and their potential applications are discussed.
• Twistor Forms on Kähler Manifolds
  
  • Moroianu Andrei
  • Semmelmann Uwe
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2003, 2, pp.823-845. Twistor forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We study twistor forms on compact Kähler manifolds and give a complete description up to special forms in the middle dimension. In particular, we show that they are closely related to Hamiltonian 2-forms. This provides the first examples of compact Kähler manifolds with non-parallel twistor forms in any even degree.
• Elementary linear algebra for advanced spectral problems
  
  • Sjöstrand Johannes
  • Zworski Maciej
  , 2003. We discuss the general method of Grushin problems, closely related to Shur complements, Feshbach projections and effective Hamiltonians, and describe various appearances in spectral theory, pdes, mathematical physics and numerical problems.
• integration on smooth rigid varieties and invariants of degenerations
  
  • Sebag Julien
  • Loeser F.
  Duke Mathematical Journal, Duke University Press, 2003, 119 (2), pp.315-344.
• Perturbations of selfadjoint operators with periodic classical flow
  
  • Sjöstrand Johannes
  , 2003. We consider non-selfadjoint perturbations of a self-adjoint $h$-pseudodifferential operator in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength $\epsilon $ of the perturbation satisfies $h^{\delta_0} <\epsilon \le \epsilon_0$ for some $\delta_0\in ]0,1/2[$ and a sufficiently small $\epsilon _0>0$. We get a complete asymptotic description of all eigenvalues in certain rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$. In particular we are able to treat the case when $\epsilon >0$ is small but independent of $h$.
• Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique
  
  • Chambert-Loir Antoine
  , 2003. In this talk, I report on three theorems concerning algebraic varieties over a field of characteristic $p>0$. a) over a finite field of cardinal $q$, two proper smooth varieties which are geometrically birational have the same number of rational points modulo $q$ (cf. Ekedahl, 1983). b) over a finite field of cardinal $q$, a proper smooth variety which is rationally chain connected, or Fano, or weakly unirational, has a number of rational points congruent to 1 modulo $q$ (Esnault, 2003). c) over an algebraic closed field of caracteristic $p>0$, the fundamental group of a proper smooth variety which is rationally chain connected, or Fano, or weakly unirational, is a finite group of order prime to $p$ (cf. Ekedahl, 1983). The common feature of the proofs is a control of the $p$-adic valuations of Frobenius and is best explained within the framework of Berthelot's rigid cohomology. I also explain its relevant properties.
• Symmetries of Contact Metric Manifolds
  
  • Belgun Florin
  • Moroianu Andrei
  • Semmelmann Uwe
  Geometriae Dedicata, Springer Verlag, 2003, 101, pp.203-216. We study the Lie algebra of infinitesimal isometries on compact Sasakian and K-contact manifolds. On a Sasakian manifold which is not a space form or 3-Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K-contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo-Riemannian quaternion-Kähler manifold after changing the sign of the metric on a maximal negative distribution. We also prove that nonregular Sasakian manifolds are not homogeneous and construct examples with cohomogeneity one. Using these results we obtain in the last section the classification of all homogeneous Sasakian manifolds. (10.1023/A:1026375212252)
  DOI : 10.1023/A:1026375212252
• À propos du groupe fondamental des variétés rationnellement connexes
  
  • Chambert-Loir Antoine
  , 2003. (On the fundamental group of rationnally connected varieties.) I show that the fundamental group of a normal variety which is rationally chain connected is finite. The proof holds in non-zero characteristic. Je démontre que le groupe fondamental d'une variété normale rationnellement connexe par cha\^{\i}nes est fini. La démonstration est valable en caractéristique différente de zéro.
• Conjecture de Bloch et nombres de Milnor
  
  • Orgogozo Fabrice
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2003, 53 (no. 6), pp.1739--1754. Nous déduisons de la formule du conducteur, conjecturée par S. Bloch, celle de P. Deligne exprimant, dans le cas d'une singularité isolée, la dimension totale des cycles évanescents en fonction du nombre de Milnor. En particulier, la formule de Deligne est établie en dimension relative un; en appendice, on généralise cet énoncé au cas d'un lieu singulier propre.
• Weakly expanding skew-products of quadratic maps
  
  • Buzzi Jerome
  • Sester Olivier
  • Tsujii Masato
  Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2003, 23 (5), pp.1401-1414. We consider quadratic skew-products over angle-doubling of the circle and prove that they admit positive Lyapunov exponents almost everywhere and an absolutely continuous invariant probability measure. This extends corresponding results of M. Viana and J. F. Alves for skew-products over the linear strongly expanding map of the circle. (10.1017/S0143385702001694)
  DOI : 10.1017/S0143385702001694
• On the role of quadratic oscillations in nonlinear Schrödinger equations
  
  • Carles Rémi
  • Fermanian Kammerer Clotilde
  • Gallagher Isabelle
  Journal of Functional Analysis, Elsevier, 2003, 203, pp.453-493. We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential. (10.1016/S0022-1236(03)00212-X)
  DOI : 10.1016/S0022-1236(03)00212-X
• Resonances associated to a closed hyperbolic trajectory in dimension 2
  
  • Sjöstrand Johannes
  Asymptotic Analysis, IOS Press, 2003, 36, pp.93-113. We consider resonances in the semi-classical limit, generated by a single closed hyperbolic orbit, for an operator on ${\bf R}^2$. We determine all such resonancess in a domain independent of the semi-classical parameter As an application we determine all resonances generated by a saddle point in a fixed disc around the critical energy.
• A nonexistence result of single peaked solutions to a supercritical nonlinear problem
  
  • Ben Ayed Mohamed
  • El Mehdi Khalil
  • Grossi Massimo
  • Rey Olivier
  Communications in Contemporary Mathematics, World Scientific Publishing, 2003, 5 (2), pp.179-195. This paper is concerned with the nonlinear elliptic problem (Pε): -Δu = up+ε, u > 0 in Ω; u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 3, p + 1 = 2n/(n - 2) is the critical Sobolev exponent and ε is a small positive parameter. In contrast with the subcritical problem (P- ε) studied by Han [11] and Rey [17], we show that (Pε) has no single peaked solution for small ε. (10.1142/S0219199703000951)
  DOI : 10.1142/S0219199703000951
• Boundary blow-up for a Brezis-Peletier problem on a singular domain
  
  • Pistoia Angela
  • Rey Olivier
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2003, 18 (3), pp.243-251.
• Hamiltonian Gromov-Witten invariants
  
  • Riera Ignasi Mundet I
  Topology, Elsevier, 2003, 42, pp.525-553. In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical equations. These equations generalize at the same time the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants. This paper is based on a part of my PhD Thesis (see math/9912150).