Publications

2014

• Infinitesimal invariants for cycles modulo algebraic equivalence and 1-cycles on Jacobians
  
  • Voisin Claire
  Algebraic Geometry, Foundation Compositio Mathematica, 2014, 1 (2), pp.140-165. We construct an infinitesimal invariant for cycles in a family with cohomology class in the total space lying in a given level of the Leray filtration. This infinitesimal invariant detects cycles modulo algebraic equivalence in the fibers. We apply this construction to the Ikeda family, which gives optimal results for the Beauville decomposition of the 1-cycle of a very general plane curve in its Jacobian. (10.14231/AG-2014-008)
  DOI : 10.14231/AG-2014-008
• Blow up for the critical gKdV equation I: dynamics near the soliton
  
  • Martel Yvan
  • Merle Frank
  • Raphaël Pierre
  Acta Mathematica, Royal Swedish Academy of Sciences, Institut Mittag-Leffler, 2014, 212 (1), pp.59-140. (10.1007/s11511-014-0109-2)
  DOI : 10.1007/s11511-014-0109-2
• Hypersurfaces quartiques de dimension 3 : non rationalité stable
  
  • Colliot-Thélène Jean-Louis
  • Pirutka Alena
  , 2014.
• Note on the counterexamples for the integral Tate conjecture over finite fields
  
  • Pirutka Alena
  • Yagita Nobuaki
  , 2014. In this note we discuss some examples of non torsion and non algebraic cohomology classes for varieties over finite fields. The approach follows the construction of Atiyah-Hirzebruch and Totaro.
• Rigueur-contraintes : mathématiques-musique
  
  • Paul Thierry
  Gazette des Mathématiciens, Société Mathématique de France, 2014, 139, pp.71-77. Nous tentons de présenter quelques idées concernant la présence de la notion de rigueur et de contraintes dans le processus de composition musicale, en lien avec l'idée de rigueur en mathématiques. Ces deux propriétés nous semblent bien être présentes dans les deux disciplines, mais à deux endroits opposés de leurs activités créatrices.
• Chow rings, decomposition of the diagonal, and the topology of families
  
  • Voisin Claire
  , 2014, 187, pp.176.
• The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings
  
  • Chenevier Gaëtan
  , 2014, 1, pp.221-285. Let G be a profinite group which is topologically finitely generated, p a prime number and d an integer. We show that the functor from rigid analytic spaces over Q_p to sets, which associates to a rigid space Y the set of continuous d-dimensional pseudocharacters G -> O(Y), is representable by a quasi-Stein rigid analytic space X, and we study its general properties. Our main tool is a theory of "determinants" extending the one of pseudocharacters but which works over an arbitrary base ring; an independent aim of this paper is to expose the main facts of this theory. The moduli space X is constructed as the generic fiber of the moduli formal scheme of continuous formal determinants on G of dimension d. As an application to number theory, this provides a framework to study the generic fibers of pseudodeformation rings (e.g. of Galois representations), especially in the "residually reducible" case, and including when p <= d.
• La construction d'Abbes et Saito pour les connexions méromorphes: aspect formel en dimension 1
  
  • Teyssier Jean-Baptiste
  Publications of the Research Institute for Mathematical Sciences, European Mathematical Society, 2014, 50, pp.10.4171/PRIMS/140. By using a blow-up construction, the nearby-cycle functor and l-adic Fourier transform, Abbes and Saito are able to define a geometric measure of wild ramification of l-adic sheaves on the generic point of any complete discrete valuation ring of equal characteristic p with perfect residue field, where p is different from l. In this paper, we adapt their construction to differential modules over the field of formal Laurent series with coefficients in any characteristic zero field K. For such a module M, we prove a formula relating Abbes and Saito's construction to the differential forms occuring in the Levelt-Turrittin decomposition of M. If K is algebraically closed, one recovers a version of Laurent's micro-characteristic cycles.
• A refinement of Izumi's theorem
  
  • Boucksom Sébastien
  • Favre Charles
  • Jonsson Mattias
  , 2014.
• Kontsevich's conjecture on an algebraic formula for vanishing cycles of local systems
  
  • Sabbah Claude
  • Saito Morihiko
  Algebraic Geometry, Foundation Compositio Mathematica, 2014, 1 (1), pp.107-130. 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 as well as its generalization 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. (10.14231/AG-2014-006)
  DOI : 10.14231/AG-2014-006
• Polynomial configurations in the primes
  
  • Le Thai Hoang
  • Wolf Julia
  International Mathematics Research Notices, Oxford University Press (OUP), 2014, 23, pp.6448-6473. The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are integers. Various generalizations of this theorem are known. Wooley and Ziegler showed that the variable m can in fact be taken to be a prime minus 1, and Tao and Ziegler showed that the Bergelson-Leibman theorem holds for subsets of the primes of positive relative upper density. Here we prove a hybrid of the latter two results, namely that the step m in the Tao-Ziegler theorem can be restricted to the set of primes minus 1. (10.1093/imrn/rnt169)
  DOI : 10.1093/imrn/rnt169
• Genus 0 characteristic numbers of tropical projective plane
  
  • Bertrand Benoît
  • Brugalle Erwan
  • Mikhalkin Grigory
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2014, 150, pp.46-104. Finding the so-called characteristic numbers of the complex projective plane $CP^2$ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of degree $d$ genus $g$ curves that pass through a certain generic configuration of points and in the same time are tangent to certain generic configuration of lines. The total number of points and lines in these two configurations is $3d-1+g$ so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when $g=0$. Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we compute genus 0 characteristic numbers of $CP^2$ in terms of open Hurwitz numbers. (10.1112/S0010437X13007409)
  DOI : 10.1112/S0010437X13007409
• Decaying Turbulence in the Generalised Burgers Equation
  
  • Boritchev Alexandre
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2014, 214, pp.331 - 357. We consider the generalised Burgers equation$$∂u/∂t + f (u) ∂u/∂x − \nu ∂^2u/∂x^2 = 0, t ≥ 0, x ∈ S^1,$$where $f$ is strongly convex and $\nu$ is small and positive. We obtain sharp estimates for Sobolev norms of u (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for the dissipation length scale and the small-scale quantities which characterise the decaying Burgers turbulence, i.e. the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell, Frisch, Lutsko and Vergassola [1]. Note that we are dealing with decaying, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval $[T_1 , T_2 ]$, where $T_1$ and $T_2$ depend only on $f$ and the initial condition, and do not depend on the viscosity. These results allow us to obtain a rigorous theory of the one-dimensional Burgers turbulence in the spirit of Kolmogorov's 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form $k^{-2}$ . These results remain valid in the inviscid limit. (10.1007/s00205-014-0766-5)
  DOI : 10.1007/s00205-014-0766-5
• Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l'École polytechnique 2006--2008
  
  • Illusie Luc
  • Laszlo Yves
  • Orgogozo Fabrice
  , 2014, Astérisque 363-364, pp.xxiv+619 pages. This book contains notes of a seminar on Ofer Gabber's work on the etale cohomology and uniformization of quasi-excellent schemes. His main results include (cf. introduction) constructibility theorems (for abelian or non-abelian coefficients), vanishing theorems (e.g. affine Lefschetz), uniformization for the "prime-to-l alteration topology", rigidity for non-abelian coefficients, a new proof of the absolute purity conjecture, duality, etc.
• Le temps de la dynamique et la dynamique du temps
  
  • Paul Thierry
  , 2014. Le temps apparait habituellement dans la résolution des équations différentielles comme un simple paramètre, le curseur d'une dynamique habitant dans un espace absolu et immuable. L'équation (forme fugue) générerait donc, pour toujours et à partir d'une donnée initiale (thème) un flot, tel une fugue qui ne s?arrêterait jamais. Les mathématiques récentes on fait voler en éclat ce côté par trop passif du temps et nous ont montré que le temps pouvait, au cours du temps, changer de statut et nécessiter une ouverture de la notion d'espace à de nouveaux paradigmes. Nous présentons plusieurs exemples d'une telle dynamique du temps et tenterons d'exhiber quelques résonances avec le temps musical.
• Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation
  
  • Bethuel Fabrice
  • Gravejat Philippe
  • Smets Didier
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2014, 64 (1), pp.19-70. We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.
• A bit of tropical geometry
  
  • Brugallé Erwan
  • Shaw Kristin
  The American Mathematical Monthly, Mathematical Association of America, 2014, 121 (7), pp.563-589. This friendly introduction to tropical geometry is meant to be accessible to first year students in mathematics. The topics discussed here are basic tropical algebra, tropical plane curves, some tropical intersections, and Viro's patchworking. Each definition is explained with concrete examples and illustrations. To a great exten, this text is an updated of a translation from a french text by the first author. There is also a newly added section highlighting new developments and perspectives on tropical geometry. In addition, the final section provides an extensive list of references on the subject. (10.4169/amer.math.monthly.121.07.563)
  DOI : 10.4169/amer.math.monthly.121.07.563
• Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II
  
  • Gérard-Varet David
  • Han-Kwan Daniel
  • Rousset Frédéric
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2014. In this paper, we study the quasineutral limit of the isothermal Euler-Poisson equation for ions, in a domain with boundary. This is a follow-up to our previous work \cite{GVHKR}, devoted to no-penetration as well as subsonic outflow boundary conditions. We focus here on the case of supersonic outflow velocities. The structure of the boundary layers and the stabilization mechanism are different. (10.5802/jep.13)
  DOI : 10.5802/jep.13
• Multi-solitons for nonlinear Klein-Gordon equations
  
  • Côte Raphaël
  • Muñoz Claudio
  Forum of Mathematics, Sigma, Cambridge University press, 2014, 2, pp.e15. In this paper we consider the existence of multi-soliton structures for the nonlinear Klein-Gordon equation (NLKG) in R^{1+d}. We prove that, independently of the unstable character of (NLKG) solitons, it is possible to construct a N-soliton family of solutions to (NLKG), of dimension 2N, globally well-defined in the energy space H^1 \times L^2 for all large positive times. The method of proof involves the generalization of previous works on supercritical NLS and gKdV equations by Martel, Merle and the first author to the wave case, where we replace the unstable mode associated to the linear NLKG operator by two generalized directions that are controlled without appealing to modulation theory. As a byproduct, we generalize the linear theory described in Grillakis-Shatah-Strauss and Duyckaerts-Merle to the case of boosted solitons, and provide new solutions to be studied using the recent Nakanishi- Schlag theory. (10.1017/fms.2014.13)
  DOI : 10.1017/fms.2014.13
• Energy partition for the linear radial wave equation
  
  • Côte Raphaël
  • Kenig Carlos
  • Schlag Wilhelm
  Mathematische Annalen, Springer Verlag, 2014, 358 (3-4), pp.573-607. We consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy relative to a light cone, as time goes to infinity. We show that the exterior energy estimate, which Duyckaerts, Merle and the second author obtained in odd dimensions, fails in even dimensions. Positive results for restricted classes of data are obtained. This is a companion paper to our two nonlinear papers with Andrew Lawrie. (10.1007/s00208-013-0970-x)
  DOI : 10.1007/s00208-013-0970-x
• On the dynamics of WKB wave functions whose phase are weak KAM solutions of H-J equation
  
  • Paul Thierry
  • Zanelli Lorenzo
  Journal of Fourier Analysis and Applications, Springer Verlag, 2014, 20 (1291-). In the framework of toroidal Pseudodifferential operators on the flat torus $\Bbb T^n := (\Bbb R / 2\pi \Bbb Z)^n$ we begin by proving the closure under composition for the class of Weyl operators $\mathrm{Op}^w_\hbar(b)$ with simbols $b \in S^m (\mathbb{T}^n \times \mathbb{R}^n)$. Subsequently, we consider $\mathrm{Op}^w_\hbar(H)$ when $H=\frac{1}{2} |\eta|^2 + V(x)$ where $V \in C^\infty (\Bbb T^n;\Bbb R)$ and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on $\Bbb T^n \times \Bbb R^n$ written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space $H^{1} (\mathbb{T}^n; \Bbb C)$ with phase functions in the class of Lipschitz continuous weak KAM solutions (of positive and negative type) of the Hamilton-Jacobi equation $\frac{1}{2} |P+ \nabla_x v_\pm (P,x)|^2 + V(x) = \bar{H}(P)$ for $P \in \ell \Bbb Z^n$ with $\ell >0$, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of $P+ \nabla_x v_\pm$.
• Toric Surfaces, K-Stability and Calabi Flow
  
  • Huang Hongnian
  Mathematische Zeitschrift, Springer, 2014, 276 (3-4), pp.953-968. Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and $\int_{\partial P} u ~ d \sigma < C_2, $ then there exists a constant $C_3$ depending only on $C_1, C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$. Moreoever, we can show that there is a constant $M > 0$ depending only on $C_1, C_2$ and $P$ such that Donaldson's $M$-condition holds for $u$. As an application, we show that if $(X,P)$ is (analytic) relative $K$-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow. (10.1007/s00209-013-1228-8)
  DOI : 10.1007/s00209-013-1228-8
• Voyage dans le non commutatif
  
  • Paul Thierry
  InFluxus, InFluxus, 2014. Nous proposons un court voyage au pays de la non commutativité. Nous présentons différents aspects des mathématiques et de la physique oú cette notion apparaît. Nous montrons comment, en particulier, elle permet de franchir un des écueils importants de la physique newtonienne en Mécanique Classique et discutons diverses situations en Mécanique Quantique.
• Solutions without any symmetry for semilinear elliptic problems
  
  • Ao Weiwei
  • Musso Monica
  • Pacard Frank
  • Wei Juncheng
  , 2014.
• Bloch's conjecture for Catanese and Barlow surfaces
  
  • Voisin Claire
  Journal of Differential Geometry, International Press, 2014, 97 (1), pp.149-175. Catanese surfaces are regular surfaces of general type with $p_g=0$. They specialize to double covers of Barlow surfaces. We prove that the $CH_0$ group of a Catanese surface is equal to $\mathbb{Z}$, which implies the same result for the Barlow surfaces. We will also give a conditional application (more precisely, assuming the variational Hodge conjecture) of the same method to the Chow motive of low degree $K3$ surfaces.