Publications

2012

• Algebraic homotopy classes of rational functions
  
  • Cazanave Christophe
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2012 (45). We compute the set of naive pointed homotopy classes of endomorphisms of the projective line P^1 over the spectrum of a field. Our computation compares well with Fabien Morel's one of the motivic pointed homotopy classes of endomorphisms of P^1: there is an a priori monoid structure on the set of naive homotopy classes such that canonical map from this monoid to group of motivic homotopy classes is a group completion.
• A note on the rank of positive closed currents
  
  • Dujardin Romain
  , 2012. In this short note we prove an estimate on the rank a.e. of the tangent (p,p) vector to a a positive closed current of bidimension (p,p) in CP^k, in terms of the dimension of its trace measure.
• Periodic points of birational maps on projective surfaces
  
  • Xie Junyi
  , 2012. We classify birational maps of projective smooth surfaces whose non-critical periodic points are Zariski dense. In particular, we show that if the first dynamical degree is greater than one, then the periodic points are Zariski dense.
• Eigenvarieties for classical groups and complex conjugations in Galois representations
  
  • Taïbi Olivier
  , 2012. The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\GL_{2n}/F$ whose multiplicative character is ''odd''. We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are ''many'' points corresponding to (quasi-)irreducible Galois representations. The recent work of James Arthur describing the automorphic spectrum for these groups is used to define these Galois representations, and also to transfer self-dual automorphic representations of the general linear group to these classical groups.
• Invariant four-forms and symmetric pairs
  
  • Moroianu Andrei
  • Semmelmann Uwe
  , 2012. We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-dimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type.
• Weakly complex homogeneous spaces
  
  • Moroianu Andrei
  • Semmelmann Uwe
  , 2012. We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.
• The supports of higher bifurcation currents
  
  • Dujardin Romain
  , 2012. Let (f_\lambda) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold \Lambda. To this data is associated a closed positive current T_1\wedge ... \wedge T_k of bidegree (k,k) on \Lambda, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c_1,..., c_k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T_1\wedge ... \wedge T_k).
• Bifurcation currents and equidistribution on parameter space
  
  • Dujardin Romain
  , 2012. In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.
• ON SOME GEOMETRY OF PROPAGATION IN DIFFRACTIVE TIME SCALES
  
  • Cheverry Christophe
  • Paul Thierry
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2012, 32 (2), pp.Pages: 499 - 538. In this article, we develop a non linear geometric optics which presents the two main following features. It is valid in diffractive times and it extends the classical approaches to the case of fast variable coefficients. In this context, we can show that the energy is transported along the rays associated with some non usual long-time hamiltonian. Our analysis needs structural assumptions and initial data suitably polrarized to be implemented. All the required conditions are met concerning a current model arising in fluid mechanics and which was the original motivation of our work. As a by product, we get results complementary to the litterature concerning the propagation of the Rossby waves which play a part in the description of large oceanic currents, like Gulf stream or Kuroshio. (10.3934/dcds.2012.32.499)
  DOI : 10.3934/dcds.2012.32.499
• On a twisted de Rham complex, II
  
  • Sabbah Claude
  , 2012. We prove an algebraic formula, conjectured by M. Kontsevich, for computing the monodromy of the vanishing cycles of a regular function on a smooth complex algebraic variety.
• Harmonic maps, second homology classes of smooth manifolds, and bounded cohomology
  
  • Ville Marina
  International Journal of Mathematics, World Scientific Publishing, 2012, 04 (06), pp.997-1006. (10.1142/S0129167X93000467)
  DOI : 10.1142/S0129167X93000467
• Harmonic morphisms from Einstein 4-manifolds to Riemann surfaces
  
  • Ville Marina
  International Journal of Mathematics, World Scientific Publishing, 2012, 14 (03), pp.327-337. If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from S4 or CP2 to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism CP2#CP2 from to S2. (10.1142/S0129167X0300179X)
  DOI : 10.1142/S0129167X0300179X
• Note on the gonality of abstract modular curves
  
  • Cadoret Anna
  , 2012, pp.89-106. (10.1007/978-3-642-23905-2_4)
  DOI : 10.1007/978-3-642-23905-2_4
• Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization
  
  • Monzner Alexandra
  • Vichery Nicolas
  • Zapolsky Frol
  Journal of modern dynamics, American Institute of Mathematical Sciences, 2012, pp.205-249. For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T^*N, and a family of functions on the space of smooth functions with compact support on T^*N. These satisfy properties analogous to those of partial quasi-morphisms and quasi-states of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T^n the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.
• Degree growth of monomial maps and McMullen's polytope algebra
  
  • Favre Charles
  • Wulcan Elizabeth
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2012, 61 (2), pp.493-524. We compute all dynamical degrees of monomial maps by interpreting them as mixed volumes of polytopes. By exploiting further the isomorphism between the polytope algebra of P. McMullen and the universal cohomology of complete toric varieties, we construct invariant positive cohomology classes when the dynamical degrees have no resonance. (10.1512/iumj.2012.61.4555)
  DOI : 10.1512/iumj.2012.61.4555
• The global existence and convergence of the Calabi flow on $\mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$
  
  • Feng Renjie
  • Huang Hongnian
  Journal of Functional Analysis, Elsevier, 2012, 263 (4), pp.1129-1146. In this note, we study the long time existence of the Calabi flow on $X = \mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvature is uniformly bounded along the Calabi flow on $X$ when the dimension is 2, partially confirming Chen's conjecture. Moreover, we show that the Calabi flow exponentially converges to the flat Kähler metric for arbitrary dimension if the curvature is uniformly bounded, partially confirming Donaldson's conjecture. (10.1016/j.jfa.2012.05.017)
  DOI : 10.1016/j.jfa.2012.05.017
• The volume of an isolated singularity
  
  • Boucksom S.
  • de Fernex Tommaso
  • Favre Charles
  Duke Mathematical Journal, Duke University Press, 2012, 161 (8), pp.1455-1520.. We introduce a notion of volume of a normal isolated singularity that generalizes Wahl's characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of non-invertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a non-invertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov's b-divisors. We define the notion of nef Weil b-divisors, and of nef envelopes of b-divisors. We relate the latter to the pull-back of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil b-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case. (10.1215/00127094-1593317)
  DOI : 10.1215/00127094-1593317
• Geometric properties of maximal psh functions
  
  • Dujardin Romain
  • Guedj Vincent
  , 2012, pp.33-52. (10.1007/978-3-642-23669-3_3)
  DOI : 10.1007/978-3-642-23669-3_3
• En hommage à Jean-Marie Souriau, quelques souvenirs
  
  • Kosmann-Schwarzbach Yvette
  Gazette des Mathématiciens, Société Mathématique de France, 2012, 133, pp.105-106.
• On the topology of fillings of contact manifolds and applications.
  
  • Oancea Alexandru
  • Viterbo Claude
  Commentarii Mathematici Helvetici, European Mathematical Society, 2012, pp.41-69. The aim of this paper is to address the following question: given a contact manifold $(\Sigma, \xi)$, what can be said about the aspherical symplectic manifolds $(W, \omega)$ bounded by $(\Sigma, \xi)$ ? We first extend a theorem of Eliashberg, Floer and McDuff to prove that under suitable assumptions the map from $H_{*}(\Sigma)$ to $H_{*}(W)$ induced by inclusion is surjective. We then apply this method in the case of contact manifolds having a contact embedding in $ {\mathbb R}^{2n}$ or in a subcritical Stein manifold. We prove in many cases that the homology of the fillings is uniquely determined. Finally we use more recent methods of symplectic topology to prove that, if a contact hypersurface has a Stein subcritical filling, then all its weakly subcritical fillings have the same homology. A number of applications are given, from obstructions to the existence of Lagrangian or contact embeddings, to the exotic nature of some contact structures.
• Multicurves and regular functions on the representation variety of a surface in SU(2)
  
  • Charles Laurent
  • Marche Julien
  Commentarii Mathematici Helvetici, European Mathematical Society, 2012, 87 (2), pp.409-431. We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier decomposition of the trace functions with respect to some torus action provided by a pants decomposition. Consequently the space of trace functions is isomorphic to the skein algebra at A=-1 of the thickened surface. (10.4171/CMH/258)
  DOI : 10.4171/CMH/258
• Configurations of flags and representations of surface groups in complex hyperbolic geometry
  
  • Marché Julien
  • Will Pierre
  Geom. Dedicata, 2012, 156, pp.49-70. In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperbolic plane H2C . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation variety of π 1(Σ).
• Uniform boundedness of p-primary torsion of abelian schemes
  
  • Cadoret Anna
  • Tamagawa Akio
  Inventiones Mathematicae, Springer Verlag, 2012, 188 (1), pp.83-125. (10.1007/s00222-011-0343-6)
  DOI : 10.1007/s00222-011-0343-6
• A uniform open image theorem for ℓ-adic representations, I
  
  • Cadoret Anna
  • Tamagawa Akio
  Duke Mathematical Journal, Duke University Press, 2012, 161 (13), pp.2605-2634. (10.1215/00127094-1812954)
  DOI : 10.1215/00127094-1812954
• Optimally small sumsets in groups IV. Counting multiplicities and the λ_G functions
  
  • Plagne Alain
  Israel Journal of Mathematics, Springer, 2012, 191 (2), pp.739-754. We continue our investigation on how small a sumset can be in a given abelian group. Here "small" takes into account not only the size of the sumset itself but also the number of elements which are repeated at least twice. A function λ_G (r, s) computing the minimal size (in this sense) of the sum of two sets with respective cardinalities r and s is introduced. (Lower and upper) bounds are obtained, which coincide in most cases. While upper bounds are obtained by constructions, lower bounds follow in particular from the use of a recent theorem by Grynkiewicz.