Vanishing theorems for associative submanifolds

Let M^7 a manifold with holonomy in G_2, and Y^3 an associative submanifold with boundary in a coassociative submanifold. In [5], the authors proved that M_{X,Y}, the moduli space of its associative deformations with boundary in the fixed X, has finite virtual dimension. Using Bochner's technique, we give a vanishing theorem that forces M_{X,Y} to be locally smooth.Comment: This new version relates the former one to results for minimal submanifold

Smooth moduli spaces of associative submanifolds

Let $M^7$ be a smooth manifold equipped with a $G_2$-structure $\phi$, and $Y^3$ be an closed compact $\phi$-associative submanifold. In \cite{McL}, R. McLean proved that the moduli space \bm_{Y,\phi} of the $\phi$-associative deformations of $Y$ has vanishing virtual dimension. In this paper, we perturb $\phi$ into a $G_2$-structure $\psi$ in order to ensure the smoothness of \bm_{Y,\psi} near $Y$. If $Y$ is allowed to have a boundary moving in a fixed coassociative submanifold $X$, it was proved in \cite{GaWi} that the moduli space \bm_{Y,X} of the associative deformations of $Y$ with boundary in $X$ has finite virtual dimension. We show here that a generic perturbation of the boundary condition $X$ into $X'$ gives the smoothness of \bm_{Y,X'}. In another direction, we use the Bochner technique to prove a vanishing theorem that forces \bm_Y or \bm_{Y,X} to be smooth near $Y$. For every case, some explicit families of examples will be given.Comment: 27 page

Rational convexity of non generic immersed lagrangian submanifolds

We prove that an immersed lagrangian submanifold in \C^n with quadratic self-tangencies is rationally convex. This generalizes former results for the embedded and the immersed transversal cases.Comment: 4 page

Percolation without FKG

We prove a Russo-Seymour-Welsh theorem for large and natural perturbative families of discrete percolation models that do not necessarily satisfy the Fortuin-Kasteleyn-Ginibre condition of positive association. In particular, we prove the box-crossing property for the antiferromagnetic Ising model with small parameter, and for certain discrete Gaussian fields with oscillating correlation function

What is the total Betti number of a random real hypersurface?

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equirepartition which we first establish. Our proofs involve H\"ormander's theory of peak sections as well as the formula of Poincar\'e-Martinelli

Exponential rarefaction of real curves with many components

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L^d inherits for every positive integer d a L^2 scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.Comment: 21 page

De l'ambiguïté dans les interrogations fermées en russe moderne

In this paper, we intend to show how ambiguity can be cleared in positive polar (yes/no) questions in modern Russian. For the purpose of our study, we will analyze three interrogative transitive predicate structures (svO[1], sVo and soV). Despite the same syntactic surface structure and the same nuclear accent placement, each of them satisfies two different types of questions. The analyses reveal that when the segmental level does not allow disambiguisation, this is achieved via various prosodic techniques: nature of nuclear accent ("neutral" accent vs. "exclusive" accent) combined with variations in pitch contour shapes and pitch range, as well as pause. [1] The capital letter indicates the accentuated term.Nous nous proposons dans le présent article de montrer comment l'ambiguïté peut être levée dans les interrogations fermées dépourvues de toute particule interrogative en russe moderne. Nous examinerons trois séquences à prédicat transitif, à accent final ou médian (svC, sVc et scV), qui correspondent chacune, pour une même structure de surface et un même choix du terme porteur de l'accent nucléaire, à deux types d'interrogations distinctes. Les analyses révèlent que lorsque le niveau segmental ne permet pas la désambiguïsation, celle-ci se fait grâce à différents procédés prosodiques : nature de l'accent nucléaire (accent "neutre" vs accent "exclusif") combinée à des variations dans la forme et dans l'amplitude du contour prosodique ainsi que présence ou absence de pause