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

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

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

Betti numbers of random nodal sets of elliptic pseudo-differential operators

Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$.Comment: 3