Approximation forte pour les variétés avec une action d’un groupe linéaire

Let $G$ be a connected linear algebraic group over a number field $k$. Let $U{\hookrightarrow}X$ be a $G$-equivariant open embedding of a $G$-homogeneous space $U$ with connected stabilizers into a smooth $G$-variety $X$. We prove that $X$ satisfies strong approximation with Brauer–Manin condition off a set $S$ of places of $k$ under either of the following hypotheses:(i)$S$ is the set of archimedean places;(ii)$S$ is a non-empty finite set and $\bar{k}^{\times }=\bar{k}[X]^{\times }$.The proof builds upon the case $X=U$, which has been the object of several works.

On the Period Map for Polarized Hyperkähler Fourfolds

We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy’s Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW (Eisenbud–Popescu–Walter) sextics. In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkähler fourfolds with Picard number 1 or 2.

On the probabilistic domain invariance

Probabilistic version of the invariance of domain for contractive field and Schauder invertibility theorem are proved. As an application, the stability of probabilistic open embedding is established.

Constructing Locally Flat Embeddings of Infinite Dimensional Manifolds without Tubular Neighborhoods

All spaces in this paper will be separable and metric. A closed embedding i: M → TV is said to be locally flat (of codimension n) if for each x0 ∈ M there is an open set U in M containing xo and an open embedding h: U X Rn → N for which h(x, 0) = i(x), for all x ∈ U.