The smooth locus in infinite-level Rapoport–Zink spaces

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

On Propagation of Sphericity of Real Analytic Hypersurfaces across Levi Degenerate Loci

A connected real analytic hypersurface M⊂Cn+1 whose Levi form is nondegenerate in at least one point—hence at every point of some Zariski-dense open subset—is locally biholomorphic to the model Heisenberg quadric pseudosphere of signature (k,n-k) in one point if and only if, at every other Levi nondegenerate point, it is also locally biholomorphic to some Heisenberg pseudosphere, possibly having a different signature (l,n-l). Up to signature, pseudosphericity then jumps across the Levi degenerate locus and in particular across the nonminimal locus, if there exists any.

Berezin quantization on generalized flag manifolds

Let $M=G/H$ be a generalized flag manifold where $G$ is a compact, connected, simply-connected Lie group with Lie algebra $\mathfrak{g}$ and $H$ is the centralizer of a torus. Let $\pi$ be a unitary irreducible representation of $G$ which is holomorphically induced from a character of $H$. Using a complex parametrization of a dense open subset of $M$, we realize $\pi$ on a Hilbert space of holomorphic functions. We give explicit expressions for the differential $d\pi$ of $\pi$ and for the Berezin symbols of $\pi (g)$ ($g\in G$) and $d\pi (X)$ ($X\in \mathfrak{g}$). In particular, we recover some results of S. Berceanu and we partially generalize a result of K. H. Neeb.

Formule du conducteur pour un caractère l-adique

LetKbe a local field of equal characteristicp>2, letXK/Kbe a smooth proper relative curve, and letbe a rank 1 smoothl-adic sheaf (l≠p) on a dense open subsetUKXK. In this paper, under some assumptions on the wild ramification of, we prove a conductor formula that computes the Swan conductor of the etale cohomology of the vanishing cycles of. Our conductor formula is a generalization of the conductor formula of Bloch, but for non-constant coefficients.

Harmonic characteristic vector fields on contact metric three-manifolds

In this paper we show that a contact metric three-manifold is a generalised (k, μ)-space on an everywhere dense open subset if and only if its characteristic vector field ξ determines a harmonic map from the manifold into its unit tangent sphere bundle equipped with the Sasaki metric. Moreover, we classify the contact metric three-manifolds whose characteristic vector field ξ is strongly normal (or equivalently, is harmonic and minimal).

On generalized albanese varieties for surfaces

Given a variety X over a field k and a dense open subset U of X, the related generalized albanese problem has two parts. First we want to classify rational maps with domain U into commutative algebraic groups, into reasonable categories, and then in each category we want to find an object α which is universal in the sense that any β in this category factors through α.

A local limit theorem for a sequence of interval transformations

Let λ > 1 be a real eigenvalue of an automorphism of the two dimensional torus. We prove that for a dense, open subset of intervals the sequence where {x} denotes the fractional part of x and χ[a, b] the characteristic function of [a, b], satisfies the local limit theorem with respect to Lebesgue measure on [0, 1].

Alternating 3-Forms and Exceptional Simple Lie Groups of Type G2

It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connected dense open subset Ψ(W) of “maximal” forms. If ψ ∈ Ψ(W) then the subgroup of AUTC(W) consisting of the invertible complex-linear transformations S such that ψ(S•, S•, S•) = ψ(•, •, •) is denoted G(ψ), and, in Proposition 3.6. we provewhere G1(ψ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G2 is defined in terms of the alternating 3-form ψ alone without the need to specify an invariant quadratic form. In the real case the result is more striking.

Local Topological Properties of Maps and Open Extensions of Maps

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

On Prime Immersions of S1 into R2

A C1-mapping ƒ from the oriented circle S1 into the oriented plane R2 such that f f’ (t) ≠ 0 for all t is called a regular immersion. We call a point p in Im f a double point if f-1(p) is a two element set with the corresponding tangent vectors being linearly independent. A regular immersion which is one-to-one except at a finite number of points whose images are double points is called a normal immersion. The work of Whitney [7], Titus [3] and Verhey [6] shows that the normal immersions form a dense open subset in the space of regular immersions with the usual C1-topology, and can be characterized up to diffeomorphic equivalence by a combinatorial invariant called the intersection sequence.