Volume 2018, Issue 745

The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C , we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C . We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.

Let D be a central simple algebra of prime degree over a field and let E be an 𝐒𝐋1⁡(D){\operatorname{\mathbf{SL}}_{1}(D)}-torsor. We determine the complete motivic decomposition of certain compactifications of E . We also compute the Chow ring of E .

Let X be an open Riemann surface. We prove an Oka property on the approximation and interpolation of continuous maps X →\to(ℂ*)2(\mathbb{C}^{*})^{2} by proper holomorphic embeddings, provided that we permit a smooth deformation of the complex structure on X outside a certain set. This generalises and strengthens a recent result of Alarcón and López. We also give a Forstnerič–Wold theorem for proper holomorphic embeddings (with respect to the given complex structure) of certain open Riemann surfaces into (ℂ*)2{(\mathbb{C}^{*})^{2}}.

In the recent articles [5, 1], it was conjectured that all rational GLn{\mathrm{GL}_{n}}-invariant functions of forms of degree d≥3{d\geq 3} on ℂn{\mathbb{C}^{n}} can be extracted, in a canonical way, from those of forms of degree n⁢(d-2){n(d-2)} by means of assigning to every form with nonvanishing discriminant the so-called associated form. The conjecture was confirmed in [5] for binary forms of degree d≤6{d\leq 6} as well as for ternary cubics. Furthermore, a weaker version of it was settled in [1] for arbitrary n and d . In the present paper, we focus on the case n=2{n=2} and establish the conjecture, in a rather explicit way, for binary forms of an arbitrary degree.

We investigate stratified-algebraic vector bundles on a real algebraic variety X . A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X . A topological vector bundle ξ on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification 𝒮{\mathcal{S}} of X such that the restriction of ξ to each stratum S in 𝒮{\mathcal{S}} is an algebraic vector bundle on S . In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which distinguish them from algebraic and topological vector bundles.

This paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup Γ in a connected Lie (or algebraic) group G , acting on suitable homogeneous spaces G / H . The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on H and acting on G /Γ. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of H in the automorphic representation on L2L^{2}(Γ ∖\setminus G ). We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.

Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C*\mathrm{C}^{*}-algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C*\mathrm{C}^{*}-algebras associated to hyperbolic groups – the C*\mathrm{C}^{*}-crossed product for the boundary action, and the reduced group C*\mathrm{C}^{*}-algebra – have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.

To an algebraic variety equipped with an involution, we associate a cycle class in the modulo two Chow group of its fixed locus. This association is functorial with respect to proper morphisms having a degree and preserving the involutions. Specialising to the exchange involution of the square of a complete variety, we obtain Rost’s degree formula in arbitrary characteristic (this formula was proved by Rost and Merkurjev in characteristic not two).

We are concerned with hypersurfaces of ℝN{\mathbb{R}^{N}} with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ℝN{\mathbb{R}^{N}} with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ℝ2{\mathbb{R}^{2}} with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

Let M be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group G . We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on T∗⁢M×GT^{\ast}M\times G with singular critical sets that were examined in [7] in order to determine the asymptotic distribution of eigenvalues of an invariant elliptic operator on M . As an immediate consequence, we deduce from this an asymptotic multiplicity formula for families of irreducible representations in L2⁢(M)\mathrm{L}^{2}(M). The improved remainder is used in [4] to prove an equivariant semiclassical Weyl law and a corresponding equivariant quantum ergodicity theorem.