Volume 2018, Issue 736

In the present paper we prove upper and lower bounds of the heat kernel for the operator Δ-∇⁡(|x|-α)⋅∇{\Delta-\nabla({|x|^{-\alpha}})\cdot\nabla} in ℝn∖{0}{\mathbb{R}^{n}\setminus\{0\}}, where α>0{\alpha>0}. We obtain these bounds from an isoperimetric inequality for a measure e-|x|-α⁢d⁢x{\mathrm{e}^{-{|x|^{-\alpha}}}dx} on ℝn∖{0}{\mathbb{R}^{n}\setminus\{0\}}. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric.

Let k be an algebraically closed field of arbitrary characteristic, let K/k{K/k} be a finitely generated field extension and let X be a separated scheme of finite type over K . For each prime ℓ{\ell}, the absolute Galois group of K acts on the ℓ{\ell}-adic étale cohomology modules of X . We prove that this family of representations varying over ℓ{\ell} is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ℓ{\ell} become linearly disjoint over a finite extension of K . In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

The purpose of this paper is to study pro excision in algebraic K -theory and cyclic homology, after Suslin–Wodzicki, Cuntz–Quillen, Cortiñas, and Geisser–Hesselholt, as well as continuity properties of André–Quillen and Hochschild homology. A key tool is first to establish the equivalence of various pro Tor vanishing conditions which appear in the literature. This allows us to prove that all ideals of commutative, Noetherian rings are pro unital in a suitable sense. We show moreover that such pro unital ideals satisfy pro excision in derived Hochschild and cyclic homology. It follows hence, and from the Suslin–Wodzicki criterion, that ideals of commutative, Noetherian rings satisfy pro excision in derived Hochschild and cyclic homology, and in algebraic K -theory. In addition, our techniques yield a strong form of the pro Hochschild–Kostant–Rosenberg theorem; an extension to general base rings of the Cuntz–Quillen excision theorem in periodic cyclic homology; a generalisation of the Feĭgin–Tsygan theorem; a short proof of pro excision in topological Hochschild and cyclic homology; and new Artin–Rees and continuity statements in André–Quillen and Hochschild homology.

We prove new kinematic formulas for tensor valuations and simplify previously known Crofton formulas by using the recently developed algebraic theory of translation invariant valuations. The heart of the paper is the computation of the Alesker–Fourier transform on the large class of spherical valuations, which is achieved by differential-geometric and representation theoretical tools. We also describe in explicit form the product and convolution of tensor valuations.

We establish the linear independence over ℚ\mathbb{Q}, in both qualitative and quantitative forms, of the four numbers 1, Li1⁡(1/z)=-log⁡(1-1/z)\operatorname{Li}_{1}(1/z)=-\log(1-1/z), Li2⁡(1/z)\operatorname{Li}_{2}(1/z) and Li2⁡(1/(1-z))\operatorname{Li}_{2}(1/(1-z)), for all integers z≥9z\geq 9 or z≤-8z\leq-8 and for rationals z=s/rz=s/r or z=1-s/rz=1-s/r with 1<r<s1<r<s, where s is large in comparison with r .

Given a generically finite morphism f from a smooth projective variety X to an abelian variety A , we show that f*⁢ωX{f_{*}\omega_{X}} is “sufficiently positive” on A . As an application, we prove that when X is of general type, the global sections of ωX2{\omega_{X}^{2}} define a generically finite map of X . We also study the structure of X when X is of general type and satisfies χ⁢(X,ωX)=0{\chi(X,\omega_{X})=0}. We formulate a conjectural characterization of such X and prove the conjecture when A has exactly three simple factors.

In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert–Samuel formula and the arithmetic Hilbert–Samuel formula. The treatments are high-dimensional generalizations of [25] and [26]. Similar results are obtained independently by Huayi Chen [7] with less explicit error terms.

Based on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of Weyl–Titchmarsh and spectral functions of 2×2{2\times 2} first order canonical systems. Our results improve several classical results and solve open problems posed by previous authors. Furthermore, they are applied to radial Dirac and radial Schrödinger operators as well as to Krein strings and generalized indefinite strings.