MATHEMATICA SCANDINAVICA https://www.mscand.dk/ en-US mscand@math.au.dk (Arne Jensen) mscand@math.au.dk (Anne Mette Christiansen) Sat, 11 Jun 2022 13:37:34 +0200 OJS 3.3.0.10 http://blogs.law.harvard.edu/tech/rss 60 The homology of the groupoid of the self-similar infinite dihedral group https://www.mscand.dk/article/view/129708 <p>We compute the $K$-theory of the $C^*$-algebra associated to the self-similar infinite dihedral group, and the homology of its associated étale groupoid. We see that the rational homology differs from the $K$-theory, strongly contradicting a conjecture posted by Matui. Moreover, we compute the abelianization of the topological full group of the groupoid associated to the self-similar infinite dihedral group.</p> Eduard Ortega, Alvaro Sanchez Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129708 Sat, 11 Jun 2022 00:00:00 +0200 Canonical subsheaves of torsionfree semistable sheaves https://www.mscand.dk/article/view/129709 <p>Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety defined over an algebraically closed field. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ also admits a filtration of subbundles for which each successive quotient is a stable vector bundle whose slope is $\mu (F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $\mu (F)$. We show that these canonical subsheaves behave well with respect to the pullback operation by étale Galois covering maps. Given a separable finite surjective map $\phi \colon Y \longrightarrow X$ between normal projective varieties, we give a criterion for the induced homomorphism of étale fundamental groups $\phi _*\colon \pi ^{\textrm {et}}_{1}(Y) \longrightarrow \pi ^{\textrm {et}}_{1}(X)$ to be surjective. The criterion in question is expressed in terms of the above mentioned unique maximal locally free subsheaf associated to the direct image $\phi _*{\mathcal O}_Y$.</p> Indranil Biswas, A. J. Parameswaran Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129709 Sat, 11 Jun 2022 00:00:00 +0200 A note on differences of composition operators acting on the Hardy space https://www.mscand.dk/article/view/129748 <p>When $\varphi$ and $\psi$ are holomorphic self-maps of the unit disk with bounded multiplicity, we give a compact characterization for the difference of composition operators $C_\varphi $ and $C_\psi $ acting on the Hardy space, which extends a recent result in the case of univalent maps by Choe et al.</p> Jineng Dai Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/129748 Sat, 11 Jun 2022 00:00:00 +0200 Direct limits of infinite-dimensional Carnot groups https://www.mscand.dk/article/view/132062 <p>We give a construction of direct limits in the category of complete metric scalable groups and provide sufficient conditions for the limit to be an infinite-dimensional Carnot group. We also prove a Rademacher-type theorem for such limits.</p> Terhi Moisala, Enrico Pasqualetto Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132062 Sat, 11 Jun 2022 00:00:00 +0200 On the Liouville and strong Liouville properties for a class of non-local operators https://www.mscand.dk/article/view/132068 <p>We prove a necessary and sufficient condition for the Liouville and strong Liouville properties of the infinitesimal generator of a Lévy process and subordinate Lévy processes. Combining our criterion with the necessary and sufficient condition obtained by Alibaud et al., we obtain a characterization of (orthogonal subgroup of) the set of zeros of the characteristic exponent of the Lévy process.</p> David Berger, René L. Schilling Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132068 Sat, 11 Jun 2022 00:00:00 +0200 Intermediate Jacobians and the slice filtration https://www.mscand.dk/article/view/132174 <p>For every $n$-dimensional smooth projective variety $X$ over ℂ, the motive $M(X)$ is expected to admit a Chow-Künneth decomposition $M_0(X)\oplus \cdots \oplus M_{2n}(X)$. Inspired by the slice filtration of $M(X)$ we propose the definitions of $M_2(X)$ and $M_{2n-2}(X)$. In our construction we use intermediate Jacobians.</p> Doosung Park Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132174 Sat, 11 Jun 2022 00:00:00 +0200 A remark on singular cohomology and sheaf cohomology https://www.mscand.dk/article/view/132191 <p>We prove a comparison isomorphism between singular cohomology and sheaf cohomology.</p> Dan Petersen Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132191 Sat, 11 Jun 2022 00:00:00 +0200 Stability of non-proper functions https://www.mscand.dk/article/view/132211 <p>The purpose of this paper is to give a sufficient condition for (strong) stability of non-proper smooth functions (with respect to the Whitney topology). We show that a Morse function is stable if it is end-trivial at any point in its discriminant, where end-triviality (which is also called local triviality at infinity) is a property concerning behavior of functions around the ends of the source manifolds. We further show that a Morse function is strongly stable if (and only if) it is quasi-proper. This result yields existence of a strongly stable but not infinitesimally stable function. Applying our result on stability, we give a sufficient condition for stability of Nash functions, and show that any Nash function becomes stable after a generic linear perturbation.</p> Kenta Hayano Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132211 Sat, 11 Jun 2022 00:00:00 +0200 The weak min-max property in Banach spaces https://www.mscand.dk/article/view/132214 <p>In this paper, we investigate the relationship between the weak min-max property and the diameter uniformity of domains in Banach spaces with dimension at least 2. As an application, we show that diameter uniform domains are invariant under relatively quasimöbius mappings.</p> Zhengyong Ouyang, Antti Rasila, Tiantian Guan Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132214 Sat, 11 Jun 2022 00:00:00 +0200 The Perfekt theory of $M$-ideals https://www.mscand.dk/article/view/132230 <p>We revisit some ideas of K.-M.&nbsp;Perfekt who has provided an elegant framework to detect the biduality between function or sequence spaces defined in terms of some $o$- respectively $O$-condition. We present new proofs under somewhat weaker assumptions than before and apply the result to Lipschitz spaces.</p> Dirk Werner Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132230 Sat, 11 Jun 2022 00:00:00 +0200 Geodesic families characterizing flat metrics on a cylinder and a plane https://www.mscand.dk/article/view/132247 <p>We prove that a complete non-compact surface contains a domain which is isometric to a pipe cylinder if all prime closed geodesics in it have the same length. As an application, we show that a flat cylinder is conjugacy rigid in the class of surfaces whose universal covering planes satisfy the divergence property. We study the divergence property from the view point of geodesic conjugacy for the Euclidean plane.</p> Nobuhiro Innami, Yoe Itokawa, Tetsuya Nagano, Katsuhiro Shiohama Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132247 Sat, 11 Jun 2022 00:00:00 +0200 Poincaré duality for tautological Chern subrings of orthogonal grassmannians https://www.mscand.dk/article/view/132376 <p>Let $X$ be an orthogonal grassmannian of a nondegenerate quadratic form $q$ over a field. Let $C$ be the subring in the Chow ring $\text {CH}(X)$ generated by the Chern classes of the tautological vector bundle on $X$. We prove Poincaré duality for $C$. For $q$ of odd dimension, the result was already known due to an identification between $C$ and the Chow ring of certain symplectic grassmannian. For $q$ of even dimension, such an identification is not available.</p> Nikita A. Karpenko, Alexander S. Merkurjev Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132376 Sat, 11 Jun 2022 00:00:00 +0200 Cover1 https://www.mscand.dk/article/view/132863 <p>Cover</p> Mathematica Scandinavica Copyright (c) 2022 MATHEMATICA SCANDINAVICA https://www.mscand.dk/article/view/132863 Sat, 11 Jun 2022 00:00:00 +0200