Journal für die reine und angewandte Mathematik

Published by De Gruyter

Volume 2019 Issue 756

Issue of Journal für die reine und angewandte Mathematik

Contents
Journal Overview

Publicly Available November 1, 2019

Frontmatter

Page range: i-iv

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July 19, 2017

𝔸-curves on log smooth varieties

Qile Chen, Yi Zhu

Page range: 1-35

Abstract

In this paper, we study 𝔸1{\mathbb{A}^{1}}-connected varieties from log geometry point of view, and prove a criterion for 𝔸1{\mathbb{A}^{1}}-connectedness. As applications, we provide many interesting examples of 𝔸1{\mathbb{A}^{1}}-connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne conjecture characterizing projective spaces and affine spaces.

June 8, 2017

Generating monotone quantities for the heat equation

Jonathan Bennett, Neal Bez

Page range: 37-63

Abstract

The purpose of this article is to expose and further develop a simple yet surprisingly far-reaching framework for generating monotone quantities for positive solutions to linear heat equations in euclidean space. This framework is intimately connected to the existence of a rich variety of algebraic closure properties of families of sub/super-solutions, and more generally solutions of systems of differential inequalities capturing log-convexity properties such as the Li–Yau gradient estimate. Various applications are discussed, including connections with the general Brascamp–Lieb inequality and the Ornstein–Uhlenbeck semigroup.

April 19, 2017

On the global Gan–Gross–Prasad conjecture for unitary groups: Approximating smooth transfer of Jacquet–Rallis

Hang Xue

Page range: 65-100

Abstract

Zhang proved the global Gan–Gross–Prasad conjecture for U⁡(n+1)×U⁡(n)\operatorname{U}(n+1)\times\operatorname{U}(n) under some local conditions [19]. One of the conditions is that the unitary groups are split at the archimedean places. We remove this assumption at the archimedean places in this paper.

April 6, 2017

Poisson manifolds of compact types (PMCT 1)

Marius Crainic, Rui Loja Fernandes, David Martínez Torres

Page range: 101-149

Abstract

This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G -spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes.

September 7, 2017

A groupoid approach to pseudodifferential calculi

Erik van Erp, Robert Yuncken

Page range: 151-182

Abstract

In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural ℝ+×{\mathbb{R}^{\times}_{+}}-action. Specifically, a properly supported semiregular distribution on M×M{M\times M} is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibers of the tangent groupoid that is homogeneous for the ℝ+×{\mathbb{R}^{\times}_{+}}-action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

January 24, 2018

Duality and socle generators for residual intersections

David Eisenbud, Bernd Ulrich

Page range: 183-226

Abstract

We prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt. Suppose that I is an ideal of codimension g in a Gorenstein ring, and J⊂I{J\subset I} is an ideal with s=g+t{s=g+t} generators such that K:=J:I{K:=J:I} has codimension s . Let I¯{{\overline{I}}} be the image of I in R¯:=R/K{{\overline{R}}:=R/K}. In the first part of the paper we prove, among other things, that under suitable hypotheses on I , the truncated Rees ring R¯⊕I¯⊕⋯⊕I¯t+1{{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules I¯u{{\overline{I}}{}^{u}} and I¯t+1-u{{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into I¯≅t+1ωR¯{{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}. In the second part of the paper we study the analogue of residue theory, and prove that, when R/K{R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of It+1/J⁢It≅ωR/K{I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

May 13, 2017

Lower semicontinuity of mass under C0 convergence and Huisken’s isoperimetric mass

Jeffrey L. Jauregui, Dan A. Lee

Page range: 227-257

Abstract

Given a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed C0{C^{0}} Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken’s isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken’s work before explaining our extension of that work. The results are all specific to three dimensions.

June 3, 2017

Gravitational instantons with faster than quadratic curvature decay (II)

Gao Chen, Xiuxiong Chen

Page range: 259-284

Abstract

This is our second paper in a series to study gravitational instantons, i.e. complete hyperkähler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: (i) The asymptotic rate of gravitational instantons to the standard models can be improved automatically. (ii) Any ALF-DkD_{k} gravitational instanton must be the Cherkis–Hitchin–Ivanov–Kapustin–Lindström–Roček metric.

April 20, 2017

Normal subgroups of invertibles and of unitaries in a C*-algebra

Leonel Robert

Page range: 285-319

Abstract

We investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a C*{\mathrm{C}^{*}}-algebra. By relating normal subgroups to closed two-sided ideals we obtain a “sandwich condition” describing all the closed normal subgroups both in the invertible and in the unitary case. We use this to prove a conjecture by Elliott and Rørdam: in a simple C*\mathrm{C}^{*}-algebra, the group of approximately inner automorphisms induced by unitaries in the connected component of the identity is topologically simple. Turning to non-closed subgroups, we show, among other things, that in a simple unital C*\mathrm{C}^{*}-algebra the commutator subgroup of the group of invertibles in the connected component of the identity is a simple group modulo its center. A similar result holds for unitaries under a mild extra assumption.

About this journal

The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of "Crelle's Journal". In the 190 years of its existence, "Crelle's Journal" has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals. It belongs to the very top mathematics periodicals, as listed in ISI's Journal Citation Report.