Published since January 1, 1826

Journal für die reine und angewandte Mathematik

ISSN: 1435-5345

Edited by: Tobias Colding, Ana Caraiani, Jun-Muk Hwang, Olivier Schiffmann, Jakob Stix, Gábor Székelyhidi

Managing Editor: Daniel Huybrechts

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.

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Supplementary Materials

Topics

Volume 2023 Issue 799

June 2023

Publicly Available June 1, 2023

Frontmatter

Page range: i-iv

Download PDF

April 20, 2023

Stable pair compactification of moduli of K3 surfaces of degree 2

Valery Alexeev, Philip Engel, Alan Thompson

Page range: 1-56

Abstract

We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable KSBA pairs ( X , ϵ ⁢ R ) (X,\epsilon R) over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.

April 20, 2023

Geometric arcs and fundamental groups of rigid spaces

Piotr Achinger, Marcin Lara, Alex Youcis

Page range: 57-107

Abstract

We develop the notion of a geometric covering of a rigid space 𝑋, which yields a larger class of covering spaces than that studied previously by de Jong. Geometric coverings are closed under disjoint unions and are étale local on 𝑋. If 𝑋 is connected, its geometric coverings form a tame infinite Galois category and hence are classified by a topological group. The definition is based on the property of lifting of “geometric arcs” and is meant to be an analogue of the notion developed for schemes by Bhatt and Scholze.

April 27, 2023

Free boundary regularity in the multiple membrane problem in the plane

Ovidiu Savin, Hui Yu

Page range: 109-154

Abstract

We study the regularity of free boundaries in the multiple elastic membrane problem in the plane. We prove the uniqueness of blow-ups, and that the free boundaries are C 1 , log {C^{1,\log}} -curves near a regular intersection point.

April 29, 2023

𝐾-invariant Hilbert modules and singular vector bundles on bounded symmetric domains

Harald Upmeier

Page range: 155-187

Abstract

We show that the “eigenbundle” (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank 𝑟 is a “singular” vector bundle (linearly fibred complex analytic space) which decomposes as a stratified sum of homogeneous vector bundles along a canonical stratification of length r + 1 r+1 . The fibres are realized in terms of representation theory on the normal space of the strata.

May 6, 2023

Hodge theory on ALG^∗ manifolds

Gao Chen, Jeff Viaclovsky, Ruobing Zhang

Page range: 189-227

Abstract

We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG ∗ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG ∗ manifolds with non-negative Ricci curvature having group Γ = { e } \Gamma=\{e\} at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG ∗ manifold. A corollary of this is vanishing of the first Betti number for any ALG ∗ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG ∗ gravitational instantons.

April 27, 2023

Constant scalar curvature Kähler metrics on ramified Galois coverings

Claudio Arezzo, Alberto Della Vedova, Yalong Shi

Page range: 229-247

Abstract

We give sufficient conditions for the existence of Kähler–Einstein and constant scalar curvature Kähler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kähler classes and the branching divisor. This result generalizes previous work on Kähler–Einstein metrics by Li and Sun [C. Li and S. Sun, Conical Kähler–Einstein metrics revisited, Comm. Math. Phys. 331 2014, 3, 927–973], and extends Chen–Cheng’s existence results for cscK metrics in [X. Chen and J. Cheng, On the constant scalar curvature Kähler metrics (II)—Existence results, J. Amer. Math. Soc. 34 2021, 4, 937–1009].

April 30, 2023

On fundamental groups of RCD spaces

Jaime Santos-Rodríguez, Sergio Zamora-Barrera

Page range: 249-286

Abstract

We obtain results about fundamental groups of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed K ∈ ℝ {K\in\mathbb{R}} , N ∈ [ 1 , ∞ ) {N\in[1,\infty)} , D > 0 {D>0} , we show the following: • There is C > 0 {C>0} such that for each RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} space X of diameter ≤ D {\leq D} , its fundamental group π 1 ⁢ ( X ) {\pi_{1}(X)} is generated by at most C elements. • There is D ~ > 0 {\tilde{D}>0} such that for each RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} space X of diameter ≤ D {\leq D} with compact universal cover X ~ {\tilde{X}} , one has diam ⁡ ( X ~ ) ≤ D ~ {\operatorname{diam}(\tilde{X})\leq\tilde{D}} . • If a sequence of RCD ∗ ⁢ ( 0 , N ) {\mathrm{RCD}^{\ast}(0,N)} spaces X i {X_{i}} of diameter ≤ D {\leq D} and rectifiable dimension n is such that their universal covers X ~ i {\tilde{X}_{i}} converge in the pointed Gromov–Hausdorff sense to a space X of rectifiable dimension n , then there is C > 0 {C>0} such that for each i , the fundamental group π 1 ⁢ ( X i ) {\pi_{1}(X_{i})} contains an abelian subgroup of index ≤ C {\leq C} . • If a sequence of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces X i {X_{i}} of diameter ≤ D {\leq D} and rectifiable dimension n is such that their universal covers X ~ i {\tilde{X}_{i}} are compact and converge in the pointed Gromov–Hausdorff sense to a space X of rectifiable dimension n , then there is C > 0 {C>0} such that for each i , the fundamental group π 1 ⁢ ( X i ) {\pi_{1}(X_{i})} contains an abelian subgroup of index ≤ C {\leq C} . • If a sequence of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces X i {X_{i}} with first Betti number ≥ r {\geq r} and rectifiable dimension n converges in the Gromov–Hausdorff sense to a compact space X of rectifiable dimension m , then the first Betti number of X is at least r + m - n {r+m-n} . The main tools are the splitting theorem by Gigli, the splitting blow-up property by Mondino and Naber, the semi-locally-simple-connectedness of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces by Wang, the isometry group structure by Guijarro and the first author, and the structure of approximate subgroups by Breuillard, Green and Tao.

May 23, 2023

Geodesic nets on non-compact Riemannian manifolds

Gregory R. Chambers, Yevgeny Liokumovich, Alexander Nabutovsky, Regina Rotman

Page range: 287-303

Abstract

A geodesic flower is a finite collection of geodesic loops based at the same point 𝑝 that satisfy the following balancing condition: the sum of all unit tangent vectors to all geodesic arcs meeting at 𝑝 is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that, in every complete non-compact manifold with locally convex ends, there exists a non-trivial geodesic flower.

Ahead of Print / Just Accepted

Ahead of Print / Just Accepted

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[3] H. Hasse, Symmetrische Matrizen im Körper der reellen Zahlen, J. reine angew. Math. 153 (1923), 12–43.
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Editorial

Editors

Tobias H. Colding; Department of Mathematics, Massachusetts Institute of Technology; 77 Massachusetts Avenue, Cambridge, MA 02139-4307; USA; email: journals@math.mit.edu

Ana Caraiani; Mathematisches Institut, Universität Bonn; Endenicher Allee 60, 53115 Bonn; Germany; email: caraiani@math.uni-bonn.de

Daniel Huybrechts; Mathematisches Institut, Universität Bonn; Endenicher Allee 60, 53115 Bonn; Germany; email: huybrech@math.uni-bonn.de

Jun-Muk Hwang; Institute for Basic Science, Center for Complex Geometry, Daejeon, Korea (South); email: jmhwang@ibs.re.kr

Olivier Schiffmann; Institut de Mathématiques d’Orsay, Faculté des Sciences d’Orsay, Bat. 307, 91405 Orsay Cedex, France; email: olivier.schiffmann@cnrs.fr

Jakob Stix; Institut für Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany; email: stix.crelle@math.uni-frankfurt.de

Gábor Székelyhidi; Mathematics Department, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208; USA; email: gaborsz@northwestern.edu

Managing Editor

Daniel Huybrechts; Mathematisches Institut, Universität Bonn; Endenicher Allee 60, 53115 Bonn; Germany; email: crelle@math.uni-bonn.de

(Deutsch)

Online ISSN: 1435-5345
Print ISSN: 0075-4102
Type: Journal
Languages: English, German, French
Publisher: De Gruyter
First published: January 1, 1826
Publication Frequency: 12 Issues per Year
Audience: Researchers interested in pure and applied mathematics

Journal für die reine und angewandte Mathematik

Frontmatter

Stable pair compactification of moduli of K3 surfaces of degree 2

Abstract

Geometric arcs and fundamental groups of rigid spaces

Abstract

Free boundary regularity in the multiple membrane problem in the plane

Abstract

𝐾-invariant Hilbert modules and singular vector bundles on bounded symmetric domains

Abstract

Hodge theory on ALG∗ manifolds

Abstract

Constant scalar curvature Kähler metrics on ramified Galois coverings

Abstract

On fundamental groups of RCD spaces

Abstract

Geodesic nets on non-compact Riemannian manifolds

Abstract

Hodge theory on ALG^∗ manifolds