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

Impact Factor: 1.809

About this journal

August Leopold Crelle (1780 - 1855)

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 2022 Issue 793

December 2022

Publicly Available December 1, 2022

Frontmatter

Page range: i-iv

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September 29, 2022

Kähler–Einstein metrics with prescribed singularities on Fano manifolds

Antonio Trusiani

Page range: 1-57

Abstract

Given a Fano manifold ( X , ω ) {(X,\omega)} , we develop a variational approach to characterize analytically the existence of Kähler–Einstein metrics with prescribed singularities, assuming that these singularities can be approximated algebraically. Moreover, we define a function α ω {\alpha_{\omega}} on the set of prescribed singularities which generalizes Tian’s α-invariant, showing that its upper lever set { α ω ⁢ ( ⋅ ) > n n + 1 } {\{\alpha_{\omega}(\,\cdot\,)>\frac{n}{n+1}\}} produces a subset of the Kähler–Einstein locus, i.e. of the locus given by all prescribed singularities that admit Kähler–Einstein metrics. In particular, we prove that many K -stable manifolds admit all possible Kähler–Einstein metrics with prescribed singularities. Conversely, we show that enough positivity of the α-invariant function at nontrivial prescribed singularities (or other conditions) implies the existence of genuine Kähler–Einstein metrics. Finally, through a continuity method we also prove the strong continuity of Kähler–Einstein metrics on curves of totally ordered prescribed singularities when the relative automorphism groups are discrete.

September 29, 2022

Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

Rémi Reboulet

Page range: 59-103

Abstract

Given a polarized projective variety ( X , L ) {(X,L)} over any non-Archimedean field, assuming continuity of envelopes, we define a metric on the space of finite-energy metrics on L , related to a construction of Darvas in the complex setting. We show that this makes finite-energy metrics on L into a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous. Our results hold in particular in all situations relevant to the study of degenerations and K-stability in complex geometry.

October 27, 2022

Hermitian K-theory via oriented Gorenstein algebras

Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Maria Yakerson

Page range: 105-142

Abstract

We show that the hermitian K-theory space of a commutative ring R can be identified, up to 𝐀 1 {\mathbf{A}^{1}} -homotopy, with the group completion of the groupoid of oriented finite Gorenstein R -algebras, i.e., finite locally free R -algebras with trivialized dualizing sheaf. We deduce that hermitian K-theory is universal among generalized motivic cohomology theories with transfers along oriented finite Gorenstein morphisms. As an application, we obtain a Hilbert scheme model for hermitian K-theory as a motivic space. We also give an application to computational complexity: we prove that 1-generic minimal border rank tensors degenerate to the big Coppersmith–Winograd tensor.

October 25, 2022

Components of symmetric wide-matrix varieties

Jan Draisma, Rob Eggermont, Azhar Farooq

Page range: 143-184

Abstract

We show that if X n {X_{n}} is a variety of c × n {c\times n} -matrices that is stable under the group Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} of column permutations and if forgetting the last column maps X n {X_{n}} into X n - 1 {X_{n-1}} , then the number of Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} -orbits on irreducible components of X n {X_{n}} is a quasipolynomial in n for all sufficiently large n . To this end, we introduce the category of affine 𝐅𝐈 𝐨𝐩 {\mathbf{FI^{op}}} -schemes of width one, review existing literature on such schemes, and establish several new structural results about them. In particular, we show that under a shift and a localisation, any width-one 𝐅𝐈 𝐨𝐩 {\mathbf{FI^{op}}} -scheme becomes of product form, where X n = Y n {X_{n}=Y^{n}} for some scheme Y in affine c -space. Furthermore, to any 𝐅𝐈 𝐨𝐩 {\mathbf{FI^{op}}} -scheme of width one we associate a component functor from the category 𝐅𝐈 {\mathbf{FI}} of finite sets with injections to the category 𝐏𝐅 {\mathbf{PF}} of finite sets with partially defined maps. We present a combinatorial model for these functors and use this model to prove that Sym ⁡ ( [ n ] ) {\operatorname{Sym}([n])} -orbits of components of X n {X_{n}} , for all n , correspond bijectively to orbits of a groupoid acting on the integral points in certain rational polyhedral cones. Using the orbit-counting lemma for groupoids and theorems on quasipolynomiality of lattice point counts, this yields our Main Theorem. We present applications of our methods to counting fixed-rank matrices with entries in a prescribed set and to counting linear codes over finite fields up to isomorphism.

October 27, 2022

Tangent curves to degenerating hypersurfaces

Lawrence Jack Barrott, Navid Nabijou

Page range: 185-224

Abstract

We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov–Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov–Witten invariants, encoding the degeneration behaviour of tangent curves. In the example of a smooth plane cubic degenerating to the toric boundary we employ localisation and tropical techniques to compute these refinements. Finally, we leverage these calculations to describe how embedded curves tangent to a smooth cubic degenerate as the cubic does; the results obtained are of a classical nature, but the proofs make essential use of logarithmic Gromov–Witten theory.

November 11, 2022

Local noncollapsing for complex Monge–Ampère equations

Bin Guo, Jian Song

Page range: 225-238

Abstract

We prove a local volume noncollapsing estimate for Kähler metrics induced from a family of complex Monge–Ampère equations, assuming a local Ricci curvature lower bound. This local volume estimate can be applied to establish various diameter and gradient estimate.

November 11, 2022

Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry

John Man Shun Ma, Ali Muhammad, Niels Martin Møller

Page range: 239-259

Abstract

In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 {\mathbb{R}^{n+1}} . First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry.

November 11, 2022

Hitting estimates on Einstein manifolds and applications

Beomjun Choi, Robert Haslhofer

Page range: 261-280

Abstract

We generalize the Benjamini–Pemantle–Peres estimate relating hitting probability and Martin capacity to the setting of manifolds with Ricci curvature bounded below. As applications we obtain: (1) a sharp estimate for the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersection estimate for two independent Brownian motions on manifolds with nonnegative Ricci curvature and positive asymptotic volume ratio. We also obtain generalizations of (1) and (2) for the manifolds with two-sided Ricci bounds and Einstein manifolds with nonzero Einstein constant.

November 23, 2022

Special values of L-functions of one-motives over function fields

Thomas H. Geisser, Takashi Suzuki

Page range: 281-304

Abstract

The purpose of this paper is to give a formula for the leading coefficient at s = 1 {s=1} of the L -function of one-motives over function fields in terms of Weil-étale cohomology, generalizing the Weil-étale version of the Birch and Swinnerton-Dyer conjecture in the authors’ previous work. As a consequence we express the Tamagawa number of a torus introduced by Ono–Oesterlé in terms of Weil-étale cohomology, and reprove their Tamagawa number formula.

Ahead of Print / Just Accepted

Ahead of Print / Just Accepted

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5-year Impact Factor	1.868
Cite Score	3.1
Impact Factor	1.809
Journal Citation Indicator	1.92
Mathematical Citation Quotient	1.44
SCImago Journal Rank	2.424
Source Normalized Impact per Paper	1.916

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Each paper should include a short but informative abstract, as well as the Mathematics Subject Classification 2010 representing the primary and secondary subjects of the article
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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