Published since January 1, 1826

Journal für die reine und angewandte Mathematik

ISSN: 1435-5345

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

Managing Editor: Daniel Huybrechts

Impact Factor: 1.713

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 787

June 2022

Publicly Available June 1, 2022

Frontmatter

Page range: i-iv

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April 20, 2022

Artin–Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge–Witt decompositions and Chow groups of quasi-F-split threefolds

Yukiyoshi Nakkajima

Page range: 1-44

Abstract

Let X/s{X/s} be a proper log smooth scheme of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field κ of characteristic p>0{p>0}. In this article we prove that the cohomology of 𝒲⁢(𝒪X){{\mathcal{W}}({\mathcal{O}}_{X})} is a finitely generated 𝒲⁢(κ){{\mathcal{W}}(\kappa)}-module if the Yobuko height of X is finite. As an application of this result, we prove that, if the Yobuko height of a proper smooth threefold Y over κ is finite, then the crystalline cohomology of Y/κ{Y/\kappa} has the Hodge–Witt decomposition and the p -primary torsion part of the Chow group of codimension 2 of Y is of finite cotype. These are nontrivial generalizations of results in [K. Joshi and C. S. Rajan, Frobenius splitting and ordinarity, Int. Math. Res. Not. IMRN 2003 2003, 2, 109–121] and [K. Joshi, Exotic torsion, Frobenius splitting and the slope spectral sequence, Canad. Math. Bull. 50 2007, 4, 567–578]. We also prove a fundamental inequality between the Artin–Mazur heights and the Yobuko height of X/s{X/s} if X/s{X/s} satisfies natural conditions.

April 20, 2022

Kazhdan–Lusztig conjecture via zastava spaces

Alexander Braverman, Michael Finkelberg, Hiraku Nakajima

Page range: 45-78

Abstract

We deduce the Kazhdan–Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category 𝒪{\mathcal{O}} from the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and 𝒲{\mathscr{W}}-algebras.

April 1, 2022

Monodromic model for Khovanov–Rozansky homology

Roman Bezrukavnikov, Kostiantyn Tolmachov

Page range: 79-124

Abstract

We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild cohomology groups (for the zero and the top degree cohomology this reduces to an earlier result of Gorsky, Hogancamp, Mellit and Nakagane). These objects turn out to be closely related to explicit character sheaves corresponding to exterior powers of the reflection representation of the Weyl group. Applying the described functors to the images of braids in the Hecke category of type A we obtain a geometric description for Khovanov–Rozansky knot homology, essentially different from the one considered earlier by Webster and Williamson.

April 20, 2022

Hyperbolic secant varieties of M-curves

Mario Kummer, Rainer Sinn

Page range: 125-162

Abstract

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1.

April 20, 2022

On the sharp lower bounds of modular invariants and fractional Dehn twist coefficients

Xiao-Lei Liu, Sheng-Li Tan

Page range: 163-195

Abstract

Modular invariants of families of curves are Arakelov invariants in arithmetic algebraic geometry. All the known uniform lower bounds of these invariants are not sharp. In this paper, we aim to give explicit lower bounds of modular invariants of families of curves, which is sharp for genus 2. According to the relation between fractional Dehn twists and modular invariants, we give the sharp lower bounds of fractional Dehn twist coefficients and classify pseudo-periodic maps with minimal coefficients for genus 2 and 3 firstly. We also obtain a rigidity property for families with minimal modular invariants, and other applications.

Open Access April 1, 2022

Degeneration of curves on some polarized toric surfaces

Karl Christ, Xiang He, Ilya Tyomkin

Page range: 197-240

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Abstract

We address the following question: Given a polarized toric surface (S,L){(S,{\mathcal{L}})}, and a general integral curve C of geometric genus g in the linear system |L|{|{\mathcal{L}}|}, do there exist degenerations of C in |L|{|{\mathcal{L}}|} to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h -transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show the irreducibility of Severi varieties of a large class of polarized toric surfaces with h -transverse polygon.

April 20, 2022

Noether–Severi inequality and equality for irregular threefolds of general type

Yong Hu, Tong Zhang

Page range: 241-273

Abstract

We prove the optimal Noether–Severi inequality that vol⁡(X)≥43⁢χ⁢(ωX){\operatorname{vol}(X)\geq\frac{4}{3}\chi(\omega_{X})} for all smooth and irregular 3-folds X of general type over ℂ{\mathbb{C}}. For those 3-folds X attaining the equality, we completely describe their canonical models and show that the topological fundamental group π1⁢(X)≃ℤ2{\pi_{1}(X)\simeq\mathbb{Z}^{2}}. As a corollary, we obtain for the same X another optimal inequality that vol⁡(X)≥43⁢ha0⁢(X,KX){\operatorname{vol}(X)\geq\frac{4}{3}h^{0}_{a}(X,K_{X})} where ha0⁢(X,KX){h^{0}_{a}(X,K_{X})} stands for the continuous rank of KX{K_{X}}, and we show that X attains this equality if and only if vol⁡(X)=43⁢χ⁢(ωX){\operatorname{vol}(X)=\frac{4}{3}\chi(\omega_{X})}.

Publicly Available June 2, 2022

Erratum to Table of Contents (J. reine angew. Math. 785 (2022), i–iv)

Page range: 275-275

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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

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; Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556, USA; email: gszekely@nd.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