Pub Date : 2026-01-01DOI: 10.1016/j.indag.2025.03.002
Luca Barbieri-Viale
A mixed Weil cohomology with values in an abelian rigid tensor category is a cohomological functor on Voevodsky’s category of motives which is satisfying Künneth formula and such that its restriction to Chow motives is a Weil cohomology. We show that the universal mixed Weil cohomology exists. Nori motives can be recovered as a universal enrichment of Betti cohomology via a localisation. This new picture is drawing some consequences with respect to the theory of mixed motives in arbitrary characteristic.
{"title":"Mixed motives","authors":"Luca Barbieri-Viale","doi":"10.1016/j.indag.2025.03.002","DOIUrl":"10.1016/j.indag.2025.03.002","url":null,"abstract":"<div><div>A mixed Weil cohomology with values in an abelian rigid tensor category is a cohomological functor on Voevodsky’s category of motives which is satisfying Künneth formula and such that its restriction to Chow motives is a Weil cohomology. We show that the universal mixed Weil cohomology exists. Nori motives can be recovered as a universal enrichment of Betti cohomology via a localisation. This new picture is drawing some consequences with respect to the theory of mixed motives in arbitrary characteristic.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 399-420"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.07.010
Ben Moonen
We show that the results proven by Deninger and Murre in their paper (Deninger and Murre, 1991) directly imply that the Chern classes of the de Rham bundle of an abelian scheme are torsion elements in the Chow ring, a result that was later proven by van der Geer. We also discuss several results about the orders of these classes.
我们证明了Deninger和Murre在他们的论文中证明的结果（Deninger和Murre， 1991）直接暗示了阿贝尔格式的de Rham束的Chern类是Chow环中的扭转元素，这一结果后来被van der Geer证明。我们还讨论了关于这些类的阶数的几个结果。
{"title":"A remark on the paper of Deninger and Murre","authors":"Ben Moonen","doi":"10.1016/j.indag.2024.07.010","DOIUrl":"10.1016/j.indag.2024.07.010","url":null,"abstract":"<div><div>We show that the results proven by Deninger and Murre in their paper (Deninger and Murre, 1991) directly imply that the Chern classes of the de Rham bundle of an abelian scheme are torsion elements in the Chow ring, a result that was later proven by van der Geer. We also discuss several results about the orders of these classes.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 149-157"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141844337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.11.004
Dinakar Ramakrishnan
Let be an elliptic curve over a number field , the abelian surface , and the F-rational albanese kernel of , which is a subgroup of the degree zero part of Chow group of zero cycles on modulo rational equivalence. The first result is that for all but a finite number of primes where has ordinary reduction, the image of in the Galois cohomology group is zero; here denotes as usual the Galois module of p-division points on . The second result is that for any prime where has good ordinary reduction, there is a finite extension of , depending on and , such that is non-zero. Much of this work was joint with Jacob Murre, and the article is dedicated to his memory.
{"title":"Global Galois symbols on E×E","authors":"Dinakar Ramakrishnan","doi":"10.1016/j.indag.2024.11.004","DOIUrl":"10.1016/j.indag.2024.11.004","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi></math></span> be an elliptic curve over a number field <span><math><mi>F</mi></math></span>, <span><math><mi>A</mi></math></span><span> the abelian surface </span><span><math><mrow><mi>E</mi><mo>×</mo><mi>E</mi></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span> the F-rational albanese kernel of <span><math><mi>A</mi></math></span>, which is a subgroup of the degree zero part of Chow group of zero cycles on <span><math><mi>A</mi></math></span> modulo rational equivalence. The first result is that for all but a finite number of primes <span><math><mi>p</mi></math></span> where <span><math><mi>E</mi></math></span> has ordinary reduction, the image of <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>/</mo><mi>p</mi></mrow></math></span> in the Galois cohomology group <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>F</mi><mo>,</mo><msup><mrow><mi>sym</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>E</mi><mrow><mo>[</mo><mi>p</mi><mo>]</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is zero; here <span><math><mrow><mi>E</mi><mrow><mo>[</mo><mi>p</mi><mo>]</mo></mrow></mrow></math></span> denotes as usual the Galois module of p-division points on <span><math><mi>E</mi></math></span>. The second result is that for any prime <span><math><mi>p</mi></math></span> where <span><math><mi>E</mi></math></span> has good ordinary reduction, there is a finite extension <span><math><mi>K</mi></math></span> of <span><math><mi>F</mi></math></span>, depending on <span><math><mi>p</mi></math></span> and <span><math><mi>E</mi></math></span>, such that <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>/</mo><mi>p</mi></mrow></math></span> is non-zero. Much of this work was joint with Jacob Murre, and the article is dedicated to his memory.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 294-302"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.10.005
Finn Bartsch, Ariyan Javanpeykar
In this short survey, we explain Parshin’s proof of the geometric Bombieri–Lang conjecture, and show that it can be used to give an alternative proof of Xie–Yuan’s recent resolution of the geometric Bombieri–Lang conjecture for projective varieties with empty special locus and admitting a finite morphism to a traceless abelian variety.
{"title":"Parshin’s method and the geometric Bombieri–Lang conjecture","authors":"Finn Bartsch, Ariyan Javanpeykar","doi":"10.1016/j.indag.2024.10.005","DOIUrl":"10.1016/j.indag.2024.10.005","url":null,"abstract":"<div><div>In this short survey, we explain Parshin’s proof of the geometric Bombieri–Lang conjecture, and show that it can be used to give an alternative proof of Xie–Yuan’s recent resolution of the geometric Bombieri–Lang conjecture for projective varieties with empty special locus and admitting a finite morphism to a traceless abelian variety.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 271-280"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2025.07.006
F. Déglise
This paper investigates the structure of generic motives and their implications for the motivic cohomology of fields. Originating in Voevodsky’s theory of motives and related to Beilinson’s vision of a motivic -structure, generic motives serve as pro-objects encoding essential information about cycles and cohomology. We present new computations of generic motives, focusing on curves and surfaces. These computations suggest a conjectural framework for morphisms of generic motives and highlight the central role of transcendental motives. We then focus on the motivic cohomology of fields, building on Borel’s rank computation of K-theory and its relation to higher regulators. We provide a direct argument for determining the weights in the -structure of the K-theory of number fields, bypassing the need for regulator maps. We show that motivic cohomology groups are often of infinite rank, typically matching the cardinality of the base field. For instance, we prove that motivic cohomology groups of and are uncountable in many bi-degrees. Despite this, we propose a conjecture that complements the Beilinson–Soulé vanishing conjecture, suggesting that the growth of motivic cohomology is more controlled than these results may initially indicate.
{"title":"Generic motives and motivic cohomology of fields","authors":"F. Déglise","doi":"10.1016/j.indag.2025.07.006","DOIUrl":"10.1016/j.indag.2025.07.006","url":null,"abstract":"<div><div>This paper investigates the structure of generic motives and their implications for the motivic cohomology of fields. Originating in Voevodsky’s theory of motives and related to Beilinson’s vision of a motivic <span><math><mi>t</mi></math></span>-structure, generic motives serve as pro-objects encoding essential information about cycles and cohomology. We present new computations of generic motives, focusing on curves and surfaces. These computations suggest a conjectural framework for morphisms of generic motives and highlight the central role of transcendental motives. We then focus on the motivic cohomology of fields, building on Borel’s rank computation of K-theory and its relation to higher regulators. We provide a direct argument for determining the weights in the <span><math><mi>λ</mi></math></span>-structure of the K-theory of number fields, bypassing the need for regulator maps. We show that motivic cohomology groups are often of infinite rank, typically matching the cardinality of the base field. For instance, we prove that motivic cohomology groups of <span><math><mi>R</mi></math></span> and <span><math><mi>ℂ</mi></math></span> are uncountable in many bi-degrees. Despite this, we propose a conjecture that complements the Beilinson–Soulé vanishing conjecture, suggesting that the growth of motivic cohomology is more controlled than these results may initially indicate.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 421-487"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.04.007
Bruno Kahn
We develop Milne’s theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne’s case-by-case approach.
{"title":"Chow–Lefschetz motives","authors":"Bruno Kahn","doi":"10.1016/j.indag.2024.04.007","DOIUrl":"10.1016/j.indag.2024.04.007","url":null,"abstract":"<div><div>We develop Milne’s theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne’s case-by-case approach.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 5-24"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.05.007
Christopher Deninger
Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space to every scheme . We also define -valued points of for every commutative ring . For normal schemes of finite type over , using we construct infinite dimensional -dynamical systems whose periodic orbits are related to the closed points of . Various aspects of these topological dynamical systems are studied. We also explain how certain -adic points of for the spectrum of a -adic local number ring are related to the points of the Fargues–Fontaine curve.
{"title":"Dynamical systems for arithmetic schemes","authors":"Christopher Deninger","doi":"10.1016/j.indag.2024.05.007","DOIUrl":"10.1016/j.indag.2024.05.007","url":null,"abstract":"<div><div>Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>rat</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> to every scheme <span><math><mi>X</mi></math></span>. We also define <span><math><mi>R</mi></math></span>-valued points <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>rat</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>rat</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> for every commutative ring <span><math><mi>R</mi></math></span>. For normal schemes <span><math><mi>X</mi></math></span> of finite type over <span><math><mrow><mi>spec</mi><mspace></mspace><mi>Z</mi></mrow></math></span>, using <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>rat</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>ℂ</mi><mo>)</mo></mrow></mrow></math></span> we construct infinite dimensional <span><math><mi>R</mi></math></span>-dynamical systems whose periodic orbits are related to the closed points of <span><math><mi>X</mi></math></span>. Various aspects of these topological dynamical systems are studied. We also explain how certain <span><math><mi>p</mi></math></span>-adic points of <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>rat</mi></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> for <span><math><mi>X</mi></math></span> the spectrum of a <span><math><mi>p</mi></math></span>-adic local number ring are related to the points of the Fargues–Fontaine curve.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 25-136"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141513686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.09.008
Luca Barbieri-Viale
Making a survey of recent constructions of universal cohomologies we suggest a new framework for a theory of motives in algebraic geometry.
通过对近年来普遍上同调构造的综述，我们提出了代数几何动机理论的一个新框架。
{"title":"Motives","authors":"Luca Barbieri-Viale","doi":"10.1016/j.indag.2024.09.008","DOIUrl":"10.1016/j.indag.2024.09.008","url":null,"abstract":"<div><div>Making a survey of recent constructions of universal cohomologies we suggest a new framework for a theory of motives in algebraic geometry.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 178-194"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.10.004
Robert Laterveer , Charles Vial
The Chow rings of hyper-Kähler varieties are conjectured to have a particularly rich structure. In this paper, we formulate a conjecture that combines the Beauville–Voisin conjecture regarding the subring generated by divisors and the Franchetta conjecture regarding generically defined cycles. As motivation, we show that this Beauville–Voisin–Franchetta conjecture for a hyper-Kähler variety follows from a combination of Grothendieck’s standard conjectures for a very general deformation of , Murre’s conjecture (D) for and the Franchetta conjecture for . As evidence, beyond the case of Fano varieties of lines on smooth cubic fourfolds, we show that this conjecture holds for codimension-2 and codimension-8 cycles on Lehn–Lehn–Sorger–van Straten eightfolds. Moreover, we establish that the subring of the Chow ring generated by primitive divisors injects into cohomology.
{"title":"The Beauville–Voisin–Franchetta conjecture and LLSS eightfolds","authors":"Robert Laterveer , Charles Vial","doi":"10.1016/j.indag.2024.10.004","DOIUrl":"10.1016/j.indag.2024.10.004","url":null,"abstract":"<div><div>The Chow rings of hyper-Kähler varieties are conjectured to have a particularly rich structure. In this paper, we formulate a conjecture that combines the Beauville–Voisin conjecture regarding the subring generated by divisors and the Franchetta conjecture regarding generically defined cycles. As motivation, we show that this Beauville–Voisin–Franchetta conjecture for a hyper-Kähler variety <span><math><mi>X</mi></math></span> follows from a combination of Grothendieck’s standard conjectures for a very general deformation of <span><math><mi>X</mi></math></span>, Murre’s conjecture (D) for <span><math><mi>X</mi></math></span> and the Franchetta conjecture for <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span>. As evidence, beyond the case of Fano varieties of lines on smooth cubic fourfolds, we show that this conjecture holds for codimension-2 and codimension-8 cycles on Lehn–Lehn–Sorger–van Straten eightfolds. Moreover, we establish that the subring of the Chow ring generated by primitive divisors injects into cohomology.</span></div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 250-270"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-01DOI: 10.1016/j.indag.2024.12.002
Claire Voisin
We introduce and study the notion of universally defined cycles of smooth varieties of dimension , and prove that they are given by polynomials in the Chern classes. A similar result is proved for universally defined cycles on products of smooth varieties. We also state a conjectural explicit form for universally defined cycles on powers of smooth varieties, and provide some steps towards establishing it.
{"title":"Universally defined cycles I","authors":"Claire Voisin","doi":"10.1016/j.indag.2024.12.002","DOIUrl":"10.1016/j.indag.2024.12.002","url":null,"abstract":"<div><div>We introduce and study the notion of universally defined cycles of smooth varieties of dimension <span><math><mi>d</mi></math></span>, and prove that they are given by polynomials in the Chern classes. A similar result is proved for universally defined cycles on products of smooth varieties. We also state a conjectural explicit form for universally defined cycles on powers of smooth varieties, and provide some steps towards establishing it.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"37 1","pages":"Pages 376-398"},"PeriodicalIF":0.8,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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