Over an algebraically closed field of characteristic , we prove that three-dimensional -factorial affine klt varieties are quasi--split. Furthermore, we show that the bound on the characteristic is optimal.
{"title":"Quasi-F-splittings in birational geometry II","authors":"Tatsuro Kawakami, Teppei Takamatsu, Hiromu Tanaka, Jakub Witaszek, Fuetaro Yobuko, Shou Yoshikawa","doi":"10.1112/plms.12593","DOIUrl":"https://doi.org/10.1112/plms.12593","url":null,"abstract":"Over an algebraically closed field of characteristic <span data-altimg=\"/cms/asset/a0d7faab-7168-4a9c-b74e-972a24c48863/plms12593-math-0001.png\"></span><math altimg=\"urn:x-wiley:00246115:media:plms12593:plms12593-math-0001\" display=\"inline\" location=\"graphic/plms12593-math-0001.png\">\u0000<semantics>\u0000<mrow>\u0000<mi>p</mi>\u0000<mo>></mo>\u0000<mn>41</mn>\u0000</mrow>\u0000$p&gt;41$</annotation>\u0000</semantics></math>, we prove that three-dimensional <span data-altimg=\"/cms/asset/cf78c1f3-24c3-4fec-9668-03f7c418e99c/plms12593-math-0002.png\"></span><math altimg=\"urn:x-wiley:00246115:media:plms12593:plms12593-math-0002\" display=\"inline\" location=\"graphic/plms12593-math-0002.png\">\u0000<semantics>\u0000<mi mathvariant=\"double-struck\">Q</mi>\u0000$mathbb {Q}$</annotation>\u0000</semantics></math>-factorial affine klt varieties are quasi-<span data-altimg=\"/cms/asset/7b306e81-fd40-4a25-ba07-064f6dd57036/plms12593-math-0003.png\"></span><math altimg=\"urn:x-wiley:00246115:media:plms12593:plms12593-math-0003\" display=\"inline\" location=\"graphic/plms12593-math-0003.png\">\u0000<semantics>\u0000<mi>F</mi>\u0000$F$</annotation>\u0000</semantics></math>-split. Furthermore, we show that the bound on the characteristic is optimal.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"25 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we exhibit two unital, separable, nuclear -algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K-theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of -algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the -algebras of stable rank one and real rank zero.
在本文中，我们展示了两个具有相同有序标度总 K 理论的稳定秩为一、实秩为零的单元、可分离、核 C∗${rm C}^*$ 格拉斯，但它们彼此并不同构，这构成了实秩为零的艾略特分类猜想的反例。因此，我们引入了一个额外的正常条件，并给出了总 K 理论的分类结果。在一般情况下，通过新的不变式--总 Cuntz 半群[2]，我们对从扩展得到的一大类 C∗${rm C}^*$ 算法进行了分类。总 Cuntz 半群区分了我们反例中的数组，它有可能分类所有稳定秩为一和实秩为零的 C∗${rm C}^*$ 数组。
{"title":"Total Cuntz semigroup, extension, and Elliott Conjecture with real rank zero","authors":"Qingnan An, Zhichao Liu","doi":"10.1112/plms.12595","DOIUrl":"https://doi.org/10.1112/plms.12595","url":null,"abstract":"In this paper, we exhibit two unital, separable, nuclear <span data-altimg=\"/cms/asset/541c34ab-4a78-46ee-abe9-3ced6d287802/plms12595-math-0002.png\"></span><math altimg=\"urn:x-wiley:00246115:media:plms12595:plms12595-math-0002\" display=\"inline\" location=\"graphic/plms12595-math-0002.png\">\u0000<semantics>\u0000<msup>\u0000<mi mathvariant=\"normal\">C</mi>\u0000<mo>∗</mo>\u0000</msup>\u0000${rm C}^*$</annotation>\u0000</semantics></math>-algebras of stable rank one and real rank zero with the same ordered scaled total K-theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K-theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of <span data-altimg=\"/cms/asset/e7e53393-6247-4338-a826-81158b3a347b/plms12595-math-0003.png\"></span><math altimg=\"urn:x-wiley:00246115:media:plms12595:plms12595-math-0003\" display=\"inline\" location=\"graphic/plms12595-math-0003.png\">\u0000<semantics>\u0000<msup>\u0000<mi mathvariant=\"normal\">C</mi>\u0000<mo>∗</mo>\u0000</msup>\u0000${rm C}^*$</annotation>\u0000</semantics></math>-algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the <span data-altimg=\"/cms/asset/557f0fef-a2cf-4ebc-946a-e8e32479e4a7/plms12595-math-0004.png\"></span><math altimg=\"urn:x-wiley:00246115:media:plms12595:plms12595-math-0004\" display=\"inline\" location=\"graphic/plms12595-math-0004.png\">\u0000<semantics>\u0000<msup>\u0000<mi mathvariant=\"normal\">C</mi>\u0000<mo>∗</mo>\u0000</msup>\u0000${rm C}^*$</annotation>\u0000</semantics></math>-algebras of stable rank one and real rank zero.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"6 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The optimal mapping properties for the (local) helical maximal function are obtained, except for endpoints. The proof relies on tools from multilinear harmonic analysis and, in particular, a localised version of the Bennett–Carbery–Tao restriction theorem.
{"title":"Off-diagonal estimates for the helical maximal function","authors":"David Beltran, Jennifer Duncan, Jonathan Hickman","doi":"10.1112/plms.12594","DOIUrl":"https://doi.org/10.1112/plms.12594","url":null,"abstract":"The optimal <span data-altimg=\"/cms/asset/360821de-57a8-46af-8e3f-181682369c83/plms12594-math-0001.png\"></span><math altimg=\"urn:x-wiley:00246115:media:plms12594:plms12594-math-0001\" display=\"inline\" location=\"graphic/plms12594-math-0001.png\">\u0000<semantics>\u0000<mrow>\u0000<msup>\u0000<mi>L</mi>\u0000<mi>p</mi>\u0000</msup>\u0000<mo>→</mo>\u0000<msup>\u0000<mi>L</mi>\u0000<mi>q</mi>\u0000</msup>\u0000</mrow>\u0000$L^p rightarrow L^q$</annotation>\u0000</semantics></math> mapping properties for the (local) helical maximal function are obtained, except for endpoints. The proof relies on tools from multilinear harmonic analysis and, in particular, a localised version of the Bennett–Carbery–Tao restriction theorem.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"122 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
There is an irreparable error in the proof of Theorem 3.26 in the above-mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem is false in a very strong sense.
{"title":"Corrigendum: Model theory of fields with virtually free group actions","authors":"Özlem Beyarslan, Piotr Kowalski","doi":"10.1112/plms.12597","DOIUrl":"https://doi.org/10.1112/plms.12597","url":null,"abstract":"There is an irreparable error in the proof of Theorem 3.26 in the above-mentioned paper and we withdraw the claim of having proved that theorem. In fact, that theorem is false in a very strong sense.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"57 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christopher Eur, Alex Fink, Matt Larson, Hunter Spink
We establish a connection between the algebraic geometry of the type permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta-matroids,” modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.
{"title":"Signed permutohedra, delta-matroids, and beyond","authors":"Christopher Eur, Alex Fink, Matt Larson, Hunter Spink","doi":"10.1112/plms.12592","DOIUrl":"https://doi.org/10.1112/plms.12592","url":null,"abstract":"We establish a connection between the algebraic geometry of the type <mjx-container aria-label=\"upper B\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper B\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/44e471ce-ada7-40ff-b8f5-eff2b25d8b7e/plms12592-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper B\" data-semantic-type=\"identifier\">B</mi>$B$</annotation></semantics></math></mjx-assistive-mml></mjx-container> permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type <mjx-container aria-label=\"upper B\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper B\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/46703655-88b5-4eb6-9bee-0971456baf6b/plms12592-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper B\" data-semantic-type=\"identifier\">B</mi>$B$</annotation></semantics></math></mjx-assistive-mml></mjx-container> generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta-matroids,” modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"19 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Tollmien–Schlichting (T-S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo, and Nguyen gave the first rigorous construction of the T-S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T-S waves of temporal mode for certain boundary layer profiles with Mach number
{"title":"Tollmien–Schlichting waves in the subsonic regime","authors":"Nader Masmoudi, Yuxi Wang, Di Wu, Zhifei Zhang","doi":"10.1112/plms.12588","DOIUrl":"https://doi.org/10.1112/plms.12588","url":null,"abstract":"The Tollmien–Schlichting (T-S) waves play a key role in the early stages of boundary layer transition. In a breakthrough work, Grenier, Guo, and Nguyen gave the first rigorous construction of the T-S waves of temporal mode for the incompressible fluid. Yang and Zhang recently made an important contribution by constructing the compressible T-S waves of temporal mode for certain boundary layer profiles with Mach number <mjx-container aria-label=\"m less than StartFraction 1 Over StartRoot 3 EndRoot EndFraction\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,5\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"inequality\" data-semantic-speech=\"m less than StartFraction 1 Over StartRoot 3 EndRoot EndFraction\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,<\" data-semantic-parent=\"6\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mfrac data-semantic-children=\"2,4\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"division\" data-semantic-type=\"fraction\"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-msqrt data-semantic-children=\"3\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"unknown\" data-semantic-type=\"sqrt\" size=\"s\"><mjx-sqrt><mjx-surd><mjx-mo><mjx-c></mjx-c></mjx-mo></mjx-surd><mjx-box style=\"padding-top: 0.164em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"4\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-box></mjx-sqrt></mjx-msqrt></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b1bd1e98-17e2-46e4-980c-ac65e953555e/plms12588-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,5\" data-semantic-content=\"1\" data-semantic-role=\"inequality\" data-semantic-speech=\"m less than StartFraction 1 Over StartRoot 3 EndRoot EndFraction\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinle","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"56 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic partial differential equations with measurable coefficients and the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt–Caffarelli–Friedman-type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole-filling technique, we present a distinct “almost-monotonicity” formula that implies Hölder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow-up and -convergence arguments.
{"title":"Liouville theorems and optimal regularity in elliptic equations","authors":"Giorgio Tortone","doi":"10.1112/plms.12587","DOIUrl":"https://doi.org/10.1112/plms.12587","url":null,"abstract":"The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic partial differential equations with measurable coefficients and the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt–Caffarelli–Friedman-type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole-filling technique, we present a distinct “almost-monotonicity” formula that implies Hölder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow-up and <mjx-container aria-label=\"upper G\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/9a491fb9-701d-48a4-b97e-d4007559c11f/plms12587-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\">G</mi>$G$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-convergence arguments.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"14 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140129616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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