Pub Date : 2024-07-26DOI: 10.1515/crelle-2024-0044
Xuemiao Chen
In this paper, we study the analytic properties of solutions to the Vafa–Witten equation over a compact Kähler manifold. Simple obstructions to the existence of nontrivial solutions are identified. The gauge theoretical compactness for the C ∗ mathbb{C}^{*} invariant locus of the moduli space is shown to behave similarly to the Hermitian Yang–Mills connections. More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions. When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers. This gives a simpler proof for Taubes’ results on rank two solutions over Kähler surfaces together with a new complex geometric interpretation. The moduli space of SU ( 2 ) mathsf{SU}(2) monopoles and some related examples are also discussed in the final section.
本文研究了紧凑凯勒流形上的瓦法-维滕方程的解的分析性质，发现了非微观解存在的简单障碍，证明了模空间的C∗mathbb{C}^{*}不变位点的量规理论紧凑性与赫米蒂杨-米尔斯连接的行为相似。最后一节还讨论了 SU ( 2 ) mathsf{SU}(2)单极的模空间和一些相关的例子。
{"title":"On Vafa–Witten equations over Kähler manifolds","authors":"Xuemiao Chen","doi":"10.1515/crelle-2024-0044","DOIUrl":"https://doi.org/10.1515/crelle-2024-0044","url":null,"abstract":"\u0000 In this paper, we study the analytic properties of solutions to the Vafa–Witten equation over a compact Kähler manifold.\u0000Simple obstructions to the existence of nontrivial solutions are identified.\u0000The gauge theoretical compactness for the \u0000 \u0000 \u0000 \u0000 C\u0000 ∗\u0000 \u0000 \u0000 \u0000 mathbb{C}^{*}\u0000 \u0000 invariant locus of the moduli space is shown to behave similarly to the Hermitian Yang–Mills connections.\u0000More generally, this holds for solutions with uniformly bounded spectral covers such as nilpotent solutions.\u0000When spectral covers are unbounded, we manage to take limits of the renormalized Higgs fields which are intrinsically characterized by the convergence of the associated spectral covers.\u0000This gives a simpler proof for Taubes’ results on rank two solutions over Kähler surfaces together with a new complex geometric interpretation.\u0000The moduli space of \u0000 \u0000 \u0000 \u0000 SU\u0000 \u0000 \u0000 (\u0000 2\u0000 )\u0000 \u0000 \u0000 \u0000 \u0000 mathsf{SU}(2)\u0000 \u0000 monopoles and some related examples are also discussed in the final section.","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"28 14","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141800751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.1515/crelle-2024-0056
Olivier Druet, Emmanuel Hebey, Frédéric Robert
We prove existence of extremal functions and compactness of the set of extremal functions for twisted sharp 𝑛-dimensional Sobolev inequalities with lower order remainder terms when n≥5ngeq 5.
我们证明了当 n ≥ 5 ngeq 5 时，具有低阶余项的扭曲尖锐𝑛维 Sobolev 不等式的极值函数的存在性和极值函数集的紧凑性。
{"title":"Extremal functions for twisted sharp Sobolev inequalities with lower order remainder terms. The high-dimensional case","authors":"Olivier Druet, Emmanuel Hebey, Frédéric Robert","doi":"10.1515/crelle-2024-0056","DOIUrl":"https://doi.org/10.1515/crelle-2024-0056","url":null,"abstract":"\u0000 <jats:p>We prove existence of extremal functions and compactness of the set of extremal functions for twisted sharp 𝑛-dimensional Sobolev inequalities with lower order remainder terms when <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>n</m:mi>\u0000 <m:mo>≥</m:mo>\u0000 <m:mn>5</m:mn>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0056_ineq_0001.png\"/>\u0000 <jats:tex-math>ngeq 5</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>.</jats:p>","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"13 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141799076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.1515/crelle-2024-0043
Alexander Gorodnik, Jialun Li, Cagri Sert
Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, Q<GQ a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety G/QG/Q with fibres given by copies of lattice quotients of a semisimple factor of 𝑄. Given a probability measure 𝜇, Zariski-dense in a copy of H=SL2(R)H=operatorname{SL}_{2}(mathbb{R}) in 𝐺, we give a description of 𝜇-
Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, Q G Q a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety G / Q G/Q with fibres given by copies of lattice quotients of a semisimple factor of 𝑄.Given a probability measure 𝜇, Zariski-dense in a copy of H = SL 2 ( R ) H=operatorname{SL}_{2}(mathbb{R}) in 𝐺, we give a description of 𝜇-stationary probability measures on 𝑋 and prove corresponding equidistribution results.Contrary to the results of Benoist–Quint corresponding to the case G = Q G=Q , the type of stationary measures that 𝜇 admits depends strongly on the position of 𝐻 relative to 𝑄.We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss.
{"title":"Stationary measures for SL2(ℝ)-actions on homogeneous bundles over flag varieties","authors":"Alexander Gorodnik, Jialun Li, Cagri Sert","doi":"10.1515/crelle-2024-0043","DOIUrl":"https://doi.org/10.1515/crelle-2024-0043","url":null,"abstract":"\u0000 <jats:p>Let 𝐺 be a real semisimple Lie group with finite centre and without compact factors, <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>Q</m:mi>\u0000 <m:mo><</m:mo>\u0000 <m:mi>G</m:mi>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0043_ineq_0001.png\"/>\u0000 <jats:tex-math>Q<G</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> a parabolic subgroup and 𝑋 a homogeneous space of 𝐺 admitting an equivariant projection on the flag variety <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>G</m:mi>\u0000 <m:mo>/</m:mo>\u0000 <m:mi>Q</m:mi>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0043_ineq_0002.png\"/>\u0000 <jats:tex-math>G/Q</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> with fibres given by copies of lattice quotients of a semisimple factor of 𝑄.\u0000Given a probability measure 𝜇, Zariski-dense in a copy of <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>H</m:mi>\u0000 <m:mo>=</m:mo>\u0000 <m:mrow>\u0000 <m:msub>\u0000 <m:mi>SL</m:mi>\u0000 <m:mn>2</m:mn>\u0000 </m:msub>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mi mathvariant=\"double-struck\">R</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0043_ineq_0003.png\"/>\u0000 <jats:tex-math>H=operatorname{SL}_{2}(mathbb{R})</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> in 𝐺, we give a description of 𝜇-","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"50 47","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141799610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-18DOI: 10.1515/crelle-2024-0045
Rebecca Bellovin, Neelima Borade, Anton Hilado, Kalyani Kansal, Heejong Lee, Brandon Levin, David Savitt, Hanneke Wiersema
Let K/QpK/mathbf{Q}_{p} be unramified. Inside the Emerton–Gee stack X2mathcal{X}_{2}, one can consider the locus of two-dimensional mod 𝑝 representations of Gal(K̄/K)mathrm{Gal}(overline{K}/K) having a crystalline lift with specified Hodge–Tate weights. We study the case where the Hodge–Tate weights a
让 K / Q p K/mathbf{Q}_{p} 是无ramified 的。在埃默顿-吉堆栈 X 2 mathcal{X}_{2} 中，我们可以考虑 Gal ( K ̄ / K ) 的二维模𝑝表示。 我们可以考虑 Gal ( K ̄ / K ) 的二维 mod 𝑝 表示的位置 我们研究了霍奇-塔特权重不规则的情况，这是希尔伯特模块形式部分权重为一条件在伽罗华表示中的类似。我们证明，如果每对权重之间的间隙以 𝑝 为界（塞尔权重的不规则类似），那么这个位置是不可还原的。
{"title":"Irregular loci in the Emerton–Gee stack for GL2","authors":"Rebecca Bellovin, Neelima Borade, Anton Hilado, Kalyani Kansal, Heejong Lee, Brandon Levin, David Savitt, Hanneke Wiersema","doi":"10.1515/crelle-2024-0045","DOIUrl":"https://doi.org/10.1515/crelle-2024-0045","url":null,"abstract":"\u0000 <jats:p>Let <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>K</m:mi>\u0000 <m:mo>/</m:mo>\u0000 <m:msub>\u0000 <m:mi mathvariant=\"bold\">Q</m:mi>\u0000 <m:mi>p</m:mi>\u0000 </m:msub>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0045_ineq_0001.png\"/>\u0000 <jats:tex-math>K/mathbf{Q}_{p}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> be unramified.\u0000Inside the Emerton–Gee stack <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mi mathvariant=\"script\">X</m:mi>\u0000 <m:mn>2</m:mn>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0045_ineq_0002.png\"/>\u0000 <jats:tex-math>mathcal{X}_{2}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, one can consider the locus of two-dimensional mod 𝑝 representations of <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>Gal</m:mi>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:mrow>\u0000 <m:mover accent=\"true\">\u0000 <m:mi>K</m:mi>\u0000 <m:mo>̄</m:mo>\u0000 </m:mover>\u0000 <m:mo>/</m:mo>\u0000 <m:mi>K</m:mi>\u0000 </m:mrow>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0045_ineq_0003.png\"/>\u0000 <jats:tex-math>mathrm{Gal}(overline{K}/K)</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> having a crystalline lift with specified Hodge–Tate weights.\u0000We study the case where the Hodge–Tate weights a","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":" 72","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141825129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-17DOI: 10.1515/crelle-2024-0048
Jeffrey S. Case
We extend Vétois’ Obata-type argument and use it to identify a closed interval InI_{n}, n≥3ngeq 3, containing zero such that if a∈Inain I_{n} and (Mn,g)
我们扩展韦托伊斯的 Obata 型论证，用它来确定一个封闭区间 I n I_{n} , n≥ 3 ngeq 3 , 其中包含零。 , n ≥ 3 ngeq 3 , containing zero such that if a ∈ I n ain I_{n} and ( M n , g ) (M^{n},g) is a compact conformally Einstein manifold with nonnegative scalar curvature and Q 4 + a σ 2 Q_{4}+asigma_{2} constant, then it is Einstein.We also relax the scalar curvature assumption to the nonnegativity of the Yamabe constant under a more restrictive assumption on 𝑎.Our results allow us to compute many Yamabe-type constants and prove sharp Sobolev inequalities on compact Einstein manifolds with nonnegative scalar curvature.In particular, we show that compact locally symmetric Einstein four-manifolds with nonnegative scalar curvature extremize the functional determinant of the conformal Laplacian, partially answering a question of Branson and Ørsted.
{"title":"The Obata–Vétois argument and its applications","authors":"Jeffrey S. Case","doi":"10.1515/crelle-2024-0048","DOIUrl":"https://doi.org/10.1515/crelle-2024-0048","url":null,"abstract":"\u0000 <jats:p>We extend Vétois’ Obata-type argument and use it to identify a closed interval <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mi>I</m:mi>\u0000 <m:mi>n</m:mi>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0048_ineq_0001.png\"/>\u0000 <jats:tex-math>I_{n}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>n</m:mi>\u0000 <m:mo>≥</m:mo>\u0000 <m:mn>3</m:mn>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0048_ineq_0002.png\"/>\u0000 <jats:tex-math>ngeq 3</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, containing zero such that if <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi>a</m:mi>\u0000 <m:mo>∈</m:mo>\u0000 <m:msub>\u0000 <m:mi>I</m:mi>\u0000 <m:mi>n</m:mi>\u0000 </m:msub>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0048_ineq_0003.png\"/>\u0000 <jats:tex-math>ain I_{n}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> and <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:msup>\u0000 <m:mi>M</m:mi>\u0000 <m:mi>n</m:mi>\u0000 </m:msup>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>g</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"29 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141640138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-04DOI: 10.1515/crelle-2024-0030
P. Caprace, A. Minasyan, Denis Osin
For each prime 𝑝 and each positive integer 𝑑, we construct the first examples of second countable, topologically simple 𝑝-adic Lie groups of dimension 𝑑 whose Lie algebras are abelian. This answers several questions of Glöckner and Caprace–Monod. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.
{"title":"Simple 𝑝-adic Lie groups with abelian Lie algebras","authors":"P. Caprace, A. Minasyan, Denis Osin","doi":"10.1515/crelle-2024-0030","DOIUrl":"https://doi.org/10.1515/crelle-2024-0030","url":null,"abstract":"\u0000 For each prime 𝑝 and each positive integer 𝑑, we construct the first examples of second countable, topologically simple 𝑝-adic Lie groups of dimension 𝑑 whose Lie algebras are abelian.\u0000This answers several questions of Glöckner and Caprace–Monod.\u0000The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"10 14","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141265668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-04DOI: 10.1515/crelle-2024-0034
Michele Pernice
This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of M̄3overline{mathcal{M}}_{3}, the moduli stack of stable curves of genus 3. In this paper, we finally compute the Chow ring of M̄3overline{mathcal{M}}_{3} with Z[1/6]mathbb{Z}[1/6]-coefficients.
本文是一系列四篇论文中的第四篇，旨在描述 M ̄ 3 overline{mathcal{M}}_{3} 的（近乎积分）Chow 环。 在本文中，我们最终计算了 M ̄ 3 的周环 Z [ 1 / 6 ]。 mathbb{Z}[1/6] -系数。
{"title":"The (almost) integral Chow ring of ℳ̅3","authors":"Michele Pernice","doi":"10.1515/crelle-2024-0034","DOIUrl":"https://doi.org/10.1515/crelle-2024-0034","url":null,"abstract":"\u0000 <jats:p>This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mover accent=\"true\">\u0000 <m:mi mathvariant=\"script\">M</m:mi>\u0000 <m:mo>̄</m:mo>\u0000 </m:mover>\u0000 <m:mn>3</m:mn>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0034_ineq_0001.png\"/>\u0000 <jats:tex-math>overline{mathcal{M}}_{3}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>, the moduli stack of stable curves of genus 3.\u0000In this paper, we finally compute the Chow ring of <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mover accent=\"true\">\u0000 <m:mi mathvariant=\"script\">M</m:mi>\u0000 <m:mo>̄</m:mo>\u0000 </m:mover>\u0000 <m:mn>3</m:mn>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0034_ineq_0001.png\"/>\u0000 <jats:tex-math>overline{mathcal{M}}_{3}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> with <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mi mathvariant=\"double-struck\">Z</m:mi>\u0000 <m:mo></m:mo>\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">[</m:mo>\u0000 <m:mrow>\u0000 <m:mn>1</m:mn>\u0000 <m:mo>/</m:mo>\u0000 <m:mn>6</m:mn>\u0000 </m:mrow>\u0000 <m:mo stretchy=\"false\">]</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0034_ineq_0003.png\"/>\u0000 <jats:tex-math>mathbb{Z}[1/6]</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-coefficients.</jats:p>","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"91 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141268054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-18DOI: 10.1515/crelle-2024-0025
Yongquan Hu, Haoran Wang
Let 𝐷 be the nonsplit quaternion algebra over Qpmathbb{Q}_{p}. We prove that a class of admissible unitary Banach space representations of D×D^{times} of global origin are topologically of finite length.
让 𝐷 是 Q p 上的非分裂四元数代数。我们证明，全局原点的 D × D^{times} 的一类可容许的单元巴纳赫空间表示在拓扑上是有限长的。
{"title":"On some 𝑝-adic and mod 𝑝 representations of quaternion algebra over ℚ𝑝","authors":"Yongquan Hu, Haoran Wang","doi":"10.1515/crelle-2024-0025","DOIUrl":"https://doi.org/10.1515/crelle-2024-0025","url":null,"abstract":"\u0000 <jats:p>Let 𝐷 be the nonsplit quaternion algebra over <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msub>\u0000 <m:mi mathvariant=\"double-struck\">Q</m:mi>\u0000 <m:mi>p</m:mi>\u0000 </m:msub>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0025_ineq_0001.png\"/>\u0000 <jats:tex-math>mathbb{Q}_{p}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>.\u0000We prove that a class of admissible unitary Banach space representations of <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mi>D</m:mi>\u0000 <m:mo>×</m:mo>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0025_ineq_0002.png\"/>\u0000 <jats:tex-math>D^{times}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> of global origin are topologically of finite length.</jats:p>","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140962076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-17DOI: 10.1515/crelle-2024-0027
Suyoung Choi, Hyeontae Jang, Mathieu Vallée
We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of ( n − 1 ) (n-1) -dimensional PL spheres with n + 4 n+4 vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number 4. This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number 4. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.
我们提出了一种适用于图形处理单元实现的高效计算算法，该算法可以从给定的输入集合中识别出符合特定面条件的所有弱伪流形。我们利用这种方法列举了环状可着色种子，从而全面描述了具有 n + 4 n+4 顶点且拥有最大布赫斯塔伯数的 ( n - 1 ) (n-1) 维 PL 球。这项研究的一个主要重点是对皮卡尔数 4 的非星形完全环状品种进行基本分类。这个分类是解决与皮卡尔数 4 的环状流形相关问题的一个有价值的工具。值得注意的是，我们确定了这些流形中哪些流形在其有理曲线空间中的最小分量数方面满足不等式内的相等。
{"title":"The characterization of (𝑛 − 1)-spheres with 𝑛 + 4 vertices having maximal Buchstaber number","authors":"Suyoung Choi, Hyeontae Jang, Mathieu Vallée","doi":"10.1515/crelle-2024-0027","DOIUrl":"https://doi.org/10.1515/crelle-2024-0027","url":null,"abstract":"\u0000 We present a computationally efficient algorithm that is suitable for graphic processing unit implementation.\u0000This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set.\u0000We employ this approach to enumerate toric colorable seeds.\u0000Consequently, we achieve a comprehensive characterization of \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 n\u0000 −\u0000 1\u0000 \u0000 )\u0000 \u0000 \u0000 \u0000 (n-1)\u0000 \u0000 -dimensional PL spheres with \u0000 \u0000 \u0000 \u0000 n\u0000 +\u0000 4\u0000 \u0000 \u0000 \u0000 n+4\u0000 \u0000 vertices that possess a maximal Buchstaber number.\u0000A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number 4.\u0000This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number 4.\u0000Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space.\u0000This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"60 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140965452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-16DOI: 10.1515/crelle-2024-0028
A. Gover, M. Gursky
Let (M4,g)(M^{4},g) be a smooth, closed, oriented anti-self-dual (ASD) four-manifold. (M4,g)(M^{4},g) is said to be unobstructed if the cokernel of the linearisation of the self-dual Weyl tensor is trivial. This condition can also be characterised as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures. It also arises in construction of ASD manifolds by twistor and gluing methods. In this article, we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed.
设 ( M 4 , g ) (M^{4},g) 是光滑、封闭、定向的反自偶（ASD）四曲面。 ( M 4 , g ) (M^{4},g) 如果自偶韦尔张量线性化的协核是微不足道的，就可以说它是无阻塞的。这个条件也可以表征为ASD变形复数的第二同调群的消失，它是理解ASD共形结构模空间局部结构的核心。
{"title":"The anti-self-dual deformation complex and a conjecture of Singer","authors":"A. Gover, M. Gursky","doi":"10.1515/crelle-2024-0028","DOIUrl":"https://doi.org/10.1515/crelle-2024-0028","url":null,"abstract":"\u0000 <jats:p>Let <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:msup>\u0000 <m:mi>M</m:mi>\u0000 <m:mn>4</m:mn>\u0000 </m:msup>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>g</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0028_ineq_0001.png\"/>\u0000 <jats:tex-math>(M^{4},g)</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> be a smooth, closed, oriented anti-self-dual (ASD) four-manifold.\u0000<jats:inline-formula>\u0000 <jats:alternatives>\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mrow>\u0000 <m:mo stretchy=\"false\">(</m:mo>\u0000 <m:msup>\u0000 <m:mi>M</m:mi>\u0000 <m:mn>4</m:mn>\u0000 </m:msup>\u0000 <m:mo>,</m:mo>\u0000 <m:mi>g</m:mi>\u0000 <m:mo stretchy=\"false\">)</m:mo>\u0000 </m:mrow>\u0000 </m:math>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_crelle-2024-0028_ineq_0001.png\"/>\u0000 <jats:tex-math>(M^{4},g)</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> is said to be <jats:italic>unobstructed</jats:italic> if the cokernel of the linearisation of the self-dual Weyl tensor is trivial.\u0000This condition can also be characterised as the vanishing of the second cohomology group of the ASD deformation complex, and is central to understanding the local structure of the moduli space of ASD conformal structures.\u0000It also arises in construction of ASD manifolds by twistor and gluing methods.\u0000In this article, we give conformally invariant conditions which imply an ASD manifold of positive Yamabe type is unobstructed.</jats:p>","PeriodicalId":508691,"journal":{"name":"Journal für die reine und angewandte Mathematik (Crelles Journal)","volume":"7 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140970182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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