Journal für die reine und angewandte Mathematik (Crelles Journal)最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}