We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O’Grady’s 1010-dimensional deformation type is trivial.
{"title":"Symplectic rigidity of O’Grady’s tenfolds","authors":"Luca Giovenzana, Annalisa Grossi, Claudio Onorati, Davide Veniani","doi":"10.1090/proc/16810","DOIUrl":"https://doi.org/10.1090/proc/16810","url":null,"abstract":"<p>We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O’Grady’s <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"10\"> <mml:semantics> <mml:mn>10</mml:mn> <mml:annotation encoding=\"application/x-tex\">10</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional deformation type is trivial.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"46 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a completely regular Hausdorff space, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote Banach spaces. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript b Baseline left-parenthesis upper X comma upper E right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C_b(X,E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-valued bounded continuous functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi>β</mml:mi> <mml:annotation encoding="application/x-tex">beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the strict topology on this space. We establish the relationship between nuclear operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper C Subscript b Baseline left-parenthesis upper X comma upper E right-parenthesis right-arrow upper F"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T:C_b(X,E)rightar
设 X X 是完全正则的豪斯多夫空间，设 E E 和 F F 表示巴拿赫空间。让 C b ( X , E ) C_b(X,E) 表示 X X 上 E E 有界连续函数的空间，让 β beta 是这个空间的严格拓扑。我们建立局部凸空间 ( C b ( X , E ) , β ) (C_b(X,E),beta ) 与巴拿赫空间 F F 之间核算子 T : C b ( X , E ) → F T:C_b(X,E)rightarrow F 之间的关系，以及它们代表的算子值博尔量。
{"title":"A characterization of nuclear operators on spaces of vector-valued continuous functions with the strict topology","authors":"Juliusz Stochmal","doi":"10.1090/proc/16805","DOIUrl":"https://doi.org/10.1090/proc/16805","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a completely regular Hausdorff space, let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=\"application/x-tex\">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=\"application/x-tex\">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote Banach spaces. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Subscript b Baseline left-parenthesis upper X comma upper E right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">C_b(X,E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the space of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=\"application/x-tex\">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-valued bounded continuous functions on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=\"application/x-tex\">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"beta\"> <mml:semantics> <mml:mi>β</mml:mi> <mml:annotation encoding=\"application/x-tex\">beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the strict topology on this space. We establish the relationship between nuclear operators <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper T colon upper C Subscript b Baseline left-parenthesis upper X comma upper E right-parenthesis right-arrow upper F\"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">→</mml:mo> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">T:C_b(X,E)rightar","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"31 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>We study the BMO-type functional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa Subscript epsilon Baseline left-parenthesis f comma double-struck upper R Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>κ</mml:mi> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">kappa _{varepsilon }(f,mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which can be used to characterize bounded variation functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of normal upper B normal upper V left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">B</mml:mi> <mml:mi mathvariant="normal">V</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">fin mathrm {BV}(mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit of this functional, taken with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal l normal o normal c Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">l</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">L^1_{mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convergence, is known to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="one fourth StartAbsoluteValue upper D f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>f</mml:mi> <mml:mro
我们研究了 BMO 型函数 κ ε ( f , R n ) kappa _{varepsilon }(f,mathbb {R}^n)，它可以用来描述有界变化函数 f∈ B V ( R n ) fin mathrm {BV}(mathbb {R}^n)。该函数的 Γ Gamma - Limit 取自 L l o c 1 L^1_{mathrm {loc}} 。 -收敛性，已知为 1 4 | D f | ( R n ) tfrac 14 |Df|(mathbb {R}^n) .我们证明，相对于 L l o c ∞ L^{infty }_{mathrm {loc}} 的 Γ Gamma - Limit 是 -convergence is [ 1 4 | D a f | ( R n ) + 1 4 | D c f | ( R n ) + 1 2 | D j f | ( R n ) 、 tfrac 14 |D^a f|(mathbb {R}^n)+tfrac 14 |D^c f|(mathbb {R}^n)+tfrac 12 |D^j f|(mathbb {R}^n), ]这与有界变化的特殊函数情况下的 "pointwise "极限一致。
{"title":"BMO-type functionals, total variation, and Γ-convergence","authors":"Panu Lahti, Quoc-Hung Nguyen","doi":"10.1090/proc/16812","DOIUrl":"https://doi.org/10.1090/proc/16812","url":null,"abstract":"<p>We study the BMO-type functional <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"kappa Subscript epsilon Baseline left-parenthesis f comma double-struck upper R Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>κ</mml:mi> <mml:mrow> <mml:mi>ε</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">kappa _{varepsilon }(f,mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which can be used to characterize bounded variation functions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f element-of normal upper B normal upper V left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant=\"normal\">B</mml:mi> <mml:mi mathvariant=\"normal\">V</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">fin mathrm {BV}(mathbb {R}^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Γ</mml:mi> <mml:annotation encoding=\"application/x-tex\">Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit of this functional, taken with respect to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L Subscript normal l normal o normal c Superscript 1\"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"normal\">l</mml:mi> <mml:mi mathvariant=\"normal\">o</mml:mi> <mml:mi mathvariant=\"normal\">c</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">L^1_{mathrm {loc}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-convergence, is known to be <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"one fourth StartAbsoluteValue upper D f EndAbsoluteValue left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mstyle displaystyle=\"false\" scriptlevel=\"0\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mrow> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> <mml:mi>f</mml:mi> <mml:mro","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"32 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141744662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
( Z ≥ 0 ) 2 (Bbb Z_{geq 0})^2 中的每个元素都是作为某个紧凑的、连通的、三维的、复数的、子满面 Z ⊂ P C N Z 的子集 Bbb P^N_{Bbb C} 的霍奇向量 ( h 3 , 0 ( Z ) , h 2 , 1 ( Z ) ) (h^{3,0}(Z),h^{2,1}(Z)) 来实现的。每个 ( x , y ) ∈ ( Z ≥ 1 ) 2 (x,y)in (Bbb Z_{geq 1})^2 with y ≤ 11 x + 8 yleq 11x+8 被证明是椭圆曲面的投影去纤积的霍奇向量，它在模数中移动。
{"title":"Hodge numbers of desingularized fiber products of elliptic surfaces","authors":"Chad Schoen","doi":"10.1090/proc/16803","DOIUrl":"https://doi.org/10.1090/proc/16803","url":null,"abstract":"<p>Each element of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis double-struck upper Z Subscript greater-than-or-equal-to 0 Baseline right-parenthesis squared\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(Bbb Z_{geq 0})^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is realized as the <italic>Hodge vector</italic> <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis h Superscript 3 comma 0 Baseline left-parenthesis upper Z right-parenthesis comma h Superscript 2 comma 1 Baseline left-parenthesis upper Z right-parenthesis right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(h^{3,0}(Z),h^{2,1}(Z))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of some compact, connected, three dimensional, complex, submanifold, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Z subset-of double-struck upper P Subscript double-struck upper C Superscript upper N\"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>⊂</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>N</mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Zsubset Bbb P^N_{Bbb C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Each <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma y right-parenthesis element-of left-parenthesis double-struck upper Z Subscript greater-than-or-equal-to 1 Baseline right-parenthesis squared\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>∈</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mrow>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"32 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141516778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christoforos Neofytidis, Hongbin Sun, Ye Tian, Shicheng Wang, Zhongzi Wang
{"title":"On the realisation problem for mapping degree sets","authors":"Christoforos Neofytidis, Hongbin Sun, Ye Tian, Shicheng Wang, Zhongzi Wang","doi":"10.1090/proc/16712","DOIUrl":"https://doi.org/10.1090/proc/16712","url":null,"abstract":"","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"38 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an equality between the Bergman metric and Carathéodry metric.
我们提出了伯格曼度量和卡拉瑟奥德里度量之间的相等关系。
{"title":"Equality between the Bergman metric and Carathéodory metric","authors":"Bo-Yong Chen, Yuanpu Xiong, Liyou Zhang","doi":"10.1090/proc/16799","DOIUrl":"https://doi.org/10.1090/proc/16799","url":null,"abstract":"<p>We present an equality between the Bergman metric and Carathéodry metric.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a fixed CM field KK with maximal totally real subfield FF, we consider isomorphism classes of dihedral Artin representations of FF which are induced from KK, distinguishing between those which are “canonically” induced from KK and those which are “noncanonically” induced from KK. The latter can arise only for Artin representations with image isomorphic to the dihedral group of order 8. We show that asymptotically, the number of noncanonically induced isomorphism classes is always comparable to and in some cases exceeds the number of canonically induced ones.
对于具有最大全实子场 F F 的固定 CM 场 K K，我们考虑从 K K 诱导的 F F 的二面体阿廷表示的同构类，区分 "规范地 "从 K K 诱导的同构类和 "非规范地 "从 K K 诱导的同构类。后者只适用于其图像与 8 阶二面群同构的 Artin 表示。我们证明，从渐近的角度看，非规范诱导同构类的数量总是与规范诱导同构类的数量相当，在某些情况下甚至超过规范诱导同构类的数量。
{"title":"Dihedral Artin representations and CM fields","authors":"David Rohrlich","doi":"10.1090/proc/16793","DOIUrl":"https://doi.org/10.1090/proc/16793","url":null,"abstract":"<p>For a fixed CM field <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with maximal totally real subfield <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=\"application/x-tex\">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we consider isomorphism classes of dihedral Artin representations of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper F\"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=\"application/x-tex\">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are induced from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, distinguishing between those which are “canonically” induced from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and those which are “noncanonically” induced from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The latter can arise only for Artin representations with image isomorphic to the dihedral group of order 8. We show that asymptotically, the number of noncanonically induced isomorphism classes is always comparable to and in some cases exceeds the number of canonically induced ones.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce Poincaré-type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure.
{"title":"A note on rearrangement Poincaré inequalities and the doubling condition","authors":"Joaquim Martín, Walter A. Ortiz","doi":"10.1090/proc/16795","DOIUrl":"https://doi.org/10.1090/proc/16795","url":null,"abstract":"<p>We introduce Poincaré-type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"63 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141531001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nikolay Bazhenov, Alexander Melnikov, Keng Meng Ng
We show that every Δ20Delta ^0_2 Polish space admits a computable topological presentation given by an effective indexing of some non-empty open sets in the space.
{"title":"Every Δ⁰₂ Polish space is computable topological","authors":"Nikolay Bazhenov, Alexander Melnikov, Keng Meng Ng","doi":"10.1090/proc/16797","DOIUrl":"https://doi.org/10.1090/proc/16797","url":null,"abstract":"<p>We show that every <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta 2 Superscript 0\"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">Delta ^0_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Polish space admits a computable topological presentation given by an effective indexing of some non-empty open sets in the space.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"25 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a closed differentiable manifold of dimension at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda 0 left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can have. We prove that for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> greater than or equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda 0 left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a Riemannian metric on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that its conformal Laplacian has exactly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> negative eigenvalues. Also, we discuss upper bounds for <inline-formula content-type="math/mathm
设 M M 是维数至少为 3 3 的闭可微分流形。设 Λ 0 ( M ) Lambda _0 (M) 是 M M 上度量的保角拉普拉卡矩的最小非正特征值个数。我们证明，对于大于或等于 Λ 0 ( M ) Lambda _0 (M) 的任何 k k，存在一个 M M 上的黎曼度量，使得它的共形拉普拉卡恰好有 k k 个负特征值。此外，我们还讨论了Λ 0 ( M ) Lambda _0 (M) 的上限。
{"title":"Negative eigenvalues of the conformal Laplacian","authors":"Guillermo Henry, Jimmy Petean","doi":"10.1090/proc/16798","DOIUrl":"https://doi.org/10.1090/proc/16798","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a closed differentiable manifold of dimension at least <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding=\"application/x-tex\">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the minimum number of non-positive eigenvalues that the conformal Laplacian of a metric on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can have. We prove that for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> greater than or equal to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda 0 left-parenthesis upper M right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">Λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Lambda _0 (M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a Riemannian metric on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that its conformal Laplacian has exactly <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> negative eigenvalues. Also, we discuss upper bounds for <inline-formula content-type=\"math/mathm","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"33 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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