{"title":"椭圆曲面去星形化纤维积的霍奇数","authors":"Chad Schoen","doi":"10.1090/proc/16803","DOIUrl":null,"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\">Z\\subset \\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> <mml:mo>≥</mml:mo> <mml:mn>1</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\">(x,y)\\in (\\Bbb Z_{\\geq 1})^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"y less-than-or-equal-to 11 x plus 8\"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>11</mml:mn> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">y\\leq 11x+8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown to be the Hodge vector of a projective desingularized fiber product of elliptic surfaces which moves in moduli.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"32 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hodge numbers of desingularized fiber products of elliptic surfaces\",\"authors\":\"Chad Schoen\",\"doi\":\"10.1090/proc/16803\",\"DOIUrl\":null,\"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\\\">Z\\\\subset \\\\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> <mml:mo>≥</mml:mo> <mml:mn>1</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\\\">(x,y)\\\\in (\\\\Bbb Z_{\\\\geq 1})^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"y less-than-or-equal-to 11 x plus 8\\\"> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>11</mml:mn> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">y\\\\leq 11x+8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown to be the Hodge vector of a projective desingularized fiber product of elliptic surfaces which moves in moduli.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"32 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-02-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16803\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16803","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
( 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 y\leq 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。
Hodge numbers of desingularized fiber products of elliptic surfaces
Each element of (Z≥0)2(\Bbb Z_{\geq 0})^2 is realized as the Hodge vector(h3,0(Z),h2,1(Z))(h^{3,0}(Z),h^{2,1}(Z)) of some compact, connected, three dimensional, complex, submanifold, Z⊂PCNZ\subset \Bbb P^N_{\Bbb C}. Each (x,y)∈(Z≥1)2(x,y)\in (\Bbb Z_{\geq 1})^2 with y≤11x+8y\leq 11x+8 is shown to be the Hodge vector of a projective desingularized fiber product of elliptic surfaces which moves in moduli.
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