Atsushi Ito, Makoto Miura, Shinnosuke Okawa, K. Ueda
{"title":"仿射线的类是Grothendieck环上的零因子:通过𝐺₂-Grassmannians","authors":"Atsushi Ito, Makoto Miura, Shinnosuke Okawa, K. Ueda","doi":"10.1090/JAG/731","DOIUrl":null,"url":null,"abstract":"<p>Motivated by [J. Algebraic Geom. 27 (2018), pp. 203–209] and [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 936–939], we show the equality <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis left-bracket upper X right-bracket minus left-bracket upper Y right-bracket right-parenthesis dot left-bracket double-struck upper A Superscript 1 Baseline right-bracket equals 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mrow>\n <mml:mo>(</mml:mo>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo stretchy=\"false\">]</mml:mo>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:mi>Y</mml:mi>\n <mml:mo stretchy=\"false\">]</mml:mo>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n <mml:mo>⋅<!-- ⋅ --></mml:mo>\n <mml:mo stretchy=\"false\">[</mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">A</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">]</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\left ( [ X ] - [ Y ] \\right ) \\cdot [ \\mathbb {A} ^{ 1 } ] = 0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in the Grothendieck ring of varieties, where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper X comma upper Y right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>Y</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">( X, Y )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G 2\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>G</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>2</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">G _{ 2 }</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2018-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/731","citationCount":"0","resultStr":"{\"title\":\"The class of the affine line is a zero divisor in the Grothendieck ring: Via 𝐺₂-Grassmannians\",\"authors\":\"Atsushi Ito, Makoto Miura, Shinnosuke Okawa, K. Ueda\",\"doi\":\"10.1090/JAG/731\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Motivated by [J. Algebraic Geom. 27 (2018), pp. 203–209] and [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 936–939], we show the equality <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis left-bracket upper X right-bracket minus left-bracket upper Y right-bracket right-parenthesis dot left-bracket double-struck upper A Superscript 1 Baseline right-bracket equals 0\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mrow>\\n <mml:mo>(</mml:mo>\\n <mml:mo stretchy=\\\"false\\\">[</mml:mo>\\n <mml:mi>X</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">]</mml:mo>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mo stretchy=\\\"false\\\">[</mml:mo>\\n <mml:mi>Y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">]</mml:mo>\\n <mml:mo>)</mml:mo>\\n </mml:mrow>\\n <mml:mo>⋅<!-- ⋅ --></mml:mo>\\n <mml:mo stretchy=\\\"false\\\">[</mml:mo>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">A</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">]</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:mn>0</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\left ( [ X ] - [ Y ] \\\\right ) \\\\cdot [ \\\\mathbb {A} ^{ 1 } ] = 0</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in the Grothendieck ring of varieties, where <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper X comma upper Y right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>X</mml:mi>\\n <mml:mo>,</mml:mo>\\n <mml:mi>Y</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">( X, Y )</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper G 2\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>G</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">G _{ 2 }</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":54887,\"journal\":{\"name\":\"Journal of Algebraic Geometry\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2018-12-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1090/JAG/731\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebraic Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/JAG/731\",\"RegionNum\":1,\"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":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/JAG/731","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The class of the affine line is a zero divisor in the Grothendieck ring: Via 𝐺₂-Grassmannians
Motivated by [J. Algebraic Geom. 27 (2018), pp. 203–209] and [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 936–939], we show the equality ([X]−[Y])⋅[A1]=0\left ( [ X ] - [ Y ] \right ) \cdot [ \mathbb {A} ^{ 1 } ] = 0 in the Grothendieck ring of varieties, where (X,Y)( X, Y ) is a pair of Calabi-Yau 3-folds cut out from the pair of Grassmannians of type G2G _{ 2 }.
期刊介绍：
The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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