{"title":"超曲面的托雷利分类定理","authors":"Dmitrii Pirozhkov","doi":"10.1112/blms.13117","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>⊂</mo>\n <msup>\n <mi>P</mi>\n <mrow>\n <mi>n</mi>\n <mo>+</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation>$X \\subset \\mathbb {P}^{n+1}$</annotation>\n </semantics></math> be a smooth Fano hypersurface of dimension <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> and degree <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>. The derived category of coherent sheaves on <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> contains an interesting subcategory called the Kuznetsov component <span></span><math>\n <semantics>\n <msub>\n <mi>A</mi>\n <mi>X</mi>\n </msub>\n <annotation>$\\mathcal {A}_X$</annotation>\n </semantics></math>. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> uniquely if <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>></mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d &gt; 3$</annotation>\n </semantics></math> or if <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>=</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d = 3$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>></mo>\n <mn>3</mn>\n </mrow>\n <annotation>$n &gt; 3$</annotation>\n </semantics></math>. This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math> divides <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>+</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$n+2$</annotation>\n </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 10","pages":"3075-3089"},"PeriodicalIF":0.8000,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A categorical Torelli theorem for hypersurfaces\",\"authors\":\"Dmitrii Pirozhkov\",\"doi\":\"10.1112/blms.13117\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mi>P</mi>\\n <mrow>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$X \\\\subset \\\\mathbb {P}^{n+1}$</annotation>\\n </semantics></math> be a smooth Fano hypersurface of dimension <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math> and degree <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math>. The derived category of coherent sheaves on <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> contains an interesting subcategory called the Kuznetsov component <span></span><math>\\n <semantics>\\n <msub>\\n <mi>A</mi>\\n <mi>X</mi>\\n </msub>\\n <annotation>$\\\\mathcal {A}_X$</annotation>\\n </semantics></math>. We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines <span></span><math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$X$</annotation>\\n </semantics></math> uniquely if <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>></mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$d &gt; 3$</annotation>\\n </semantics></math> or if <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>=</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$d = 3$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>></mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$n &gt; 3$</annotation>\\n </semantics></math>. This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math> divides <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>+</mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation>$n+2$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 10\",\"pages\":\"3075-3089\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-07-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13117\",\"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":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13117","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let be a smooth Fano hypersurface of dimension and degree . The derived category of coherent sheaves on contains an interesting subcategory called the Kuznetsov component . We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines uniquely if or if and . This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that divides .
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