A categorical Torelli theorem for hypersurfaces

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-07-17 DOI:10.1112/blms.13117

Dmitrii Pirozhkov

{"title":"A categorical Torelli theorem for hypersurfaces","authors":"Dmitrii Pirozhkov","doi":"10.1112/blms.13117","DOIUrl":null,"url":null,"abstract":"Let <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 <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> and degree <math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>. The derived category of coherent sheaves on <math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> contains an interesting subcategory called the Kuznetsov component <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 <math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math> uniquely if <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 <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 <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 <math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math> divides <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>.","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}

引用次数: 0

Abstract

Let $X \subset P^{n + 1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$ . The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component $A_{X}$ . We show that this subcategory, together with a certain autoequivalence called the rotation functor, determines $X$ uniquely if $d > 3$ or if $d = 3$ and $n > 3$ . This generalizes a result by Huybrechts and Rennemo, who proved the same statement under the additional assumption that $d$ divides $n + 2$ .