{"title":"A gap theorem for minimal log discrepancies of noncanonical singularities in dimension three","authors":"Chen Jiang","doi":"10.1090/jag/759","DOIUrl":null,"url":null,"abstract":"<p>We show that there exists a positive real number <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta greater-than 0\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>δ<!-- δ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:mn>0</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\delta >0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> such that for any normal quasi-projective <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Q</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-Gorenstein <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\n <mml:semantics>\n <mml:mn>3</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-fold <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> has worse than canonical singularities, that is, the minimal log discrepancy of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is less than <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\">\n <mml:semantics>\n <mml:mn>1</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, then the minimal log discrepancy of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is not greater than <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1 minus delta\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>1</mml:mn>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi>δ<!-- δ --></mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">1-\\delta</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. As applications, we show that the set of all noncanonical klt Calabi–Yau <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\n <mml:semantics>\n <mml:mn>3</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\n <mml:semantics>\n <mml:mn>3</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-folds are bounded from above.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2019-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"25","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/759","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 25
Abstract
We show that there exists a positive real number δ>0\delta >0 such that for any normal quasi-projective Q\mathbb {Q}-Gorenstein 33-fold XX, if XX has worse than canonical singularities, that is, the minimal log discrepancy of XX is less than 11, then the minimal log discrepancy of XX is not greater than 1−δ1-\delta. As applications, we show that the set of all noncanonical klt Calabi–Yau 33-folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 33-folds are bounded from above.
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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