三维非正则奇点最小对数差的一个间隙定理

IF 0.9 1区数学 Q2 MATHEMATICS

Journal of Algebraic Geometry Pub Date : 2019-04-21 DOI:10.1090/jag/759

Chen Jiang

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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":"{\"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</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</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>. 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引用次数: 25

摘要

我们证明了存在一个正实数δ>0δ>0，使得对于任何正规的拟投影Q\mathbb｛Q｝-Gorenstein 3 3重X X X，如果X X具有比规范奇点更差的奇异性，即X X的最小对数偏差小于11，则X X的最小对数偏差不大于1−δ。作为应用，我们证明了所有非正则klt-Calabi–Yau 3-折叠的集合是有界模触发器，并且所有klt-Calobi–Yau3-折叠的全局索引是从上有界的。

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A gap theorem for minimal log discrepancies of noncanonical singularities in dimension three

We show that there exists a positive real number δ > 0 \delta >0 such that for any normal quasi-projective Q \mathbb {Q} -Gorenstein 3 3 -fold X X , if X X has worse than canonical singularities, that is, the minimal log discrepancy of X X is less than 1 1 , then the minimal log discrepancy of X X is not greater than 1 − δ 1-\delta . As applications, we show that the set of all noncanonical klt Calabi–Yau 3 3 -folds are bounded modulo flops, and the global indices of all klt Calabi–Yau 3 3 -folds are bounded from above.

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来源期刊

Journal of Algebraic Geometry 数学-数学

CiteScore

2.70

自引率

5.60%

发文量

审稿时长

>12 weeks

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.