{"title":"奇异辛flops与阮上同调","authors":"Bohui Chen , An-Min Li , Qi Zhang , Guosong Zhao","doi":"10.1016/j.top.2009.03.001","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient <span><span><span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∣</mo><mi>x</mi><mi>y</mi><mo>−</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></msup><mo>+</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mo>/</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mo>−</mo><mi>a</mi><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mtext>,</mtext><mspace></mspace><mi>r</mi><mo>≥</mo><mn>1</mn><mtext>,</mtext></math></span></span></span> which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span> be two symplectic orbifolds connected by such a flop. We study orbifold Gromov–Witten invariants of exceptional classes on <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span> and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.</p></div>","PeriodicalId":54424,"journal":{"name":"Topology","volume":"48 1","pages":"Pages 1-22"},"PeriodicalIF":0.0000,"publicationDate":"2009-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.top.2009.03.001","citationCount":"13","resultStr":"{\"title\":\"Singular symplectic flops and Ruan cohomology\",\"authors\":\"Bohui Chen , An-Min Li , Qi Zhang , Guosong Zhao\",\"doi\":\"10.1016/j.top.2009.03.001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient <span><span><span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∣</mo><mi>x</mi><mi>y</mi><mo>−</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></msup><mo>+</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>0</mn><mo>}</mo></mrow><mo>/</mo><msub><mrow><mi>μ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mo>−</mo><mi>a</mi><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mtext>,</mtext><mspace></mspace><mi>r</mi><mo>≥</mo><mn>1</mn><mtext>,</mtext></math></span></span></span> which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span> be two symplectic orbifolds connected by such a flop. We study orbifold Gromov–Witten invariants of exceptional classes on <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span> and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.</p></div>\",\"PeriodicalId\":54424,\"journal\":{\"name\":\"Topology\",\"volume\":\"48 1\",\"pages\":\"Pages 1-22\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2009-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.top.2009.03.001\",\"citationCount\":\"13\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Topology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0040938309000020\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0040938309000020","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let and be two symplectic orbifolds connected by such a flop. We study orbifold Gromov–Witten invariants of exceptional classes on and and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.
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