{"title":"关于Higgs丛模的Kirwan映射","authors":"Emily Cliff, T. Nevins, Shi-ying Shen","doi":"10.14231/AG-2021-011","DOIUrl":null,"url":null,"abstract":"Let $C$ be a smooth complex projective curve and $G$ a connected complex reductive group. We prove that if the center $Z(G)$ of $G$ is disconnected, then the Kirwan map $H^*\\big(\\operatorname{Bun}(G,C),\\mathbb{Q}\\big)\\rightarrow H^*\\big(\\mathcal{M}_{\\operatorname{Higgs}}^{\\operatorname{ss}},\\mathbb{Q}\\big)$ from the cohomology of the moduli stack of $G$-bundles to the moduli stack of semistable $G$-Higgs bundles, fails to be surjective: more precisely, the \"variant cohomology\" (and variant intersection cohomology) of the stack $\\mathcal{M}_{\\operatorname{Higgs}}^{\\operatorname{ss}}$ of semistable $G$-Higgs bundles, is always nontrivial. We also show that the image of the pullback map $H^*\\big(M_{\\operatorname{Higgs}}^{\\operatorname{ss}},\\mathbb{Q}\\big)\\rightarrow H^*\\big(\\mathcal{M}_{\\operatorname{Higgs}}^{\\operatorname{ss}},\\mathbb{Q}\\big)$, from the cohomology of the moduli space of semistable $G$-Higgs bundles to the stack of semistable $G$-Higgs bundles, cannot be contained in the image of the Kirwan map. The proof uses a Borel-Quillen--style localization result for equivariant cohomology of stacks to reduce to an explicit construction and calculation.","PeriodicalId":48564,"journal":{"name":"Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On the Kirwan map for moduli of Higgs bundles\",\"authors\":\"Emily Cliff, T. Nevins, Shi-ying Shen\",\"doi\":\"10.14231/AG-2021-011\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $C$ be a smooth complex projective curve and $G$ a connected complex reductive group. We prove that if the center $Z(G)$ of $G$ is disconnected, then the Kirwan map $H^*\\\\big(\\\\operatorname{Bun}(G,C),\\\\mathbb{Q}\\\\big)\\\\rightarrow H^*\\\\big(\\\\mathcal{M}_{\\\\operatorname{Higgs}}^{\\\\operatorname{ss}},\\\\mathbb{Q}\\\\big)$ from the cohomology of the moduli stack of $G$-bundles to the moduli stack of semistable $G$-Higgs bundles, fails to be surjective: more precisely, the \\\"variant cohomology\\\" (and variant intersection cohomology) of the stack $\\\\mathcal{M}_{\\\\operatorname{Higgs}}^{\\\\operatorname{ss}}$ of semistable $G$-Higgs bundles, is always nontrivial. We also show that the image of the pullback map $H^*\\\\big(M_{\\\\operatorname{Higgs}}^{\\\\operatorname{ss}},\\\\mathbb{Q}\\\\big)\\\\rightarrow H^*\\\\big(\\\\mathcal{M}_{\\\\operatorname{Higgs}}^{\\\\operatorname{ss}},\\\\mathbb{Q}\\\\big)$, from the cohomology of the moduli space of semistable $G$-Higgs bundles to the stack of semistable $G$-Higgs bundles, cannot be contained in the image of the Kirwan map. The proof uses a Borel-Quillen--style localization result for equivariant cohomology of stacks to reduce to an explicit construction and calculation.\",\"PeriodicalId\":48564,\"journal\":{\"name\":\"Algebraic Geometry\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2018-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebraic Geometry\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.14231/AG-2021-011\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14231/AG-2021-011","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let $C$ be a smooth complex projective curve and $G$ a connected complex reductive group. We prove that if the center $Z(G)$ of $G$ is disconnected, then the Kirwan map $H^*\big(\operatorname{Bun}(G,C),\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$ from the cohomology of the moduli stack of $G$-bundles to the moduli stack of semistable $G$-Higgs bundles, fails to be surjective: more precisely, the "variant cohomology" (and variant intersection cohomology) of the stack $\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}}$ of semistable $G$-Higgs bundles, is always nontrivial. We also show that the image of the pullback map $H^*\big(M_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$, from the cohomology of the moduli space of semistable $G$-Higgs bundles to the stack of semistable $G$-Higgs bundles, cannot be contained in the image of the Kirwan map. The proof uses a Borel-Quillen--style localization result for equivariant cohomology of stacks to reduce to an explicit construction and calculation.
期刊介绍:
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