{"title":"对称商堆栈的同调积分性","authors":"Lucien Hennecart","doi":"arxiv-2408.15786","DOIUrl":null,"url":null,"abstract":"In this paper, we establish the sheafified version of the cohomological\nintegrality conjecture for stacks obtained as a quotient of a smooth affine\nsymmetric algebraic variety by a reductive algebraic group equipped with an\ninvariant function. A crucial step is the definition of the BPS sheaf as a\ncomplex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf\nwhen the situation arises from a smooth affine weakly symplectic algebraic\nvariety with a weak moment map. This situation gives local models for 1-Artin\nderived stacks with self-dual cotangent complex. We then apply these results to\nprove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore\nhomology of $0$-shifted symplectic stacks (or more generally, derived stacks\nwith self-dual cotangent complex) having a proper good moduli space. One\nstriking application is the purity of the Borel--Moore homology of the moduli\nstack of principal Higgs bundles over a smooth projective curve for a reductive\ngroup.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"60 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Cohomological integrality for symmetric quotient stacks\",\"authors\":\"Lucien Hennecart\",\"doi\":\"arxiv-2408.15786\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we establish the sheafified version of the cohomological\\nintegrality conjecture for stacks obtained as a quotient of a smooth affine\\nsymmetric algebraic variety by a reductive algebraic group equipped with an\\ninvariant function. A crucial step is the definition of the BPS sheaf as a\\ncomplex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf\\nwhen the situation arises from a smooth affine weakly symplectic algebraic\\nvariety with a weak moment map. This situation gives local models for 1-Artin\\nderived stacks with self-dual cotangent complex. We then apply these results to\\nprove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore\\nhomology of $0$-shifted symplectic stacks (or more generally, derived stacks\\nwith self-dual cotangent complex) having a proper good moduli space. One\\nstriking application is the purity of the Borel--Moore homology of the moduli\\nstack of principal Higgs bundles over a smooth projective curve for a reductive\\ngroup.\",\"PeriodicalId\":501063,\"journal\":{\"name\":\"arXiv - MATH - Algebraic Geometry\",\"volume\":\"60 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Algebraic Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.15786\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.15786","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Cohomological integrality for symmetric quotient stacks
In this paper, we establish the sheafified version of the cohomological
integrality conjecture for stacks obtained as a quotient of a smooth affine
symmetric algebraic variety by a reductive algebraic group equipped with an
invariant function. A crucial step is the definition of the BPS sheaf as a
complex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf
when the situation arises from a smooth affine weakly symplectic algebraic
variety with a weak moment map. This situation gives local models for 1-Artin
derived stacks with self-dual cotangent complex. We then apply these results to
prove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore
homology of $0$-shifted symplectic stacks (or more generally, derived stacks
with self-dual cotangent complex) having a proper good moduli space. One
striking application is the purity of the Borel--Moore homology of the moduli
stack of principal Higgs bundles over a smooth projective curve for a reductive
group.