[{"external_id":{"arxiv":["2212.11836"]},"project":[{"grant_number":"P35847","_id":"34b2c9cb-11ca-11ed-8bc3-a50ba74ca4a3","name":"Geometry of the tip of the global nilpotent cone"},{"grant_number":"26525","_id":"34cd0f74-11ca-11ed-8bc3-bf0492a14a24","name":"Topology of open smooth varieties with a torus action"}],"OA_place":"publisher","intvolume":"         9","publication_identifier":{"eissn":["2491-6765"]},"title":"Spectrum of equivariant cohomology as a fixed point scheme","tmp":{"image":"/images/cc_by_sa.png","name":"Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","short":"CC BY-SA (4.0)"},"license":"https://creativecommons.org/licenses/by-sa/4.0/","scopus_import":"1","acknowledgement":"The first author was supported by an FWF grant “Geometry of the top of the nilpotent cone” number P 35847. The second author was supported by an Austrian Academy of Sciences DOC Fellowship “Topology of open smooth varieties with a torus action”. ","day":"03","related_material":{"record":[{"id":"17157","relation":"earlier_version","status":"public"}]},"month":"02","date_published":"2025-02-03T00:00:00Z","status":"public","article_processing_charge":"Yes","oa_version":"Published Version","volume":9,"type":"journal_article","author":[{"full_name":"Hausel, Tamás","orcid":"0000-0002-9582-2634","last_name":"Hausel","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","first_name":"Tamás"},{"first_name":"Kamil P","id":"85A07246-A8BF-11E9-B4FA-D9E3E5697425","last_name":"Rychlewicz","full_name":"Rychlewicz, Kamil P"}],"citation":{"ista":"Hausel T, Rychlewicz KP. 2025. Spectrum of equivariant cohomology as a fixed point scheme. Epijournal de Geometrie Algebrique. 9, 1.","apa":"Hausel, T., &#38; Rychlewicz, K. P. (2025). Spectrum of equivariant cohomology as a fixed point scheme. <i>Epijournal de Geometrie Algebrique</i>. EPI Sciences. <a href=\"https://doi.org/10.46298/epiga.2025.12591\">https://doi.org/10.46298/epiga.2025.12591</a>","short":"T. Hausel, K.P. Rychlewicz, Epijournal de Geometrie Algebrique 9 (2025).","chicago":"Hausel, Tamás, and Kamil P Rychlewicz. “Spectrum of Equivariant Cohomology as a Fixed Point Scheme.” <i>Epijournal de Geometrie Algebrique</i>. EPI Sciences, 2025. <a href=\"https://doi.org/10.46298/epiga.2025.12591\">https://doi.org/10.46298/epiga.2025.12591</a>.","mla":"Hausel, Tamás, and Kamil P. Rychlewicz. “Spectrum of Equivariant Cohomology as a Fixed Point Scheme.” <i>Epijournal de Geometrie Algebrique</i>, vol. 9, 1, EPI Sciences, 2025, doi:<a href=\"https://doi.org/10.46298/epiga.2025.12591\">10.46298/epiga.2025.12591</a>.","ieee":"T. Hausel and K. P. Rychlewicz, “Spectrum of equivariant cohomology as a fixed point scheme,” <i>Epijournal de Geometrie Algebrique</i>, vol. 9. EPI Sciences, 2025.","ama":"Hausel T, Rychlewicz KP. Spectrum of equivariant cohomology as a fixed point scheme. <i>Epijournal de Geometrie Algebrique</i>. 2025;9. doi:<a href=\"https://doi.org/10.46298/epiga.2025.12591\">10.46298/epiga.2025.12591</a>"},"department":[{"_id":"TaHa"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"has_accepted_license":"1","date_created":"2025-02-23T23:01:56Z","doi":"10.46298/epiga.2025.12591","OA_type":"gold","_id":"19071","DOAJ_listed":"1","article_type":"original","year":"2025","article_number":"1","publication_status":"published","abstract":[{"lang":"eng","text":"An action of a complex reductive group G on a smooth projective variety X is regular when all regular unipotent elements in G act with finitely many fixed points. Then the complex G\r\n-equivariant cohomology ring of X is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties."}],"file_date_updated":"2025-02-25T06:53:27Z","quality_controlled":"1","ddc":["510"],"arxiv":1,"publication":"Epijournal de Geometrie Algebrique","language":[{"iso":"eng"}],"publisher":"EPI Sciences","file":[{"access_level":"open_access","success":1,"checksum":"3915c6f117461502f7103878460428df","file_name":"2025_Epiga_Hausel.pdf","date_updated":"2025-02-25T06:53:27Z","relation":"main_file","file_id":"19085","content_type":"application/pdf","date_created":"2025-02-25T06:53:27Z","file_size":3276395,"creator":"dernst"}],"corr_author":"1","date_updated":"2025-04-15T06:31:58Z"},{"oa_version":"Published Version","article_processing_charge":"No","date_published":"2024-06-25T00:00:00Z","status":"public","month":"06","related_material":{"record":[{"id":"17157","relation":"part_of_dissertation","status":"public"}]},"day":"25","department":[{"_id":"TaHa"},{"_id":"GradSch"}],"citation":{"chicago":"Rychlewicz, Kamil P. “Equivariant Cohomology and Rings of Functions.” Institute of Science and Technology Austria, 2024. <a href=\"https://doi.org/10.15479/at:ista:17156\">https://doi.org/10.15479/at:ista:17156</a>.","short":"K.P. Rychlewicz, Equivariant Cohomology and Rings of Functions, Institute of Science and Technology Austria, 2024.","apa":"Rychlewicz, K. P. (2024). <i>Equivariant cohomology and rings of functions</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:17156\">https://doi.org/10.15479/at:ista:17156</a>","ista":"Rychlewicz KP. 2024. Equivariant cohomology and rings of functions. Institute of Science and Technology Austria.","ama":"Rychlewicz KP. Equivariant cohomology and rings of functions. 2024. doi:<a href=\"https://doi.org/10.15479/at:ista:17156\">10.15479/at:ista:17156</a>","ieee":"K. P. Rychlewicz, “Equivariant cohomology and rings of functions,” Institute of Science and Technology Austria, 2024.","mla":"Rychlewicz, Kamil P. <i>Equivariant Cohomology and Rings of Functions</i>. Institute of Science and Technology Austria, 2024, doi:<a href=\"https://doi.org/10.15479/at:ista:17156\">10.15479/at:ista:17156</a>."},"type":"dissertation","author":[{"last_name":"Rychlewicz","first_name":"Kamil P","id":"85A07246-A8BF-11E9-B4FA-D9E3E5697425","full_name":"Rychlewicz, Kamil P"}],"alternative_title":["ISTA Thesis"],"publication_identifier":{"issn":["2663-337X"]},"title":"Equivariant cohomology and rings of functions","OA_place":"publisher","page":"117","project":[{"_id":"34cd0f74-11ca-11ed-8bc3-bf0492a14a24","name":"Topology of open smooth varieties with a torus action","grant_number":"26525"}],"tmp":{"short":"CC BY-NC-SA (4.0)","image":"/images/cc_by_nc_sa.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)"},"license":"https://creativecommons.org/licenses/by-nc-sa/4.0/","ddc":["516"],"file_date_updated":"2024-06-26T21:00:14Z","abstract":[{"lang":"eng","text":"This dissertation is the summary of the author’s work, concerning the relations between\r\ncohomology rings of algebraic varieties and rings of functions on zero schemes and fixed\r\npoint schemes. For most of the thesis, the focus is on smooth complex varieties with\r\nan action of a principally paired group, e.g. a parabolic subgroup of a reductive group.\r\nThe fundamental theorem 5.2.11 from co-authored article [66] says that if the principal\r\nnilpotent has a unique zero, then the zero scheme over the Kostant section is isomorphic\r\nto the spectrum of the equivariant cohomology ring, remembering the grading in terms of\r\na C^* action. A similar statement is proved also for the G-invariant functions on the total\r\nzero scheme over the whole Lie algebra. Additionally, we are able to prove an analogous\r\nresult for the GKM spaces, which poses the question on a joint generalisation.\r\nWe also tackle the situation of a singular variety. As long as it is embedded in a smooth\r\nvariety with regular action, we are able to study its cohomology as well by means of\r\nthe zero scheme. In case of e.g. Schubert varieties this determines the cohomology ring\r\ncompletely. In largest generality, this allows us to see a significant part of the cohomology\r\nring.\r\nWe also show (Theorem 6.2.1) that the cohomology ring of spherical varieties appears as\r\nthe ring of functions on the zero scheme. The computational aspect is not easy, but one\r\ncan hope that this can bring some concrete information about such cohomology rings.\r\nLastly, the K-theory conjecture 6.3.1 is studied, with some results attained for GKM\r\nspaces.\r\nThe thesis includes also an introduction to group actions on algebraic varieties. In\r\nparticular, the vector fields associated to the actions are extensively studied. We also\r\nprovide a version of the Kostant section for arbitrary principally paired group, which\r\nparametrises the regular orbits in the Lie algebra of an algebraic group. Before proving\r\nthe main theorem, we also include a historical overview of the field. In particular we bring\r\ntogether the results of Akyildiz, Carrell and Lieberman on non-equivariant cohomology\r\nrings."}],"publication_status":"published","date_updated":"2026-04-07T12:55:46Z","corr_author":"1","file":[{"file_id":"17179","content_type":"application/zip","relation":"source_file","file_size":2761814,"date_created":"2024-06-26T20:56:27Z","creator":"krychlew","date_updated":"2024-06-26T21:00:14Z","access_level":"closed","checksum":"1610063569f5452f8a5acef728c2fc26","file_name":"thesis.zip"},{"date_updated":"2024-06-26T20:58:24Z","access_level":"open_access","file_name":"thesis.pdf","checksum":"7bbadb1fbc9ed2a1ecf54597f88af99c","date_created":"2024-06-26T20:58:24Z","file_size":3695952,"creator":"krychlew","file_id":"17180","content_type":"application/pdf","relation":"main_file"}],"supervisor":[{"full_name":"Hausel, Tamás","orcid":"0000-0002-9582-2634","last_name":"Hausel","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","first_name":"Tamás"}],"publisher":"Institute of Science and Technology Austria","language":[{"iso":"eng"}],"doi":"10.15479/at:ista:17156","date_created":"2024-06-23T15:07:06Z","has_accepted_license":"1","oa":1,"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","year":"2024","keyword":["equivariant cohomology","zero schemes","algebraic groups","Lie algebras"],"degree_awarded":"PhD","_id":"17156"},{"type":"preprint","date_updated":"2026-04-07T12:55:46Z","author":[{"last_name":"Hausel","first_name":"Tamás","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","full_name":"Hausel, Tamás","orcid":"0000-0002-9582-2634"},{"full_name":"Rychlewicz, Kamil P","last_name":"Rychlewicz","first_name":"Kamil P","id":"85A07246-A8BF-11E9-B4FA-D9E3E5697425"}],"department":[{"_id":"GradSch"},{"_id":"TaHa"}],"citation":{"ama":"Hausel T, Rychlewicz KP. Spectrum of equivariant cohomology as a fixed point scheme. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2212.11836\">10.48550/arXiv.2212.11836</a>","ieee":"T. Hausel and K. P. Rychlewicz, “Spectrum of equivariant cohomology as a fixed point scheme,” <i>arXiv</i>. .","mla":"Hausel, Tamás, and Kamil P. Rychlewicz. “Spectrum of Equivariant Cohomology as a Fixed Point Scheme.” <i>ArXiv</i>, 2212.11836, doi:<a href=\"https://doi.org/10.48550/arXiv.2212.11836\">10.48550/arXiv.2212.11836</a>.","short":"T. Hausel, K.P. Rychlewicz, ArXiv (n.d.).","chicago":"Hausel, Tamás, and Kamil P Rychlewicz. “Spectrum of Equivariant Cohomology as a Fixed Point Scheme.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2212.11836\">https://doi.org/10.48550/arXiv.2212.11836</a>.","apa":"Hausel, T., &#38; Rychlewicz, K. P. (n.d.). Spectrum of equivariant cohomology as a fixed point scheme. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2212.11836\">https://doi.org/10.48550/arXiv.2212.11836</a>","ista":"Hausel T, Rychlewicz KP. Spectrum of equivariant cohomology as a fixed point scheme. arXiv, 2212.11836."},"publication":"arXiv","language":[{"iso":"eng"}],"article_processing_charge":"No","status":"public","date_published":"2022-12-22T00:00:00Z","arxiv":1,"oa_version":"Preprint","article_number":"2212.11836","month":"12","abstract":[{"lang":"eng","text":"An action of a complex reductive group G on a smooth projective variety X is regular when all regular unipotent elements in G act with finitely many fixed points. Then the complex G-equivariant cohomology ring of X is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties."}],"publication_status":"draft","related_material":{"record":[{"relation":"later_version","id":"19071","status":"public"},{"relation":"dissertation_contains","id":"17156","status":"public"}]},"day":"22","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.11836"}],"year":"2022","_id":"17157","date_created":"2024-06-23T15:01:27Z","doi":"10.48550/arXiv.2212.11836","title":"Spectrum of equivariant cohomology as a fixed point scheme","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["2212.11836"]},"OA_place":"repository","oa":1},{"author":[{"full_name":"Rychlewicz, Kamil P","last_name":"Rychlewicz","first_name":"Kamil P","id":"85A07246-A8BF-11E9-B4FA-D9E3E5697425"}],"type":"journal_article","department":[{"_id":"TaHa"}],"citation":{"mla":"Rychlewicz, Kamil P. “The Positivity of Local Equivariant Hirzebruch Class for Toric Varieties.” <i>Bulletin of the London Mathematical Society</i>, vol. 53, no. 2, Wiley, 2021, pp. 560–74, doi:<a href=\"https://doi.org/10.1112/blms.12442\">10.1112/blms.12442</a>.","ieee":"K. P. Rychlewicz, “The positivity of local equivariant Hirzebruch class for toric varieties,” <i>Bulletin of the London Mathematical Society</i>, vol. 53, no. 2. Wiley, pp. 560–574, 2021.","ama":"Rychlewicz KP. The positivity of local equivariant Hirzebruch class for toric varieties. <i>Bulletin of the London Mathematical Society</i>. 2021;53(2):560-574. doi:<a href=\"https://doi.org/10.1112/blms.12442\">10.1112/blms.12442</a>","ista":"Rychlewicz KP. 2021. The positivity of local equivariant Hirzebruch class for toric varieties. Bulletin of the London Mathematical Society. 53(2), 560–574.","apa":"Rychlewicz, K. P. (2021). The positivity of local equivariant Hirzebruch class for toric varieties. <i>Bulletin of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/blms.12442\">https://doi.org/10.1112/blms.12442</a>","short":"K.P. Rychlewicz, Bulletin of the London Mathematical Society 53 (2021) 560–574.","chicago":"Rychlewicz, Kamil P. “The Positivity of Local Equivariant Hirzebruch Class for Toric Varieties.” <i>Bulletin of the London Mathematical Society</i>. Wiley, 2021. <a href=\"https://doi.org/10.1112/blms.12442\">https://doi.org/10.1112/blms.12442</a>."},"volume":53,"article_processing_charge":"No","date_published":"2021-04-01T00:00:00Z","status":"public","oa_version":"Preprint","month":"04","day":"01","scopus_import":"1","issue":"2","intvolume":"        53","publication_identifier":{"issn":["0024-6093"],"eissn":["1469-2120"]},"title":"The positivity of local equivariant Hirzebruch class for toric varieties","external_id":{"arxiv":["1910.10435"],"isi":["000594805800001"]},"page":"560-574","corr_author":"1","date_updated":"2024-10-09T20:59:03Z","publication":"Bulletin of the London Mathematical Society","publisher":"Wiley","language":[{"iso":"eng"}],"quality_controlled":"1","arxiv":1,"isi":1,"publication_status":"published","abstract":[{"text":"The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties). The generalization for singular varieties is due to Brasselet–Schürmann–Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are recalled. After this review part, we prove the positivity of local Hirzebruch classes for all toric varieties, thus proving false the alleged counterexample given by Weber.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.10435"}],"article_type":"original","year":"2021","_id":"6965","date_created":"2019-10-24T08:04:09Z","doi":"10.1112/blms.12442","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1}]