---
DOAJ_listed: '1'
OA_place: publisher
OA_type: gold
_id: '19071'
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."
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”. '
article_number: '1'
article_processing_charge: Yes
article_type: original
arxiv: 1
author:
- first_name: Tamás
  full_name: Hausel, Tamás
  id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
  last_name: Hausel
  orcid: 0000-0002-9582-2634
- first_name: Kamil P
  full_name: Rychlewicz, Kamil P
  id: 85A07246-A8BF-11E9-B4FA-D9E3E5697425
  last_name: Rychlewicz
citation:
  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>
  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>
  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>.
  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.
  ista: Hausel T, Rychlewicz KP. 2025. Spectrum of equivariant cohomology as a fixed
    point scheme. Epijournal de Geometrie Algebrique. 9, 1.
  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>.
  short: T. Hausel, K.P. Rychlewicz, Epijournal de Geometrie Algebrique 9 (2025).
corr_author: '1'
date_created: 2025-02-23T23:01:56Z
date_published: 2025-02-03T00:00:00Z
date_updated: 2025-04-15T06:31:58Z
day: '03'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.46298/epiga.2025.12591
external_id:
  arxiv:
  - '2212.11836'
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  file_size: 3276395
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  success: 1
file_date_updated: 2025-02-25T06:53:27Z
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intvolume: '         9'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-sa/4.0/
month: '02'
oa: 1
oa_version: Published Version
project:
- _id: 34b2c9cb-11ca-11ed-8bc3-a50ba74ca4a3
  grant_number: P35847
  name: Geometry of the tip of the global nilpotent cone
- _id: 34cd0f74-11ca-11ed-8bc3-bf0492a14a24
  grant_number: '26525'
  name: Topology of open smooth varieties with a torus action
publication: Epijournal de Geometrie Algebrique
publication_identifier:
  eissn:
  - 2491-6765
publication_status: published
publisher: EPI Sciences
quality_controlled: '1'
related_material:
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scopus_import: '1'
status: public
title: Spectrum of equivariant cohomology as a fixed point scheme
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OA_place: publisher
_id: '17156'
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."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Kamil P
  full_name: Rychlewicz, Kamil P
  id: 85A07246-A8BF-11E9-B4FA-D9E3E5697425
  last_name: Rychlewicz
citation:
  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>
  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>
  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>.
  ieee: K. P. Rychlewicz, “Equivariant cohomology and rings of functions,” Institute
    of Science and Technology Austria, 2024.
  ista: Rychlewicz KP. 2024. Equivariant cohomology and rings of functions. Institute
    of Science and Technology Austria.
  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>.
  short: K.P. Rychlewicz, Equivariant Cohomology and Rings of Functions, Institute
    of Science and Technology Austria, 2024.
corr_author: '1'
date_created: 2024-06-23T15:07:06Z
date_published: 2024-06-25T00:00:00Z
date_updated: 2026-04-07T12:55:46Z
day: '25'
ddc:
- '516'
degree_awarded: PhD
department:
- _id: TaHa
- _id: GradSch
doi: 10.15479/at:ista:17156
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  file_size: 2761814
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  date_created: 2024-06-26T20:58:24Z
  date_updated: 2024-06-26T20:58:24Z
  file_id: '17180'
  file_name: thesis.pdf
  file_size: 3695952
  relation: main_file
file_date_updated: 2024-06-26T21:00:14Z
has_accepted_license: '1'
keyword:
- equivariant cohomology
- zero schemes
- algebraic groups
- Lie algebras
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-sa/4.0/
month: '06'
oa: 1
oa_version: Published Version
page: '117'
project:
- _id: 34cd0f74-11ca-11ed-8bc3-bf0492a14a24
  grant_number: '26525'
  name: Topology of open smooth varieties with a torus action
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
  - id: '17157'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Tamás
  full_name: Hausel, Tamás
  id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
  last_name: Hausel
  orcid: 0000-0002-9582-2634
title: Equivariant cohomology and rings of functions
tmp:
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  legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
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    BY-NC-SA 4.0)
  short: CC BY-NC-SA (4.0)
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2024'
...
---
OA_place: repository
_id: '17157'
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.
article_number: '2212.11836'
article_processing_charge: No
arxiv: 1
author:
- first_name: Tamás
  full_name: Hausel, Tamás
  id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
  last_name: Hausel
  orcid: 0000-0002-9582-2634
- first_name: Kamil P
  full_name: Rychlewicz, Kamil P
  id: 85A07246-A8BF-11E9-B4FA-D9E3E5697425
  last_name: Rychlewicz
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>
  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>
  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>.
  ieee: T. Hausel and K. P. Rychlewicz, “Spectrum of equivariant cohomology as a fixed
    point scheme,” <i>arXiv</i>. .
  ista: Hausel T, Rychlewicz KP. Spectrum of equivariant cohomology as a fixed point
    scheme. arXiv, 2212.11836.
  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.).
date_created: 2024-06-23T15:01:27Z
date_published: 2022-12-22T00:00:00Z
date_updated: 2026-04-07T12:55:46Z
day: '22'
department:
- _id: GradSch
- _id: TaHa
doi: 10.48550/arXiv.2212.11836
external_id:
  arxiv:
  - '2212.11836'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2212.11836
month: '12'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '19071'
    relation: later_version
    status: public
  - id: '17156'
    relation: dissertation_contains
    status: public
status: public
title: Spectrum of equivariant cohomology as a fixed point scheme
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...
---
_id: '6965'
abstract:
- lang: eng
  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.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Kamil P
  full_name: Rychlewicz, Kamil P
  id: 85A07246-A8BF-11E9-B4FA-D9E3E5697425
  last_name: Rychlewicz
citation:
  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>
  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>
  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>.
  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.
  ista: Rychlewicz KP. 2021. The positivity of local equivariant Hirzebruch class
    for toric varieties. Bulletin of the London Mathematical Society. 53(2), 560–574.
  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>.
  short: K.P. Rychlewicz, Bulletin of the London Mathematical Society 53 (2021) 560–574.
corr_author: '1'
date_created: 2019-10-24T08:04:09Z
date_published: 2021-04-01T00:00:00Z
date_updated: 2024-10-09T20:59:03Z
day: '01'
department:
- _id: TaHa
doi: 10.1112/blms.12442
external_id:
  arxiv:
  - '1910.10435'
  isi:
  - '000594805800001'
intvolume: '        53'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1910.10435
month: '04'
oa: 1
oa_version: Preprint
page: 560-574
publication: Bulletin of the London Mathematical Society
publication_identifier:
  eissn:
  - 1469-2120
  issn:
  - 0024-6093
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: The positivity of local equivariant Hirzebruch class for toric varieties
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 53
year: '2021'
...