@article{19071,
  abstract     = {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.},
  author       = {Hausel, Tamás and Rychlewicz, Kamil P},
  issn         = {2491-6765},
  journal      = {Epijournal de Geometrie Algebrique},
  publisher    = {EPI Sciences},
  title        = {{Spectrum of equivariant cohomology as a fixed point scheme}},
  doi          = {10.46298/epiga.2025.12591},
  volume       = {9},
  year         = {2025},
}

@phdthesis{17156,
  abstract     = {This dissertation is the summary of the author’s work, concerning the relations between
cohomology rings of algebraic varieties and rings of functions on zero schemes and fixed
point schemes. For most of the thesis, the focus is on smooth complex varieties with
an action of a principally paired group, e.g. a parabolic subgroup of a reductive group.
The fundamental theorem 5.2.11 from co-authored article [66] says that if the principal
nilpotent has a unique zero, then the zero scheme over the Kostant section is isomorphic
to the spectrum of the equivariant cohomology ring, remembering the grading in terms of
a C^* action. A similar statement is proved also for the G-invariant functions on the total
zero scheme over the whole Lie algebra. Additionally, we are able to prove an analogous
result for the GKM spaces, which poses the question on a joint generalisation.
We also tackle the situation of a singular variety. As long as it is embedded in a smooth
variety with regular action, we are able to study its cohomology as well by means of
the zero scheme. In case of e.g. Schubert varieties this determines the cohomology ring
completely. In largest generality, this allows us to see a significant part of the cohomology
ring.
We also show (Theorem 6.2.1) that the cohomology ring of spherical varieties appears as
the ring of functions on the zero scheme. The computational aspect is not easy, but one
can hope that this can bring some concrete information about such cohomology rings.
Lastly, the K-theory conjecture 6.3.1 is studied, with some results attained for GKM
spaces.
The thesis includes also an introduction to group actions on algebraic varieties. In
particular, the vector fields associated to the actions are extensively studied. We also
provide a version of the Kostant section for arbitrary principally paired group, which
parametrises the regular orbits in the Lie algebra of an algebraic group. Before proving
the main theorem, we also include a historical overview of the field. In particular we bring
together the results of Akyildiz, Carrell and Lieberman on non-equivariant cohomology
rings.},
  author       = {Rychlewicz, Kamil P},
  issn         = {2663-337X},
  keywords     = {equivariant cohomology, zero schemes, algebraic groups, Lie algebras},
  pages        = {117},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Equivariant cohomology and rings of functions}},
  doi          = {10.15479/at:ista:17156},
  year         = {2024},
}

@unpublished{17157,
  abstract     = {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.},
  author       = {Hausel, Tamás and Rychlewicz, Kamil P},
  booktitle    = {arXiv},
  title        = {{Spectrum of equivariant cohomology as a fixed point scheme}},
  doi          = {10.48550/arXiv.2212.11836},
  year         = {2022},
}

@article{6965,
  abstract     = {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.},
  author       = {Rychlewicz, Kamil P},
  issn         = {1469-2120},
  journal      = {Bulletin of the London Mathematical Society},
  number       = {2},
  pages        = {560--574},
  publisher    = {Wiley},
  title        = {{The positivity of local equivariant Hirzebruch class for toric varieties}},
  doi          = {10.1112/blms.12442},
  volume       = {53},
  year         = {2021},
}