MARC details
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03143ntm a22003257a 4500 |
| 003 - CONTROL NUMBER IDENTIFIER |
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AT-ISTA |
| 005 - DATE AND TIME OF LATEST TRANSACTION |
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20250915140007.0 |
| 008 - FIXED-LENGTH DATA ELEMENTS--GENERAL INFORMATION |
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250915s2024 au ||||| m||| 00| 0 eng d |
| 040 ## - CATALOGING SOURCE |
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ISTA |
| 100 ## - MAIN ENTRY--PERSONAL NAME |
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| Personal name |
Rychlewicz, Kamil |
| 9 (RLIN) |
1084235 |
| 245 ## - TITLE STATEMENT |
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| Title |
Equivariant cohomology and rings of functions |
| 260 ## - PUBLICATION, DISTRIBUTION, ETC. (IMPRINT) |
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| Name of publisher, distributor, etc. |
Institute of Science and Technology Austria |
| Date of publication, distribution, etc. |
2024 |
| 500 ## - GENERAL NOTE |
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| General note |
Thesis |
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| Formatted contents note |
Abstract |
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Acknowledgements |
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About the Author |
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List of Collaborators and Publications |
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Table of Contents |
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List of Figures |
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1 Introduction |
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2 Background material |
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3 Regular elements and Konstant sections |
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4 Zero schemes and ordinary cohomology |
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5 Zero scheme as the spectrum of equivariant cohomology |
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6 Further directions and open problems |
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Bibliography |
| 520 ## - SUMMARY, ETC. |
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| Summary, etc. |
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. |
| 856 ## - ELECTRONIC LOCATION AND ACCESS |
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| Uniform Resource Identifier |
<a href="https://doi.org/10.15479/at:ista:17156">https://doi.org/10.15479/at:ista:17156</a> |
| 942 ## - ADDED ENTRY ELEMENTS (KOHA) |
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| Source of classification or shelving scheme |
Dewey Decimal Classification |