Studying Deformations of Fuchsian Representations with Higgs Bundles
This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular in...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2019 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2019
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210065 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Studying Deformations of Fuchsian Representations with Higgs Bundles / B. Collier // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 34 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862740146565152768 |
|---|---|
| author | Collier, B. |
| author_facet | Collier, B. |
| citation_txt | Studying Deformations of Fuchsian Representations with Higgs Bundles / B. Collier // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 34 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichmüller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples, which are our motivating objects. In particular, we do not prove any of the foundational theorems; rather, we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) that define the bridge between Higgs bundles and the character variety. For this side of the story, we refer the reader to the survey article of Q. Li [arXiv:1809.05747].
|
| first_indexed | 2025-12-07T21:24:15Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210065 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:15Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Collier, B. 2025-12-02T09:35:17Z 2019 Studying Deformations of Fuchsian Representations with Higgs Bundles / B. Collier // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 34 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14D20; 14F45; 14H60 arXiv: 1809.06786 https://nasplib.isofts.kiev.ua/handle/123456789/210065 https://doi.org/10.3842/SIGMA.2019.010 This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichmüller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples, which are our motivating objects. In particular, we do not prove any of the foundational theorems; rather, we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) that define the bridge between Higgs bundles and the character variety. For this side of the story, we refer the reader to the survey article of Q. Li [arXiv:1809.05747]. The article is roughly based on a three-hour mini-course given by the author at the University of Illinois at Chicago in June 2018. I would like to thank Qiongling Li for her complementary minicourse. I would also like to thank all the participants of the mini-courses for their enthusiasm and interest. The author is funded by a National Science Foundation Mathematical Sciences Postdoctoral Fellowship, NSF MSPRF no. 1604263. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Studying Deformations of Fuchsian Representations with Higgs Bundles Article published earlier |
| spellingShingle | Studying Deformations of Fuchsian Representations with Higgs Bundles Collier, B. |
| title | Studying Deformations of Fuchsian Representations with Higgs Bundles |
| title_full | Studying Deformations of Fuchsian Representations with Higgs Bundles |
| title_fullStr | Studying Deformations of Fuchsian Representations with Higgs Bundles |
| title_full_unstemmed | Studying Deformations of Fuchsian Representations with Higgs Bundles |
| title_short | Studying Deformations of Fuchsian Representations with Higgs Bundles |
| title_sort | studying deformations of fuchsian representations with higgs bundles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210065 |
| work_keys_str_mv | AT collierb studyingdeformationsoffuchsianrepresentationswithhiggsbundles |