Arithmetic geometry of character varieties with regular monodromy
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913685317877760 |
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| author | Kamgarpour, Masoud Nam, GyeongHyeon Puskás, Anna |
| author_facet | Kamgarpour, Masoud Nam, GyeongHyeon Puskás, Anna |
| contents | We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_02171 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Arithmetic geometry of character varieties with regular monodromy Kamgarpour, Masoud Nam, GyeongHyeon Puskás, Anna Representation Theory Algebraic Geometry We study character varieties arising as moduli of representations of an orientable surface group into a reductive group $G$. We first show that if $G/Z$ acts freely on the representation variety, then both the representation variety and the character variety are smooth and equidimensional. Next, we count points on a family of smooth character varieties; namely, those involving both regular semisimple and regular unipotent monodromy. In particular, we show that these varieties are polynomial count and obtain an explicit expression for their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine certain topological invariants of these varieties such as the Euler characteristic and the number of connected components. As an application, we give an example of a cohomologically rigid representation which is not physically rigid. |
| title | Arithmetic geometry of character varieties with regular monodromy |
| topic | Representation Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2209.02171 |