Mathematics > Algebraic Geometry
arXiv:2606.26767 (math)
[Submitted on 25 Jun 2026]
Abstract:These lecture notes contain the material presented at one of the mini-courses of the Workshop on Character Varieties and Higgs Bundles held in Liberia, Guanacaste, Costa Rica, in August 2025. They also contain some exercises for the students attending the conference.
This manuscript contains the basic ideas and constructions about $e$-polynomials in character varieties and the state of the art of certain research in the field, plus some new further directions. We introduce mixed Hodge structures and $e$-polynomials, together with a series of arithmetic (counting points over finite fields) and geometric (stratification into parabolic types) techniques to compute them. We include a complete example of the calculation of the $e$-polynomial for the ${\rm GL}_3$-character variety of the free group. Finally, we extend the geometric stratification into parabolic types to a general reductive group $G$ to obtain explicit motivic expressions for the $G$-character varieties, and reduce certain topological mirror symmetry conjectures for these moduli spaces.
| Comments: | 40 pages, 1 figure. To appear in Revista de Matemática: Teoría y Aplicaciones |
| Subjects: | Algebraic Geometry (math.AG); Representation Theory (math.RT) |
| MSC classes: | Primary: 14M35. Secondary: 14L24, 14D20, 14C15, 16G99, 17B22, 32S35 |
| Cite as: | arXiv:2606.26767 [math.AG] |
| (or arXiv:2606.26767v1 [math.AG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.26767 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Alfonso Zamora [view email]
[v1]
Thu, 25 Jun 2026 08:49:56 UTC (51 KB)
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