Cohomology of non-generic character stacks

Tommaso Scognamiglio

doi:10.5802/jep.278

Cohomology of non-generic character stacks
[Cohomologie des champs de caractères non génériques]

Tommaso Scognamiglio ¹

¹ Université Paris Cité/IMJ-PRG, Campus Grands Moulins, 8 Pl. Aurélie Nemours, 75013 Paris

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1287-1371.

Résumé
Abstract

On étudie la cohomologie (à support compact) des champs de caractères pour les surfaces de Riemann épointées avec monodromies locales semi-simples fixées. Dans le cas de monodromies locales génériques, la cohomologie de ces champs de caractères a été étudiée dans [23, 39]. Dans cet article, on étend les résultats et la conjecture de [23] au cas non générique. En particulier, on calcule la E-série et on donne une formule conjecturale pour la série mixte de Poincaré. On démontre cette conjecture dans le cas de la droite projective privée de $4$ points.

We study (compactly supported) cohomology of character stacks of punctured Riemann surface with prescribed semisimple local monodromies at punctures. In the case of generic local monodromies, the cohomology of these character stacks has been studied in [23, 39]. In this paper we extend the results and conjectures of [23] to the non-generic case. In particular we compute the E-series and give a conjectural formula for the mixed Poincaré series. We prove our conjecture in the case of the projective line with $4$ punctures.

Reçu le : 2023-12-15
Accepté le : 2024-10-15
Publié le : 2024-10-24

DOI : 10.5802/jep.278

Classification : 14M35, 14D23
Keywords: Character varieties, compactly supported cohomology, moduli stacks
Mot clés : Variétés de caractères, cohomologie à support compact, champs de modules

Affiliations des auteurs :

Tommaso Scognamiglio ¹

¹ Université Paris Cité/IMJ-PRG, Campus Grands Moulins, 8 Pl. Aurélie Nemours, 75013 Paris

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Tommaso Scognamiglio. Cohomology of non-generic character stacks. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 1287-1371. doi : 10.5802/jep.278. https://jep.centre-mersenne.org/articles/10.5802/jep.278/

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