P=W conjectures for character varieties with symplectic resolution

Camilla Felisetti; Mirko Mauri

doi:10.5802/jep.196

P=W conjectures for character varieties with symplectic resolution
[Les conjectures P=W pour les variétés de caractères ayant une résolution symplectique]

Camilla Felisetti ¹ ; Mirko Mauri ²

¹ University of Trento, Department of mathematics Via Sommarive 14, 38123 Povo (TN), Italy
² University of Michigan East Hall, 530 Church St, Ann Arbor 48109, Michigan, USA

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 853-905.

Résumé
Abstract

On établit les conjectures P=W et PI=WI pour les variétés de caractères avec groupe structurel ${GL}_{n}$ et ${SL}_{n}$ qui admettent une résolution symplectique, c’est-à-dire pour le genre $1$ en rang arbitraire, et le genre $2$ en rang $2$ . On formule la conjecture P=W pour une résolution et on la prouve pour les résolutions symplectiques. Pour la démonstration on fait appel à la topologie des modifications birationnelles et quasi-étales des espaces de modules de fibrés de Higgs. Pour cela, on démontre des résultats auxiliaires d’intérêt indépendant, comme la construction d’une compactification relative de l’espace de modules de Hodge pour les groupes algébriques réductifs, ou la théorie de l’intersection de certains cycles lagrangiens singuliers. En particulier, on étudie en détail un espace de modules des fibrés de Higgs qui est une spécialisation de la variété symplectique holomorphe irréductible singulière de type O’Grady $6$ .

We establish P=W and PI=WI conjectures for character varieties with structural group ${GL}_{n}$ and ${SL}_{n}$ which admit a symplectic resolution, i.e., for genus $1$ and arbitrary rank, and genus $2$ and rank $2$ . We formulate the P=W conjecture for a resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is a specialization of the singular irreducible holomorphic symplectic variety of type O’Grady $6$ .

Reçu le : 2020-10-24
Accepté le : 2022-05-14
Publié le : 2022-05-30

DOI : 10.5802/jep.196

Classification : 14D20, 53D30, 14D22, 14E15, 32S35, 55N33
Keywords: P=W conjecture, intersection cohomology, Higgs bundles
Mot clés : Conjecture P=W, cohomologie d’intersection, fibrés de Higgs

Affiliations des auteurs :

Camilla Felisetti ¹ ; Mirko Mauri ²

¹ University of Trento, Department of mathematics Via Sommarive 14, 38123 Povo (TN), Italy
² University of Michigan East Hall, 530 Church St, Ann Arbor 48109, Michigan, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {P=W conjectures for character varieties with symplectic resolution},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Camilla Felisetti; Mirko Mauri. P=W conjectures for character varieties with symplectic resolution. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 853-905. doi : 10.5802/jep.196. https://jep.centre-mersenne.org/articles/10.5802/jep.196/

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