Intersection cohomology of character varieties for punctured Riemann surfaces

Ballandras, Mathieu

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 141-198

We study intersection cohomology of character varieties for punctured Riemann surfaces with prescribed monodromies around the punctures. Relying on a previous result from Mellit [Mel20a] for semisimple monodromies we compute the intersection cohomology of character varieties with monodromies of any Jordan type. This proves the Poincaré polynomial specialization of a conjecture from Letellier [Let15].

Nous étudions la cohomologie d’intersection des variétés de caractères des surfaces de Riemann épointées, la monodromie autour des points enlevés étant fixée. En nous appuyant sur un résultat de Mellit [Mel20a] pour des monodromies semi-simples, nous calculons la cohomologie d’intersection des variétés de caractères avec des monodromies ayant un type de Jordan quelconque. Ceci prouve la spécialisation au polynôme de Poincaré d’une conjecture de Letellier [Let15].

Classification : 14M35, 14F43
Keywords: Character varieties, intersection cohomology, parabolic Higgs bundles
Mots-clés : Variétés de caractères, cohomologie d’intersection, fibrés de Higgs paraboliques

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