Intersection cohomology of character varieties for punctured Riemann surfaces

Abstract

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]. Résumé (Cohomologie d'intersection des variétés de caractères des surfaces de Riemann épointées)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].

On the cohomology of characters stacks for non-orientable surfaces

“…, m i,s i respectively. Let Q = (I, Ω) be the following star-shaped quiver with g loops on the central vertex • [1,1] • [1,2] . .…”

“…In this case, the unipotent characters, by Lemma 5.3.1 (2), are in bijection with the multipartitions λ ∈ P n 1 × • • • × P nr and we denote by R λ the associated irreducible unipotent character.…”

Comet-shaped quiver varieties, Weyl group actions, and modified Kostka polynomials

“…In [Bal22] we compute Poincaré polynomial for compactly supported intersection cohomology of the character varieties M C . This allows to prove the following theorem which is the Poincaré polynomial specialization of Conjecture 1.2.…”

“…Let S be a regular semisimple conjugacy class such that the 4-tuple C ′ = (C 1 , C 2 , S, C 4 ) is generic. By [Bal22] and (50), the Poincaré polynomial of the character variety M C ′ is the same as the Poincaré polynomial of the variety M µ,ν . But the variety M C ′ is affine, hence its compactly supported intersection cohomology vanishes in degree strictly smaller than its dimension and only positive power of t appear in the expression.…”

“…This allows to describe the intersection cohomology of the partial resolutions, together with its Weyl group action, in terms of intersection cohomology of character varieties. The Poincaré polynomials for intersection cohomology of character varieties are computed in [Bal22]. The formula for (twisted) Poincaré polynomials of the resolutions are therefore a consequence of the Poincaré polynomial specialization of [Let13, Proposition 5.7].…”