Cell decompositions of character varieties

Abstract

We establish curious Lefschetz property for generic character varieties of Riemann surfaces conjectured by Hausel, Letellier and Rodriguez-Villegas. Our main tool applies directly in the case when there is at least one puncture where the local monodromy has distinct eigenvalues. We pass to a vector bundle over the character variety, which is then decomposed into cells, which look like vector bundles over varieties associated to braids by Shende-Treumann-Zaslow. These varieties are in turn decomposed into cells that look like $({\mathbb{C}}^{\ast }{)}^{d-2k}\times {\mathbb{C}}^k$. The curious Lefschetz property is shown to hold on each cell, and therefore holds for the character variety. To deduce the general case, we introduce a fictitious puncture with trivial monodromy, and show that the cohomology of the character variety where one puncture has trivial monodromy is isomorphic to the sign component of the $S_n$ action on the cohomology for the character variety where trivial monodromy is replaced by regular semisimple monodromy. This involves an argument with the Grothendieck-Springer sheaf, and analysis of how the cohomology of the character variety varies when the eigenvalues are moved around.

arXiv commentcomments are very welcome

References in corpus (2)

• Weyl group actions on the Springer sheaf
• Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

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• E-Polynomials of Generic ${\text{GL}}_n⋊<\sigma >$-Character Varieties: Branched Case
• Algebra and geometry of link homology