  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 $(\mathbb{C}^*)^{d-2k}\times \mathbb{C}^k$. 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.
