Toric stratifications of character varieties

Abstract We establish the 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 stratified into strata which look like vector bundles over varieties associated to positive braids. These varieties are in turn stratified into strata that look like $\mathbf {C}^{*d-2k}\times \mathbf {C}^{k}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:math> . The curious Lefschetz property is shown to hold on each stratum, 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 $S_{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> 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.