A candidate description of the cohomology ring of genus-zero character varieties is proposed in terms of modules on the Hilbert scheme of C^2, supported by small-rank computations but left as a conjecture.
de Cataldo and Luca Migliorini, The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bulletin of the American Mathematical Society 46 (2009), no
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Cohomology rings of character varieties
A candidate description of the cohomology ring of genus-zero character varieties is proposed in terms of modules on the Hilbert scheme of C^2, supported by small-rank computations but left as a conjecture.