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3. Geometry of character varieties

    1. Stable cohomology of character varieties

          Fix an algebraic group $G$, and consider the relative character variety of $G$ local systems for the universal curve over $\mathcal{M}_g$, denoted $\mathcal{U}_{G,g}$ (such that the total monodromy is Zariski dense in $G$).

      Add remark

      Problem 3.1.

      [Aaron Landesman] Prove (or disprove) that the cohomology of $\mathcal{U}_{G,g}$ stabilizes with respect to $g$ and compute its stable value.

      A conjecture for the value of the putative stable cohomology should include Chern classes. Does it depend on anything else? What about integral cohomology?
          Aaron Landesman conjectures that when $G$ is finite, each component of this space stabilize to the cohomology of $\mathcal{M}_g$.

      As some examples, when $G$ is trivial, one just gets the usual stable cohomology of $\mathcal{M}_g. When $G$ is cyclic abelian, I believe it was proven in Andy Putman’s recent article [https://arxiv.org/abs/2209.06183].
        1. Remark. [Aaron Landesman] The relation to actions on character varieties is as follows: One can also look at the character variety $\mathrm{LocSys}_{G,g}$ of a single genus $g$ curve (so $\mathrm{LocSys}_{G,g}$ is a fiber of $\mathcal{U}_{G,g}$ over a point of $\mathcal{M}_g$.) Then, the Leray spectral sequence gives $$ H^i(\mathrm{Mod}_g, H^j(\mathrm{LocSys}_{G,g})) \implies H^{i+j}(\mathcal{U}_g). $$ So understanding this problem can be framed as understanding the mapping class group action on the cohomology of character varieties.

            • Log-Calabi-Yau compactifications of character varieties

                  Quotienting the moduli space of $\lambda$-connections by the canonical algebraic $\mathrm{G}_m$-action alows one to construct a canonical compactification of the Moduli space of flat connections. Hence the Riemann-Hilbert correspondence realizes this as an *analytic* compactification of the character variety.

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              Problem 3.2.

              Do there exist “canonical” algebraic compactifications of the character variety $M(\Sigma_{g,n}, r)$ of a punctured surface? It is a conjecture (of Simpson?) that $M(\Sigma_{g,n}, r)$ admits a compactification as a log-Calabi-Yau variety. Prove subcases of this conjecture.
                  A theorem of Junho Whang covers the case $r = 2$ when $\Sigma$ has at least one puncture i.e. produces a log-Calabi-Yau compactification of $M(\Sigma_{g,n}, 2)$ for $n \ge 1$.

                  Cite this as: AimPL: Motives and mapping class groups, available at http://aimpl.org/motivesandmcg.