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2. Period maps and families of abelian varieties

    1. Examples of MCG-finite irreducible representations of surface groups not arising from abelian varieties


      Add remark

      Problem 2.1.

      Can we find an examples of an irreducible representation $\rho : \pi_1(\Sigma_{g,n}) \to \GL_r(\mathbb{C})$ with finite orbit under $\mathrm{Mod}(\Sigma_{g,n})$ that do not arise from families of abelian varieties?

      We say that a representation “arises from a family of abelian varieties” if there exists a complex structure on $S = \Sigma_{g,n}$ such that there is an Zariski open $U \subset S$ and a family of abelian varities $A \to U$ such that the representation restricted to $U$ is a subquotient of the Gauss-Manin local system of $A \to U$.

        • Geometric local systems on $\mathcal{M}_g$ and rigidity


          Add remark

          Problem 2.2.

          Consider a local system $\mathbb{L}$ on $\mathcal{M}_g$ (or a space with some level structure) or equivalently a representation of $\mathrm{Mod}(\Sigma_g)$ (or correspondingly a representation of some finite index subgroup). Suppose $\mathbb{L}$ underlies a variation of Hodge structures. Is the period map $\mathcal{M}_g \to D / \Gamma$ rigid in some sense? We can ask: is ther period map

          (1) isolated in its mapping space

          (2) infinitesimally rigid

          (3) is it actually unique as a nonconstant map.
              There are some big monodromy results in the work of Sawin-Landesman-Litt. These can perhaps be used with a theorem of Peters to get infinitesimal rigidity results.

            • Higher Prym period maps

                  Let $G$ be a finite group. Let $\mathcal{M}' \to \mathcal{M}_g$ be the universal space of $G$-covers of curves of genus $g$. Given a $G$-cover $\varphi : C_h \to C_g$ we get an exact sequence of abelain varieties

              $$ 0 \to \mathrm{Prym}(\varphi) \to \mathrm{Jac}(C_h) \to \mathrm{Jac}(C_g) \to 0 $$

              The map $\varphi \mapsto \mathrm{Prym}(\varphi)$ induces a period map $\mathcal{M}' \to D / \Gamma$ (to the moduli space of polarized abelian varieties).

              A Theorem of Serván shows that for $G = \mathbb{Z} / 2 \mathbb{Z}$ and $h = 2g - 1$ then the homolomorphic Prym map $\mathcal{M}' \to \mathcal{A}_{g-1}$ is the unique nonconstant holomorphic map to $\mathcal{A}_k$ for $k < g$. Farb proved that the Torelli map $\mathcal{M}_g \to \mathcal{A}_g$ is unique amoung all nonconstant maps to $\mathcal{A}_h$ for $h \le g$.

              Add remark

              Problem 2.3.

              Is the higher Prym map unique in general?
                  The tools in the previous problem can show it is infinitesimally rigid.

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