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5. Monodromy of surface bundles

    1. Geometric proof of a theorem of McMullen–Imayoshi-Shiga

          Imayoshi-Shiga (1999) [CITE] proved the following theorem based of the work of McMullen. Let $X \to B$ be a smooth proper family of curves of genus $g \ge 2$ parametrized by a (possibly open) curve $B$ with $\chi(B) < 0$. If the family is not isotrivial then the monodromy representation is irreducible in a homotopy-theoretic sense. Precisely, any simple closed curve $C \subset X_s$ fixed under monodromy must bound a disk.

      Add remark

      Problem 5.05.

      Is there a proof of this theorem using primarily algebraic geometry?


        • Add remark

          Problem 5.1.

          Does the theorem still hold if we drop the adjective “simple” from the closed curve?

            •     We can ask about arithmetic analogs of the above questions.

              Add remark

              Problem 5.15.

              Let $X / K$ be a smooth curve over a field $K$ which is one of the following:

              (1) a finite field $\mathbb{F}_q$

              (2) a number field $K$

              (3) the function field of a curve over $\mathbb{F}_q$

              given a $K$-point $x \in X$ and hence an action of $\mathrm{Gal}_K$ on the geometric fundamental group $\pi_1^{et}(X_{\overline{K}}, \bar{x})$ are there any fixed elements (except possibly when $K$ is a function field and $X$ is isotrivial over $K$).

                • Conjecture of Katzarkov-Pantov-Simpson


                  Add remark

                  Problem 5.2.

                  Does the monodromy representation of a nonisotrivial family $\mathcal{C} \to B$ of curves of genus $g \ge 2$ (perhaps one arising from a sufficiently ample Lefschetz pencil on a surface) have a dense orbit on the character variety of the fiber?

                  More precisely, the monodromy group $G \subset \mathrm{Mod}(\Sigma_g)$ of self-diffeomorphisms of the fiber acts on the character varieties $M(\Sigma_g, H)$ where $H$ is any reductive group. Does this action always have a Zariski dense orbit if the family is not isotrivial?
                      Katzarkov-Pantov-Simpson prove that when $H = \mathrm{GL}_{2r+1}(\mathbb{C})$ and $\mathcal{C} \to B$ arises as a sufficiently ample Lefschetz pencil on a surface $X$ with $H^1(X, \mathbb{Q}) = 0$ then there is indeed always a Zariski dense orbit.

                    • New examples of finite orbits of homomorphism $\pi_1(\Sigma_{g,n}) \to \mathrm{Mod}(\Sigma_{h})$


                      Add remark

                      Problem 5.25.

                      What are some examples of finite orbits under the mapping class group $\mathrm{Mod}(\Sigma_{g,n})$ of homomorphisms (considered up to conjugation) $\pi_1(\Sigma_{g,n}) \to \mathrm{Mod}(\Sigma_h)$. Which arise from algebraic surface bundles over an $n$-punctured curve?
                          Some examples arise from Kodaira-Parshin families

                        • Isotrivial isogeny factors in families of Jacobians


                          Add remark

                          Problem 5.3.

                          Let $\mathcal{C} \to B$ be a nonisotrivial family of proper curves of genus $g$. What is the maximum dimension of an isotrivial isogeny factor in the Jacobian fibration of $\mathcal{C} \to B$?

                            • Counterexamples to the Putman-Wieland conjecture

                                  The Putman-Wieland conjecture predicts that if $\Sigma_{g',n'} \to \Sigma_{g,n}$ is an unramified finite $H$-cover of punctured surfaces then the induced action of a finite index subgroup $\Gamma \subset \mathrm{Mod}_{g,n+1}$ on $H_1(\Sigma_{g',n'}, \mathbb{C})$ has no fixed vectors. The action is given as follows. Consider the canonical action of $\mathrm{Mod}_{g,n+1}$ on $\pi_1(\Sigma_{g,n}, x)$ for some fixed basepoint. Let $\Gamma$ be the finite index subgroup stabilizing the surjection $\pi_1(\Sigma_{g,n}, x) \to H$ hence $\Gamma$ acts on the kernel which is $\pi_1(\Sigma_{g',n'}, x')$. This induces an action on its abelianization $H_1(\Sigma_{g',n'}, \mathbb{C})$.

                              For $g = 2$ the Putman-Wieland conjecture is known to fail [MR4549123].

                              Add remark

                              Problem 5.35.

                              Can we classify all counterexamples to the Putman-Wieland conjecture in genus $g = 2$?

                                • Hyperbolicity of surface bundles


                                  Add remark

                                  Problem 5.4.

                                  [Bena Tshishiku] Are there examples in higher dimension given as surface bundles over a surface? Can these be compact examples? In particular, for $g,h \ge 2$ does there exist an $\Sigma_g$-bundle over $\Sigma_h$ $$ \Sigma_g \to E \to \Sigma_h $$ having one of the (increasingly weaker) properties:

                                  (1) $E$ is a hyperbolic $4$-manifold

                                  (2) $E$ has negative Riemannian curvature

                                  (3) $E$ is Gromov hyperbolic

                                  (4) $E$ is atoroidal

                                    • Hyperbolicity of surface bundles


                                      Add remark

                                      Problem 5.45.

                                      [Bena Tshishiku] Are there examples in higher dimension given as surface bundles over a surface? Can these be compact examples? In particular, for $g,h \ge 2$ does there exist an $\Sigma_g$-bundle over $\Sigma_h$ $$ \Sigma_g \to E \to \Sigma_h $$ having one of the (increasingly weaker) properties:

                                      (1) $E$ is a hyperbolic $4$-manifold

                                      (2) $E$ has negative Riemannian curvature

                                      (3) $E$ is Gromov hyperbolic

                                      (4) $E$ is atoroidal

                                        • Hyperbolicity of surface bundles


                                          Add remark

                                          Problem 5.5.

                                          [Bena Tshishiku] Are there examples in higher dimension given as surface bundles over a surface? Can these be compact examples? In particular, for $g,h \ge 2$ does there exist an $\Sigma_g$-bundle over $\Sigma_h$ $$ \Sigma_g \to E \to \Sigma_h $$ having one of the (increasingly weaker) properties:

                                          (1) $E$ is a hyperbolic $4$-manifold

                                          (2) $E$ has negative Riemannian curvature

                                          (3) $E$ is Gromov hyperbolic

                                          (4) $E$ is atoroidal

                                            • Hyperbolic surface bundles over surfaces

                                                  Thurston gave the following example of a hyperbolic $3$-manifold fibered over $S^1$. Consider $N = S^3 \setminus K$ where $K$ is the figure-eight knot. Then there is a map $N \to S^1$ whose fibers are punctured torii $\Sigma_{1,1}$. This fits into the following general story. For $\phi \in \mathrm{Mod}(\Sigma_{g,n})$ let $M_{\phi}$ be the self-gluing of $\Sigma_{g,n} \times [0,1]$ of the two boundary components via $\phi$. This is fibered over $S^1$ in $\Sigma_{g,n}$ surfaces.

                                              Theorem (Thurston): $M_{\phi}$ is hyperbolic iff $M_\phi$ is atoroidal ($\pi_1(M_{\phi})$ does not contain $\mathbb{Z}^2$) iff $\phi$ is pseudo-Anosov.

                                              It turns out that $M_\phi = N$ for $$ \phi = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} $$ which is pseudo-Anosov.

                                              Add remark

                                              Problem 5.55.

                                              Are there examples in higher dimension given as surface bundles over a surface? Can these be compact examples? In particular, for $g,h \ge 2$ does there exist an $\Sigma_g$-bundle over $\Sigma_h$ $$ \Sigma_g \to E \to \Sigma_h $$ having one of the (increasingly weaker) properties:

                                              (1) $E$ is a hyperbolic $4$-manifold

                                              (2) $E$ has negative Riemannian curvature

                                              (3) $E$ is Gromov hyperbolic

                                              (4) $E$ is atoroidal

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