4 problems

Existence conjecture. There is a Kodaira fibration with no topological section. The source motivates this by the universal family of curves, which has no topological sections, but…

Weak topological section conjecture. The fibration $f$ admits an algebraic section if and only if its short exact sequence of fundamental groups splits. This is introduced as a con…

McMullen's geometric Shafarevich conjecture. For a given base $C$, there are only finitely many Kodaira fibrations $f:S\to C$ with fiber genus $g(S_b)=g$. This finiteness statement…

Let $X_{2,2}$ be the example cited in the source, and let $S$ be the surface constructed above, with realisation genus $\tilde{b}=9$, $\tilde{f}=4$, and realisation signature…