Given $E\to B$ a non-isotrivial (compact) Riemann surface-bundle (of genus $g>1$) between two complex manifolds and $E'\to E$ is a finite branched cover. Then is the composition map $E'\to E\to B$ still non-isotrivial?



Is the composition of a finite branched cover and a non-isotrivial Riemann surface bundle still non-isotrivial
If $E'$ is isotrivial, then the fibers $E'\to B$ are all isomorphic to a single fiber, so the Jacobians of every fiber are isomorphic to a single Jacobian abelian variety $A$. Then the Jacobian of every fiber of $E \to B$ is isomorphic to a quotient of $A$. But $A$ has only countably many distinct quotients, up to isomorphism, while by the Torelli theorem a non-isotrivial family of Riemann surfaces has uncountably many Jacobians.


Probably there's a deformation theory argument that also works here.


