A well-known theorem of Grauert and Fischer states that a smooth proper family of complex manifolds is a locally trivial fibration as soon as all the fibers are isomorphic. It is also easy to obtain an algebro-geometric variant of this; cf. Lemma 1.3 in Buium's Differential Algebra and Diophantine Geometry.
For a complete (generically smooth) family of curves $X/B$, isotriviality (take this to mean that all smooth fibers are isomorphic) is thus a priori equivalent to having a quasi-finite dominant map $B' \to B$ over which $X \times_B B'$ is a product family. Now, for families of positive genus, it is often taken for granted that $B' \to B$ may be taken to be surjective (that is, finite). Why is this, or where is it proved? This of course is false for $\mathbb{P}^1$-bundles: if there were a finite cover $f: C \to \mathbb{P}^1$ under which $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(1)) \to \mathbb{P}^1$ pulled back to the product bundle $C \times \mathbb{P}^1$, then the line bundles $f^* \mathcal{O}_{\mathbb{P}^1}$ and $f^*\mathcal{O}_{\mathbb{P}^1}(1)$ would be isomorphic, which they are not: one is ample, the other is not.
For elliptic surfaces $E/B$ we can see it as follows. First, following a finite base change attaining stable reduction, we may assume that the family is smooth. Consider then the line bundle $\omega := 0^* \Omega_{E/B}^1$ on $B$. We know that $\omega^{\otimes 12}$ is trivial: it has the nowhere vanishing global section $\Delta$. It follows that there is a finite etale covering $p :B' \to B$ of degree dividing $12$ with $p^*\omega \cong \mathcal{O}_{B'}$ trivial. Then it is easy to see that $E \times_B B' \to B'$ splits (cf. section 3.2 of Ulmer's survey Elliptic curves over function fields).
The question: If $X/B$ is a family of abelian varieties or of curves of genus $> 0$, with all members isomorphic (a smooth isotrivial morphism), is there a finite etale covering $B' \to B$ such that $X \times_B B'$ is a product family? Does the same statement hold for smooth isotrivial families of projective varieties of non-negative Kodaira dimension?
Best Answer
Let me work out the case of an isotrivial family $X\rightarrow B$ of curves -- the same argument applies to abelian varieties. The point is that the moduli space $\mathscr{M}_g(n)$ of curves $C$ of genus $g$ with a level $n$ structure (that is, a symplectic isomorphism $(\mathbb{Z}/n)^{2g}\stackrel{\sim}{\rightarrow }H^1(C,\mathbb{Z}/n)$) is a fine moduli space for $g\geq 2$, $n\geq 3$. Let $B'\rightarrow B$ be a finite étale cover such that the local system $H^1(X_b,\mathbb{Z}/n)_{b\in B}$ becomes trivial on $B'$. Since the curves $X_b$ are all isomorphic, the classifying map $B'\rightarrow \mathscr{M}_g(n)$ must be constant; since $\mathscr{M}_g(n)$ is a fine moduli space, this implies that the pull back of the fibration $X\rightarrow B$ to $B'$ is trivial.