# When are isotrivial families split by a finite base-change?

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?

• This is a question where the answer is different in the analytic and algebraic settings. There is an analytic smooth isotrivial family of elliptic curves over $P^1$, the Hodge surface. Jul 29, 2014 at 15:40

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.

• Thank you! It seems that the same should hold for smooth isotrivial families of varieties of non-negative Kodaira dimension? Jul 24, 2014 at 20:53
• It seems likely, but I am not completely sure: you might get into trouble with automorphisms acting trivially in cohomology, for instance.
– abx
Jul 25, 2014 at 6:54
• At least in characteristic 0, I think it should be fine for all non-uniruled varieties. Of course everything is fine when the automorphism group is finite (or can be "naturally" reduced to a finite group). That leaves varieties whose automorphism group contains a positive-dimensional (connected) linear group. But all such groups are (geometrically) rational. Thus the orbits are unirational. So the variety is uniruled. Jul 25, 2014 at 12:44

This is false for abelian varieties, as eluded by Ari. The moduli of polarized abelian varieties is a DM stack, as they have only finitely many automorphisms preserving the polarization, but some abelian varieties have infinite order automorphisms, which must permute the polarizations. In particular, the product of a non-CM elliptic curve with itself has automorphism group $GL_2(\mathbb Z)$. An arbitrary automorphism can be realized as monodromy over the base a nodal curve by taking the trivial family over the normalization and gluing by the automorphism.

But you may consider a nodal curve pathological. A normal variety has profinite étale fundamental group and cannot have such exotic automorphisms as monodromy. An abelian variety over a normal base must support a polarization.

• Also, since analytic fundamental groups are discrete, this allows the creation of an analytic bundles of abelian varieties over a smooth complete base (say, an elliptic curve) that is isotrivial but not trivial on a finite cover. These is a very different example than the Hodge surface. Jul 31, 2014 at 23:52

If $X \to B$ is an isotrivial family of (smooth projective) varieties over an algebraically closed field $k$ and $F$ is a fiber of $X \to B$, the family trivializes over the Isom-scheme Isom$(X, B \times_k F) \to B$ (for tautological reasons). Thus, to answer your question (as Jason Starr says), an isotrivial family trivializes over a finite etale base change if the Isom-scheme above is finite etale. Thus, if $k$ is of characteristic zero, it suffices for your purposes that the above Isom-scheme is finite over $B$. As Isom-schemes are usually affine (at least in the cases you are interested in), it suffices for the Isom-scheme to be proper. This translates into a statement about lifting $K$-automorphisms on the generic fibre of $X\to B$ uniquely to $R$-automorphisms (via the valuative criterion) with $K$ the function field of a dvr $R$. The latter statement was proven by Matsusaka and Mumford for "one-parameter" polarized families of varieties of non-negative Kodaira dimension; see Theorem 2 in http://dash.harvard.edu/handle/1/3450065 . This answers your question positively for (polarized) families when $B$ is one-dimensional.
it is shown that if the fibres of the family $X \to B$ are canonically polarized (i.e., have ample canonical bundle), and $B$ is a curve, the Isom-scheme is finite etale. Their result is slightly more general and even allows $B$ to be higher-dimensional under the condition that $B$ fibres smoothly over a $\dim B -1$-dimensional variety. It might be possible to extend the arguments in the proof of Lemma 7.3 to your case.