I'm curious whether the following is true:

Question 1:Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to V$ isotrivial?

Specifically I'm curious about the case when $V=\mathbb{A}^1_\mathbb{C}$.

I **think** the answer to Question 1 is yes by the following line of reasoning. Deligne shows in Theorie de Hodge II that $\mathscr{A}$ is characterized by $(R^1f^\text{an}_\ast\underline{\mathbb{Z}})^\vee$ as polarizable variation of Hodge structure of type $\{(-1,0),(0,-1)\}$. Now, in our case we know that $(R^1f_\ast^\text{an}\underline{\mathbb{Z}})^\vee$ is a $\mathbb{Z}$-local system and so trivial as such. But, of course, there is no *a priori* reason to believe that the Hodge filtration is constant.

That said, in this article (see the remarks following Theorem 11) any $\mathbb{Q}$-VHS on a simply connected compactifiable complex manifold is actually constant (as a VHS). In particular, this should imply (since every smooth variety is compactifiable by resolution of singularities), by Deligne's theorem that there is a constant family $\underline{A}$ on $V$ and an isogeny $\varphi:\underline{A}\to\mathscr{A}$. But, $\varphi$ must be $\underline{\mathbb{Z}/n\mathbb{Z}}$ and so $\mathscr{A}$ must be $\underline{A/(\mathbb{Z}/n\mathbb{Z})}$ as desired.

Is this correct?

It then leads to the following natural questions

Question 2:Is every $\mathbb{Z}$-VHS on a simply connected compactifiable (perhaps even algebraic) manifold constant?

and

Question 3:Let $V/\mathbb{C}$ be a smooth connected variety such that $\pi^1_{\acute{e}\text{t}}(V)=0$. Then, is every abelian scheme $f:\mathscr{A}\to V$ isotrivial?

I apologize if these are silly questions.

Also, of note, there must be something truly algebraic happening here as the 'universal family' $\mathscr{E}/\mathbb{H}$ shows.

Thanks!

ordinaryabelian varieties over a smooth projective curve becomes trivial after a finite etale cover. In particular, every family of ordinary abelian varieties over a proper variety (no assumptions on $\pi_1$) is isotrivial. $\endgroup$ – Piotr Achinger Apr 30 '16 at 7:24