I'm curious whether the following is true:



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> **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?
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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](https://www-fourier.ujf-grenoble.fr/~peters/Articles/PubSteen2.pdf) 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



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



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> **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?
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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!



Families of abelian varieties on the line (or more generally simply connected varieties)
Q2 should be yes for polarized VHS by a rigidity theorem of Schmid [theorem 7.24, Variations of Hodge structure, Inventiones 1973], which says roughly that a PVHS is determined by the Hodge structure at a fibre plus monodromy. Q1 should follow from this by Deligne's equivalence that you stated. Q3 is OK also. The point is that a group with trivial profinite completion will only have trivial representations into $GL\_n(\mathbb{Z})$, since the target is residually finite.


**Addendum** Let me give an elementary argument that a family of princ. polarized abelian varieties over the affine line $\mathbb{C}$ is trivial.
Given such a family, we get a holomorphic map $f$ from $\mathbb{C}$ to the Siegel upper half plane. This is a bounded domain, so you can separate points using bounded holomorphic functions. After pulling these back to $\mathbb{C}$, they must all be constant. Therefore $f$ is constant. 


