Let $X$ and $Y$ be *smooth* varieties over the field of complex numbers $\bf C$
(smooth integral separated schemes of finite type over $\bf C$). Let
$$f\colon X\to Y$$
be a surjective morphism such that
for any closed point $y\in Y$, the schematic fibre $f^{-1}(y)\subset X$
is isomorphic to the affine space ${\Bbb A}\_{\bf C}^{n(y)}$.
Moreover, assume that the morphism $f$ is *smooth* (which is equivalent to the assumption that $n(y)$
is the constant function $n(y)=n$, where $n=\dim X-\dim Y$).


Consider the real $C^\infty$-manifolds $X^\infty=X({\bf C})$ and $Y^\infty=Y({\bf C})$ and the induced $C^\infty$-map
$$f^\infty\colon X^\infty\to Y^\infty.$$
Since $f$ is smooth, the map $f^\infty$ is a submersion, that is, for any $x\in X^\infty$, the differential
$$d\_x f\colon T\_x(X)\to T\_{f(x)}Y$$
is surjective. Moreover, each fibre of $f^\infty$ is diffeomorphic to ${\bf R}^{2n}$.
By Corollary 31 of [G. Meigniez, Submersions, fibrations and bundles,
Trans. Amer. Math. Soc. 354 (2002), no. 9, 3771-3787](https://www.ams.org/journals/tran/2002-354-09/S0002-9947-02-02972-0/S0002-9947-02-02972-0.pdf),
the map $f^\infty$ is a *locally trivial fibre bundle of $C^\infty$-manifolds*, that is, for any $y\in Y^\infty$
there exists an open neighborhood ${\mathcal U}\_y$ of $y$ in $Y^\infty$
such that $f^{-1}({\mathcal U}\_y)\simeq {\bf R}^{2n}\times {\mathcal U}\_y$,
where $\simeq$ denotes a $C^\infty$-diffeomorphism compatible with the projections onto ${\mathcal U}\_y$.



> 
> **Question 1.** Does it follow that the morphism $f$
> is a locally trivial fibre bundle in the étale topology,
> that is, for any closed point $y\in Y$
> there exists an étale open neighborhood $ U\_y\to Y$ of $y$ such that
> $$X\times\_Y U\_y\simeq {\Bbb A}\_{\bf C}^n\times\_{\bf C} U\_y\,,$$
> where $\simeq$ denotes an isomorphism of $\bf C$-varieties
> compatible with the projections onto $U\_y$ ?
> 
> 
> 



> 
> **Question 2.** Is $f$ a locally trivial fibre bundle in the flat topology?
> 
> 
> 


Consider the complex analytic manifolds $X^{\rm an}=X({\bf C})$, $Y^{\rm an}=Y({\bf C})$
and the induced complex analytic morphism
$$f^{\rm an}\colon X^{\rm an}\to Y^{\rm an}.$$



> 
> **Question 3.** Is $f^{\rm an}\colon X^{\rm an}\to Y^{\rm an}$
> a locally trivial fibre bundle of complex analytic manifolds,
> that is, for any $y\in Y^{\rm an}$
> there exists an open neighborhood ${\mathcal U}\_y$ of $y$ in $Y^{\rm an}$
> such that $(f^{\rm an})^{-1}({\mathcal U}\_y)\simeq {\bf C}^n\times {\mathcal U}\_y$,
> where $\simeq$ denotes an analytic isomorphism compatible with the projections onto ${\mathcal U}\_y$ ?
> 
> 
> 



Smooth morphism of smooth varieties with fibres isomorphic to an affine space
Regarding Question 1, it seems to be an open problem, known as a variant of Dolgachev–Weisfeiler Conjecture. The article [$\mathbb{A}^2$-fibrations between affine spaces are $\mathbb{A}^2$-trivial](https://arxiv.org/pdf/1704.04440.pdf) (A. Dubouloz) shows that an $\mathbb{A}^2$-fibration $f\colon X\to S$ is étale-locally trivial if and only if $\Omega^1\_{X/S}$ is a pullback of a locally-free sheaf $\mathcal{E}$ on $S$. Similar questions are also mentioned in [Vénéreau polynomials and related fiber bundles](https://www.sciencedirect.com/science/article/pii/S0022404904000325) (S. Kaliman, M. Zaidenberg), page 276. Perhaps some experts can answer this question in greater detail.


