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, 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 ${\mathcal U}_y$ ?


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 (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 (S. Kaliman, M. Zaidenberg), page 276. Perhaps some experts can answer this question in greater detail.

  • $\begingroup$ I am probably mistaken, but the paper by Dubouloz you mention seems to be interested in local trivilaity in the Zarisi topology, not in the étale topology. $\endgroup$ – Libli May 9 at 18:48
  • $\begingroup$ @Libli I'm not an expert, but looking at the theorem in the introduction and Definitions 1 and 2, it seems that the result treats etale-local triviality, no? In any case, Zariski local triviality is even stronger, so gives a better answer to Q1? $\endgroup$ – Piotr Achinger May 9 at 18:56
  • $\begingroup$ @PiotrAtchinger I must have a problem with my English today but I read the following : "Summing up, if an $\mathbb{A}^n$-fibration $\pi :V \longrightarrow X$ over an affine scheme $X$ is a Zariski locally trivial $\mathbb{A}^n$-bundle, then its relative cotangent sheaf is induced from $X$. Our main result, which can be summarized as follows, implies in particular that the converse holds for $\mathbb{A}^2$-fibrations over smooth affine schemes". $\endgroup$ – Libli May 9 at 19:01
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    $\begingroup$ Locally trivial on the source, but not on the target! $\endgroup$ – Piotr Achinger May 9 at 19:06
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    $\begingroup$ Good find! The first thing the paper says is that the paper considers affine morphisms. Is it even clear that a morphism $f$ as in the question is affine? $\endgroup$ – Johan May 9 at 21:21

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