# Smooth morphism of smooth varieties with fibres isomorphic to an affine space

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.
• @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". May 9 at 19:01
• 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? May 9 at 21:21