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1. Let $X$ be a smooth irreducible $\Bbb C$-variety, on which the algebraic $\Bbb C$-group $G={\bf G}_{a,{\Bbb C}}$ (the additive group) acts freely on the right: $$ X\times _{\Bbb C} G\to X,\quad (x,g)\mapsto x\cdot g.$$ Assume that there exists a surjective morphism onto a smooth $\Bbb C$-variety $Y$ \begin{equation}\label{e:*} \varphi\colon X\to Y \tag{$*$} \end{equation} whose fibres are the orbits of $G$ in $X$. Then the morphism $\varphi$ is smooth, from which one can deduce that $\varphi$ induces a locally trivial fibre bundle (in the usual topology) of $C^\infty$-manifolds $$\varphi\colon X(\Bbb C)\to Y(\Bbb C).$$

Question 1. Does it follow that $(*)$ is locally trivial in the flat topology, that is, a $Y$-torsor under $G$? In other words, is the morphism $$ X\times_Y G\to X\times_Y X,\quad (x,g)\mapsto (x,x\cdot g) $$ an isomorphism of $\Bbb C$-varieties?

2. Assume that $(*)$ is a torsor. Since $H^1({\Bbb C}(Y),G)=\{1\}$, we know that $(*)$ admits a rational section.

Question 2. Does $(*)$ admit a regular section? In other words, does there exist a regular map (morphism) $s\colon Y\to X$ such that $\varphi\circ s={\rm id}_Y\,$?

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For 1, yes. In fact, any smooth morphism of varieties admits a section locally in the etale topology everywhere.

Proof: A generic hypersurface section is smooth of dimension one lower over any particular point. Repeat until the relative dimension is zero.

For 2, no. Any variety $Y$ with $H^1 (Y, \mathcal O_Y) \neq 0$ is a counterexample, as every class gives a torsor $X$.

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  • $\begingroup$ Thank you, Will, for a prompt answer! Could you please add details to the proof for 1? A generic hypersurface section of what? $\endgroup$ – Mikhail Borovoi Apr 29 at 16:51
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    $\begingroup$ @MikhailBorovoi of $X$. For example, take $y\in Y$, $x \in X$ with $\varphi(x)=y$, and choose any function on (a neighborhood of $x$ in) $X$ vanishing at $x$ and whose first derivative is nonzero on the relative tangent space of $X$ over $Y$. The vanishing locus of this function is still smooth over $Y$ at the point $x$, thus smooth over $Y$ in a neighborhood of $x$. $\endgroup$ – Will Sawin Apr 29 at 17:28
  • $\begingroup$ Does it follow that a morphism $\varphi$ as in my question induces a locally trivial fibre bundle of complex analytic manifolds (with the usual topology)? $\endgroup$ – Mikhail Borovoi May 1 at 18:05
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    $\begingroup$ @MikhailBorovoi Yes. Locally trivializable in the etale topology implies locally trivializable in the flat, complex analytic, and several other topologies. In fact, the comparison between etale and Zariski cohomology for a quasicoherent sheaf implies that it is locally trivializable in the Zariski topology as well. The only subtlety I can think of, which applies equally to all these topologies, is that you want the schematic fibers to be orbits, i.e. you want the fibers to be reduced. $\endgroup$ – Will Sawin May 1 at 18:14
  • $\begingroup$ Thank you! Is your argument for "In fact, any smooth morphism of varieties admits a section locally in the etale topology everywhere" written somewhere, so that I could refer? $\endgroup$ – Mikhail Borovoi May 1 at 18:30

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