Affine (or Stein) tubular neighbourhood theorem Fix an embedding $X\subset Y$ of smooth complex affine varieties, or Stein manifolds.
I would guess that in general there is no analytic neighbourhood $X\subset U\subset Y$ with a holomorphic retraction $U\to X$. But does anyone know a counterexample? (If not, a proof of the existence of $U$?)
 A: My original interest in this problem was the following question.
Given a holomorphic bundle $E$ on an affine/Stein scheme $X$ inside any scheme/analytic space $Y$ of finite embedding dimension, is it the restriction of a bundle on a Stein neighbourhood $U\subset Y$ of $X$?
Smooth case. If $X$ and $Y$ are both smooth, then Corollary 1 in Siu's paper now gives this, by pulling $E$ back along the holomorphic retraction.
Reduced case. If $X$ and $Y$ are only reduced then Mohan Ramachandran's comment below solves this. In more detail: $Y$ admits a triangulation with $X$ as a subcomplex, so there is an open neighbourhood $U'$ of $X$ which is a retract, giving a continuous (not usually holomorphic) projection $\pi:U'\to X$.
Now by Siu's theorem there is a Stein neighbourhood $X\subset U\subset U'$ on which we have the topological bundle $\pi^*E|_U$. By Grauert's Oka principle there is a holomorphic structure on this which restricts to the given one on $\pi^*E|_X=E$.
$X$ nonreduced. Here we apply the above result to $E|_{X^{red}}$ to get a bundle $\widetilde E$ on $U\subset Y$ with $\widetilde E|_{X^{red}}\cong E|_{X^{red}}$. But I think this means $\widetilde E|_X\cong E$ since there is at most one extension of $E|_{X^{red}}$ from $X^{red}$ to $X$ when $X$ is Stein.
$Y$ nonreduced. Finally for general $Y$ we may assume it is Stein by replacing it by Siu's Stein neighbourhood of $X^{red}$. Then we can embed in some $\mathbb C^N$, find a holomorphic extension $\widetilde E$ there, then restrict back to $Y$.
So thanks to Richard Lärkäng and Mohan Ramachandran I think the result is probably true. Surely there's a reference for it somewhere?
A: A theorem of Siu says there in fact exists such a retraction, see Corollary 1 in
Every Stein Subvariety Admits a Stein Neighborhood, Inventiones math. 38, 89-100 (1976).
