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 $U$?)



Affine (or Stein) tubular neighbourhood theorem
A theorem of Siu says there in fact exists such a retraction, see Corollary 1 in
[Every Stein Subvariety Admits a Stein Neighborhood](https://link.springer.com/article/10.1007/BF01390170), Inventiones math. 38, 89-100 (1976).


