Let $Y$ be an affine open subset of a locally noetherian scheme $X$. Then, $X \setminus Y$ has pure codimension one [EGAIV$_4$, Cor. 21.12.7]. Moreover, if $X$ is proper and of finite type over a field $k$, and $\dim X \ge 2$, then $X \setminus Y$ is connected [[Hartshorne][1], Cor. II.6.2].

My question is the following:

> If $X$ is a complex projective variety of dimension $\ge 2$, and $Y$ is an open
> subset that is Stein, then is it still true that $X \setminus Y$ is
> connected?

The fact that $X \setminus Y$ is of pure codimension one is [[Neeman][2], Prop. 3.4], and is a consequence of Hartog's theorem. Jing Zhang cites this proposition for the statement that $X \setminus Y$ is connected in [Algebraic Stein Varieties][3], specifically in the proof of Thm. 2.8, but it is not clear to me how Neeman's proposition implies this result.

It could also be true that the algebraic proof when $Y$ is affine works in the case when $Y$ is Stein instead. But Hartshorne (and Goodman in his [original article][4]) use theorems about blow-ups from Nagata's [paper on compactifications][5], and I don't know if these carry over to the analytic case.


  [1]: http://www.ams.org/mathscinet-getitem?mr=282977
  [2]: http://www.ams.org/mathscinet-getitem?mr=932296
  [3]: http://www.ams.org/mathscinet-getitem?mr=2424914
  [4]: http://www.ams.org/mathscinet-getitem?mr=242843
  [5]: http://www.ams.org/mathscinet-getitem?mr=142549