3
$\begingroup$

Let $X$ be a Stein manifold, $K$ be a compact subset of $X$. Consider the following conditions:

1.$K$ admit an open neighborhood basis in $X$ whose members are Stein;

2.$K=\cap_{j\ge 1}V_j$, where $V_1\supset V_2\supset V_3\dots$ is a decreasing sequence of Stein opens in $X$;

3.there exists a family of Stein opens $\{U_i\}_{i\in I}$ in $X$ with $K=\cap_{i\in I}U_i$.

We have that 1 implies 2, and 2 implies 3. I wonder if the three conditions are equivalent.

The question arises because I have seen in difference papers that they all bear the same names: holomorphic set, $S_{\delta}$, Stein compactum, Stein compact, etc.

Improve this question
$\endgroup$
3
  • $\begingroup$ For Stein neighborhood, do you specific means a strongly pseudoconvex open domain? $\endgroup$
    – Liding Yao
    Commented Feb 26 at 5:28
  • 1
    $\begingroup$ Dear @LidingYao, No. A connected open subset of $\mathbb{C}^n$ is Stein iff it is pseudoconvex. I guess that there is a pseudoconvex domain in $\mathbb{C}^n$ which is not strongly pseudoconvex. $\endgroup$
    – Doug Liu
    Commented Feb 27 at 9:18
  • 1
    $\begingroup$ Once you consider an pseudoconvex open neighborhood $U=\{z:\rho(z)<0\}$ of $K$, you can certainly shrink the neighborhood, say considering $\{z:\rho(z)+\epsilon|z|^2<0\}$ for small enough $\epsilon>0$. This is going to give you a strongly pseudoconvex neighborhood. That's why I ask what convention you take for the definitions. $\endgroup$
    – Liding Yao
    Commented Feb 27 at 17:38

0

Reset to default

You must log in to answer this question.