Complex hypersurface globally defined

Question

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ?

Answer 1

No, it is not true that given a hypersurface $A\subset M$ of a Stein manifold there exists a global holomorphic function $f\in \mathcal O(M)$ whose zero locus is $A$.
The relevant notion is that of divisor: the hypersurface $A$ defines a holomorphic (= positive) divisor and your question is whether that divisor is solvable i.e. is the zero set of some global holomorphic function.
This is called the Cousin-II problem and it has a nice soluton in terms of cohomology:
The divisor $A$ defines a line bundle $\mathcal O(A)$ and the divisor has a global equation $f\in \mathcal O(M)$ if and only if the line bundle has zero first Chern class : $c_1(\mathcal O(A))=0\in H^2(M,\mathbb Z)$.
One can show that every Stein manifold with $H^2(M,\mathbb Z)\neq0 $ actually has a non solvable positive divisor.

All this is beautifully explained in L. Kaup-B.Kaup's excellent book and the last sentence comes from their Proposition 54.6.

Edit
A completely explicit but a bit complicated example of a hypersurface without global equation in a Stein manifold is given in this book by Grauert-Fritzche on pages 172-174.