Let $X$ be a compact, Kähler surface (so $\dim X = 4$) and let $f:X\to \mathbb{CP}^1$ be an elliptic fibration, in particular $f$ is holomorphic.
Suppose that $p\in \mathbb{CP}^1$ is a singular value of $f$ and consider a closed ball $D\subset \mathbb{CP}^1$ centered at $p$, we take $D$ small enough that $\partial D\cap CritValues(f) = \emptyset$ .
Then $f^{-1}(D)$ is a compact 4-manifold with boundary having a Kähler structure,
Q1: Is $f^{-1}(D)$ a Stein domain? Recall that a Stein manifold is a proper complex submanifold $M$ of $\mathbb C^n$ and a Stein domain is a region of $M$ cut out by $g\leq c$ where $g:M\to \mathbb{R}$ is a J-convex proper Morse function and $c$ is a regular value.
My approach: from the classification of complex surfaces we know that $X$, is projective, i.e. embeds in $\mathbb {CP}^n$ , hence, by intersectig $X$ with an affine chart of $\mathbb {CP}^n$ if we can find an open ball $B \subset \mathbb {CP}^1$ containing $D\subset B$, such that $f^{-1}(B)\subset \mathbb C^n$ is a properly embedded complex submanifold. At this point I would take $M=f^{-1}(B)$ and as $J$-convex function $g:M\to \mathbb{R}$ $g(x) = |f(x)|^2$.
Is this approach correct?
Assume $f^{-1}(D) \subset \mathbb{C}^n$. Then $$ f^{-1}(b) \subset \mathbb{C}^n $$ is a holomorphic embedding. It induces a notrivial holomorphic morphism from a smooth Riemann surface to $\mathbb{C}^n$. But any holomorphic function on a compact Riemann surface is constant, a contradiction.