Fix $n\ge 1$, we work over $\Bbb C$.
Let $U_k:={(u_k,v_k)\in \Bbb C^2: |u_kv_k|<1}$ for $k\in \Bbb Z$.
Glue $U_k$ and $U_{k+1}$ by identifying $(u_{k+1},v_{k+1})$ with $((v_k)^{-1},u_k(v_k)^2)$.
Let the infinite cyclic group act on the manifold obtained from above identification by $(u_k,v_k) \to (u_{k+n},v_{k+n})$. The quotient $Z_n$ admits a fibration $Z_n \to \Delta(s)$ given by $(u_k,v_k) \mapsto s=u_kv_k$.
Is $Z_n$ a Kähler surface?
Locally we can see that the fiber over $0\in \Delta$ is a singularity of type $I_n$, but away from $0\in \Delta$ the fiber is a smooth elliptic curve (whose lattice depends on $s\in \Delta$). This result shows that $Z_n$ is locally conformally Kähler, but can we hope for $Z_n$ to be Kähler? (my original goal was to see if we can speak about stability of vector bundles over such surfaces).