The L²-cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff

We give an example of a pseudoconvex domain in a complex manifold whose (Formula presented.)-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in (Formula presented.), hence Stein. This implies that for (Formula presented.), the usual Dolbeault cohomology with respect to smooth forms vanishes in degree (Formula presented.). But the (Formula presented.)-Cauchy–Riemann operator on the domain does not have closed range on (Formula presented.)-forms and consequently its (Formula presented.)-Dolbeault cohomology is not Hausdorff.