The $$L^2$$ L 2 -cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff

Abstract

We give an example of a pseudoconvex domain in a complex manifold whose \(L^2\)-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 \({\mathbb {C}}^2\), hence Stein. This implies that for \(q>0\), the usual Dolbeault cohomology with respect to smooth forms vanishes in degree \((p,q)\). But the \(L^2\)-Cauchy–Riemann operator on the domain does not have closed range on \((2,1)\)-forms and consequently its \(L^2\)-Dolbeault cohomology is not Hausdorff.