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 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. * c are again closed, densely defined, unbounded operators on L 2 * , * (Ω). The Complex Laplacian is the operator = ∂∂ * + ∂ * ∂, and its inverse (modulo kernel) is the ∂-Neumann