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

TL;DR

This paper presents an example of a Stein domain with non-Hausdorff $L^2$-cohomology, challenging assumptions about the relationship between Stein domains and the topological properties of their $L^2$-cohomology.

Contribution

It constructs a specific Stein domain where the $L^2$-Dolbeault cohomology is non-Hausdorff, illustrating that Steinness does not guarantee Hausdorff $L^2$-cohomology.

Findings

The domain is biholomorphic to a product in $C^2$ and is Stein.

The usual Dolbeault cohomology vanishes in positive degrees.

The $L^2$-Cauchy-Riemann operator does not have closed range on certain forms.

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.