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arXiv · 1305.5924

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

Debraj Chakrabarti·Mei-Chi Shaw
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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.

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Debraj Chakrabarti, Mei-Chi Shaw. 2015-03-01. The $L^2$-cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff. https://arxiv.org/abs/1305.5924

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