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

2013/05/25 by Debraj Chakrabarti, Chakrabarti, Debraj, Mei-Chi Shaw +1 · 1 citation
Mathematics · #32C35 #32V25 #32W10 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1305.5924

openalex publication_date 2013/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give an example of a pseudoconvex domain in a complex manifold whose L²-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 ℂ², hence Stein. This implies that for q>0, the usual Dolbeault cohomology with respect to smooth forms vanishes in degree (p,q). But the L²-Cauchy-Riemann operator on the domain does not have closed range on (2,1)-forms and consequently its L²-Dolbeault cohomology is not Hausdorff.

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