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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 Hausd"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.5924","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2013-05-25T14:01:58Z","cross_cats_sorted":[],"title_canon_sha256":"6858db271d023fb1b44e6b8f7e12472998f206b855fe815ee4bb13f8db4cccca","abstract_canon_sha256":"fb4e38035c5a89d239dcbdfa31182c55570c8c01b2f48571ff67202c88fea3d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:01.118938Z","signature_b64":"/Ls9ksoyb/MyBoKLPJnXsu7GhEIarhdfetjM7M4MzAPJqey7r76VRk5OwGuwkf5e0Elkoh3j6yYoyW9evYu6Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"363ecf57a5196de45ad445cbf8f144ceab1b60c59453ec7db29a9ee7c67c2147","last_reissued_at":"2026-05-18T02:26:01.118384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:01.118384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $L^2$-cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Debraj Chakrabarti, Mei-Chi Shaw","submitted_at":"2013-05-25T14:01:58Z","abstract_excerpt":"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. 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