{"paper":{"title":"$L^2$-theory for the $\\overline\\partial$-operator on compact complex spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Jean Ruppenthal","submitted_at":"2010-04-02T22:03:37Z","abstract_excerpt":"Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\\overline\\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an $L^2$-resolution for the Grauert-Riemenschneider canonical sheaf $\\mathcal{K}_X$. As an application, we obtain a Grauert-Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If $X$ is a Gorenstein space with canonical singularities, then we get also an $L^2$-representation of the flabby cohomology of the structure sheaf $\\mathc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0396","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}