{"paper":{"title":"Generic vector fields on isolated complex hypersurface germs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The GSV-index of a holomorphic vector field on an isolated hypersurface singularity is at least 1 plus or minus the Tjurina number, with equality exactly when the field extends to a nondegenerate zero in ambient space.","cross_cats":[],"primary_cat":"math.AG","authors_text":"Diogo da Silva Machado, Jose Seade","submitted_at":"2026-05-09T23:04:46Z","abstract_excerpt":"We study holomorphic vector fields on isolated hypersurface singularities and derive global obstructions to the existence of holomorphic vector fields on compact singular varieties. For a hypersurface germ $(V,0)$ with an isolated singularity, we characterize the generic elements in the space of holomorphic vector fields with isolated singularity in terms of the GSV-index. Letting $\\tau(V,0)$ denote the Tjurina-Greuel number, we prove that the minimal possible index is bounded below by $1+(-1)^{\\dim(V)}\\tau(V,0)$. We further prove that equality holds if the vector field admits an extension to "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the minimal possible index is bounded below by 1+(-1)^{dim(V)}τ(V,0). We further prove that equality holds if and only if the vector field admits an extension to C^{n+1} with a nondegenerate singularity at 0, and that such extensions form an open dense subset.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The space of holomorphic vector fields with isolated singularities on the germ is equipped with a topology in which open-dense subsets are meaningful, and the hypersurface germ has an isolated singularity at 0.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For an isolated hypersurface singularity, generic holomorphic vector fields with isolated zeros have GSV-index at least 1 + (-1)^dim(V) * τ(V,0), with equality exactly when the field extends to a nondegenerate zero in ambient space.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The GSV-index of a holomorphic vector field on an isolated hypersurface singularity is at least 1 plus or minus the Tjurina number, with equality exactly when the field extends to a nondegenerate zero in ambient space.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"56082946e47f9999bc76e0fb38136b39b348fcca0c27cf25df7451153016a09a"},"source":{"id":"2605.09210","kind":"arxiv","version":2},"verdict":{"id":"6b1dd7cf-0467-40da-b0c4-0735f8a78e31","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T02:47:30.499348Z","strongest_claim":"We prove that the minimal possible index is bounded below by 1+(-1)^{dim(V)}τ(V,0). We further prove that equality holds if and only if the vector field admits an extension to C^{n+1} with a nondegenerate singularity at 0, and that such extensions form an open dense subset.","one_line_summary":"For an isolated hypersurface singularity, generic holomorphic vector fields with isolated zeros have GSV-index at least 1 + (-1)^dim(V) * τ(V,0), with equality exactly when the field extends to a nondegenerate zero in ambient space.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The space of holomorphic vector fields with isolated singularities on the germ is equipped with a topology in which open-dense subsets are meaningful, and the hypersurface germ has an isolated singularity at 0.","pith_extraction_headline":"The GSV-index of a holomorphic vector field on an isolated hypersurface singularity is at least 1 plus or minus the Tjurina number, with equality exactly when the field extends to a nondegenerate zero in ambient space."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.09210/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T08:02:09.776489Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T20:35:13.838428Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T13:31:18.242555Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:28:18.756693Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"64d52bb591d3cad627bd89c4a225ac3569411667b657c1d8c68c7f8a1b270739"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"45830f3ed5294a47892e63e4c12c4cea77bd883c98bdb6cad354816c63e92f50"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}