{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1066",
  "content_hash": null,
  "slug": "generic-kahler-laplacian-simple-spectrum-claim-status-20260731",
  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-31. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample. Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual.",
  "relevance": "Defines the dated frontier and the exact remainder that research on this problem must resolve.",
  "relevance_source": "recorded",
  "body": "UNKNOWN as of 2026-07-31. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.\n\nA complete resolution must satisfy this condition: Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/32810/are-the-eigenvalues-of-the-laplacian-of-a-generic-k%C3%A4hler-metric-simple",
      "locator": "See dataset.references[0] for the exact external source and locator."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/32810/are-the-eigenvalues-of-the-laplacian-of-a-generic-k%C3%A4hler-metric-simple",
    "locator": "See dataset.references[0] for the exact external source and locator."
  },
  "relations": [
    {
      "slug": "R1065",
      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "generic-kahler-laplacian-simple-spectrum",
      "title": "generic kahler laplacian simple spectrum",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}