{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0175",
  "title": "Complex Sectional Curvature Blow-Up under Circle Cheeger Collapse",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An AIM problem asks whether complex sectional curvatures remain uniformly bounded below in a collapsing circle Cheeger deformation. At a fixed point whose normal circle representation contains rotation blocks of speeds a,b > 0, we derive an exact fixed-point formula. A totally isotropic complex two-plane has K_C(g_ε) = K_C(g) − ab/ε². Thus every fixed component of real codimension at least four forces complex sectional curvature to diverge to −∞. For the standard weight-(1,1) action on the unit round S⁴, the value at either fixed pole is 1 − ε⁻². With only one rotation block, the singular curvature operator is instead rank one and positive semidefinite. The AIM workshop report records the corresponding negative curvature-operator phenomenon but does not identify a complex decomposable direction; the observation here is that its two real negative directions combine into a decomposable totally isotropic bivector. The note is unrefereed and makes no priority claim for this elementary calculation.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.DG"
  ],
  "keywords": [
    "Cheeger deformation",
    "complex sectional curvature",
    "collapse",
    "circle actions",
    "curvature operators",
    "AIM-GEOMETRY-0175",
    "open mathematics",
    "mathematical proof",
    "math.DG"
  ],
  "manuscript_version_date": "2026-09-02",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0175/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0175/paper.pdf?v=067c13cbc631",
  "doi": "10.5281/zenodo.22245130",
  "zenodo_record_url": "https://zenodo.org/records/22245130",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}