{"paper":{"title":"Homological Mirror Symmetry for Conic Bundle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For the conic bundle mirror of a toric Fano orbifold's canonical bundle, the wrapped microlocal sheaf category on the skeleton equals the coherent sheaves on the space minus its anti-canonical divisor.","cross_cats":["math.SG"],"primary_cat":"math.AG","authors_text":"Bohan Fang, Peng Zhou, Yuze Sun","submitted_at":"2026-05-15T15:13:56Z","abstract_excerpt":"We study the homological mirror symmetry statement where A-side is the conic bundle Hori--Vafa mirror $\\mathcal{Y} = \\{uv = f(z)\\} \\subset \\mathbb{C}^2 \\times (\\mathbb{C}^\\ast)^n$ for a Laurent polynomial $f$ in $(\\mathbb{C}^\\ast)^n$, and B-side is some a toric Calabi--Yau $(n+2)$-fold with a smooth anti-canonical divisor removed $\\mathcal{X}^\\circ = \\mathcal{X} \\setminus w^{-1}(-1)$. We show that when $\\mathcal{X}$ is the canonical bundle of a toric Fano $n$-orbifold $S$ and $f$ is its Givental superpotential, the strong deformation retraction skeleton $\\mathsf{L}$ of $\\mathcal{Y}$ in the sen"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"When X is the canonical bundle of a toric Fano n-orbifold S and f is its Givental superpotential, the strong deformation retraction skeleton L of Y has a Weinstein neighborhood U such that the wrapped microlocal sheaf category μSh^w_L(L) ≅ Coh(X^∘). This proves a microlocal categorical version of the SYZ mirror.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The paper assumes that the skeleton L from the RSTZ construction admits a Weinstein neighborhood U in which the wrapped microlocal sheaf category can be defined and that the specific choice of f as the Givental superpotential ensures the isomorphism holds, as stated in the conditions for the equivalence.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves that the wrapped microlocal sheaf category μSh^w_L(L) is equivalent to Coh(X^∘) for conic bundle mirrors of toric Calabi-Yau (n+2)-folds under given conditions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For the conic bundle mirror of a toric Fano orbifold's canonical bundle, the wrapped microlocal sheaf category on the skeleton equals the coherent sheaves on the space minus its anti-canonical divisor.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9a98e82de251fd1e045d6d83329bb4bea216549328145b4b4a3dc4143139623e"},"source":{"id":"2605.16040","kind":"arxiv","version":1},"verdict":{"id":"44a7923f-34a9-4e1b-82c0-98ac4cdc7875","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T19:22:44.500030Z","strongest_claim":"When X is the canonical bundle of a toric Fano n-orbifold S and f is its Givental superpotential, the strong deformation retraction skeleton L of Y has a Weinstein neighborhood U such that the wrapped microlocal sheaf category μSh^w_L(L) ≅ Coh(X^∘). This proves a microlocal categorical version of the SYZ mirror.","one_line_summary":"Proves that the wrapped microlocal sheaf category μSh^w_L(L) is equivalent to Coh(X^∘) for conic bundle mirrors of toric Calabi-Yau (n+2)-folds under given conditions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The paper assumes that the skeleton L from the RSTZ construction admits a Weinstein neighborhood U in which the wrapped microlocal sheaf category can be defined and that the specific choice of f as the Givental superpotential ensures the isomorphism holds, as stated in the conditions for the equivalence.","pith_extraction_headline":"For the conic bundle mirror of a toric Fano orbifold's canonical bundle, the wrapped microlocal sheaf category on the skeleton equals the coherent sheaves on the space minus its anti-canonical divisor."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16040/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T19:31:19.009156Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:31:02.612050Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:41.561124Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T16:41:55.536252Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"789c3c2cf8a58c2aed2a89a00f0834ded4eea75e6e39057d4899bf40575d471f"},"references":{"count":34,"sample":[{"doi":"","year":null,"title":"Publications math","work_id":"2d77a68c-8bb0-4b83-bde3-f71ba153c374","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"and Chen, Linda and Smith, Gregory G","work_id":"5d4ad860-e10a-4d2d-a339-dfb418b25d28","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"International Mathematics Research Notices , volume =","work_id":"88c97f92-597c-49b5-92f6-4be82e714f82","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.48550/arxiv.2504.15696","year":2025,"title":"2025 , eprint =","work_id":"c734ea71-f50c-438c-9277-efabeb10952b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Gammage, Benjamin and Le, Ian , title =. 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