{"paper":{"title":"Extended operational Chow group and Lefschetz (1,1)-theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Ananyo Dan, Inder Kaur","submitted_at":"2025-06-16T08:19:53Z","abstract_excerpt":"Let $X$ be a singular, projective variety. For every $p>0$, $H^{2p}(X,\\mathbb{Q})$ is equipped with a mixed Hodge structure. The elements of $\\mathrm{Gr}^W_{2p}H^{2p}(X,\\mathbb{Q}) \\cap H^{p,p} \\mathrm{Gr}^W_{2p}H^{2p}(X,\\mathbb{C})$ will be called Hodge (p,p)-classes.The purpose of this article, is to study the Bloch-Gillet-Soul\\'{e} (BGS) cycle class map from the $p$-th operational Chow group $A^p(X)$ to the space of $(p,p)$-Hodge classes. We show that if $p=1$ and $X$ is a normal surface with at worst rational singularities, then the BGS cycle class map is surjective. This extends the Lefsc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.13220","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.13220/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}