{"paper":{"title":"Hodge decomposition and Hard Lefschetz Condition on almost K\\\"{a}hler manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Teng Huang, Weiwei Wang","submitted_at":"2025-06-06T05:17:14Z","abstract_excerpt":"In this article, we discuss the spaces of harmonic forms $\\mathcal{H}^{\\bullet}_{d}$ over a closed almost K\\\"{a}hler manifold $(X, J,\\omega)$. We show that if the almost complex structure $J$ on the almost K\\\"{a}hler manifold $X$ is not too non-integrable in some sense, then the spaces $\\mathcal{H}^{\\bullet}_{d}$ have the Hodge decomposition $\\mathcal{H}^{k}_{d}=\\oplus_{p+q=k}\\mathcal{H}^{p,q}_{d}$. As a consequence, the not too non-integrable almost complex structure $J$ is complex $C^{\\infty}$-pure-and-full, and the Hard Lefschetz Condition (HLC) on $\\mathcal{H}^{\\bullet}_{d}$ is satisfied. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.06402","kind":"arxiv","version":1},"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/2506.06402/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"}