{"paper":{"title":"Symplectic solvmanifolds not satisfying the hard-Lefschetz condition","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Adri\\'an Andrada, Agust\\'in Garrone","submitted_at":"2025-05-12T22:56:05Z","abstract_excerpt":"For Lie groups $G$ of the form $G = \\R^k \\ltimes_{\\phi} \\R^m$, with $k + m$ even, a result of H. Kasuya shows that if the action $\\phi:\\R^k \\to \\mathrm{Aut}(\\R^m)$ is semisimple then any symplectic solvmanifold $(\\Gamma \\backslash G, \\omega)$ satisfies the hard-Lefschetz condition for any symplectic form. In this article, we prove the converse in the case $k = 1$ and $G$ completely solvable: no symplectic form on such a solvmanifold satisfies the hard-Lefschetz condition if $\\phi$ is not semisimple; moreover, we show that the failure occurs either at degree $1$ or at degree $2$ in cohomology, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.08113","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/2505.08113/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"}