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In this paper, we will show that if $\\mu$ is constant on $\\Lambda_{2k}$ for each $k$, then $(M,\\omega)$ satisfies the hard Lefschetz property. In particular, if $(M,\\omega)$ admits a self-indexing moment map, i.e. $\\mu(p) = 2k$ for every $p \\in \\Lambda_{2k}$ and $k=0,1,\\cdots,n,$ then $(M,\\omega)$ satisfies the hard Lefsc"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1312.6512","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2013-12-23T10:43:36Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"09d600e605675b26cff5b81c655c6982b77951a0f3bc65f2027dffb71ca083fa","abstract_canon_sha256":"45ff9cc137718f3aeb27c3db89c3f5a34dfa1e5ee9c5d81cb360ef1a26bcb5c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:58.040305Z","signature_b64":"RDJbkSMyzBK9DzaMw23bK7mT1RbCk3BUpMEDPGgmpz0vmmppIB6P1ohVtQC7tLvrS+ENJFxoS54GJmLyY4pBAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f4d4a5eb69cd4fa8795bff357d834280335f40ac6a04a29dc576acef1d3572ed","last_reissued_at":"2026-05-18T03:03:58.039432Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:58.039432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hamiltonian circle action with self-indexing moment map","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.SG","authors_text":"Min Kyu Kim, Yunhyung Cho","submitted_at":"2013-12-23T10:43:36Z","abstract_excerpt":"Let $(M,\\omega)$ be a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points and let $\\mu : M \\rightarrow \\R$ be a corresponding moment map. 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