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We first prove that the action is Hamiltonian if and only if $b_2^+(M_{\\red})=1$, where $M_{\\red}$ is any reduced space with respect to $\\mu$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\\red}) \\geq 2$. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if $M^{S^1}$ consists of surfaces, then the number $k$ of fixed su"},"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":"1005.0193","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2010-05-03T06:21:36Z","cross_cats_sorted":[],"title_canon_sha256":"8f326d8a81ea0fb5eab167554a478f21255284ee1cf3b3f7e721618d596ec21d","abstract_canon_sha256":"7685cc89ed1ef71de14ec2c1659947259cac81b4f56e5751fa168e0a2236dd5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:33.719506Z","signature_b64":"cjd0Igc36NjNHA4tC6XxFSt2y3Smrv+sxK29rozWFJTqTvhKRRD+m59tcR/9l9hgID1NQdISKMt+WYUK3i+PBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1243705fe0558c835240ed5b52c780def4ff9f6533bd7f6f25cf5e6e9303572f","last_reissued_at":"2026-05-18T01:16:33.718866Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:33.718866Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semifree Hamiltonian circle actions on 6-dimensional symplectic manifolds with non-isolated fixed point set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Dong Youp Suh, Taekgyu Hwang, Yunhyung Cho","submitted_at":"2010-05-03T06:21:36Z","abstract_excerpt":"Let $(M, \\omega)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \\neq \\emptyset$ and $\\dim M^{S^1} \\leq 2$. Assume that $\\omega$ is integral with a generalized moment map $\\mu$. We first prove that the action is Hamiltonian if and only if $b_2^+(M_{\\red})=1$, where $M_{\\red}$ is any reduced space with respect to $\\mu$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\\red}) \\geq 2$. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if $M^{S^1}$ consists of surfaces, then the number $k$ of fixed su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0193","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1005.0193","created_at":"2026-05-18T01:16:33.718969+00:00"},{"alias_kind":"arxiv_version","alias_value":"1005.0193v5","created_at":"2026-05-18T01:16:33.718969+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0193","created_at":"2026-05-18T01:16:33.718969+00:00"},{"alias_kind":"pith_short_12","alias_value":"CJBXAX7AKWGI","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_16","alias_value":"CJBXAX7AKWGIGUSA","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_8","alias_value":"CJBXAX7A","created_at":"2026-05-18T12:26:06.534383+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33","json":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33.json","graph_json":"https://pith.science/api/pith-number/CJBXAX7AKWGIGUSA5VNVFR4A33/graph.json","events_json":"https://pith.science/api/pith-number/CJBXAX7AKWGIGUSA5VNVFR4A33/events.json","paper":"https://pith.science/paper/CJBXAX7A"},"agent_actions":{"view_html":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33","download_json":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33.json","view_paper":"https://pith.science/paper/CJBXAX7A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1005.0193&json=true","fetch_graph":"https://pith.science/api/pith-number/CJBXAX7AKWGIGUSA5VNVFR4A33/graph.json","fetch_events":"https://pith.science/api/pith-number/CJBXAX7AKWGIGUSA5VNVFR4A33/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33/action/storage_attestation","attest_author":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33/action/author_attestation","sign_citation":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33/action/citation_signature","submit_replication":"https://pith.science/pith/CJBXAX7AKWGIGUSA5VNVFR4A33/action/replication_record"}},"created_at":"2026-05-18T01:16:33.718969+00:00","updated_at":"2026-05-18T01:16:33.718969+00:00"}