{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:MKNNWGPKNQGVEOZGI6BTBXYP7G","short_pith_number":"pith:MKNNWGPK","canonical_record":{"source":{"id":"1205.4608","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-05-21T14:13:00Z","cross_cats_sorted":["math.GR","math.RT"],"title_canon_sha256":"7e228b41e9fe6dc5bc9002fb24b04421568f845d25af4ec94568846367185c10","abstract_canon_sha256":"73d188b436daee87caabcf702e27dac27343279d81b581fe32bfba4b9439a2d4"},"schema_version":"1.0"},"canonical_sha256":"629adb19ea6c0d523b26478330df0ff9a2b76c53d63c0b47d0b18a21712e30f2","source":{"kind":"arxiv","id":"1205.4608","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4608","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4608v3","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4608","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"pith_short_12","alias_value":"MKNNWGPKNQGV","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MKNNWGPKNQGVEOZG","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MKNNWGPK","created_at":"2026-05-18T12:27:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:MKNNWGPKNQGVEOZGI6BTBXYP7G","target":"record","payload":{"canonical_record":{"source":{"id":"1205.4608","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-05-21T14:13:00Z","cross_cats_sorted":["math.GR","math.RT"],"title_canon_sha256":"7e228b41e9fe6dc5bc9002fb24b04421568f845d25af4ec94568846367185c10","abstract_canon_sha256":"73d188b436daee87caabcf702e27dac27343279d81b581fe32bfba4b9439a2d4"},"schema_version":"1.0"},"canonical_sha256":"629adb19ea6c0d523b26478330df0ff9a2b76c53d63c0b47d0b18a21712e30f2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:20.816665Z","signature_b64":"pgkxl9cXZwicZvrlKS33YRgEaf4OPZAi5ZvYWOrdUOHDNdSGMNXDylXIZu++wlEMQEcKMC6JK+IgqNj62wcvCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"629adb19ea6c0d523b26478330df0ff9a2b76c53d63c0b47d0b18a21712e30f2","last_reissued_at":"2026-05-18T03:21:20.816193Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:20.816193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1205.4608","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O0xfcN0mNLeslxjIlgZU6NhwNG5OkI1dkG/ZZKCYMxI5DajrYR7FAcGx2maLKEuMK2i0vkGpI4v3KgAomesVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:41:11.457395Z"},"content_sha256":"10490254808306d49493f44c4b237fbb9ffa543231b1fc31dc6c725ffbcc4c61","schema_version":"1.0","event_id":"sha256:10490254808306d49493f44c4b237fbb9ffa543231b1fc31dc6c725ffbcc4c61"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:MKNNWGPKNQGVEOZGI6BTBXYP7G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Koszul complex of a moment map","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.RT"],"primary_cat":"math.SG","authors_text":"Gerald W. Schwarz, Hans-Christian Herbig","submitted_at":"2012-05-21T14:13:00Z","abstract_excerpt":"Let $K\\to U(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\\rho\\colon V\\to\\mathfrak k^*$. We have the Koszul complex ${\\mathcal K}(\\rho,\\mathcal C^\\infty(V))$ of the component functions $\\rho_1,...,\\rho_k$ of $\\rho$. Let $G=K_{\\mathbb C}$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on $\\rho^{-1}(0)$ if and only if $G\\to\\GL(V)$ is 1-large, a concept introduced in earlier work of the second author. Now let $M$ be a symplectic manifold with a Hamiltonian action of $K$. Let $\\rho$ be "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4608","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ArJd0tXo3G8EZvI3yO0GVj1K6wyR6L+yUm34fKeYBJn1PaLldVGAV7h7ykIjZY+E6o1EO2Ec76lDBmpsoN+ADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:41:11.457791Z"},"content_sha256":"35e9bec3d78af882941818172fe049664e93973fa466e74e09982365cbab95ea","schema_version":"1.0","event_id":"sha256:35e9bec3d78af882941818172fe049664e93973fa466e74e09982365cbab95ea"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MKNNWGPKNQGVEOZGI6BTBXYP7G/bundle.json","state_url":"https://pith.science/pith/MKNNWGPKNQGVEOZGI6BTBXYP7G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MKNNWGPKNQGVEOZGI6BTBXYP7G/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T14:41:11Z","links":{"resolver":"https://pith.science/pith/MKNNWGPKNQGVEOZGI6BTBXYP7G","bundle":"https://pith.science/pith/MKNNWGPKNQGVEOZGI6BTBXYP7G/bundle.json","state":"https://pith.science/pith/MKNNWGPKNQGVEOZGI6BTBXYP7G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MKNNWGPKNQGVEOZGI6BTBXYP7G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:MKNNWGPKNQGVEOZGI6BTBXYP7G","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"73d188b436daee87caabcf702e27dac27343279d81b581fe32bfba4b9439a2d4","cross_cats_sorted":["math.GR","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-05-21T14:13:00Z","title_canon_sha256":"7e228b41e9fe6dc5bc9002fb24b04421568f845d25af4ec94568846367185c10"},"schema_version":"1.0","source":{"id":"1205.4608","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.4608","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"arxiv_version","alias_value":"1205.4608v3","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.4608","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"pith_short_12","alias_value":"MKNNWGPKNQGV","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"MKNNWGPKNQGVEOZG","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"MKNNWGPK","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:35e9bec3d78af882941818172fe049664e93973fa466e74e09982365cbab95ea","target":"graph","created_at":"2026-05-18T03:21:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $K\\to U(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\\rho\\colon V\\to\\mathfrak k^*$. We have the Koszul complex ${\\mathcal K}(\\rho,\\mathcal C^\\infty(V))$ of the component functions $\\rho_1,...,\\rho_k$ of $\\rho$. Let $G=K_{\\mathbb C}$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on $\\rho^{-1}(0)$ if and only if $G\\to\\GL(V)$ is 1-large, a concept introduced in earlier work of the second author. Now let $M$ be a symplectic manifold with a Hamiltonian action of $K$. Let $\\rho$ be ","authors_text":"Gerald W. Schwarz, Hans-Christian Herbig","cross_cats":["math.GR","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-05-21T14:13:00Z","title":"The Koszul complex of a moment map"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.4608","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:10490254808306d49493f44c4b237fbb9ffa543231b1fc31dc6c725ffbcc4c61","target":"record","created_at":"2026-05-18T03:21:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"73d188b436daee87caabcf702e27dac27343279d81b581fe32bfba4b9439a2d4","cross_cats_sorted":["math.GR","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2012-05-21T14:13:00Z","title_canon_sha256":"7e228b41e9fe6dc5bc9002fb24b04421568f845d25af4ec94568846367185c10"},"schema_version":"1.0","source":{"id":"1205.4608","kind":"arxiv","version":3}},"canonical_sha256":"629adb19ea6c0d523b26478330df0ff9a2b76c53d63c0b47d0b18a21712e30f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"629adb19ea6c0d523b26478330df0ff9a2b76c53d63c0b47d0b18a21712e30f2","first_computed_at":"2026-05-18T03:21:20.816193Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:20.816193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pgkxl9cXZwicZvrlKS33YRgEaf4OPZAi5ZvYWOrdUOHDNdSGMNXDylXIZu++wlEMQEcKMC6JK+IgqNj62wcvCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:20.816665Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.4608","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:10490254808306d49493f44c4b237fbb9ffa543231b1fc31dc6c725ffbcc4c61","sha256:35e9bec3d78af882941818172fe049664e93973fa466e74e09982365cbab95ea"],"state_sha256":"36ca976cc8837cbdc9aef46900968681e793dda8f68d3633a73055ab8fdc3d95"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0pxibR7Z0x94L2VGGhQBc+jFNFbenXcpJRbCwXRpu4NOFu4A8jfUOsEZYFEL3uRWewNl/ntY0j44ROc9fZkLBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:41:11.460791Z","bundle_sha256":"e92558289e517c06861e697422b4e71f9b64b833a397c6a2ac1b8b43266e5a40"}}