{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:BBALBSBRKV3DJL654XDVG4WRB5","short_pith_number":"pith:BBALBSBR","canonical_record":{"source":{"id":"math/0611491","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2006-11-16T09:15:55Z","cross_cats_sorted":[],"title_canon_sha256":"2a577c559ff04a55c09feba941a9fefe8e1d3dae1169a2018ffb46fbd1325393","abstract_canon_sha256":"3ba5665735c56c5538a2bbf806ccd52abf67571ea24a9985aef62734745126ab"},"schema_version":"1.0"},"canonical_sha256":"0840b0c831557634afdde5c75372d10f77abddd7344159e878547141a81a1f02","source":{"kind":"arxiv","id":"math/0611491","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611491","created_at":"2026-05-18T03:02:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611491v2","created_at":"2026-05-18T03:02:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611491","created_at":"2026-05-18T03:02:31Z"},{"alias_kind":"pith_short_12","alias_value":"BBALBSBRKV3D","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"BBALBSBRKV3DJL65","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"BBALBSBR","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:BBALBSBRKV3DJL654XDVG4WRB5","target":"record","payload":{"canonical_record":{"source":{"id":"math/0611491","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2006-11-16T09:15:55Z","cross_cats_sorted":[],"title_canon_sha256":"2a577c559ff04a55c09feba941a9fefe8e1d3dae1169a2018ffb46fbd1325393","abstract_canon_sha256":"3ba5665735c56c5538a2bbf806ccd52abf67571ea24a9985aef62734745126ab"},"schema_version":"1.0"},"canonical_sha256":"0840b0c831557634afdde5c75372d10f77abddd7344159e878547141a81a1f02","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:31.105014Z","signature_b64":"NpJ+EyV+C3DOiycoB38mIJiYYCoreYZL7C11cMBt4Kh0mBF5lzSluRLECnopWhC1GD7tyXwxY6CBj7tXHq6WDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0840b0c831557634afdde5c75372d10f77abddd7344159e878547141a81a1f02","last_reissued_at":"2026-05-18T03:02:31.104350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:31.104350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0611491","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:02:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6AD8nSXUqdcujyMbfFl03bF5fC1aW+eNpazyit19rxAjo1nXKpIppnT/fRjsZc9WrVcjRSM/sPjv7OKYJu3bDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T18:28:22.549391Z"},"content_sha256":"f442e3c06d61c41c89aae600249f679d8b24bd5c238dd2ebefa408f0b00bd0f1","schema_version":"1.0","event_id":"sha256:f442e3c06d61c41c89aae600249f679d8b24bd5c238dd2ebefa408f0b00bd0f1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:BBALBSBRKV3DJL654XDVG4WRB5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stratifications with respect to actions of real reductive groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Gerald W. Schwarz, Henrik Stoetzel, Peter Heinzner","submitted_at":"2006-11-16T09:15:55Z","abstract_excerpt":"We study the action of a real reductive group G on a real submanifold X of a K\"ahler manifold Z. We suppose that the action of G extends holomorphically to an action of a complex reductive group and is Hamiltonian with respect to a compatible maximal compact subgroup of the complex reductive group. There is a corresponding gradient map obtained from a Cartan decomposition of G. We obtain a Morse like function on X. Associated to its critical points are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds of X which are called pre-strata"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611491","kind":"arxiv","version":2},"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":"2026-05-18T03:02:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3tWCQRjdi970oOzaeH+EkfSRBe3Wmg9QWmO8Su3M43Z3tKaHCOAEfaNQA89piNqdVsetZLUxmtLry4jPxxkLAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T18:28:22.550042Z"},"content_sha256":"6f3b4e9df483f0888db8a21c96b2a12075e12c888ef118beecff0b91e805515a","schema_version":"1.0","event_id":"sha256:6f3b4e9df483f0888db8a21c96b2a12075e12c888ef118beecff0b91e805515a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BBALBSBRKV3DJL654XDVG4WRB5/bundle.json","state_url":"https://pith.science/pith/BBALBSBRKV3DJL654XDVG4WRB5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BBALBSBRKV3DJL654XDVG4WRB5/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":"2026-05-27T18:28:22Z","links":{"resolver":"https://pith.science/pith/BBALBSBRKV3DJL654XDVG4WRB5","bundle":"https://pith.science/pith/BBALBSBRKV3DJL654XDVG4WRB5/bundle.json","state":"https://pith.science/pith/BBALBSBRKV3DJL654XDVG4WRB5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BBALBSBRKV3DJL654XDVG4WRB5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:BBALBSBRKV3DJL654XDVG4WRB5","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":"3ba5665735c56c5538a2bbf806ccd52abf67571ea24a9985aef62734745126ab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2006-11-16T09:15:55Z","title_canon_sha256":"2a577c559ff04a55c09feba941a9fefe8e1d3dae1169a2018ffb46fbd1325393"},"schema_version":"1.0","source":{"id":"math/0611491","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0611491","created_at":"2026-05-18T03:02:31Z"},{"alias_kind":"arxiv_version","alias_value":"math/0611491v2","created_at":"2026-05-18T03:02:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0611491","created_at":"2026-05-18T03:02:31Z"},{"alias_kind":"pith_short_12","alias_value":"BBALBSBRKV3D","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"BBALBSBRKV3DJL65","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"BBALBSBR","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:6f3b4e9df483f0888db8a21c96b2a12075e12c888ef118beecff0b91e805515a","target":"graph","created_at":"2026-05-18T03:02:31Z","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":"We study the action of a real reductive group G on a real submanifold X of a K\"ahler manifold Z. We suppose that the action of G extends holomorphically to an action of a complex reductive group and is Hamiltonian with respect to a compatible maximal compact subgroup of the complex reductive group. There is a corresponding gradient map obtained from a Cartan decomposition of G. We obtain a Morse like function on X. Associated to its critical points are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds of X which are called pre-strata","authors_text":"Gerald W. Schwarz, Henrik Stoetzel, Peter Heinzner","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2006-11-16T09:15:55Z","title":"Stratifications with respect to actions of real reductive groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0611491","kind":"arxiv","version":2},"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:f442e3c06d61c41c89aae600249f679d8b24bd5c238dd2ebefa408f0b00bd0f1","target":"record","created_at":"2026-05-18T03:02:31Z","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":"3ba5665735c56c5538a2bbf806ccd52abf67571ea24a9985aef62734745126ab","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2006-11-16T09:15:55Z","title_canon_sha256":"2a577c559ff04a55c09feba941a9fefe8e1d3dae1169a2018ffb46fbd1325393"},"schema_version":"1.0","source":{"id":"math/0611491","kind":"arxiv","version":2}},"canonical_sha256":"0840b0c831557634afdde5c75372d10f77abddd7344159e878547141a81a1f02","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0840b0c831557634afdde5c75372d10f77abddd7344159e878547141a81a1f02","first_computed_at":"2026-05-18T03:02:31.104350Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:31.104350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NpJ+EyV+C3DOiycoB38mIJiYYCoreYZL7C11cMBt4Kh0mBF5lzSluRLECnopWhC1GD7tyXwxY6CBj7tXHq6WDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:31.105014Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0611491","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f442e3c06d61c41c89aae600249f679d8b24bd5c238dd2ebefa408f0b00bd0f1","sha256:6f3b4e9df483f0888db8a21c96b2a12075e12c888ef118beecff0b91e805515a"],"state_sha256":"c689731bc1f6d90e0aecbbd11b52bd165f5145e0cac7771c16f5e7d3503a16ed"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gZfDFn1ruEzekVltHlFlMxY+dxyOIhH5DhaTolNQftEg4xptFQ+3GrSO5caE6YpgE2nuvsb7v2+A5/av1qWQAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T18:28:22.553514Z","bundle_sha256":"b65a53727433151feba20ddcd7639743bf6f447b5a81ed568f31481862ea031c"}}