{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:W7ZQHDH5NVYHZPUOPRL4YNXDA2","short_pith_number":"pith:W7ZQHDH5","canonical_record":{"source":{"id":"2302.14032","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2023-01-28T06:52:02Z","cross_cats_sorted":[],"title_canon_sha256":"839d070165868ab963d15f85773edfe69ba9672528f0915f856949a008fa8362","abstract_canon_sha256":"d7f3b1e93ea1f54019481b6cbf121d383ebbb0e924fbc5e18aa09bbd1dbf530c"},"schema_version":"1.0"},"canonical_sha256":"b7f3038cfd6d707cbe8e7c57cc36e306b18f64fb2faa25d7e2cb6c1c513b06ef","source":{"kind":"arxiv","id":"2302.14032","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2302.14032","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"arxiv_version","alias_value":"2302.14032v4","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.14032","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"pith_short_12","alias_value":"W7ZQHDH5NVYH","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"pith_short_16","alias_value":"W7ZQHDH5NVYHZPUO","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"pith_short_8","alias_value":"W7ZQHDH5","created_at":"2026-05-29T00:04:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:W7ZQHDH5NVYHZPUOPRL4YNXDA2","target":"record","payload":{"canonical_record":{"source":{"id":"2302.14032","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2023-01-28T06:52:02Z","cross_cats_sorted":[],"title_canon_sha256":"839d070165868ab963d15f85773edfe69ba9672528f0915f856949a008fa8362","abstract_canon_sha256":"d7f3b1e93ea1f54019481b6cbf121d383ebbb0e924fbc5e18aa09bbd1dbf530c"},"schema_version":"1.0"},"canonical_sha256":"b7f3038cfd6d707cbe8e7c57cc36e306b18f64fb2faa25d7e2cb6c1c513b06ef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-29T00:04:12.141654Z","signature_b64":"ykpbCvzz3ok4y7jXnjbqHbgwIh+LRMwwbZjT1HTeSt1F0aI2+4awomooXA6/pssxgh3kP9X8m/TYo38N3nYDDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b7f3038cfd6d707cbe8e7c57cc36e306b18f64fb2faa25d7e2cb6c1c513b06ef","last_reissued_at":"2026-05-29T00:04:12.139900Z","signature_status":"signed_v1","first_computed_at":"2026-05-29T00:04:12.139900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2302.14032","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-29T00:04:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q7Av4rdgxOB2ZEOKL+d+JIZ5GIYb0XKTHp0GvKWFDlIofv9fHYniBB1Z/urqKliHiIc3KE9nyTVeLzuRHhjNCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:41:08.237402Z"},"content_sha256":"4acbf6c7d22732161d32099bb5ae872ed4cab290492aab13ad0002f834b7c0c0","schema_version":"1.0","event_id":"sha256:4acbf6c7d22732161d32099bb5ae872ed4cab290492aab13ad0002f834b7c0c0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:W7ZQHDH5NVYHZPUOPRL4YNXDA2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$L^{2}$-Hodge theory on Complete Almost K\\\"{a}hler Manifolds and the Hopf Conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Pan Zhang, Qiang Tan, Teng Huang","submitted_at":"2023-01-28T06:52:02Z","abstract_excerpt":"In this article, we develop an $L^{2}$-Hodge theory on complete $2n$-dimensional almost K\\\"{a}hler manifolds $(X,\\omega)$. In the first part, we establish several identities for various Laplacians, generalized Hodge and Serre dualities, a generalized Hard Lefschetz duality, and a Lefschetz decomposition, all restricted to the space $\\ker{\\Delta_{\\partial}}\\cap\\ker{\\Delta_{\\bar{\\partial}}}$ of forms of pure bidegree. In the second part, as applications of these identities, we prove vanishing theorems for $L^{2}$-harmonic $(p,q)$-forms on $X$ under some growth assumptions on the K\\\"{a}her form $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.14032","kind":"arxiv","version":4},"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/2302.14032/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-29T00:04:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MUZRfxr1u1drqly4ZjEMrV6ML43TxAkWek8RWDFCtuBJdVwXi8mo+VnWpq7Omd+bkiJ77o/F4zqkzLzVd5YCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T05:41:08.237783Z"},"content_sha256":"735d32a10ba5bb8669c17ec8ef2d28e41a7a7b0c6dae5552992f2fdd7328b38f","schema_version":"1.0","event_id":"sha256:735d32a10ba5bb8669c17ec8ef2d28e41a7a7b0c6dae5552992f2fdd7328b38f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W7ZQHDH5NVYHZPUOPRL4YNXDA2/bundle.json","state_url":"https://pith.science/pith/W7ZQHDH5NVYHZPUOPRL4YNXDA2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W7ZQHDH5NVYHZPUOPRL4YNXDA2/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":"2026-06-03T05:41:08Z","links":{"resolver":"https://pith.science/pith/W7ZQHDH5NVYHZPUOPRL4YNXDA2","bundle":"https://pith.science/pith/W7ZQHDH5NVYHZPUOPRL4YNXDA2/bundle.json","state":"https://pith.science/pith/W7ZQHDH5NVYHZPUOPRL4YNXDA2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W7ZQHDH5NVYHZPUOPRL4YNXDA2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:W7ZQHDH5NVYHZPUOPRL4YNXDA2","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":"d7f3b1e93ea1f54019481b6cbf121d383ebbb0e924fbc5e18aa09bbd1dbf530c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2023-01-28T06:52:02Z","title_canon_sha256":"839d070165868ab963d15f85773edfe69ba9672528f0915f856949a008fa8362"},"schema_version":"1.0","source":{"id":"2302.14032","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2302.14032","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"arxiv_version","alias_value":"2302.14032v4","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2302.14032","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"pith_short_12","alias_value":"W7ZQHDH5NVYH","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"pith_short_16","alias_value":"W7ZQHDH5NVYHZPUO","created_at":"2026-05-29T00:04:12Z"},{"alias_kind":"pith_short_8","alias_value":"W7ZQHDH5","created_at":"2026-05-29T00:04:12Z"}],"graph_snapshots":[{"event_id":"sha256:735d32a10ba5bb8669c17ec8ef2d28e41a7a7b0c6dae5552992f2fdd7328b38f","target":"graph","created_at":"2026-05-29T00:04:12Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2302.14032/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, we develop an $L^{2}$-Hodge theory on complete $2n$-dimensional almost K\\\"{a}hler manifolds $(X,\\omega)$. In the first part, we establish several identities for various Laplacians, generalized Hodge and Serre dualities, a generalized Hard Lefschetz duality, and a Lefschetz decomposition, all restricted to the space $\\ker{\\Delta_{\\partial}}\\cap\\ker{\\Delta_{\\bar{\\partial}}}$ of forms of pure bidegree. In the second part, as applications of these identities, we prove vanishing theorems for $L^{2}$-harmonic $(p,q)$-forms on $X$ under some growth assumptions on the K\\\"{a}her form $","authors_text":"Pan Zhang, Qiang Tan, Teng Huang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2023-01-28T06:52:02Z","title":"$L^{2}$-Hodge theory on Complete Almost K\\\"{a}hler Manifolds and the Hopf Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2302.14032","kind":"arxiv","version":4},"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":"sha256:4acbf6c7d22732161d32099bb5ae872ed4cab290492aab13ad0002f834b7c0c0","target":"record","created_at":"2026-05-29T00:04:12Z","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":"d7f3b1e93ea1f54019481b6cbf121d383ebbb0e924fbc5e18aa09bbd1dbf530c","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2023-01-28T06:52:02Z","title_canon_sha256":"839d070165868ab963d15f85773edfe69ba9672528f0915f856949a008fa8362"},"schema_version":"1.0","source":{"id":"2302.14032","kind":"arxiv","version":4}},"canonical_sha256":"b7f3038cfd6d707cbe8e7c57cc36e306b18f64fb2faa25d7e2cb6c1c513b06ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b7f3038cfd6d707cbe8e7c57cc36e306b18f64fb2faa25d7e2cb6c1c513b06ef","first_computed_at":"2026-05-29T00:04:12.139900Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-29T00:04:12.139900Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ykpbCvzz3ok4y7jXnjbqHbgwIh+LRMwwbZjT1HTeSt1F0aI2+4awomooXA6/pssxgh3kP9X8m/TYo38N3nYDDQ==","signature_status":"signed_v1","signed_at":"2026-05-29T00:04:12.141654Z","signed_message":"canonical_sha256_bytes"},"source_id":"2302.14032","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4acbf6c7d22732161d32099bb5ae872ed4cab290492aab13ad0002f834b7c0c0","sha256:735d32a10ba5bb8669c17ec8ef2d28e41a7a7b0c6dae5552992f2fdd7328b38f"],"state_sha256":"228892546927b0bff73c15d310933cd289397236ff926dff76eebebd66098c9b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q30qZpaW7Z5o6Lac88bHVzW9DV4QmVzc7qXvhJVHtsRMMlH8pcTYvyiQjm9sGhmX4JE22tdXMgdFSu5LeeH4Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T05:41:08.240029Z","bundle_sha256":"371cae88e8a897fd564cab2eabdf1d5bb398cd0334a8b89d9f9b781aaadfc179"}}