{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:HB5T7ZFXPXZOPJXSI37VZBPYJD","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":"05c75b7f5eab4f4f77b3a993d287299934857db3abb3d312273688d267ed7378","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-12-07T05:50:40Z","title_canon_sha256":"3f26a17e90a0ed00fe1439ba247d64f57dca924a6e9ae5d3bd218bedb0007c75"},"schema_version":"1.0","source":{"id":"1112.1479","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.1479","created_at":"2026-05-18T02:53:05Z"},{"alias_kind":"arxiv_version","alias_value":"1112.1479v4","created_at":"2026-05-18T02:53:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.1479","created_at":"2026-05-18T02:53:05Z"},{"alias_kind":"pith_short_12","alias_value":"HB5T7ZFXPXZO","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HB5T7ZFXPXZOPJXS","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HB5T7ZFX","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:8534989dadc8cca856a69859c0606182df76bd4b1bc6d8e00b917c2011be6cd6","target":"graph","created_at":"2026-05-18T02:53:05Z","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 $(M^n, g)$ be a compact K\\\"ahler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact K\\\"ahler manifold $N^k$ with $c_1 < 0$. This confirms a conjecture of Yau. As a corollary, for any compact K\\\"ahler manifold with nonpositive bisectional curvature, the Kodaira dimension is equal to the maximal rank of the Ricci tensor. We also prove a global splitting result under the assumption of certain immersed complex submanifolds.","authors_text":"Gang Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-12-07T05:50:40Z","title":"Compact K\\\"ahler manifolds with nonpositive bisectional curvature"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1479","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":"sha256:5e577053dcf87f8a9245b3a8abf79b535cce35adfe400e6924204490332aaa99","target":"record","created_at":"2026-05-18T02:53:05Z","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":"05c75b7f5eab4f4f77b3a993d287299934857db3abb3d312273688d267ed7378","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-12-07T05:50:40Z","title_canon_sha256":"3f26a17e90a0ed00fe1439ba247d64f57dca924a6e9ae5d3bd218bedb0007c75"},"schema_version":"1.0","source":{"id":"1112.1479","kind":"arxiv","version":4}},"canonical_sha256":"387b3fe4b77df2e7a6f246ff5c85f848ddf5a888b8c1cdac574d19ce051582b5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"387b3fe4b77df2e7a6f246ff5c85f848ddf5a888b8c1cdac574d19ce051582b5","first_computed_at":"2026-05-18T02:53:05.019720Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:05.019720Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"slNjXNjhRXXve+uMI46Xhw9yBPIeYPdlyMqmYt2zc15LIA33FHHvvhM6KUbDqNZlleh/TkHNJKjxXdaYG0LvCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:05.020340Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.1479","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e577053dcf87f8a9245b3a8abf79b535cce35adfe400e6924204490332aaa99","sha256:8534989dadc8cca856a69859c0606182df76bd4b1bc6d8e00b917c2011be6cd6"],"state_sha256":"4b06ac073bf6f53dea5ff0c3d8a8358a4643ffaecad88da8031f23cb71a10c3b"}