{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:DLRODLQYGLK4ILXKOZE64DYWD4","short_pith_number":"pith:DLRODLQY","canonical_record":{"source":{"id":"1303.4975","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-03-20T16:10:01Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"90f0e4a318f76aba8780308ed54b8d6a7da5ca40bd193e5d6648687c328fc9fa","abstract_canon_sha256":"7c87c3a306c33474e657fc7d642ab95574331d8c43fd91b6c11587825a337ac3"},"schema_version":"1.0"},"canonical_sha256":"1ae2e1ae1832d5c42eea7649ee0f161f1f7ca727824210e17abbcecdf2edf264","source":{"kind":"arxiv","id":"1303.4975","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.4975","created_at":"2026-05-18T02:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"1303.4975v2","created_at":"2026-05-18T02:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.4975","created_at":"2026-05-18T02:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"DLRODLQYGLK4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DLRODLQYGLK4ILXK","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DLRODLQY","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:DLRODLQYGLK4ILXKOZE64DYWD4","target":"record","payload":{"canonical_record":{"source":{"id":"1303.4975","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-03-20T16:10:01Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"90f0e4a318f76aba8780308ed54b8d6a7da5ca40bd193e5d6648687c328fc9fa","abstract_canon_sha256":"7c87c3a306c33474e657fc7d642ab95574331d8c43fd91b6c11587825a337ac3"},"schema_version":"1.0"},"canonical_sha256":"1ae2e1ae1832d5c42eea7649ee0f161f1f7ca727824210e17abbcecdf2edf264","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:41.213011Z","signature_b64":"SaKgfbeSDU3MEBcnefTBl4ni+ocyeYKxYh0Fp4oDXT15M+aUleekVWXhzCCaYFbKva2d+99EJKhgQ/pN0SMbAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ae2e1ae1832d5c42eea7649ee0f161f1f7ca727824210e17abbcecdf2edf264","last_reissued_at":"2026-05-18T02:18:41.212307Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:41.212307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1303.4975","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":"2026-05-18T02:18:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZYBse5CS7NiC4kJ6n+HKCNcDWPdgd/cSrN/w30+qXVxqnbsIQSWQxc5VSIQkf83VdsNzKt53zVBOLCcanFFqDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:18:15.650825Z"},"content_sha256":"7e9a2b1d9b05f9daecb5591b61bb45265715d291a0e940c0262d3a00391aaa2d","schema_version":"1.0","event_id":"sha256:7e9a2b1d9b05f9daecb5591b61bb45265715d291a0e940c0262d3a00391aaa2d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:DLRODLQYGLK4ILXKOZE64DYWD4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in K\\\"ahler geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DG","authors_text":"Bo Berndtsson","submitted_at":"2013-03-20T16:10:01Z","abstract_excerpt":"For $\\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \\int_X e^{-\\phi}. $$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As a consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem for K\\\"ahler - Einstein metrics. A generalization of this theorem to 'twisted' K\\\"ahler-Einstein metrics and some classes of manifolds that sat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4975","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":"2026-05-18T02:18:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YmCupaKc2DGEic+cjuUGgPaWYDsgq+VMyZq1fL6vUQtFHLHAGBBVycoEAfhSc5k3NNDh8W3y0ci6w2Anq27kBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T23:18:15.651495Z"},"content_sha256":"d7bbffc27240e6520bbcfd5ba7157c42746436f21f6d4dfad15ff769b6c3ab57","schema_version":"1.0","event_id":"sha256:d7bbffc27240e6520bbcfd5ba7157c42746436f21f6d4dfad15ff769b6c3ab57"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DLRODLQYGLK4ILXKOZE64DYWD4/bundle.json","state_url":"https://pith.science/pith/DLRODLQYGLK4ILXKOZE64DYWD4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DLRODLQYGLK4ILXKOZE64DYWD4/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":"2026-05-27T23:18:15Z","links":{"resolver":"https://pith.science/pith/DLRODLQYGLK4ILXKOZE64DYWD4","bundle":"https://pith.science/pith/DLRODLQYGLK4ILXKOZE64DYWD4/bundle.json","state":"https://pith.science/pith/DLRODLQYGLK4ILXKOZE64DYWD4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DLRODLQYGLK4ILXKOZE64DYWD4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:DLRODLQYGLK4ILXKOZE64DYWD4","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":"7c87c3a306c33474e657fc7d642ab95574331d8c43fd91b6c11587825a337ac3","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-03-20T16:10:01Z","title_canon_sha256":"90f0e4a318f76aba8780308ed54b8d6a7da5ca40bd193e5d6648687c328fc9fa"},"schema_version":"1.0","source":{"id":"1303.4975","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.4975","created_at":"2026-05-18T02:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"1303.4975v2","created_at":"2026-05-18T02:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.4975","created_at":"2026-05-18T02:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"DLRODLQYGLK4","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"DLRODLQYGLK4ILXK","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"DLRODLQY","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:d7bbffc27240e6520bbcfd5ba7157c42746436f21f6d4dfad15ff769b6c3ab57","target":"graph","created_at":"2026-05-18T02:18:41Z","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":"For $\\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \\int_X e^{-\\phi}. $$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As a consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem for K\\\"ahler - Einstein metrics. A generalization of this theorem to 'twisted' K\\\"ahler-Einstein metrics and some classes of manifolds that sat","authors_text":"Bo Berndtsson","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-03-20T16:10:01Z","title":"A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in K\\\"ahler geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4975","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":"sha256:7e9a2b1d9b05f9daecb5591b61bb45265715d291a0e940c0262d3a00391aaa2d","target":"record","created_at":"2026-05-18T02:18:41Z","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":"7c87c3a306c33474e657fc7d642ab95574331d8c43fd91b6c11587825a337ac3","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-03-20T16:10:01Z","title_canon_sha256":"90f0e4a318f76aba8780308ed54b8d6a7da5ca40bd193e5d6648687c328fc9fa"},"schema_version":"1.0","source":{"id":"1303.4975","kind":"arxiv","version":2}},"canonical_sha256":"1ae2e1ae1832d5c42eea7649ee0f161f1f7ca727824210e17abbcecdf2edf264","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1ae2e1ae1832d5c42eea7649ee0f161f1f7ca727824210e17abbcecdf2edf264","first_computed_at":"2026-05-18T02:18:41.212307Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:41.212307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SaKgfbeSDU3MEBcnefTBl4ni+ocyeYKxYh0Fp4oDXT15M+aUleekVWXhzCCaYFbKva2d+99EJKhgQ/pN0SMbAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:41.213011Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.4975","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7e9a2b1d9b05f9daecb5591b61bb45265715d291a0e940c0262d3a00391aaa2d","sha256:d7bbffc27240e6520bbcfd5ba7157c42746436f21f6d4dfad15ff769b6c3ab57"],"state_sha256":"2f0b82dc2c48df3c5a7f60831cf3b7de8c011b7b701a3132bdc752bc8ca5e0d3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xmZw9lcEJPd/HZZFOgyWRqEiAHLP9uPnhkBJrW7W1kr94nTdrGqV5XL41bSdmPcHY44zmNuEt6a3BWMeL5PvDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T23:18:15.654786Z","bundle_sha256":"43a6da87e6da61d2bfb9a3025de11d0a59e11133d5292702ba8b5c6c153bb83c"}}