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We show that after iteration and equivariant birational contractions, an quasi-amplified endomorphism will descend to an amplified endomorphism.\n  As an application, when $X$ is Hyperk\\\"ahler, $f$ is quasi-amplified if and only if it is of positive entropy. In both cases, $f$ has Zariski dense periodic points. When $X$ is an abelian variety, we give and compare several cohomologic"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.07089","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-01-21T21:40:20Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f2ea9c6bdba7cfb012eb5343b17f684d2dc3895d0a493b774877f12615a0c69a","abstract_canon_sha256":"ec3007b77d8b84059af4ddcf9705594c96e8805d1961398959193624bf7bb0b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:04:32.081198Z","signature_b64":"ugZGsQyJL/1Hbc4T6Ts3/Q8znFsj8S0gLus0X6ZCEj2yYDexD7cJgDCIGR9CegH3tpa8pEJhsGbhpIMYs0uYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e14c0976bf5f75405b75c95f3ccf4ec3613bd42623a86b512c5ae3d4b776e095","last_reissued_at":"2026-07-05T11:04:32.080710Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:04:32.080710Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On endomorphisms of projective varieties with numerically trivial canonical divisors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AG","authors_text":"Sheng Meng","submitted_at":"2019-01-21T21:40:20Z","abstract_excerpt":"Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\\to X$ is amplified (resp.~quasi-amplified) if $f^*D-D$ is ample (resp.~big) for some Cartier divisor $D$. We show that after iteration and equivariant birational contractions, an quasi-amplified endomorphism will descend to an amplified endomorphism.\n  As an application, when $X$ is Hyperk\\\"ahler, $f$ is quasi-amplified if and only if it is of positive entropy. In both cases, $f$ has Zariski dense periodic points. When $X$ is an abelian variety, we give and compare several cohomologic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.07089","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1901.07089/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.07089","created_at":"2026-07-05T11:04:32.080766+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.07089v2","created_at":"2026-07-05T11:04:32.080766+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.07089","created_at":"2026-07-05T11:04:32.080766+00:00"},{"alias_kind":"pith_short_12","alias_value":"4FGAS5V7L52U","created_at":"2026-07-05T11:04:32.080766+00:00"},{"alias_kind":"pith_short_16","alias_value":"4FGAS5V7L52UAW3V","created_at":"2026-07-05T11:04:32.080766+00:00"},{"alias_kind":"pith_short_8","alias_value":"4FGAS5V7","created_at":"2026-07-05T11:04:32.080766+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.01605","citing_title":"Kawaguchi-Silverman conjecture for certain surjective endomorphisms","ref_index":26,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN","json":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN.json","graph_json":"https://pith.science/api/pith-number/4FGAS5V7L52UAW3VZFPTZT2OYN/graph.json","events_json":"https://pith.science/api/pith-number/4FGAS5V7L52UAW3VZFPTZT2OYN/events.json","paper":"https://pith.science/paper/4FGAS5V7"},"agent_actions":{"view_html":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN","download_json":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN.json","view_paper":"https://pith.science/paper/4FGAS5V7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.07089&json=true","fetch_graph":"https://pith.science/api/pith-number/4FGAS5V7L52UAW3VZFPTZT2OYN/graph.json","fetch_events":"https://pith.science/api/pith-number/4FGAS5V7L52UAW3VZFPTZT2OYN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN/action/storage_attestation","attest_author":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN/action/author_attestation","sign_citation":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN/action/citation_signature","submit_replication":"https://pith.science/pith/4FGAS5V7L52UAW3VZFPTZT2OYN/action/replication_record"}},"created_at":"2026-07-05T11:04:32.080766+00:00","updated_at":"2026-07-05T11:04:32.080766+00:00"}