{"paper":{"title":"Polarized endomorphisms of normal projective threefolds in arbitrary characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AG","authors_text":"De-Qi Zhang, Paolo Cascini, Sheng Meng","submitted_at":"2017-10-05T07:38:26Z","abstract_excerpt":"Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \\ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\\ast H \\sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$.\n  Suppose $f$ is separable and $X$ is $\\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01903","kind":"arxiv","version":3},"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/1710.01903/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"}