{"paper":{"title":"$P$-trivial MMP, Zariski decompositions and minimal models for generalised pairs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Zhengyu Hu","submitted_at":"2025-01-13T18:36:42Z","abstract_excerpt":"We develop a theory of $P$-trivial MMP whose each step is $P$-trivial for a given nef divisor $P$. As an application, we prove that, given a projective generalised klt pair $(X,B+M)$ with data $M'$ being just a nef $\\mathbb{R}$-divisor, if $K_X+B+M$ birationally has a Nakayama-Zariski decomposition with nef positive part, and either if $M'$ or the positive part is log numerically effective, then it has a minimal model. Furthermore, we prove this for generalised lc pairs in dimension $3$. This is a generalisation of the main theorem of [Birkar-Hu14]. We also prove some related results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.07551","kind":"arxiv","version":1},"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/2501.07551/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"}