{"paper":{"title":"The finite degree formula for normalized volumes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.DG"],"primary_cat":"math.AG","authors_text":"Zhiyu Liu","submitted_at":"2026-07-29T15:25:56Z","abstract_excerpt":"Let $f\\colon \\big(x\\in (X, \\Delta_X)\\big)\\to \\big(y\\in (Y, \\Delta_Y)\\big)$ be a finite surjective morphism between klt singularities such that $K_X+\\Delta_X=f^*(K_Y+\\Delta_Y)$. We show that the normalized volumes satisfy \\[\\widehat{\\mathrm{vol}}(x, X, \\Delta_X)=\\mathrm{deg}(f)\\cdot \\widehat{\\mathrm{vol}}(y, Y, \\Delta_Y).\\] This proves a conjecture in [LLX20, Zhu25, XZ26a]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27032","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/2607.27032/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"}