{"paper":{"title":"Hierarchies in relative Picard-Lefschetz theory","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["hep-th","math.AT","math.MP"],"primary_cat":"math-ph","authors_text":"Erik Panzer, Marko Berghoff","submitted_at":"2022-12-12T16:12:49Z","abstract_excerpt":"We prove a relative version of the Picard-Lefschetz theorem, describing the variation of relative homology groups $H_d(Y_t \\setminus A_t,B_t\\setminus A_t)$ in the fibers of a smooth fiber bundle $Y \\to T$ of complex manifolds with $A\\cup B \\subset Y$ transverse. From this we derive the vanishing of certain iterated variations, a system of constraints dubbed \"hierarchy\". As applications, we rederive the known analytic structure of Aomoto polylogarithms and massive one loop Feynman integrals. Moreover, we introduce the \"simple type\" to prove hierarchy constraints in degenerate cases where the Pi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.06661","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/2212.06661/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"}