{"paper":{"title":"Generalized Brieskorn Modules I: Convergent (a,b)-modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Generalized Brieskorn modules receive a full description through convergent asymptotic expansions of Nilsson class.","cross_cats":["math.CV"],"primary_cat":"math.AG","authors_text":"Daniel Barlet (IUF, IECL), UL","submitted_at":"2023-07-10T07:57:24Z","abstract_excerpt":"This paper is the first one of two papers whose goal is to give a converse to the main result of my previous paper [6], so to prove the existence of multiple poles for the distribution |f|2$\\lambda$ with an hypothesis on a Higher Bernstein Polynomial of the (a,b)-module generated by the germ $\\omega$$\\in$$\\Omega$n+1 0 of a given holomorphic volum form. Note that, even for the existence of a simple pole this converse is already new. One difficulty to prove such a result comes from the use of the formal completion in f of the Brieskorn module of the holomorphic germ f\\,: (Cn+1 ,0) $\\rightarrow$("},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain a full description of generalized Brieskorn-modules in terms of (convergent) asymptotics expansions of Nilsson class which will be used as a starting point in part II.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The formal completion in f of the Brieskorn module does not give access to the cohomology of the Milnor fiber of f, which by definition is outside the zero set of f; this necessitates the introduction of generalized Brieskorn modules.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Develops the theory of convergent generalized Brieskorn modules, including semi-simple filtration, with a full description via convergent asymptotic expansions of Nilsson class and explicit relation to the nilpotent filtration of monodromy.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized Brieskorn modules receive a full description through convergent asymptotic expansions of Nilsson class.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f2f3c87ba8c4493ad7d2bf0587cfdb4976c3a393ef15c14025b9d04153943571"},"source":{"id":"2307.04395","kind":"arxiv","version":3},"verdict":{"id":"ad5d951f-e3f5-413b-8a05-69d1ef9f3b1c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-24T07:26:52.417385Z","strongest_claim":"We obtain a full description of generalized Brieskorn-modules in terms of (convergent) asymptotics expansions of Nilsson class which will be used as a starting point in part II.","one_line_summary":"Develops the theory of convergent generalized Brieskorn modules, including semi-simple filtration, with a full description via convergent asymptotic expansions of Nilsson class and explicit relation to the nilpotent filtration of monodromy.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The formal completion in f of the Brieskorn module does not give access to the cohomology of the Milnor fiber of f, which by definition is outside the zero set of f; this necessitates the introduction of generalized Brieskorn modules.","pith_extraction_headline":"Generalized Brieskorn modules receive a full description through convergent asymptotic expansions of Nilsson class."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.04395/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"}