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Then the only interesting non-trivial Alexander modules of $\\U$ and resp. $M$ appear in the middle degree $n$. We revisit the mixed Hodge structures on these Alexander modules and study their associated spectral pairs (or equivariant mixed Hodge numbers). We obtain upper bounds for the spectral pairs of the $n$-th Alexander module of $\\U$, which can be viewed as a Hodge-theor"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1607.05521","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-07-19T11:15:37Z","cross_cats_sorted":[],"title_canon_sha256":"5e70b21c6fb2d4273e37cef3d8a59fc5c9a2064846301f6d0cb2760d21e142e9","abstract_canon_sha256":"2f60258089669affeb47d7c1b93b84604b18c4c97d61708b72ad7efb30547ca7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:46.659883Z","signature_b64":"3tTMlfEyoy+g2WCgor6gP+JtCEgiQL4zUc5ns9cIveNl4t95U94/cxCy3O8JXIdJ0dUO8PsR3vBSiDJEweRfAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7e0975a0585f8354b3254693ecce2af4ea2d2b842f43c321f27748c4798c425","last_reissued_at":"2026-05-18T01:10:46.659370Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:46.659370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral pairs, Alexander modules, and boundary manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Laurentiu Maxim, Yongqiang Liu","submitted_at":"2016-07-19T11:15:37Z","abstract_excerpt":"Let $f: \\CN \\rightarrow \\C $ be a reduced polynomial map, with $D=f^{-1}(0)$, $\\U=\\CN \\setminus D$ and boundary manifold $M=\\partial \\U$. 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