{"paper":{"title":"Numerical spectrums control Cohomological spectrums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Junyi Xie","submitted_at":"2024-12-02T07:35:48Z","abstract_excerpt":"Let $X$ be a smooth irreducible projective variety over a field $\\mathbf{k}$ of dimension $d.$ Let $\\tau: \\mathbb{Q}_l\\to \\mathbb{C}$ be any field embedding. Let $f: X\\to X$ be a surjective endomorphism. We show that for every $i=0,\\dots,2d$, the spectral radius of $f^*$ on the numerical group $N^i(X)\\otimes \\mathbb{R}$ and on the $l$-adic cohomology group $H^{2i}(X_{\\overline{\\mathbf{k}}},\\mathbb{Q}_l)\\otimes \\mathbb{C}$ are the same. As a consequence, if $f$ is $q$-polarized for some $q>1$, we show that the norm of every eigenvalue of $f^*$ on the $j$-th cohomology group is $q^{j/2}$ for all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.01216","kind":"arxiv","version":2},"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/2412.01216/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"}