{"paper":{"title":"Von Neumann dimension, Hodge index theorem and geometric applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DG","authors_text":"Francesco Bei","submitted_at":"2017-11-07T15:51:59Z","abstract_excerpt":"This note contains a reformulation of the Hodge index theorem within the framework of Atiyah's $L^2$-index theory. More precisely, given a compact K\\\"ahler manifold $(M,h)$ of even complex dimension $2m$, we prove that $$\\sigma(M)=\\sum_{p,q=0}^{2m}(-1)^ph_{(2),\\Gamma}^{p,q}(M)$$ where $\\sigma(M)$ is the signature of $M$ and $h_{(2),\\Gamma}^{p,q}(M)$ are the $L^2$-Hodge numbers of $M$ with respect to a Galois covering having $\\Gamma$ as group of Deck transformations. Likewise we also prove an $L^2$-version of the Fr\\\"olicher index theorem. Afterwards we give some applications of these two theor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.02571","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":""},"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"}