{"paper":{"title":"Holomorphic injectivity and the Hopf map","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AG","authors_text":"Frederico Xavier, Scott Nollet","submitted_at":"2005-01-13T08:36:30Z","abstract_excerpt":"We give sharp conditions on a local biholomorphism $F:X \\to \\mathbb C^{n}$ which ensure global injectivity. For $n \\geq 2$, such a map is injective if for each complex line $l \\subset \\mathbb C^{n}$, the pre-image $F^{-1}(l)$ embeds holomorphically as a connected domain into $\\mathbb C \\mathbb P^{1}$, the embedding being unique up to M\\\"obius transformation. In particular, $F$ is injective if the pre-image of every complex line is connected and conformal to $\\mathbb C$. The proof uses the topological fact that the natural map $\\mathbb R \\mathbb P^{2n-1} \\to \\mathbb C \\mathbb P^{n-1}$ associate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501196","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":""},"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"}