{"paper":{"title":"The Complex Structures on $S^{2n}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jianwei Zhou","submitted_at":"2006-08-15T01:39:12Z","abstract_excerpt":"Let $\\widetilde{\\cal J}(S^{2n})$ be the set of orthogonal complex structures on\n  $TS^{2n}$. We show that the twistor space $\\widetilde{\\cal J}(S^{2n})$ is a Kaehler manifold. Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\\colon\\; S^{2n}\\to \\widetilde{\\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$. We also show that there is no complex structure in a neighborhood of the space $\\widetilde{\\cal J}(S^{2n})$. The method is to stud"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0608368","kind":"arxiv","version":5},"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"}