{"paper":{"title":"Skew-symmetric complex matrices, pure spinors, the twistor space of the conformal $2n$-sphere, and the Fano variety of linear $n$-folds of a non-singular complex quadric hypersurface in $\\mathbb{P}^{2n+1}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.DG","authors_text":"Alberto Verjovsky, Elsa Puente","submitted_at":"2011-11-14T22:15:55Z","abstract_excerpt":"For $n \\geq 1$, the twistor space $\\mathfrak{Z}(\\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\\mathfrak{Z}(\\mathbb{S}^{2n})$ with $n \\geq 3$, in terms of what we have called {\\it generalised complex orthogonal Stiefel manifolds} of $\\mathbb{C}^{n+1}$. In particular, the twistor space $\\mathfrak{Z}(\\mathbb{S}^{6})$ is biholomorphic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.3382","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"}