most recent 30 from http://mathoverflow.net 2013-05-22T10:40:00Z http://mathoverflow.net/feeds/question/69961 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69961/locally-euclidean-varieties Qfwfq 2011-07-10T22:06:46Z 2011-07-10T22:06:46Z

<p>Differentiable manifolds can be described in terms of local charts to open subsets of $\mathbb{R}^n$ and transition functions that are diffeomorphisms. Trying to put $\mathbb{A}^n$ (over an algebraically closed field, say) in place of $\mathbb{R}^n$ and isomorphisms of open subvarieties of $\mathbb{A}^n$ in place of local charts of course will not reproduce all the algebraic varieties, even the nonsingular ones, as they have a rich local structure (the stalk of the structure sheaf at a point fully determines its birational equivalence class etc.).</p> <p>My question is </p> <blockquote> <p>What about the varieties that <em>can</em> be described in this way? At least for the complete ones, is there a characterization or a classification?</p> </blockquote> <p>Projective spaces and grassmannians belong to this class. Clearly they must be smooth, and birational to $\mathbb{P}^n$ ... </p>