What are the Compact Symmetric Kahler Algebraic Varieties? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:46:45Z http://mathoverflow.net/feeds/question/45748 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45748/what-are-the-compact-symmetric-kahler-algebraic-varieties What are the Compact Symmetric Kahler Algebraic Varieties? Abtan Massini 2010-11-11T21:35:46Z 2010-11-11T22:48:09Z <p>Here are some direct questions at the interface of algebraic and differential geometry:</p> <p>(1) Is there an easy characterisation of those affine algebraic varieties which are Kahler?</p> <p>(2) Is there an easy characterisation of those affine algebraic varieties which are symmetric spaces?</p> <p>(3) Is there an easy characterisation of those affine algebraic varieties which are both? (From the first comment below, it seems that we can rephrase this question as: which affine algebraic varieties are symmetric?)</p> <p>(4) What happens if I then also require compactness?</p> http://mathoverflow.net/questions/45748/what-are-the-compact-symmetric-kahler-algebraic-varieties/45753#45753 Answer by Tobias Hartnick for What are the Compact Symmetric Kahler Algebraic Varieties? Tobias Hartnick 2010-11-11T22:02:12Z 2010-11-11T22:02:12Z <p>If we ignore the trivial case of the affine line, then irreducible symmetric spaces come in pairs compact - non-compact. The compact ones are naturally projective varieties, while the non-compact ones are affine varieties. Thus question (4) is problematic, unless you mean "locally symmetric" or a more general notion of symmetric space than I understand here (i.e. "globally symmetric Riemannian symmetric space"). As far as non-compact symmetric spaces are concerned, they are Kähler if and only if they are biholomorphic to a bounded symmetric domain. Equivalently, there exists a compact quotient with non-trivial H^2 or, equivalently, the point stabilizer of the automorphism group has infinite center... I could give many more characterizations, but I do not quite see what you are after, so maybe you can provide more detailed information?</p>