Here are some direct questions at the interface of algebraic and differential geometry:


(1) Is there an easy characterisation of those affine algebraic varieties which are Kahler?


(2) Is there an easy characterisation of those affine algebraic varieties which are symmetric spaces?


(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?)


(4) What happens if I then also require compactness?



What are the Compact Symmetric Kahler Algebraic Varieties?
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?


