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?