# What are the Compact Symmetric Kahler Algebraic Varieties?

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?

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Smooth affine varieties are always Kahler: restriction of Euclidean metric from $\mathbb{C}^n$. Smooth projective varieties are always Kahler: restriction of Fubini-Study. –  Qfwfq Nov 11 '10 at 22:05
Some comments: (1) All Stein spaces (effectively: closed submanifolds of $\mathbb C^n$) are Kahler. I don't imagine the converse holds. (2) Elie Cartan classified all Riemannian symmetric spaces based on their homotopy groups. If I recall correctly, then having a complex structure is equivalent to having a holonomy group contained in $U(n)$. (3) I don't think so, see (1). (4) Same response for (2), extremely difficult for (1). In dimension 1, everything is Kahler. In dimension 2, having even 1st Betti number is equivalent to being Kahler. For dim > 2, this is a completely open question. –  Gunnar Magnusson Nov 11 '10 at 22:07
Compact affine algebraic varieties are unions of finitely many points, by the maximum modulus principle. –  David Speyer Nov 11 '10 at 22:08
Oh, wait, you only want algebraic varieties? Then everything is Kahler as soon as it is smooth. As unknown said, affine varieties are Kahler because $\mathbb C^n$ is Kahler, and projective varieties are Kahler because $\mathbb P^n$ is Kahler. This becomes a difficult question when you consider arbitrary complex manifolds. –  Gunnar Magnusson Nov 11 '10 at 22:10
@Abtan: As a complex algebraic variety the sphere is not affine. –  babubba Nov 11 '10 at 23:07
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