# Projective homogeneous varieties of small dimension

I am interested in smooth complex projective varieties $X$ which are homogeneous as complex manifolds.

If $\dim X=1$, $X$ must be isomorphic to $P^1$ or an elliptic curve $E$, since curves of higher genus has a finite automorphism group.

In dimension 2, I can think of $P^2$, ruled surfaces over $P^1$ or $E$ and abelian surfaces.

Are there more? Or more presicely, have such $X$ been classified in dimension 2? For example, I suspect that being a homogeneous variety has implications for the Kodaira dimension of $X$.

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The definition of parabolic is a bit technical, but to give you some feeling for the notion, let me mention that given a finite dimensional vector space $V$, a parabolic subgroup of $GL(V)$ is a subgroup fixing some (incomplete) flag $V_0 \subsetneq V_1\subsetneq ...\subsetneq V_s\subsetneq ...\subsetneq V_r=V \quad$ [incomplete means that you don't require $dimV_s=s$]
Thanks for the answer. My question was indeed just about projective surfaces, where I know the Kodaira classification. In the Tits paper you linked to, he mentions that the only homogeneous surfaces are $P^2$, abelian varieties, and a space he calls $E(1,1)$ which is a quotient (is this $P^1\times P^1$)? Is this interpretation correct? – Bonanza Sep 30 '11 at 0:31
Dear trony, the varieties of type $E(1,1)$ are fibrations with basis $\mathbb P^1$ and elliptic curves as fibers. Beware that $\mathbb P^1\times \mathbb P^1$ is not listed in the classification because Tits classifies "irreducible spaces", that is varieties which are not products. – Georges Elencwajg Sep 30 '11 at 6:45