# 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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## 1 Answer

The study of homogeneous varieties is a vast part of algebraic geometry, notably of algebraic group theory.
You certainly know quite a few homogeneous projective varieties. As you said there are abelian varieties, which are groups, but also quotients of groups like projective spaces, grassmannians and flag varieties.
The quotients here are quotients of affine algebraic groups by subgroups which are not normal, hence the quotient varieties are not groups. On the other hand, since you want the quotient to be projective, you have to divide out by a parabolic subgroup.
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 $dimV_s=s$]

As to dimension two, Tits ( the celebrated Belgian Abel prize laureate) has classified not only the projective homogeneous surfaces but also the compact holomorphic ones here ( Potters has generalized this classification to almost homogeneous compact holomorphic surfaces there)

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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
I see. This answers my question completely. Thank you very much! –  Bonanza Sep 30 '11 at 8:42