Principal Bundles over Complex Projective Varieties

Question

For various reasons, I'm interested in working with complex projective varieties that are also principal bundles. I began by looking at projective spaces themselves $\mathbb{CP}^n = SU(n+1)/U(n)$, then generalised to Grassmannian spaces $Gr(n,d) = U(n)/U(d) \times U(n-d)$, and finally to flag manifolds $U(n)/U(k_1) \times \cdots \times U(k_m)$. I am now looking for other examples of projective varieties that are also principal bundles. Does anyone know of any?

EDIT: Sorry for asking such an ill-posed question, I didn't realize I had made so many tacit assumptions. What I am looking for are principal $G$ bundles $\pi:P \to X$ such that $G$ and $P$ are both compact Lie groups (ideally matrix groups), and $X$ is a projective variety. As far as I understand it is not always possible to construct such a bundle for a projective variety, one obstruction being the absence of a continuous symmetry group.

Answer 1

You probably mean that you are interested in principal bundles on a projective variety $X$ with fiber being a compact Lie group $G$. If you are interested in the topological classification of such bundles, then this is a classical problem, and the bundles are classified by homotopy classes of mappings from $X$ to the classifying space $BG$ (the structure of a projective variety on $X$ is not important here, only its topology). A lot is known about the classification of such mappings; in particular, this problem gives rise to the theory of characteristic classes, and (in the case of $G=U(n)$, i.e. complex vector bundles) to topological $K$-theory, which describe invariants of such bundles. Wikipedia contains lots of information and references on this.

Another problem is to look at bundles with an additional structure, usually a connection. For example, if $X$ is a curve, it is interesting to look at bundles with a flat connection. These bundles form an interesting moduli space, which is the same as the space of stable holomorphic bundles with fiber being the complexification $G_{\Bbb C}$ of $G$, for simple simply connected $G$ (the Narasimhan-Seshadri theorem). If $G=U(1)$, this moduli space is just the Jacobian of $X$. There is an analog of the Narasimhan-Seshadri theorem to higher dimensions, which is called the Donaldson-Uhlenbeck-Yau theorem.

Answer 2

As I interpret your question, you're looking for principal bundles where the base is projective, and both the fiber and total space are both compact groups.

There's an obvious class of these, which is taking any compact group $P$ and modding out by a Levi subgroup $G$ (I'm trying to match the notation in your question. I apologize to the Lie theorists in the audience for using the most confusing notation ever) [EDIT: in the original, this had read "containing the maximal torus" since I'd forgotten that there are non-Levi subgroups which contain the maximal torus]. This quotient will always be a projective variety.

I believe that one can prove that this is the only way of getting a quotient which is projective (note that the complexification of $P$ acts on the quotient, since it is compact, and the complexification of the Lie algebra acts; now use the Borel fixed point theorem). [EDIT: it seems that this isn't true. For example, elliptic curves exist.]

It's possible that there's some strange way of making $G$ act on $P$ that's not a subgroup, but I think the above are the right class of examples generalizing the Grassmannian. [EDIT: this paragraph at least is vague enough to not be false, but as pointed out in comments, there are other examples]