Q:

Principal Bundles over Complex Projective Varieties 

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.

A:

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. 

