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?

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.