Complete flag bundle

Let $E\to X$ be a smooth complex vector bundle of rank $n\ge0$ . Its complete flag bundle $p:\operatorname{Fl}(E)\to X$ has fiber at $x$ the set of chains of complex vector subspaces

0=V_0\subset V_1\subset\cdots\subset V_n=E_x, \qquad \dim_{\mathbb C}V_j=j.

The standard flag manifold $\operatorname{Fl}(\mathbb C^n)$ is the smooth homogeneous space $\operatorname{GL}(n,\mathbb C)/B$ , where $B$ is the closed subgroup of invertible upper-triangular matrices, the stabilizer of the coordinate flag. Local trivializations of $E$ identify $\operatorname{Fl}(E)|_U$ with $U\times\operatorname{Fl}(\mathbb C^n)$ ; these charts specify the topology and smooth structure. Equivalently,

\operatorname{Fl}(E)=\operatorname{Fr}(E)\times_{\operatorname{GL}(n,\mathbb C)}\operatorname{Fl}(\mathbb C^n)

is the associated bundle for the natural left action on flags and the right action on the complex frame bundle. The projection sends a flag in $E_x$ to $x$ . For $n=0$ , the unique empty flag gives $\operatorname{Fl}(E)=X$ .

Tautological filtration

The pulled-back bundle $p^*E$ has subbundles $F_j$ whose fiber at a flag $(V_\bullet)$ is $V_j$ . The quotients $L_j=F_j/F_{j-1}$ are complex line bundles. A Hermitian metric splits this filtration smoothly, while the filtration and quotients require no chosen metric.