Flag bundle

Definition

In algebraic geometry, let ${\displaystyle X}$ be a scheme, let ${\displaystyle {\mathcal {E}}}$ be a vector bundle of rank ${\displaystyle n}$ on ${\displaystyle X}$, and fix integers $${\displaystyle 0<d_{1}<d_{2}<\cdots <d_{r}<n.}$$ The flag bundle, or relative flag variety, of type ${\displaystyle (d_{1},\ldots ,d_{r})}$ associated with ${\displaystyle {\mathcal {E}}}$ is the ${\displaystyle X}$-scheme $${\displaystyle p\colon \operatorname {Fl} _{d_{1},\ldots ,d_{r}}({\mathcal {E}})\longrightarrow X}$$ whose fiber over a point ${\displaystyle x\in X}$ is the flag variety parametrizing chains $${\displaystyle V_{d_{1}}\subset V_{d_{2}}\subset \cdots \subset V_{d_{r}}\subset {\mathcal {E}}(x),\qquad \dim _{\kappa (x)}V_{d_{i}}=d_{i},}$$ where ${\displaystyle {\mathcal {E}}(x)={\mathcal {E}}_{x}\otimes _{{\mathcal {O}}_{X,x}}\kappa (x)}$ is the fiber of ${\displaystyle {\mathcal {E}}}$ at ${\displaystyle x}$.

Universal property

The flag bundle represents the functor which assigns to an ${\displaystyle X}$-scheme ${\displaystyle f\colon T\to X}$ the set of flags of subbundles $${\displaystyle 0\subset {\mathcal {S}}_{d_{1}}\subset {\mathcal {S}}_{d_{2}}\subset \cdots \subset {\mathcal {S}}_{d_{r}}\subset f^{*}{\mathcal {E}},}$$ where ${\displaystyle {\mathcal {S}}_{d_{i}}}$ has rank ${\displaystyle d_{i}}$. Here, a subbundle is understood to be locally a direct summand, equivalently, to have a locally free quotient.

Consequently, on ${\displaystyle \operatorname {Fl} _{d_{1},\ldots ,d_{r}}({\mathcal {E}})}$ there is a tautological flag $${\displaystyle 0\subset {\mathcal {U}}_{d_{1}}\subset {\mathcal {U}}_{d_{2}}\subset \cdots \subset {\mathcal {U}}_{d_{r}}\subset p^{*}{\mathcal {E}},}$$ with ${\displaystyle \operatorname {rank} {\mathcal {U}}_{d_{i}}=d_{i}}$. Every family of flags over an ${\displaystyle X}$-scheme is obtained uniquely by pulling back this tautological flag.

Some authors define flag bundles using chains of locally free quotients rather than subbundles. The two conventions are equivalent after passing to dual bundles and reversing the sequence of ranks.

Construction

Flag bundles can be constructed as iterated Grassmann bundles. Set ${\displaystyle d_{0}=0}$, ${\displaystyle X_{0}=X}$, and ${\displaystyle {\mathcal {U}}_{d_{0}}=0}$. Suppose that ${\displaystyle X_{i-1}}$ and the universal rank-${\displaystyle d_{i-1}}$ subbundle ${\displaystyle {\mathcal {U}}_{d_{i-1}}\subset {\mathcal {E}}_{X_{i-1}}}$ have been constructed. Define $${\displaystyle X_{i}=\operatorname {Gr} _{d_{i}-d_{i-1}}\left({\mathcal {E}}_{X_{i-1}}/{\mathcal {U}}_{d_{i-1}}\right).}$$ The universal subbundle of the quotient has an inverse image ${\displaystyle {\mathcal {U}}_{d_{i}}\subset {\mathcal {E}}_{X_{i}}}$. After ${\displaystyle r}$ steps one obtains $${\displaystyle X_{r}\cong \operatorname {Fl} _{d_{1},\ldots ,d_{r}}({\mathcal {E}}).}$$

For the complete flag bundle, where ${\displaystyle d_{i}=i}$ for ${\displaystyle 1\leq i\leq n-1}$, each step chooses a line in a quotient bundle. Thus, the complete flag bundle is a tower of projective bundles: $${\displaystyle \operatorname {Fl} ({\mathcal {E}})\longrightarrow \operatorname {Fl} _{1,\ldots ,n-2}({\mathcal {E}})\longrightarrow \cdots \longrightarrow \mathbf {P} ({\mathcal {E}})\longrightarrow X.}$$

Basic properties

Formation of the flag bundle commutes with arbitrary base change. If ${\displaystyle g\colon Y\to X}$, then there is a canonical isomorphism $${\displaystyle \operatorname {Fl} _{d_{1},\ldots ,d_{r}}(g^{*}{\mathcal {E}})\cong Y\times _{X}\operatorname {Fl} _{d_{1},\ldots ,d_{r}}({\mathcal {E}}).}$$

On an open subset ${\displaystyle U\subset X}$ over which ${\displaystyle {\mathcal {E}}}$ is trivial, the flag bundle is a product $${\displaystyle \operatorname {Fl} _{d_{1},\ldots ,d_{r}}({\mathcal {E}})|_{U}\cong U\times \operatorname {Fl} _{d_{1},\ldots ,d_{r}}(\mathbb {Z} ^{n}).}$$ In particular, the morphism ${\displaystyle p}$ is smooth and projective. Its relative dimension is $${\displaystyle \sum _{i=1}^{r}(d_{i}-d_{i-1})(n-d_{i}),\qquad d_{0}=0.}$$

Equivalently, if ${\displaystyle \operatorname {Fr} ({\mathcal {E}})}$ is the principal ${\displaystyle \operatorname {GL} _{n}}$-bundle of frames of ${\displaystyle {\mathcal {E}}}$ and ${\displaystyle P\subset \operatorname {GL} _{n}}$ is the parabolic subgroup stabilizing a standard flag of type ${\displaystyle (d_{1},\ldots ,d_{r})}$, then $${\displaystyle \operatorname {Fl} _{d_{1},\ldots ,d_{r}}({\mathcal {E}})\cong \operatorname {Fr} ({\mathcal {E}})\times ^{\operatorname {GL} _{n}}\operatorname {GL} _{n}/P.}$$

Special cases

If ${\displaystyle X=\operatorname {Spec} k}$ is a point, the flag bundle is the usual flag variety of a vector space. If ${\displaystyle r=1}$, it is the Grassmann bundle ${\displaystyle \operatorname {Gr} _{d_{1}}({\mathcal {E}})}$. Under the convention that ${\displaystyle \mathbf {P} ({\mathcal {E}})}$ parametrizes lines, the case ${\displaystyle d_{1}=1}$ is the projective bundle ${\displaystyle \mathbf {P} ({\mathcal {E}})}$.

The type ${\displaystyle (1,2,\ldots ,n-1)}$ gives the complete flag bundle ${\displaystyle \operatorname {Fl} ({\mathcal {E}})}$; all other types are called partial flag bundles.

Sections and fixed flags

By the universal property, a section $${\displaystyle s\colon X\longrightarrow \operatorname {Fl} _{d_{1},\ldots ,d_{r}}({\mathcal {E}})}$$ is equivalent to a flag of subbundles $${\displaystyle 0\subset {\mathcal {E}}_{d_{1}}\subset \cdots \subset {\mathcal {E}}_{d_{r}}\subset {\mathcal {E}}}$$ of the prescribed ranks. Thus, a fixed flag of subbundles is not required to define the flag bundle; rather, it gives a section of it.

A fixed flag may also be used to impose incidence or rank conditions on the tautological flag. The resulting closed subschemes are relative Schubert varieties, and their pullbacks by sections give degeneracy loci.

Splitting principle

On the complete flag bundle, put ${\displaystyle {\mathcal {U}}_{0}=0}$ and ${\displaystyle {\mathcal {U}}_{n}=p^{*}{\mathcal {E}}}$. The successive quotients $${\displaystyle {\mathcal {L}}_{i}={\mathcal {U}}_{i}/{\mathcal {U}}_{i-1},\qquad 1\leq i\leq n,}$$ are line bundles. Hence ${\displaystyle p^{*}{\mathcal {E}}}$ has a canonical filtration with line-bundle quotients, and its total Chern class satisfies $${\displaystyle c(p^{*}{\mathcal {E}})=\prod _{i=1}^{n}\left(1+c_{1}({\mathcal {L}}_{i})\right).}$$ This is one geometric form of the splitting principle and is a principal application of complete flag bundles in intersection theory.

See also

• Flag variety
• Grassmann bundle
• Projective bundle
• Schubert variety
• Splitting principle

References