Definition
In algebraic geometry, let be a scheme, let be a vector bundle of rank on , and fix integers
The flag bundle, or relative flag variety, of type associated with is the -scheme
whose fiber over a point is the flag variety parametrizing chains
where is the fiber of at .
Universal property
The flag bundle represents the functor which assigns to an -scheme the set of flags of subbundles
where has rank . Here, a subbundle is understood to be locally a direct summand, equivalently, to have a locally free quotient.
Consequently, on there is a tautological flag
with . Every family of flags over an -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 , , and . Suppose that and the universal rank- subbundle have been constructed. Define
The universal subbundle of the quotient has an inverse image . After steps one obtains
For the complete flag bundle, where for , each step chooses a line in a quotient bundle. Thus, the complete flag bundle is a tower of projective bundles:
Basic properties
Formation of the flag bundle commutes with arbitrary base change. If , then there is a canonical isomorphism
On an open subset over which is trivial, the flag bundle is a product
In particular, the morphism is smooth and projective. Its relative dimension is
Equivalently, if is the principal -bundle of frames of and is the parabolic subgroup stabilizing a standard flag of type , then
Special cases
If is a point, the flag bundle is the usual flag variety of a vector space. If , it is the Grassmann bundle . Under the convention that parametrizes lines, the case is the projective bundle .
The type gives the complete flag bundle ; all other types are called partial flag bundles.
Sections and fixed flags
By the universal property, a section
is equivalent to a flag of subbundles
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 and . The successive quotients
are line bundles. Hence has a canonical filtration with line-bundle quotients, and its total Chern class satisfies
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