I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able to find such a presentation online. Is it known? And, if so, is it 'basic/common' knowledge in the field?


For example, I think it is possible to get to that description using the degeneracy sets of sections, which seems to be a pretty standard thing.


**EDIT**


I am only working with flag varieties flags of linear subspaces of $\mathbb{C}^n$ ($U(n)/U(n\_1)\times\cdots\times U(n\_k)$).



Geometric interpretation of Chern classes over flag manifolds
You mentioned the degeneracy subset definition. First lets re-establish the Grassman case as an example. Let $BU(n)=Gr\_n(\mathbb{R}^{\infty})$ and take $e\_i=(0,\dots 1, \dots )$ in the $i$-th position. Then define vectorfields of the canonical bundle by $v\_i=pr\_{V}(e\_i)$ at point $V$. Then the degenerancy set is $X=\{V| det(pr\_V(e\_i))\neq 0\}$. The fact that this is "generic" is easily checked, so that $c\_1=[X]$. Now since $v\_i$ are always orthogonal, this is simply gonna be the cap product, by transverality of $\{V| det(pr\_V(e\_i))\neq 0\}$, or simply $c\_i=[\{V|rank(pr\_V(e\_k))<i, k\le i\}]$, or the set such that the first $i$ $v\_i$ are linearly dependant. This is easier to see in the more familiar setting of $BU(1)$. We have that $H^1(BU(1))=H^1(\mathbb{C}P^{\infty})=Hom(H\_1(\mathbb{C}P^{\infty}), \mathbb{Z})=[\mathbb{C}P^{\infty-1}]^\*$ and likewise, $H^i(BU(1))=[\mathbb{C}P^{\infty-i}]^\*$, where these terms denote the limit of the inclusiosn $\mathbb{C}P^{n-i}\to \mathbb{C}P^n$. The situation for Stiefel-Whitney classes is similar.


Now for the general flag manifold, say $F=F(d\_1,\dots d\_i, n)$. Then take $v\_j=pr\_{V\_i}(e\_j)$, which gives a deneracy set of the dependant sets of the $v\_j$, and so on, and we can proceed exactly as in the Grassmanian case getting $c\_l=\{V|rank(v\_j)<l\}$. This noteably agrees on the dot with the canonical maps involving other flag manifolds. This gives exactly the result we want, since the map $F\to BU(d\_i)$ induces the top tautological bundle, and thus the Chern class is just the pullback of the original above class. Similar results hold for all other tautological bundles on the flag.


