# Geometric interpretation of Chern classes over flag manifolds

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 $\mathbb{C}^n$ ($U(n)/U(n_1)\times\cdots\times U(n_k)$).

• You might want to consult Chapter 14 of Fulton's "Intersection Theory". Even though the title of the chapter only mentions Grassmannians, Fulton does describe cycles on flag manifolds. Fulton's filtered Thom-Porteous formula should give a geometric interpretation to Chern classes of the subquotients of the universal flag of vector bundles on the flag manifolds. Jul 9, 2015 at 14:05

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.
• I assume (by definition of dependency sets together with your notation) that $c_i=D_i\in H_{n-(k+1-i)}$, where $n$ is the dimension of the manifold and $k$ is the rank of the bundle, so for bigger $i$ we should get bigger sets in homology. This however seems to be in contradiction with some of your findings. I believe the notation might be inconsistent. However I understand the procedure and will work on it now. Jul 10, 2015 at 13:59