I'm sorry for the stupid question, but..


I need to check that zero-scheme of general global section of homogeneous vector bundle over G/P (G simple algebraic group over C, say Spin, P its maximal parabolic subgroup) is non-empty. Is there an easy way to do this?


For now it seems to me that the only way to do this is to check that r-th chern class of my bundle E of rank r is nonzero. So, how do I do this?


I'm afraid I don't know any generalized schubert calculus, and not much of the intersection theory, so down-to-earth answers, or, alternatively, readable references, would be most appreciated.



how to compute chern classes of homogeneous vector bundles
It's easier to compute its $T$-equivariant $r$th Chern class $c\_r(E) \in H^\*\_T(G/P)$, and use the maps $H^\*\_T(G/P) \twoheadrightarrow H^\ast(G/P)$ and $H\_T^\*(G/P) \hookrightarrow \oplus\_{W/W\_P} H\_T^\*(pt)$.


The latter comes from restricting the equivariant Chern class to the $T$-fixed points on $G/P$ (here $T$ is a maximal torus). The ring $H\_T^\*(pt)$ is the symmetric algebra on the weight lattice $T^\ast$, and $c\_r(E|\_{wP/P}) = w\cdot ($the $r$th elementary symmetric function of the $T$-weights on the vector bundle's base fiber, over $P/P \in G/P)$.


Now, how to show that's not zero, in ordinary cohomology... well, let's pull it back to $G/B$, first, and the cohomology injects so it's enough to work up there. On $\oplus\_W H\_T^\ast(pt) = [$functions $f : W \rightarrow H^\ast\_T(pt)]$ there is a BGG/Demazure operator $\partial\_\alpha$ for each simple root $\alpha$, taking $f$ to $w \mapsto (f(w) - f(w r\_\alpha))/\alpha$. (It comes from pushing the class on $G/B$ to $G/P\_\alpha$, then pulling it back.)


Your $c\_r$ will be nonzero iff for some sequence of $r$ simple roots $\alpha\_1,\ldots,\alpha\_r$, this $\partial\_{\alpha\_1}\cdots \partial\_{\alpha\_r} c\_r$ (now necessarily a constant function on $W$) is not zero.


Not knowing your vector bundle, I can't advise you what sequence of simple roots to use, other than that it should give a reduced word in $W$. EDIT: you can choose them greedily, applying $\partial\_\alpha$ and trying to stay nonzero. If you get to a point (before getting to a constant) where all $\partial\_\alpha$ give zero, then your Euler class is zero.


One place this combinatorial approach to equivariant cohomology of $G/P$ is laid out is my paper with Terry Tao on the equivariant cohomology of Grassmannians.


