Let $X$ be a smooth projective variety of dimension $n$. Take the bundle $TX \oplus Sym^2(TX)$ over $X$ where $Sym^2(TX)$ is the second symmetric product of the tangent space. The Grassmannian bundle $Gr(n,TX \oplus Sym^2(TX))$ has a canonical section, namely $TX$.

My question is: what is the Poincare dual of this section in the cohomology ring of the Grassmannian bundle?

The cohomology ring is

$H^*(Gr(n,TX \oplus Sym^2(TX)))=H(X)[c_1,\ldots, c_n,d_1,\ldots, d_{{n+1 \choose 2}}]/$

$(1+c_1+\ldots +c_n)(1+d_1+\ldots +d_{{n+1 \choose 2}})=c(TX \oplus Sym^2(TX))$

This is probably a trivial question I am a bit confused about it now.

`s and $\alpha_j$`

s are the Chern roots of the $d$`s and $c$'s respectively, i.e $\prod(1+\beta_i)=1+d_1+\ldots$ – Gergely Berczi Jan 26 '11 at 12:18