In W. Fulton, R. K. Lazarsfeld – Positive polyonomials for vector bundles, Ann. of Math. 118 (1983) 35-60, they prove the positivities of the top Chern class of a rank $r$ ample vector bundle $E$ over a smooth complex projective variety $X$ of dimension $n\leq r$ (Theorem in Appendix B), that is $\displaystyle\int_Xc_n(E)>0$.
Let $\mathbb{P}\equiv\mathbb{P}\left(E^{\vee}\right)$, by hypothesis $\xi=c_1\left(\mathcal{O}_{\mathbb{P}}(1)\right)$ is the Chern class of an ample line bundle, and by construction
$$
\xi^r-\xi^{r-1}\pi^{*}c_1(E)+\dotsc+(-1)^n\xi^{r-n}\pi^{*}c_n(E)=0\in H^{2r}\left(\mathbb{P},\mathbb{Z}\right)
$$
where $\pi:\mathbb{P}(E)\to X$ is the obvious projection.
They consider the class
$$
\alpha=\xi^{n-1}-\xi^{n-2}\pi^{*}c_1(E)+\dotsc+(-1)^{n-1}\pi^{*}c_{n-1}(E)\in H^{2(n-1)}(\mathbb{P},\mathbb{Z})
$$
and state $\pi_{*}\left(\left(\xi^{r-n}\smile\alpha\right)\frown[\mathbb{P}]\right)=[X]$; where $[\cdot]$ is the fundamental class of $\cdot$ in homology, $\smile$ is the cup product in cohomology, and $\frown$ is the cap product between cohomology and homology.
I do not object on the rightness of this equality; the point is: I am not able to figure out the geometric meaning of this equality!
Roughly speaking: how can I compute this?
I am sorry for the poor clarity, but I am very confused by all this.
Best Answer
You can compute this by the projection formula. $$ \pi_* (\xi^{r-1-k} . \pi^* c_k(E)) = (\pi_* \xi^{r-1-k}) . c_k(E) $$ Note $\xi^{r-1-k} \in H^{2(r-1-k)}(\mathbb{P}) \cong H_{2(n+k)}(\mathbb{P})$, so when $k \geq 1$, the pushforward class $\pi_* \xi^{r-1-k} = 0$. Therefore the equality claimed is just $\pi_* \xi^{r-1} = [X]$. Geometrically the class $\xi^{r-1}$ is the class of a point in each fibre $\mathbb{P}^{r-1} = \mathbb{P}(E_x^{\vee})$, so pushing this class to $X$ you get the fundamental class of $X$.