Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{P}^n$ such that $\mathbb{P}(E)=\mathbb{P}^n\times\mathbb{P}^m$ as bundles over $\mathbb{P}^n$ (for example, $E$ might be $\mathbb{P}^n\times\mathbb{C}^{m+1}$).

Let $\mathcal{Q}$ be the universal quotient bundle on $\mathbb{P}^n\times\mathbb{P}^m$ of rank $m$, coming from the short exact sequence $$ \begin{align} 0\to \mathcal{O}_{\mathbb{P}(E)}(-1)\to \pi^*(E)\to \mathcal{Q}\to 0. \end{align} $$

Let $l$ be a line in $\mathbb{P}^n$. Then, in Chow groups, $\pi ^*([l]) = [l\times \mathbb{P}^m]\in CH_{m+1}(\mathbb{P}^n\times\mathbb{P}^m)$.

Question: In $CH_1(\mathbb{P}^n\times\mathbb{P}^m)$, do we have $c_m(\mathcal{Q})\cap[l\times \mathbb{P}^m] = [l\times\{*\}]$?,

where $c_m(\_)$ denotes the $m$-th chern class, and $*$ denotes any point in $\mathbb{P}^m$?

I believe that they should not be equal (e.g. taking $m=n=1$ I feel thwy should not be equal) , but I can't prove it concretely. Any help/reference would be highly appreciated.

Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{P}^n$ such that $\mathbb{P}(E)=\mathbb{P}^n\times\mathbb{P}^m$ as bundles over $\mathbb{P}^n$ (for example, $E$ might be $\mathbb{P}^n\times\mathbb{C}^{m+1}$).

Let $\mathcal{Q}$ be the universal quotient bundle on $\mathbb{P}^n\times\mathbb{P}^m$ of rank $m$, coming from the short exact sequence $$ \begin{align} 0\to \mathcal{O}_{\mathbb{P}(E)}(-1)\to \pi^*(E)\to \mathcal{Q}\to 0. \end{align} $$

Let $l$ be a line in $\mathbb{P}^n$. Then, in Chow groups, $\pi ^*([l]) = [l\times \mathbb{P}^m]\in CH_{m+1}(\mathbb{P}^n\times\mathbb{P}^m)$.

Question: In $CH_1(\mathbb{P}^n\times\mathbb{P}^m)$, do we have $c_m(\mathcal{Q})\cap[l\times \mathbb{P}^m] = [l\times\{*\}]$?,

where $c_m(\_)$ denotes the $m$-th chern class, and $*$ denotes any point in $\mathbb{P}^m$?

I believe that they should not be equal (e.g. taking $m=n=1$, I feel they should not be equal) , but I can't prove it concretely. Any help/reference would be highly appreciated.

Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{P}^n$ such that $\mathbb{P}(E)=\mathbb{P}^n\times\mathbb{P}^m$ (for example, $E$ might be $\mathbb{P}^n\times\mathbb{C}^{m+1}$).

Let $\mathcal{Q}$ be the universal quotient bundle on $\mathbb{P}^n\times\mathbb{P}^m$ of rank $m$, coming from the short exact sequence $$ \begin{align} 0\to \mathcal{O}_{\mathbb{P}(E)}(-1)\to \pi^*(E)\to \mathcal{Q}\to 0. \end{align} $$

Let $l$ be a line in $\mathbb{P}^n$. Then, in Chow groups, $\pi ^*([l]) = [l\times \mathbb{P}^m]\in CH_{m+1}(\mathbb{P}^n\times\mathbb{P}^m)$.

Question: In $CH_1(\mathbb{P}^n\times\mathbb{P}^m)$, do we have $c_m(\mathcal{Q})\cap[l\times \mathbb{P}^m] = [l\times\{*\}]$?,

where $c_m(\_)$ denotes the $m$-th chern class, and $*$ denotes any point in $\mathbb{P}^m$?

Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{P}^n$ such that $\mathbb{P}(E)=\mathbb{P}^n\times\mathbb{P}^m$ as bundles over $\mathbb{P}^n$ (for example, $E$ might be $\mathbb{P}^n\times\mathbb{C}^{m+1}$).

Let $\mathcal{Q}$ be the universal quotient bundle on $\mathbb{P}^n\times\mathbb{P}^m$ of rank $m$, coming from the short exact sequence $$ \begin{align} 0\to \mathcal{O}_{\mathbb{P}(E)}(-1)\to \pi^*(E)\to \mathcal{Q}\to 0. \end{align} $$

Let $l$ be a line in $\mathbb{P}^n$. Then, in Chow groups, $\pi ^*([l]) = [l\times \mathbb{P}^m]\in CH_{m+1}(\mathbb{P}^n\times\mathbb{P}^m)$.

Question: In $CH_1(\mathbb{P}^n\times\mathbb{P}^m)$, do we have $c_m(\mathcal{Q})\cap[l\times \mathbb{P}^m] = [l\times\{*\}]$?,

where $c_m(\_)$ denotes the $m$-th chern class, and $*$ denotes any point in $\mathbb{P}^m$?

I believe that they should not be equal (e.g. taking $m=n=1$ I feel thwy should not be equal) , but I can't prove it concretely. Any help/reference would be highly appreciated.