From the tautological exact sequence and the Whitney formula in the <Chern class> formalism,
$$
c(\xi)=\frac{c(p^*E)}{c(\mathcal O(-1))}
=p^*c(E)(1+\zeta+\zeta^2+\cdots).
$$
Thus $c_i(\xi)$ is a sum of terms $\zeta^j p^*c_{i-j}(E)$ with $0\leq j\leq i$. The projective-bundle pushforward satisfies
$$
p_*(\zeta^j\cap p^*\gamma)=0\quad(j<e),
\qquad
p_*(\zeta^e\cap p^*\gamma)=\gamma.
$$
The projection formula now makes every term vanish when $i<e$. For $i=e$, only the term $\zeta^e p^*\alpha$ survives, giving
$$
p_*(c_i(\xi)\cap p^*\alpha)=
\begin{cases}
\alpha,&i=e,\\
0,&i<e.
\end{cases}
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 162 / 2 / iii / Solution

{c}
{title2=$c_i(E)$}
{wiki}

Chern classes are characteristic classes of vector bundles satisfying functoriality and the Whitney product formula $c(E)=c(E')c(E'')$ for a short exact sequence.


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 Incoming links: Chern class