The <Euler sequence> on $\mathbb P^m$ is
$$
0\longrightarrow\mathcal O_{\mathbb P^m}
\longrightarrow\mathcal O_{\mathbb P^m}(1)^{\oplus(m+1)}
\longrightarrow T\mathbb P^m\longrightarrow0.
$$
The Whitney product formula for the <Total Chern class> therefore gives
$$
c(T\mathbb P^m)=\frac{(1+H)^{m+1}}1=(1+H)^{m+1}
$$
in $A^*(\mathbb P^m)=\mathbb Z[H]/(H^{m+1})$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 162 / 3 / i / Solution

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On projective space, the Euler sequence is
$$
0\to\mathcal O\to\mathcal O(1)^{\oplus(m+1)}\to T\mathbb P^m\to0.
$$


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