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

Let $\zeta =c_1({\mathcal{O}}_{\mathbb{P}(E)}(1))$. The projective bundle formula for Chow groups is the mapFlat pullback raises dimension by $e$, and intersection with ${\zeta }^i$ lowers it by $i$, so every summand has dimension $k$.

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Compactify the total space of $E$ as $\overline{E}=\mathbb{P}(E\oplus {\mathcal{O}}_X)$; its hyperplane at infinity is $\mathbb{P}(E)$ and its open complement is $E$. Under the projective bundle formula for $\overline{E}$, the pushforward from that hyperplane spans the summands containing positive powers of $\zeta$. The localization quotient therefore retains the zeroth summandwhose restriction to $E$ is precisely the vector-bundle pullback ${\pi }^{\ast }:A_k(X)\to A_{k+e+1}(E)$. It survives injectively. In fact this proves the stronger homotopy invariance of Chow groups: ${\pi }^{\ast }$ is an isomorphism.

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From the tautological exact sequence and the Whitney formula in the Chern class formalism,Thus $c_i(\xi )$ is a sum of terms ${\zeta }^j{p}^{\ast }{c}_{i-j}(E)$ with $0\le j\le i$. The projective-bundle pushforward satisfiesThe projection formula now makes every term vanish when $i<e$. For $i=e$, only the term ${\zeta }^e{p}^{\ast }\alpha$ survives, giving

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Write $S(\zeta )={\sum }_{j\ge 0}{\zeta }^j$. By the projective bundle formula, it is enough to verify the identity on a basis element $\beta ={\zeta }^q\cap p^{\ast }\alpha$, $0\le q\le e$. Expandand use the pushforward identities obtained from part iii. Multiplication by $c(p^{\ast }E)S(\zeta )$ is the inverse triangular operation, becauseIt recovers ${\zeta }^q{p}^{\ast }\alpha$ in dimension $k$. By linearity, every $\beta \in A_k(\mathbb{P}(E))$ satisfies

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