= Chern classes classify rank-d complex vector bundles over CPd
{c}
{title2=$E\to\mathbb{CP}^d$}
Two rank-$d$ complex vector bundles over $\mathbb{CP}^d$ are isomorphic exactly when their Chern classes agree. Equality of Chern classes gives equality of their <Chern characters>; the Chern character is injective here because $K^0(\mathbb{CP}^d)$ is torsion-free. The bundles are therefore stably isomorphic, and <stable cancellation for complex vector bundles> applies because $\mathbb{CP}^d$ has real dimension $2d$.
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