= Chern classes of a tautological-line complement
{c}
{title2=$c_k(S^\perp)=\sum_{j=0}^{k}(-x)^j p^*c_{k-j}(E),\quad x=c_1(S)$}
For the line-projectivization of a rank-$n$ complex bundle, $p^*E=S\oplus S^\perp$. The <Whitney sum formula for Chern classes> gives $c_t(S^\perp)=p^*c_t(E)/(1+xt)$ as a formal power series. Comparing coefficients proves the formula. Since $S^\perp$ has rank $n-1$, its degree-$n$ <Chern class> vanishes and yields the monic projective-bundle relation.
// Target: geometry-and-topology.bigb
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