= Mod-two restriction from complex to real projective space
{title2=$i^*h=a^2$}
Let $i:\mathbb{RP}^d\hookrightarrow\mathbb{CP}^d$ be the standard inclusion. If $h=c_1(\mathcal O(1))\bmod2$ and $a=w_1(\gamma_d)$, then
$$
i^*h=a^2.
$$
The restricted complex <tautological bundle> is the <complexification of a real vector bundle> $\gamma_d\otimes_{\mathbb R}\mathbb C$. Its underlying real bundle is $\gamma_d\oplus\gamma_d$. The <Whitney product formula for Stiefel–Whitney classes> gives total class $(1+a)^2=1+a^2$, while the <Stiefel–Whitney class of the underlying real bundle of a complex line> identifies its second class with the reduced <First Chern class>. Dualizing changes the integral sign but has no effect modulo two. Thus the full <induced map on cohomology> sends $h^j$ to $a^{2j}$.
Back to article page
