Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 2 / c / Solution

We induct on $n$. The claim is immediate for ${\mathbb{CP}}^0$. In the stated cover, $U=U_0\cong {\mathbb{C}}^n$ is contractible. The homotopydeformation retracts $V$ onto the hyperplane $z_0=0$, which is ${\mathbb{CP}}^{n-1}$. Moreover,deformation retracts onto $S^{2n-1}$.

The Mayer--Vietoris sequence for the de Rham complex containsFor odd $i>1$, the group on the right vanishes by contractibility and the induction hypothesis, while the group on the left vanishes because $i-1$ is positive and even and is neither $0$ nor $2n-1$. Thus the middle group vanishes. For $i=1$, the preceding mapis surjective because all three spaces are connected, so the connecting map into $H^1({\mathbb{CP}}^n)$ is zero. Therefore all odd de Rham cohomology groups of ${\mathbb{CP}}^n$ vanish.