How do we compute the even cohomology $H^{2i}(Q)$ of the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$?

Furthermore, what is the curvature form $\Omega$ on $Q$, i.e. what is a reasonable Kähler metric for $Q$? Given this, how do we compute $\chi(S^n,Q)=\int_{S^n}ch(Q)Td(S^n)$?

Thanks!