Consider the affine hyperquadric $Q:=\biggl\{(z\_1,...,z\_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum\_{i=1}^{n+1}z\_i^2=1\biggr\}\cong TS^n$.



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> What is a reasonable Kähler metric for $Q$ (induced by the pullback of the metric from the ambient space $\mathbb{C}^{n+1})$?
>  Furthermore, how do we explicitly calculate the curvature form $\Omega$ on $Q$? Hence, compute the Chern classes of $Q$. Given this, how do we find $\chi(Q,\mathcal{O}\_Q)$?
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Thanks in advance!



How do we compute the even cohomology $H^{2i}(Q)$ of the affine hyperquadric?
I'm going to answer questions 1 and 2.


1. You can restrict the Kaehler metric from the ambient affine space to get a Kaehler metric on this hypersurface if you want (there are, of course, many others).
2. The "curvature form" definition of Chern classes is not usually very useful for calculations. Instead, as you have helpfully pointed out, the quadric is diffeomorphic to the cotangent bundle of $S^n$. This deformation retracts onto the zero section, so you might as well compute the Chern classes of $T(T^\*S^n)|\_{S^n}$ (living in $H^\*(S^n)$). The tangent bundle of $T^\*S^n$ restricted to the zero section is $T^\*S^n \oplus TS^n$ and, since the $S^n$ is totally real (corresponds to the real locus under your diffeomorphism), this is the complexification of $T^\*S^n$. Therefore the Chern classes of the quadric n-fold agree up to a sign with the Pontryagin classes of $S^n$. In particular, you get $c\_{n/2}=\pm 1\in Z=H^n(S^n)$ whenever $n/2$ is even (e.g. $S^4$, $S^8$, etc ) and all other Chern classes zero (because they live in cohomology groups which vanish).


