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Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.

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 $\chi(Q,\mathcal{O}_Q)$?

If you down-vote, please explain why so I can improve the question. Thanks in advance!

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.

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 $\chi(Q,\mathcal{O}_Q)$?

Thanks in advance!

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.

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 compute $\chi(Q,\mathcal{O}_Q)$?

If you down-vote, please explain why so I can improve the question. Thanks!

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.

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 $\chi(Q,\mathcal{O}_Q)$?

If you down-vote, please explain why so I can improve the question. Thanks in advance!

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.

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

If you down-vote, please explain why so I can improve the question. Thanks!

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.

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 compute $\chi(Q,\mathcal{O}_Q)$?

If you down-vote, please explain why so I can improve the question. Thanks!