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Denote by $V$ the vanishing locus of the affine quadric $z_1^2+\dots+z_n^2-1$ in $\mathbb{C}^n$. Clearly $V$ is a complex manifold. Let $g:V\to V$ be an automorphism of $V$ (a biholomorphism, not necessarily algebraic). As any holomorphic function on $V$ extends to a holomorphic function on $\mathbb{C}^n$, we can extend $g$ to a map $g=(g_1,\dots,g_n):\mathbb{C}^n\to\mathbb{C}^n$ (not unique). Does the fact that $g$ restricted to $V$ is an automorphism of $V$ implies that we can find an extension $g$ (Thanks Nate!) such that the equation $g_1^2+\dots+g_n^2=z_1^2+\dots+z_n^2$ holds globally on $\mathbb{C}^n$? Anyone knows a proof or a counterexample?

Maybe another question is that can we always extend an automorphism of $V$ to an automorphism of $\mathbb{C}^n$?

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    $\begingroup$ I think this is not true. Set $g: V \to V$ to be the identity, and define $g_1, \ldots, g_n$ by setting $g_1(z) := z_1 + \left(z_1^2 + \cdots + z_n^2 - 1\right)$ and $g_i(z) := z_i$ for $i > 1$. Am I missing something? $\endgroup$
    – Nathaniel Bottman
    Commented Oct 23, 2013 at 13:30
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    $\begingroup$ There is something I do not understand. Take $z_1^2+z_2^2-1=0$ in $\mathbb{C}^2$ and consider the inversion with respect to the point $(1,1)$. Then we obtain the automorphism of $V$ given by $$z_1 \to 1 + \frac{z_1-1}{3-2z_1-2z_2}, \quad z_2 \to 1 + \frac{z_2-1}{3-2z_1-2z_2}.$$ It seems to me that it does not extend to an automorphism of $\mathbb{C}^2$. Am I missing something? $\endgroup$
    – Francesco Polizzi
    Commented Oct 23, 2013 at 13:51
  • $\begingroup$ I also do not understand. For n = 1 isn't 1/z holomorphic on V but not on $\mathbb{C}$? $\endgroup$
    – emperordali
    Commented Oct 23, 2013 at 13:52
  • $\begingroup$ To Polizzi: I guess your map is not defined at the points $(\frac{3}{4}+\frac{1}{4}i,\frac{3}{4}-\frac{1}{4}i)$ and $(\frac{3}{4}-\frac{1}{4}i,\frac{3}{4}+\frac{1}{4}i)$. $\endgroup$
    – Piojo
    Commented Oct 23, 2013 at 14:29
  • $\begingroup$ To David: For n=1, 1/z=z is the extension you want. $\endgroup$
    – Piojo
    Commented Oct 23, 2013 at 14:30

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