Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.

I searched all over the internet but I can't find an example of a calculation of the volume for a projective subvariety $X \subset \mathbb{P}^n$ (we can assume it to be smooth of course).

For example given the projective plane $\mathbb{P}^2$ with coordinates $[x,y,z]$ and the smooth conic $$C:(xy-z^2=0) \subset \mathbb{P}^2$$ does somebody know how to compute the Volume $\textit{Vol}(C)$? Or maybe a reference where a similar computation is carried out? For simplicity we can assume that the ground field is $\mathbb{C}$.

Thanks in advance.

  • 6
    $\begingroup$ This is nicely treated in §5.C of Mumford's Algebraic geometry I: complex projective varieties (Springer). $\endgroup$
    – abx
    Oct 13, 2020 at 16:35
  • 1
    $\begingroup$ Thank you very much @abx $\endgroup$
    – gigi
    Oct 13, 2020 at 17:23
  • $\begingroup$ You are welcome. $\endgroup$
    – abx
    Oct 13, 2020 at 18:02


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