I understand that $CP^1$ is the Riemann Sphere. I guess all its discrete symmetries were known for a long time and well-classified. (But suggestions or good references where this is worked out in a neat way would be appreciated.)

My question is: I want to know about the discrete symmetries of $CP^2$ and more generally, also $CP^n$. Is there any place these are worked out in a simple way?

Are there any classical references specifically about the complex projective plane?

  • 2
    $\begingroup$ What do you mean by 'discrete' symmetries? The usual symmetries of $\mathbb{C}P^1$ are the Mobius transformations, that form the Lie (i.e. non-discrete) group $PGL(2,\mathbb C)$. In general the symmetry group of $\mathbb{C}P^n$ is $PGL(n+1,\mathbb C)$. $\endgroup$ Aug 8, 2019 at 19:17
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    $\begingroup$ Are you asking for a classification of discrete subgroups of $\mathrm{PGL}(n+1,\mathbb{C})$? $\endgroup$
    – Qfwfq
    Aug 8, 2019 at 19:53
  • $\begingroup$ I guess I want to know all the discrete subgroups of PGL(n+1,C), and possibly interpret them geometrically, like regular polyhedra. $\endgroup$
    – guest78
    Aug 11, 2019 at 13:21
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    $\begingroup$ Every finite group appears as automorphisms of some complex projective space, so you need a more precise question. $\endgroup$
    – Ben McKay
    Aug 13, 2019 at 10:07
  • $\begingroup$ Possible duplicate of The finite subgroups of SU(n) $\endgroup$ Sep 13, 2019 at 3:22


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