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On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

Is this true for $\mathbb {CP}^n$?

The answer should be only one involution for even $n$, two of them for odd $n$.

The proof I have in mind uses the invariance of $z_i \mapsto M_{ij}\bar z_j$ under $U^{-1}MU^*$ together with the facts $MM^*=\pm 1$ and $MM^\dagger=1$ to get the conclusion, where $M^*$ is complex conjugate, while $M^\dagger$ is hermitian conjugate, i.e. complex conjugate and transpose, so it depends on the choice of complex structure.

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

For $\mathbb {CP}^n$, there should be one involution for even $n$, two for odd $n$.

As suggested: in homogeneous coordinates, let us write our involution as $z \to \overline {Az}$ for some $A \in GL_{n+1}(C)$. Then we have $\overline{A} A = \lambda id$ for some real $\lambda$. By rescaling $\lambda$ and taking determinant, we see that for even $n$ only $\overline{A}A=1$ is possible (class $\tau$ above), while for odd $n$ also $\overline{A}A=-1$ is possible (so also $\eta$ above).

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

is this true for $\mathbb {CP}^n$?

The answer should be only one involution for even $n$, two of them for odd $n$.

The proof I have in mind uses the invariance of $z_i \mapsto M_{ij}\bar z_j$ under $U^{-1}MU^*$ together with the facts $MM^*=\pm 1$ and $MM^\dagger=1$ to get the conclusion, where $M^*$ is complex conjugate, while $M^\dagger$ is hermitian conjugate, i.e. complex conjugate and transpose, so it depends on the choice of Fubini-Study metric on $\mathbb{CP}^n$.

QUESTION: Is it possible to prove the statement without reference to the metric?

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

Is this true for $\mathbb {CP}^n$?

The answer should be only one involution for even $n$, two of them for odd $n$.

The proof I have in mind uses the invariance of $z_i \mapsto M_{ij}\bar z_j$ under $U^{-1}MU^*$ together with the facts $MM^*=\pm 1$ and $MM^\dagger=1$ to get the conclusion, where $M^*$ is complex conjugate, while $M^\dagger$ is hermitian conjugate, i.e. complex conjugate and transpose, so it depends on the choice of complex structure.

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

is this true for $\mathbb {CP}^n$?

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto [\bar w, -\bar z]\ ?$$

is this true for $\mathbb {CP}^n$?

The answer should be only one involution for even $n$, two of them for odd $n$.

The proof I have in mind uses the invariance of $z_i \mapsto M_{ij}\bar z_j$ under $U^{-1}MU^*$ together with the facts $MM^*=\pm 1$ and $MM^\dagger=1$ to get the conclusion, where $M^*$ is complex conjugate, while $M^\dagger$ is hermitian conjugate, i.e. complex conjugate and transpose, so it depends on the choice of Fubini-Study metric on $\mathbb{CP}^n$.

QUESTION: Is it possible to prove the statement without reference to the metric?