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

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