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I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto \frac{1}{\overline{z}}, $$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$.

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in fact $z\mapsto \overline{z}$ becomes $z\mapsto -\overline{z}$ if you replace $z$ with $iz$. And yes, there are just two involutions, up to change of coordinate. – user8268 Sep 19 '12 at 7:27

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