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The title describes completely the question.

For n=1 it is an easy exercise. For n=2 the statement is still true, and depends on the classification of birational involutions of P^2 (see e.g. For n=3 and above I could not find any useful hint in the literature.

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In the specific case I'm interested in an additional condition is true: the involution is homotopic (through birational automorphism) to an automorphism with a fixed point. I don't know if it helps. – user175348 Mar 17 '10 at 12:49
Maybe I am missing something, but it seems to me that $(x,y)\mapsto(x+1/y,-y)$ is a counterexample. – user2035 Mar 31 '10 at 8:31

The answer is no. There are plenty of counterexamples, for example the map given by "a-fortiori", which is $(x,y)\mapsto (x+1/y,-y)$, which can be generalised to any dimension.

The good question is probably to look at birational involutions of $\mathbb{P}^n$.

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