Let $X$ be a projective complex manifold and $G$ a finite group. Assume that $G$ acts on $X$ holomorphically and freely. Is it true that any birational map $\phi \in Bir(X/G)$ lifts to some birational map $\tilde{\phi}\in Bir(X)$? If this is not true, what kind of condition should one impose on the manifold or the action?
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This is all about the function fields, if I understand the question. Let $L$ be the function field of $X$ and let $K$ be the function field of $X/G$. $L$ is a finite Galois extension of $K$. A typical automorphism of $K$ will not extend to an automorphism of $L$. For example, if $G$ has order $2$ and $L=K(\sqrt f)$ then the automorphism would have to take $f$ to $f$ times a square. |
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