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Let $X$ and $Y$ be (smooth projective connected) varieties over $\mathbf{C}$.

Let $\pi:X\to Y$ be a finite surjective flat morphism.

Does this induce (by base change) a map $\mathrm{Aut}(Y) \to \mathrm{Aut}(X)$?

I think it does. Given an automorphism $\sigma:Y\to Y$, the base change via $\pi:X\to Y$ gives an automorphism of $X$.

My real question is as follows:

Is $\mathrm{Aut}(Y)\to \mathrm{Aut}(X)$ injective?

If not, under which hypotheses is $\mathrm{Aut}(Y)\to \mathrm{Aut}(X)$ injective? Does $\pi$ etale do the trick?

What if $\dim X=\dim Y =1$?

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I don't think that there is any (canonical) map $Aut(Y) \to Aut(X)$ of the kind you presume exists.

E.g. if $X$ is a curve of genus $g \geq 2$ and $Y$ is $\mathbb P^1$, then $Aut(X)$ is finite (often trivial), while $Aut(Y)$ equals $PGL_2(\mathbb C)$, which is simple. What is the map that you have in mind? (In any case, whatever it it is, it won't be injective.)

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    @Harry: Dear Harry, certainly the fibre product you write down will be isomorphic to $Y$ as a variety, but it *won't* usually be isomorphic to $Y$ as an $X$-variety, i.e. we can't find an isomorphism of it with $Y$ which lies over the automorphism $\sigma$ of $X$. As the simplest examples show (higher genus curves mapping to $\mathbb P^1$, higher genus curves mapping to genus one curves, etc.), automorphisms typically don't lift through finite flat maps. Regards,2012-06-18