# automorphisms of varieties with respect to a cover

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.)
Let me first motivate my question. We know that, for each hyperbolic curve $X$, the inequality $\# \mathrm{Aut}(X) \leq 84(g-1)$ holds. Now, forgetting for a moment that we already know this, suppose that $X$ is a curve for which this inequality holds. If $Y$ is hyperbolic of genus $g$ (same genus as $X$!) and $Y\to X$ is a finite morphism, can we conclude that the same inequality holds for $\# \mathrm{Aut} (Y)$? So, back to the question. Let $\pi:Y\to X$ be finite (I reversed $X$ and $Y$...). Let $\sigma:X\to X$ be an autom. We can make a Cartesian diagram and base change. This gives a – Harry Jun 18 '12 at 14:28
morphism $Y\times_{X,\sigma} X\to Y$. I probably did something stupid and identified $Y\times_{X,\sigma} X$ with $Y$ and said this base changed morphism is an automorphism. This would be then the map I had in mind. – Harry Jun 18 '12 at 14:29
@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, – Matt E Jun 18 '12 at 15:06