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I know there is a plethora of literature on how to construct quotients by groups, but my situation is quite particular, so I would appreciate if you could give me some hints or bibliographical references.

I'm interested in the following question: let $X$ be an algebraic variety defined over an embedded number field $k \hookrightarrow \mathbb{C}$. Assume that $X$ is smooth and projective and that it comes with the action of a finite group $G$. Then the quotient $X/G$ exists.

$\textbf{First}$: what is the best reference to learn the construction?

$\textbf{Second}$: what are the (scheme theoretic) properties of the "projection" $\pi: X \to Y$?

For instance, is it true that the direct image of the constant sheaf

$\pi_\ast \mathbb{C}_X$

on $X(\mathbb{C})$ is a local system on a certain open subset $U \subset Y$ excluding the singularities of $Y$?

Is it still true that the direct image by $\pi$ of a regular singular connection is still regular singular?

Thanks for your help !

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In the case of curves, the group acts on the function field of the curve, and taking invariants (the fixed field, as in Galois theory) yields the function field of the quotient curve. –  Thom Tyrrell Nov 16 '12 at 21:03
    
Thanks Thom! I'm more interested in the case when $X$ is of dimension at least 2. For instance, for curves the quotient is always regular (it is normal because otherwise the normalization would be a better quotient and normal implies regular for curves over a field). –  loren Nov 16 '12 at 21:10

1 Answer 1

Since $G$ is finite and $X$ is projective, you can easily check that any point has an open affine n.h. that is preserved by the $G$-action. Thus $X$ can be covered by open affines compatibly with the $G$-action, say $X = $ union of the $U$s.

Then to compute $X/G$, we can instead compute the various $U/G$, and then glue these together.

If $U = $ Spec $A$, then the $G$-action on $U$ is equivalent to a $G$-action on $A$, and $U/G = $ Spec $A^G$. (You can take this as a definition, but you can also check that it makes intuitive sense.)

The morphism $X \to X/G$ is a finite morphism, and the answer to your sheaf-theoretic questions will be the same as for any finite morphism. (I don't think that finite morphisms arising as a quotient in this way are particularly special.)

For example, if $G$ acts faithfully on $X$ (which you can assume WLOG), then there will be an open subset $V$ of $X$ on which $G$ acts freely, and $V \to V/G$ will be etale. The pushforward of the constant sheaf under a finite etale morphism is indeed a local system, and so $\pi_*\mathbb C$ will be a local system when restricted to $V/G$.

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That's a great answer, Matt. Thanks a lot –  loren Nov 16 '12 at 21:48

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