quotients of smooth projective varieties by finite groups 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 !
 A: 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 $V/G$.
