Let $G$ be a group, say finitely presented as $\langle x\_1,\ldots,x\_k|r\_1,\ldots,r\_\ell\rangle$. Fix $n\geq 1$ a natural number. Then there exists a scheme $V\_G(n)$ contained in $GL(n)^k$ given by the relations. This scheme parameterizes $n$ dimensional representations of $G$.


Now, I've known this scheme since I first started learning algebraic geometry (one of the first examples shown to me of an algebraic set was $V\_{S\_3}(2)$) but I've never found a good reference for this. So my first question is:



```
Is there a good reference for the geometry of schemes of representations?

```

Now, I have some much more specific questions. The main one being a point I'd been wondering about idly and tangentially since reading about some open problems related to the Calogero-Moser Integrable System:


Are there natural conditions on $G$ that will guarantee that $V\_G(n)$ be smooth? Reduced? Now, this is on the affine variety, I know that the projective closure will generally be singular, but in the case of $V\_{S\_3}(2)$, I know that the affine variety defined above is actually smooth, of four irreducible components.


Finally, for any $G$ and $n$, we have $V\_G(n)\subset V\_G(n+1)$ (By taking the subscheme where the extra row and column are zeros, except on the diagonal, where it is 1). We can take the limit and get an ind-scheme, $V\_G$. What is the relationship between $V\_G$ and the category $Rep(G)$? Can the latter be realized as a category of sheaves on the former? I know nothing here, and as I said, most of these questions are the result of idle speculation while reading about something else.


Edit: It occurs to me that as defined, $V\_G(n)$ and $V\_G$ may not be invariants of $G$, but really of the presentation. So two things to add: one, $V\_G(n)$ is intended to really be the scheme $Hom(G,GL(n))$ (there's some issues I want to sweep under the rug with finitely generated infinite groups here, which is part of why I was thinking in presentations), and second, the situations that I'm thinking of are often the data of group with a presentation, so for that situation, $V\_G(n)$ as defined should be good enough.



Schemes of Representations of Groups
Charlie, as Dmitri pointed out there is a big difference between compact Kaehler and non-Kaehler manifolds as far as the structure of the representation varieties of their fundamental groups are concerned. 


By the way, by a theorem of Taubes, every finitely presentable group is the fundamental of a compact complex three dimensional complex manifold so you don't really get any restriction by saying that you want your group to be the fundamental group of a complex manifold. Taubes' manifolds however are constructed as twistor spaces of anti-self dual real 4-manifolds and are never Kaehler. The condition that your group can be realized as the fundamental group of a compact Kaehler manifold is a serious condition and puts many constraints on the representation variety. 


Another comment is that the representation scheme does not depend on the presentation of your group. You only need to know that the group is finitely generated.


There is a huge literature on this subject. I will list just a few of the landmarks:


1. A very nice classical source is the Lubotzky-Magid book ["Varieties of representations of finitely generated groups"](http://books.google.com/books?id=3UC5EKcYyjIC&dq=Lubotzky,+Magid&printsec=frontcover&source=bl&ots=wama5qwKgo&sig=gaPTJVGlEoY_IWqw3WigW3YylXI&hl=en&ei=7e87S-fUL9XNlAe5yrWaBw&sa=X&oi=book_result&ct=result&resnum=2&ved=0CAoQ6AEwAQ#v=onepage&q=&f=false)
2. The paper of Goldman-Milson that Dmitri mentioned is a must-read.
3. There are two fundamental papers of Simpson that I mentioned in this [MO post](https://mathoverflow.net/questions/7212/moduli-space-of-flat-bundles/7221#7221).
4. On the topic of Kaehler fundamental groups you can start with this [book](http://www.ams.org/cgi-bin/bookstore/booksearch?pg1=&s1=&op1=ADJ&co1=AND&pg2=&s2=&op2=ADJ&co2=AND&pg3=CN&op3=ADJ&s3=amoros&co3=AND&pg4=CN&op4=ADJ&s4=burger&co4=AND&pg5=AS&op5=ADJ&s5=&co5=AND&pg6=CMSC&s6=&op6=or&co6=AND&pg7=%40PYR&arg1=eq&s7=&fn=100&d=BOOK&p=1&u=&r=0&l=100&f=S) by Amoros et al. and with this [article](http://www.google.com/url?sa=t&source=web&ct=res&cd=1&ved=0CAcQFjAA&url=http%3A%2F%2Fwww.msri.org%2Fpublications%2Fbooks%2FBook28%2Ffiles%2Farapura.pdf&ei=BPQ7S-KQH8fVlAf0s62SBw&usg=AFQjCNGLzeHE9DBEPc3rssgguUacavDbBw) by Arapura.


