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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.

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3 Answers 3

up vote 7 down vote accepted

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"

  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.

  4. On the topic of Kaehler fundamental groups you can start with this book by Amoros et al. and with this article by Arapura.

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Ahh, I've got those Simpson papers in my pile for the stuff on Higgs bundles on higher dimensional varieties. Just moved them up my list, and will definitely be checking out the others. –  Charles Siegel Dec 31 '09 at 2:57

This is a collection of remarks about your question. I will treat the question as a question about repersentations in GL(n,R) or GL(n,C).

Then first (simple) remak is that for every finite group the repesenation vairety is smooth. I would not be able to give some other non-trivial (infinite) examples apart from free group and patological examples, say when your group is an infinite simple group, so it does not have representations at all (such group will be non-linear), or combinations of whose.

Second, there is an extencive theory of repersentation variteties of fundamental groups of Kahler manifolds. One refference is Goldman and Milson The deformation theory of representations of fundamental groups of compact Kahler manifolds.

67/PMIHES_1988_67_43_0/PMIHES_1988_67_43_0.pdf">http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1988_67/PMIHES_1988_67_43_0/PMIHES_1988_67_43_0.pdf

If you track this article on mathscinet, you will get a lot relevant literature. They show in particular that singularites of such varieties are quadratic (theorem 1). Though I did not read this paper, and don't know how readable it is.

It you conisder a simplest infinite group, say fundamental group of a genus $g>1$ surface, its repersentation variety will be singular at the trivial representation (it seems to me this will be quite a common situation) but it will be non-sigular at all points that correspond to irreducible repersentations.

Finally, there are some examples of "terrbile" non-quadratic sinuglarites, for example for repesentation varieties of fundametnal groups of three-dimensional hyperbolic manfiolds (again at the trival repesentation). This is contained in an article of Ghys (cool aricle but in French:) Notice that that 3-dimesnional hyperbolic group is a fundamental group of a Complex but Non-Kahler manifold (this is the main point of the article of Ghys).

Déformations des structures complexes sur les espaces homogènes de SL(2,C). (Reine Angew. Math. 468 (1995), 113-138

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This is nothing like a complete answer, but some kind of extended comment:

Firstly, it is usual to quotient out by the conjugation action of $GL(n)$ (so that one is really parameterizing isomorphism classes of representations).

Secondly, if $\rho$ is some fixed representation, giving a point of $V_G(n)$, and $End(\rho)$ is the space of $k$-linear endomorphisms of $\rho$, with its natural $G$-action, then $H^1(G,End(\rho))$ computes the tangent space of $V_G(n)$ at $\rho$, and the obstructions to deformations lie in $H^2(G,End(\rho))$. So if the latter space vanishes, then $V_G(n)$ is smooth at $\rho$.

As for references, in the case when $G$ is a Galois group (which gives a slightly different, but closely related framework, to the one of your question), much can be found in Mazur's original article on deformations of Galois representations: ``Deforming Galois representations''. Of course, there is a huge subsequent literature.

Topologists frequently consider representation varieties in the case when $G$ is a fundamental group. There is a big literature in this case as well.

EDIT: If $G$ is finite, then it has only finitely many reps. of a given dimension, and the representation variety will be a finite set of points.

I should add that quotienting out by conjugation, or not, doesn't make too much difference. If you don't take this quotient then you are just considering representations with a fixed basis, and including this basis doesn't really give additional information about the representation. (In some contexts, for example working near a point $\rho$ that is reducible, there are technical advantages to keeping this information, but only for the purposes of avoiding working with stacks.)

To find more literature, I would google ``representation varieties''.

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I'm actually explicitly not quotienting by conjugation above (my intuition, that I haven't checked, is that that should collapse things a lot, down to a point for each irrep of dim $\leq n$ and then a point for each direct sum of irreps, at least in the case of finite groups.) Now, the case of $\pi_1(X)$ for $X$ a smooth complex variety is actually the case I'm thinking about, can you give a few references for this case? You seem to know a bit more than I do about it. –  Charles Siegel Dec 30 '09 at 23:19

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