# Sheaf (Gieseker) compactification of moduli space of vector bundles

I am given to understand that the moduli space $M_k^G$ of $G$ vector bundles with second Chern class $c_2=k$ over an algebraic curve/variety (for me a Riemann surface is enough/projective space for varieties) is a non-compact space. I am familiar with the Uhlenbeck compactification $$\widetilde{M}_k^G = M_{k}^G \cup (M_{k-1}^G \times \mathbb{R}^4) \cup (M_{k-2}^G \times \mathrm{Sym}^2\mathbb{R}^4) \cup \ldots \cup \mathrm{Sym}^k\mathbb{R}^4$$ Apparently, there is another kind of compactification using sheaves. Vector bundles are locally free sheaves so I am trying to understand, in other words, the moduli space of locally free sheaves. Now, if we include more general sheaves, not necessarily locally free, we get the moduli space of coherent (torsion free) sheaves.

Is this statement correct? What is the precise idea behind this compactification, what is its relation to the Uhlenbeck one and is it true that this compact moduli space is now singular? And what if we have a smooth surface, e.g. $\mathbb{P}^n$ or any of the Hirzebruch surfaces?

I would appreciate very much any reference on this "sheaf" compactification.

P.S. I think this is referred to as Gieseker compactification of the moduli space of sheaves but a Google search wont give me much info and Huybrecht's book is way to hard and lengthy.

This is perhaps more of a comment, but it's too long.

I'm coming from the algebraic side of things, but it seems to me that there are several issues here. Be warned that when I say curve I mean a (smooth) algebraic curve/Riemann surface, and when I say surface I mean a (smooth) complex surface. All my vector bundles are algebraic/holomorphic.

1. Don't you need to mention some type of stability (Gieseker or slope, say) to get some reasonable moduli space?
2. On a curve, a torsion-free sheaf is automatically locally free. (Basically by the structure theorem for modules over a PID.) So, in the curve case moduli of torsion-free sheaves = moduli of vector bundles.
3. Moduli of stable bundles are typically not compact, but if you allow (S-equivalence classes of) semi-stable bundles then the resulting spaces are compact. This will typically introduce singularities.
4. Everything with Gieseker semistability is really concerning bundles on a surface or higher dimensional variety. In the curve case, Gieseker semistability reduces to ordinary slope-semistability.
5. You're mentioning that the 2nd Chern class is fixed. Bundles on curves don't have 2nd Chern classes; you would fix the degree(/first Chern class) if you care about curves/Riemann surfaces.

A more gentle book than Huybrechts/Lehn would be Le Potier's "Lectures on Vector Bundles."

• Correction to 2: You need a smooth curve. – user1504 Jun 28 '17 at 10:42
• Yes, I need (semi)stability conditions indeed. The Gieseker moduli space is about (semi)stable sheaves' moduli space, right? Indeed, I know about Le Potier's book but I cannot find it anywhere unfortunately. Finally, I have to admit that indeed I did not know that torsion free sheaves on an alg. curve are automatically free. But, then, what happens in the case of surfaces? – Marion Jun 28 '17 at 11:27
• @user1504: Sorry, smoothness was implicit. – Jack Huizenga Jun 29 '17 at 1:34
• @Marion: My impression is that basically all the subtleties you are talking about only arise once you are talking about surfaces. And then you need to read Le Potier or Huybrechts/Lehn. Yes the Gieseker moduli space is what algebraic geometers just call the moduli space of semistable sheaves. In AG the Uhlenbeck compactification is sometimes called the Donaldson-Uhlenbeck-Yau compactification by slope-semistable sheaves. – Jack Huizenga Jun 29 '17 at 1:36
• FWIW, there's also some interesting phenomena that arise when the curve is nodal (as tends to happen in familes). The geometry tends towards combinatorial complexity rather than obstruction and singularity. – user1504 Jun 29 '17 at 2:53