The good moduli space people seem to be getting closer to the "soft GIT dream". Let's egg them on!

GIT is a funny beast: when you first learn about it, you are enthralled. Then you are disgusted. Why, you ask, should I give a crap about linearized actions of reductive groups when all I want to do is enjoy this moduli problem? It gets worse when you learn about stacks and realize that a lot of classical arguments that get tied into all sorts of groupical junk when chained to GIT have perfectly understandable explanations living purely on the moduli stack. And yet GIT does something totally amazing: it gives you a quasi-projective scheme that corepresents the stack in the category of algebraic spaces. Sometimes these schemes have nicer properties than they might deserve -- witness the fact that the GIT space of semistable vector bundles on a curve is locally factorial (but only at strictly semistable points -- elsewhere it's smooth, and the stack is always smooth everywhere). Only a fool would throw all of this love away. One might even try to figure out a way to characterize this love, purely in terms of the map from the stack to the scheme.

I played with this idea briefly (for approximately 60 minutes one day) in graduate school. Call a stack "reductive" if no quasi-coherent sheaf has higher cohomology. This implies a lot of stuff (e.g., closed substacks are determined by the global functions that vanish along them). A stupid example is given by taking the stack quotient of a reductive group acting on an affine scheme; by the definition of GIT, such things cover any GIT quotient stack (and their rings of invariants give an affine open cover of the GIT quotient). You start to wonder if you can corepresent a reductive stack by the Spec of its ring of global functions, etc., etc.

Luckily for me, a little while after I stopped "thinking" about this Jarod Alper (independently) worked out a lot of things in this direction in much greater depth (starting with his thesis) and arrived at a concept of "good moduli space", which (roughly speaking) is a map X \to M from a stack to a space that behaves like an orbit space and for which no quasi-coherent sheaf has higher direct images (so X is "relatively reductive" over M). GIT quotients are the canonical example of this. The constant question since this theory was first developed was how much of the GIT picture one can recreate from this more intrinsic notion. Related to this: is every stack with a good moduli space, if not itself a quotient, etale-locally a quotient? (You might view this as good or bad, depending upon your inner child.) This is vacuous for DM stacks, but it is rather interesting for Artin stacks in general. (It is vaguely similar to something you can find in various places -- e.g., Vistoli's thesis -- about DM stacks being finite group quotients etale-locally over their coarse moduli spaces.)

A sticking point in this last question has always been whether a GAGA-like statement holds. How would one fantasize about making a local quotient structure? Fix a point of the good moduli space and consider the closed residual gerbe Z of the stack above it. Let's work with things of finite type over an algebraically closed field and assume that we're near a closed point, so that this closed residual gerbe is just BG for a reductive G. (G is reductive because no G-representation has cohomology, by our assumption!) Now the point mapping to BG is a G-torsor, and the coho assumption implies that this torsor admits infinitesimal deformations to arbitrary order around Z. If only we could algebraize it, we would then be able to use the standard finite presentation trick to puff it out to a G-torsor etale-locally, i.e., we would get an etale-local quotient-stack structure! BUT HOW DO WE ALGEBRAIZE TORSORS OVER HIGHLY NON-SEPARATED "THINGIES"?!

This paper shows, very roughly, that being able to do this is basically equivalent to the resolution property for the stack pulled back to the complete local ring of the good moduli space at the point in question. Of course, the resolution property is highly non-trivial, but one might think it is some kind of demented replacement for the original quasi-projective linearization, except that it is more local. (It is also familiar from classical GAGA, as one uses resolutions by sums of O(n) as a starting point.) Food for thought!

This is far from the only question one can ask about these spaces. Look at the collected works of Alper and his coauthors for lots more.
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