The n-Category Café

Hello Urs; I just went through and cleaned it up a bit and added a few things…

And actually the seminar is being run by me (also from Berkeley, but not a Teichner student) and two other MPI grad students.

Sounds very 90’s to me.

In re:

abstract argument generally, fine moduli spaces do not exist if the objects to be classified have nontrivial automorphisms. This allows to build families of objects that are locally trivial but globally not and no finite moduli space will be able to represent that.

this argument makes use of the fact that if we have a moduli space, then the preshesaf we started with must actually be a sheaf (with respect to a subcanonical topology that is implicitly assumed, probably). so we know that we should be able to compute it from gluing its local assignments. But locally our locally trivial family looks, well, trivial, so it all looks the same to our sheaf. So it will just try to glue the trivial object to itself, which is not what we actually have.

Of course vector bundles do have automophisms and are patched together from local trivial ones, so what am I missing?

also in what sense do you wnat a `finite’ moduli space?

It appears that nonexistence of automorphisms is a necessary condition for existence of a fine moduli space. What additional condition on a moduli problem in algebraic geometry will make sure that a coarse moduli space is in fact a fine moduli space?

You say that your argument merely proves that the $j$-line is not a fine moduli space for elliptic curves, and not that a fine moduli space does not exist.

The $j$-line is a coarse moduli space for the moduli problem of elliptic curves. Therefore if a fine moduli space exists, it has to be the $j$-line itself. So if the $j$-line is not a fine moduli space, then in fact there is no fine moduli space.

What is the problem with this argument?

You were saying that the problem with non-properness is the lack of integration and poincare duality.

However taking cohomology with compact supports, is there not a statement of poincare duality for noncompact manifolds?

Two easy remarks, and some propaganda:

> It appears that nonexistence of automorphisms is a necessary condition
> for existence of a fine moduli space.

This is not correct (as has already been pointed out). The easiest
example is N, the set of natural numbers, which is a fine moduli space
for the moduli problem of classifying finite sets up to isomorphism.
Yet finite sets (except the empty set and the singleton) have
non-trivial automorphisms.

> You say that your argument merely proves that the $j$-line is not a
> fine moduli space for elliptic curves, and not that a fine moduli
> space does not exist.
>
> The $j$-line is a coarse moduli space for the moduli problem of
> elliptic curves. Therefore if a fine moduli space exists, it has to
> be the $j$-line itself. So if the $j$-line is not a fine moduli
> space, then in fact there is no fine moduli space.
>
> What is the problem with this argument?

This argument is correct, since coarse moduli spaces are characterised
by a universal property, that fine moduli spaces share.

Both arguments can be found (exercise 15 p.19 and §0.2.9, resp.) in

[Kock-Vainsencher, “An invitation to quantum cohomology:
Kontsevich’s formula for rational plane curves”, Birkhäuser, 2006]

that I take the opportunity to advertise as an elementary introduction
to Gromov-Witten theory. (Admittedly this book is a bit 90’ish.)

Cheers,
Joachim.

> For me, a moduli problem is a certain functor from a category into
> Sets. And a fine moduli space exists if this functor is
> representable – and then by Yoneda lemma there will be a universal
> object. So which is the category? Which is the functor? And what
> is the universal object?

Sorry for leaving out the details:

A family of finite sets is by definition a set map E -> B with
finite fibres. An isomorphism of two such is a fibrewise iso.

The moduli functor is F : Set^{op} -> Set sending a set B to
the set of isomorphism classes of families with base B.

Given a family E -> B the classifying map B -> N sends b to
the cardinality of E_b. Conversely, given a map B -> N, you
construct a family E -> B by pullback of the universal family
(the family N’ -> N whose fibre over n is of cardinality n).
This establishes the bijection F(B) = Hom(B,N).

> There is no fine moduli space for classifying finite sets, but N is a
> fine moduli space for classifying totally ordered finite sets. If you
> try to sheafify the naive finite set functor, you will find that you
> are actually classifying finite degree covering spaces.

Sorry again for the confusion. There is no contradiction: the
topology on Set is just the epi topology, so finite degree covering
‘space’ is the same thing as family of finite sets.

The classical notion of fine moduli space in algebraic geometry is
about Set-valued functors. It does not follow from being a fine
moduli space in this sense that it is also represents the
corresponding groupoid-valued functor.

I think the moral of the finite-set example is that the classical
notion of fine moduli space in algebraic geometry does not correspond
exactly to what a category theorist think of: a category theorist may
tend to rather think of a fine moduli space as existing when the
grupoid-valued functor is homotopy equivalent to the set-valued
functor (pi_0 of the groupoid-valued functor), which is amounts
precisely to saying ‘no non-trivial automorphisms’. Algebraic
geometers are responsible for confusing category theorists like this,
since they often write in their papers that ‘automorphisms prevent
fine moduli spaces from existing’. So what I am pointing out is,
first of all that that statement is not true, and second, that you
might as well forget about formulating moduli problems in terms of
set-valued functors. Just stick to groupoid-valued (or
higher-groupoid-valued), which is most likely what readers of this
café do anyway.

Cheers,
Joachim.

I have started to incorporate a discussion of the slogan

Objecs with automorphisms don’t have fine moduli spaces.

and its subtleties into the entry $n$Lab:moduli space in the chaper

… because they have automorphisms.

Am not entirely happy with that yet, but this is what I could do with the given time and energy this morning and it’s a start. Maybe somebody feels like expanding on it.

Feeling emboldened by Schreiber’s kind encouragement, let me again venture into my first question(automorphisms and fine/coarse moduli spaces).

In the n-lab page on Deligne-Mumford, the following appears.

Deligne-Mumford stacks correspond to moduli problems in which the objects being parametrized have finite automorphism groups.

Also, the talk mentioned a few problematic examples in which the automorphism group was infinite.

Therefore, is it true that for a moduli problem in which the stack is Deligne-Mumford, and where there are no automorphisms, existence of a coarse moduli space would imply the existence of a fine moduli space?

I look forward very much to hear more details about the derived
viewpoint on virtual classes!

Meanwhile here are some down-to-earth geometric examples (plane cubics,
and conics on the quintic three-fold) to illustrate the need of the
virtual fundamental class.

Although the moduli space of stable maps is sometimes referred to as a
compactifiaction of the space of maps, in analogy with the
Deligne-Mumford compactification of the moduli space of curves, in
fact it typically has boundary components of higher dimension than the
space it was supposed to compactify!

Take for example Mbar_{1,0}(P^2,3). It ought to be a compactification
of the space of degree-3 maps from genus-1 curves to P^2, and indeed
one of its components has a Zariski open subset birational to the P^9
of all plane cubics. But there is also a ‘boundary component’ of
higher dimension, namely the boundary component consisting of maps
whose domain is a genus-1 curve glued to a nodal rational curve: the
nodal curve maps to a rational cubic in P^2, while the g=1 component
contracts to a point on that nodal cubic. This boundary component has
dimension 10: namely, there are 8 parameters to specify the image
nodal cubic, 1 paramenter to determine the point to which the g=1
component contracts, and finally there is 1 paramenter for the
j-invariant for the g=1 component. The topological fundamental class
lives in dimension 10 so it is rather useless to integrate against if
all your cohomology classes are codimension 9 — which is the
expected dimension. The virtual fundamental class always lives in the
expected dimension.

(The expected dimension is often the one you would expect(!) from
naive counts like the above. More formally it can be computed as dim
H^0(C,N_f), where f:C\to P^2 is a moduli point (with normal bundle
N_f) such that H^1(C,N_f)=0 (this is to say that the first order
infinitesimal deformations are unobstructed).)

The situation is analogous (possibly in fact a special case of) the
standard situation in intersection theory when a section of a vector
bundle is not regular: its zero locus is then of too high dimension
and is of little use to intersect against. The correct class to
work with is then the top Chern class of the vector bundle (cf.
[Fulton] ch.14), which could be called the virtual class of the zero
locus.

In the example above, I don’t know right now if the virtual class in
fact appears as a top Chern class of a vector bundle — I think it
should, because the excess is just a variation of the standard example
Mbar_{1,1}(X,0) mentioned by A.J., and in that example it is true that
the virtual class appears as a top Chern class: there is a so-called
obstruction bundle which in this case is the dual of the Hodge bundle
from the factor Mbar_{1,1} tensored with the tangent bundle from X.
(The Hodge bundle is the direct image bundle of the canonical bundle
of the universal curve, hence of rank g, hence just a line bundle in
this case.) The virtual fundamental class is the top Chern class of
the obstruction bundle (cap the topological fundamental class).
In this case, dim Mbar_{1,1}(X,0) = 1 + dim X, and the obstruction
bundle has rank dim X, hence the virtual class has dimension 1.

Perhaps it should be mentioned also that the moduli space of maps can
have components of too high dimension even before it is
‘compactified’, and even without involving contracting curves. A
famous example is M_{0,0}(Q,d) (no bar needed for this argument) where
Q is a quintic three-fold. Let’s say d=2, so we are talking about
conics on the quintic three-fold. Since Q has trivial canonical class
it follows that the expected dimension is always 0 (i.e. in every
degree there ought to be a finite number of rational curves on Q).
But now, M_{0,0}(Q,2) is a space of maps, not a space of curves, and
for every one of the famous 2875 lines on Q there is a 2-dimensional
family of double covers of the line, which clearly count as stable
degree-2 maps, so M_{0,0}(Q,2) contains 2875 components of dimension
2, in contrast to the virtual dimension 0.

Cheers,
Joachim.

After the general overview last time, here more introductory details on elliptic curves and their moduli spaces/stacks:

A Seminar on a Survey of Elliptic Cohomology – elliptic curves

I am dreaming that some kind soul will at some point enjoy going through this and increasing his or her $n$Lab page number by creating separate entries for the keywords appearing and copy-and-pasting the material there.

That’s anyway what I think the purpose of these typed up notes should be: raw material for further entries.