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So sometime ago in my math education I discovered that many mathematicians were interested in moduli problems. Not long after I got the sense that when mathematicians ran across a non compact moduli they would really like to compactify it.

My question is, why are people so eager to compactify things? I know compactness is a great property of a space to have because it often makes other results much easier to prove. So I think my question is better stated as: what are some examples of nice/good/cool results related to a moduli spaces that were (only) able to be proved once there was a compactification of the space?

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6 Answers 6

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The answers here are all excellent examples of things that can only be proved once a moduli space is compactified. I would like to add a perhaps more basic reason for compactifying moduli spaces, involving something simpler than theoretical applications such as defining enumerative invariants. The moral is the following:

If you study families of geometric objects then either you are almost certain to encounter the boundary of the moduli space, or you must have some very good reason to rule it out.

For example, to find a non-trivial compact family of smooth complex curves is actually quite awkward and such families are very rare. (The first examples were due to Atiyah and Kodaira.) From this point of view the "ubiquity of the compactification" amounts to the fact that the boundary divisor of singular curves in the compactified moduli space is positive in a certain sense, so it intersects almost all curves in the moduli space. It is this positivity of the boundary which forces us to study it!

Some more examples explain - I hope! - the way compactification enters when considering pseudoholomorphic curves as in Gromov-Witten theory, without ever coming close to trying to define an enumerative invariant. Just by looking at a conic in $\mathbb{CP}^2$, which degenerates into two lines, one sees that when moving a pseudoholomorphic curve around, one is almost certain to encounter bubbling, unless one has a very good reason to know otherwise. Understanding how to compactify the moduli space, we see that this bubbling phenomenon is the main thing which can go wrong. What is interesting here is that often one tries to prove this compactification is not actually necessary, by ruling out bubbling somehow. Two examples follow - taken from Gromov's original use of pseudoholomorphic curves in his Inventiones paper - which exploit this idea.

Firstly, Gromov's proof of his non-squeezing theorem. Here the key point in the argument is that one can find a certain pseudoholomorphic disc for a standard almost complex structure on $\mathbb{C}^n$. One would like to know that as one deforms the almost complex structure the disc persists so that we have such a disc for a special non-standard almost complex structure. It is standard in this kind of "continuity method" that you can always deform the disc for a little while because the problem is elliptic. But to push the deformation indefinitely you need to show compactness - why doesn't the disc break up? Thanks to our knowledge of the compactification of the moduli space, we understand that the only thing that can go wrong is bubbling and in this case bubbles cannot form because the symplectic structure is exact.

The second example is of the following type: suppose one knows the existence of one pseudoholomorphic curve in a symplectic manifold; then one can try and use it to investigate the ambient space, moving it around and trying to sweep out as much of the space as possible. In this way you can prove, for example, that any symplectic structure on $\mathbb{CP}^2$ which admits a symplectic sphere with self-intersection 1 must be the standard symplectic structure. The reason is you can find an almost complex structure which makes this sphere a pseudoholomorphic curve. Then you move the curve around until is sweeps out the whole space, doing it carefully enough to give a symplectomorphism with the standard $\mathbb{CP}^2$. Here you can push the curve wherever you want because it wont break. Bubbles can't form because the curve has symplectic area 1 and so there is no "spare area" to make bubbles with.

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There are several reasons for compactifying of a space in general. But there are two good reasons in my mind:

To do intersection theory on a space it is good to work with a compact space. For instance, the projective line is a natural compactification of the affine line. you loose the information about a point on the affine line via homology: you can move that point to infinity on the projective line, the result on the affine line is the disappearance of the point. So, there is not a well-defined notion of degree for divisors on the affine line. This generalizes easily to higher dimensional cases. We can't prove the Bezout theorem on non-compact spaces since it is not true.

The second reason is more about moduli spaces:

Points on a moduli space correspond to some objects in a category $C$ which are being parametrized by the moduli space. When there is a notion of continuos change of objects of $C$ we get a topology on the moduli space. The moduli space is not compact when it doesn't contain some limits of families of the objects of $C$. So, a compactification should be considered as adding a class of natural limits of the objects of $C$. We get different compactification if we give different interpretation of points on the moduli space. For example $M_{0,5}$, the moduli space of smooth pointed curves of genus zero with 5 points is an open subset of $\mathbb{P}^1 \times \mathbb{P}^1$. Its Deligne-Mumford compactification $\overline{M}_{0,5}$, which is $\mathbb{P}^1 \times \mathbb{P}^1$ blown-up at three points is not just $\mathbb{P}^1 \times \mathbb{P}^1$.

The second space doesn't give a flat family of stable curves of genus zero with 5 disjoint sections in the smooth locus of the fibers as an extension of the universal family of curves over $M_{0,5}$.

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We can often cast enumerative geometry questions in terms of intersection theory in some moduli space. To make this work, one needs to have something like a fundamental class for the moduli space. To get something like a fundamental class one usually tries to find a nice (geometrically meaningful) compactification. Then, pairing against the fundamental class is computed by integrating over the whole space - and there you see why it is important/useful to have a compact space.

A nice elementary example of the benefits of compactifying is provided by Bezout's theorem on counting the intersections of curves in $P^2$. It says that the number of intersection points (counted with multiplicity) is equal to the product of the degrees. If you instead try to work with curves in affine space $A^2$ then it is more complicated to count the intersection points. In this example, I am thinking of $A^2 \subset P^2$ as a simple example of compactifying a moduli space.

I don't know the details of this one, but I recall that the irreducibility of the moduli space of curves in any characteristic was proven by Deligne and Mumford via the introduction of the famous Deligne-Mumford compactification.

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Take Seiberg-Witten theory as an example: There you consider the solution space of some differential equation (motivated by physics) on a 4-manifold modulo gauge transformations. It can be shown that this quotient space is indeed a compact, orientable smooth manifold if the equations are perturbed properly. The dimension of this space can be calculated from the indices of the operators involved and taking its fundamental class (which only exists because the space is compact!) yields a diffeomorphism invariant of 4-manifolds, which is kind of a cool thing to have.

Compare this to Donaldson theory, which has the same goals, but here it is much harder to construct the invariants, precisely because you need to compactify the moduli space, which is a difficult task.

If you are interested in this, I recommend (more or less) the book by Nicolaescu "Notes on Seiberg-Witten Theory".

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Another great book on Seiberg-Witten is John Morgan's book "The Seiberg Witten equations and applications to the topology of smooth four-manifolds". It's far shorter than Nicolaescu's book, which brings of course different advantages and disadvantages. –  Joel Fine Apr 21 '10 at 11:26

Thurston's compactification of Teichmueller space gives an elegant proof of the Nielsen--Thurston classification of surface automorphisms. The compactified Teichmueller space of the surface of genus g is homeomorphic to a ((6g-6)-dimensional) ball, on which an automorphism $\phi$ acts. By Brouwer's Fixed Point Theorem, $\phi$ fixes a point $x$.

  • If $x$ is an interior point then $\phi$ is an isometry of some hyperbolic structure, and it's easy to deduce that $\phi$ is periodic.

  • If $x$ is a point on the boundary then it corresponds to some measured lamination. If the lamination has a closed leaf then $x$ is reducible.

  • Finally, if the lamination has no closed leaves then $\phi$ is pseudo-Anosov.

Admittedly, it is possible to give a proof of the classification that doesn't use the compactification. But why would you want to?

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Well, Bers's proof of Thurston's theorem doesn't exactly use the compactification (though it certainly is implicit), and I personally prefer Bers's approach. But I agree with your larger point that understanding Thurston's compactification of Teichmuller space is the key to understanding many phenomena in low-dimensional geometry/topology. –  Andy Putman Apr 21 '10 at 1:51

Where invariants are constructed as integrals over a moduli space (as in Gromov-Witten theory) the compactness is needed to know that the answer will be well defined.

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I have often wondered, though, whether it is possible to have a "Gromov-Witten theory" for, say, integrals of compactly supported things over the uncompactified moduli space(s). –  Kevin H. Lin Apr 20 '10 at 17:08
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@Kevin. In GW theory, integrating a top degree cohom class is of course counting the number of curves which satisfy certain properties. For a particular choice of almost complex structure, these points may well be located away from the boundary - your class is represented by a compactly supported form. But the problem is that for arbitrary deformations of the almost complex structure the points will inevitably end up on the boundary. Demanding a representative exist with compact support isn't a property which is independent of the choice of almost complex structure. –  Joel Fine Apr 21 '10 at 9:32

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