What is the official definition of $\mathcal{M}_g$ as an orbifold, and how much can I ignore it?

Question

There is a well-known description of $\mathcal{M}_g$ as $\mathcal{T}_g/\Gamma$ where $\mathcal{T}_g$ is the Teichmuller space and $\Gamma$ is the mapping class group. Teichmuller space is homeomorphic to a ball. But the quotient here is not exactly a manifold because $\Gamma$ has fixed points.

Let's just consider the case $g = 1$, where $\mathcal{T}_g = \mathbf{H}$ and $\Gamma = \text{PSL}(2,\mathbf{Z})$. The quotient $\mathbf{H}/\text{PSL}(2,\mathbf{Z})$ has two orbifold points.

Most talks I've seen about this topic ignore the orbifold behavior and pretend that $\mathcal{M}_g$ is a manifold. It seems the general sentiment is that this does not hurt us for most things we are interested in. But I would like to actually understand what the space really is at these points.

So my general question is:

What is the official definition of an orbifold? What is, for instance, a vector bundle (such as the Hodge bundle) on an orbifold?

Is there a general reference somewhere that tells us what kinds of things about manifolds can be imported freely over to orbifolds, and which things require more delicate analysis?

Explicit descriptions in the case of the modular curve above would be very welcome.

Answer 1

Since you seem to be mainly interested in $\mathcal{M_g}$, let me suggest a "quick and dirty" approach based on the following fact:

$\mathcal{M}_g$ is quotient of a nonsingular algebraic variety $\tilde M_g$(and particular complex manifold) by a finite group $G$

Idea: Take $\Gamma(3)\subset \Gamma$ to be the preimage of congruence group $\{M\in Sp_{2g}(\mathbb{Z})\mid M\equiv I \mod 3\}$ under the "Torelli" homomorphism $\Gamma\to Sp_{2g}(\mathbb{Z})$. $\Gamma(3)$ can be shown to act freely on Teichmuller space, so the quotient $\tilde M_g$ is nonsingular with action by $G=Sp_{2g}(\mathbb{Z}/3)$, such that $\mathcal{M}_g$ is the quotient. (It helps to think about the $g=1$ case first.)

This makes a lot of things fairly straightforward. For example, a vector bundle on $\mathcal{M}_g$ can be understood to mean a $G$-equivariant vector bundle of $\tilde M_g$ etc.. [Added Since $\tilde M_g$ is a so called fine moduli space, it has a universal family of curves. To this we can associate a Hodge bundle in the usual way. Since it's naturally $G$-equivariant, it yields the Hodge bundle of $\mathcal{M}_g$.] More general orbifolds are only locally quotients, which makes the foundations more complicated, as people have already explained.

Answer 2

Since in the comments you seem drawn to the "charts" approach to orbifolds, here are some resources which give definitions and more (there is no "official definition" as far as I know):

1. Joan Porti has some nice slides on orbifolds.
2. Here is a link to Thurston's notes on the topic.
3. Here is a link to a book by Adem, Leida, Ruan which discusses, among many other things, orbifold vector bundles.
4. Here is a link to the first paper defining orbifolds ("V-manifolds") by Sakake: On a generalization of a notion of manifold (1956).

With regard to your specific query about moduli space, you can read all about it in A Primer on Mapping Class Groups. For example, they prove:

"For $g \ge 1$, the space $\mathcal{M}_g$ is an aspherical orbifold and is finitely covered by an aspherical manifold."

I would also recommend reading some answers to past MO questions about orbifolds, with regard to your question "how much can you ignore it":

1. Orbifold fundamental group in terms of loops?
2. What is meant by smooth orbifold?
3. How should one understand orbifold fundamental groups?
4. What tools cannot work for orbifolds?

Lastly, Suhyoung Choi has a paper Geometric structures on Orbifolds and Holonomy Representations, which utilizes Thurston's point-of-view (which might interest you).

Answer 3

Not being a moduli space expert, I will give an answer to the more general question. To me an orbifold is a way to recover the action of a group $G$ on a space $X$ in terms of the quotient object $G\backslash X$.

In the nicest possible case, $G$ acts freely, properly, and cocompactly on $X$. From the quotient $G\backslash X$ we can recover $G = \pi_1(G\backslash X)$ and recover the action of $G$ on $X$ via the deck transformations of $\pi_1(G\backslash X)$ on $X$.

When the the action is no longer free, but the action is still nice, e.g. the action of the modular group on Teichmuller space, then information is lost in the quotient $G\backslash X$. An orbifold structure on $G\backslash X$ is the extra structure required to recover the action of $G$ on $X$. Every point of $G\backslash X$ is either regular if its lifts in $X$ have trivial stabilizers, or singular if its lifts have non-trivial stabilizers. The orbifold structure on $G\backslash X$ is the following extra information: at each singular point $x$ we record the action of the stabilizer of a lift $G_{\tilde x}$ on a small neighborhood $\tilde x \in \tilde U \subset X$. This group $G_{\tilde x}$ is called the isotropy group of $x$.

This extra local information will be enough to patch together all the local actions in order to recover the orbifold universal cover $X \twoheadrightarrow G\backslash X$. You asked about what can be ignored. Here is a list of increasing detail:

1. Simply the quotient $G\backslash X$.
2. $G\backslash X$ where we record the order of the isotropy groups of each singular point.
3. $G\backslash X$ where we record the isomorphism type of each isotropy group of each singular point.
4. The full orbifold: where we record the how the isotropy groups act on small neighbourhoods in $X$.

In my experience, sometimes just level 2 of detail is sufficient depending on what I need to prove.

In the case of $\mathcal M_g$ the points represent shapes of the underlying surface (with fat and skinny bits.) By Nielsen realization all finite groups fix points, so in the full orbifold quotient the isotropy groups represent symmetries of a given shape. For example if your surface is very "lumpy", then it shouldn't have any symmetries so the shape should represent a regular point in $\mathcal M_g$. On the other hand a very symmetric shape would correspond to a singular point in $\mathcal M_g$ because you can flip it, reflect it, etc and the symmetry group will be represented by the isotropy group.