# Algebraic and Analytic Compactifications of Moduli Spaces

Communicated by *Notices* Associate Editor Steven Sam

## The Tour Ahead

The basic objects of algebraic geometry, such as subvarieties of a projective space, are defined by polynomial equations. The seemingly innocuous observation that one can vary the coefficients of these equations leads at once to unexpectedly deep questions:

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When are objects with distinct coefficients equivalent?

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What types of geometric objects appear if those coefficients move “towards infinity”?

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Can we make sense of “equivalence classes at infinity”?

Searching for answers leads to the discovery of *moduli spaces* and their compactifications, parametrizing equivalence classes of said objects.

The construction of compact moduli spaces and the study of their geometry amounts nowadays to a busy and central neighborhood of algebraic geometry. Any vibrant district in an old city, of course, has too many landmarks to visit, and the first job of a tour guide is to curate a selection of sites and routes — including multiple routes to the same site for the different perspectives they afford. Our tour today has three main stops: elliptic curves, Picard curves (together with “points on a line,” their alter ego), and a brief panoramic glimpse of the general theory.

As for the routes, we first approach the elliptic curve example along the unswerving path that compactifies algebro-geometric moduli spaces with “limiting” algebro-geometric invariants. The way is straight, but entails scaling a brick wall to discover what is meant by “limits.” Our subsequent turn down 19th-century vennels will unveil a connection as old as algebraic geometry itself: associated to our algebraic varieties are analytic *periods*, producing a *period map* from our moduli space to a classifying space for periods. While lacking the ideological consistency of the former route, the “limits” of this one are more conceptually straightforward — with Calculus providing a door in the wall.

For the Picard curve example, we reverse the order of these two approaches; this example is important because it is the simplest one where there are *multiple* natural compactifications of both sorts. Both examples have two very nice features, besides involving objects one can draw on paper. First, the period map is close to being an isomorphism and is inverted by *modular forms*, an observation going back in the elliptic case to work of Weierstraß. Second, the period map extends to isomorphisms of the various algebraic and analytic compactifications.

These examples will illustrate techniques and methods in moduli theory, preparing the stage for our last stop, high above the city. From there we shall be able to see a vague outline of the modern definition of moduli spaces, as well as various algebro-geometric (GIT, KSBA, K-stable) and Hodge-theoretic (Baily-Borel, toroidal, etc.) compactifications of the same moduli space. The aim of our brief journey is to travel towards understanding their differences — and especially their spectacular coincidences.

## 1. Elliptic Curves

With a rich history going back to Abel, Jacobi, and Weierstraß in their guise as complex 1-tori (think of the surface of a donut), elliptic curves are central objects in many areas of mathematics, from cryptography to complex analysis. At this first stop on our tour, we’ll use their moduli space to illustrate constructions such as geometric quotients and the period map (and its inversion).

### 1.1. Algebraic perspective

Our starting point is the fact, first hinted at by Jacobi in 1834, that any complex 1-torus can be realized as a *smooth plane cubic* — that is, an algebraic curve defined as the zero-locus in of a homogeneous polynomial of degree three in 3 variables

with certain conditions on the coefficients to guarantee the smoothness of Conversely, for an algebraic geometer, it is natural to approach the set of elliptic curves via a suitable quotient of the set of all such cubics. .

*Without* the smoothness requirement, the coefficients of such equations comprise all ordered 10-tuples of complex numbers not all zero, defined up to scaling (by , Since an equation ). is determined uniquely by its coefficients up to scaling, we conclude that the set of *all* plane cubics can be identified with the set of complex points of the projective space (Outside the open set parametrizing smooth cubics, there are . different flavors of singular cubics as displayed in Figure 1.)

The fact that the set of all plane cubics is itself a complex algebraic variety is not a coincidence! Instead, it is our first encounter with one of the most important objects in moduli theory: *the Hilbert scheme*. Indeed, to keep track of the complex solutions of polynomial equations within projective space, we need to fix an invariant known as the Hilbert polynomial. This polynomial records geometric information about our solutions such as their dimension and degree. It was shown in 1961 by Grothendieck that there exists an algebro-geometric space (that is, a projective scheme) which parametrizes all the closed complex solutions of polynomial equations in with Hilbert polynomial In our particular example, all plane cubics have Hilbert polynomial equal to . and the associated Hilbert scheme is , Every point in . corresponds to a curve with this Hilbert polynomial and vice versa.

At this juncture the reader will point out that is certainly not the sought-for “moduli space of elliptic curves,” because it includes singular cubics. But the open subset parametrizing smooth cubics is not the solution either. The reason is that given an elliptic curve defined by the equation we can use an invertible linear change of coordinates , with to obtain another equation , The elliptic curve defined by this second equation is isomorphic as a complex variety to the first one, and yet the Hilbert scheme tells us that they are different objects. .

The critical observation here is that if we are parametrizing varieties with a fixed Hilbert polynomial, then we want to account for the automorphisms of the ambient projective space. In our example the ambient space of elliptic curves is and the group associated to linear automorphisms has dimension 8. Therefore, of the 9 degrees of freedom associated with the coefficients of the equation only ,*one* is intrinsic to the geometry of the elliptic curve, while the other eight are related to linear changes of coordinates in the ambient space.

The relation between elliptic curves and is tight; two plane cubics -orbits and represent isomorphic elliptic curves (algebraically *or* complex analytically) if and only if for some Therefore, the set of elliptic curves up to isomorphism can be identified with . of smooth plane cubics. Although the set of such orbits exists as a topological space, it is not at all obvious that this space is itself a complex variety. We arrive at one of the most delicate problems in algebraic geometry: -orbits*Given the action of a linear group on a variety does there exist an algebro-geometric space parametrizing the , in -orbits ?*

The correct framework for constructing quotients within algebraic geometry is given by *Geometric Invariant Theory (GIT)*, initiated by Mumford in 1969 17. One of the key results from GIT is the existence of an open locus called the stable locus and a well-defined geometric quotient, which in our case is

One of the first exercises in GIT is to show that is the locus parametrizing *smooth* plane cubics, see 17, Example 7.12. The fact that the quotient is geometric means that every point of corresponds to a unique of a smooth cubic. Therefore, we arrive at our first (and almost correct!) example of a moduli space: -orbit is the “moduli space” of elliptic curves up to isomorphism.

We also arrive at the crux of our problem: is a non-compact variety. Is there a natural compact (projective) variety that contains it and that parametrizes a larger class of algebraic varieties? It will be tempting to consider a naive quotient of by for constructing a compactification of However, these naive quotients are usually either of the wrong dimension or yield a non-Hausdorff topological space. Here we use a second key result from GIT: There exists a (usually larger) open set . called the semistable locus, containing , and admitting a well-defined “categorical quotient.” In our case this is

a complex projective variety compactifying

To understand the geometry of the above compactification, we recall that

The categorical quotient is not the naive one: the points of

### 1.2. Analytic perspective

Turning to a complex-analytic perspective, we recall that an elliptic curve

Up to scale, there is a unique holomorphic form *period ratio*

This is closely related to the classical theory of *modular forms*. Here “modular” refers to the moduli of complex 1-tori

satisfies

with period ratio

The key point is that any

over

While the choice of

For any *cusps*, over which the elliptic fiber degenerates to a cycle of

Now consider an algebraic realization *monodromy group* generated by *all* loops in *period map*

### 1.3. First spectacular coincidence

In Section 1.1, we used plane cubics and GIT techniques to construct a compactification

We arrive then to a natural question: *Are any of the Hodge theoretic compactifications * The answer turns out to be yes! From our previous discussion about the invariant

Moreover, isomorphisms of this kind (that is, between Hodge theoretic and geometric compactifications) also exist among other geometric realizations of the elliptic curves. For instance, by keeping track of the ordered 2-torsion subgroup and the Legendre family, one can show that

## 2. Picard Curves and Points in a Line

As we begin to dig into this second example (did we mention that the tour includes amateur archaeological activities?), we shall unearth several ideas that are central for constructing compact moduli spaces. They include the use of finite covers to associate periods to varieties without periods, the use of limits of periods to refine compactifications, and the first example of “stable pairs.”

### 2.1. Analytic perspective

We begin this time with the complex-analytic point of view. Though ordered collections of

Let

to

and

Fix ^{1}

^{1}

and that for

with

and its

The Picard modular formsFootnote^{2} describing their inverses are none other than the

^{2}

Here