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# Curve-Counting and Mirror Symmetry

Communicated by *Notices* Associate Editor Han-Bom Moon

Curve-counting is a subject that dates back hundreds, even thousands, of years. Broadly speaking, its goal is to answer questions about the number of curves in some ambient space that satisfy prescribed conditions, such as the following:

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How many conics pass through five given points in the plane?

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How many lines pass through four given lines in three-space?

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How many lines lie on the quintic threefold

in ?

The answer to the first of these questions was known to the ancient Greeks: given five (sufficiently general) points in there is exactly one conic that passes through all five of them. The method by which the ancient Greeks would have arrived at this answer is by an explicit construction, given the coordinates of the five points, of the conic that passes through them. ,

The modern perspective on curve-counting is somewhat different. Rather than seeking explicit constructions of the curves being counted—which can be unnecessarily cumbersome if our ultimate goal is simply enumeration—one instead searches for answers that are deformation-invariant: for example, a count of conics through five given points that remains unchanged if the five points are slightly varied. This property not only allows us to answer entire families of questions simultaneously (not just “how many conics pass through *these* five points?” but “how many conics pass through *any* five general points?”), but it also introduces the possibility of answering a difficult question by deforming it to a simpler one.

For instance, suppose one wishes to answer the second question posed at the beginning of the article: how many lines pass through four given lines in If the answer to this question is deformation-invariant, then we can deform our lines until they meet in pairs, so that ? and At this point, a bit of reflection is enough to see that there are exactly two lines passing through all four of our original lines: one that joins . and and one where the plane spanned by , and meets the plane spanned by and .

The deformation-invariance of enumerations like this one was first proposed by Hermann Schubert in the 1870s under the name *Prinzip der Erhaltung der Anzahl*, or *principle of convseration of number* *Bulletin of the AMS* in 1902) as “rigorous foundation of Schubert’s enumerative calculus.”

The solution to Hilbert’s fifteenth problem came in the second half of the twentieth century, with the twin developments of moduli spaces and intersection theory. A moduli space, roughly speaking, is a geometric space (often a variety or manifold) in which each point corresponds to an object of some type being studied. For example, someone wishing to study the number of conics passing through five points in the plane might form a moduli space in which each point corresponds to a plane conic. From this perspective, the conics passing through a given point form a subvariety of and the original enumerative question is reinterpreted as a count of the number of intersection points of the five corresponding subvarieties. The advantage of this reframing is that it allows curve-counting questions to be attacked via the tools of intersection theory, a deep mathematical subject studying the structure of intersections within an ambient variety that was developed (in large part with precisely the application to Hilbert’s fifteenth problem in mind) over many decades in the early twentieth century. ,

Through the lens of intersection theory, one can see more clearly the sense in which curve counts are—or are not—deformation-invariant. First, an intersection theory problem generally only has a deformation-invariant answer when one works over an algebraically closed field (for instance, rather than and within a compact ambient variety. To illustrate the first of these limitations, consider the intersection of the parabola ) and the line in These curves intersect in two points for . but for , the intersection points are only visible if we work not in , but in In fact, so long as we count intersections “with multiplicity,” the parabola . and the line in meet in exactly two points for any choice of .

To see why compactness is necessary, consider the question “in how many points do two lines in intersect?” The answer to this question can change when the lines are deformed, because the lines can become parallel, which effectively means that their point of intersection has “fallen off” the noncompact ambient space To avoid this phenomenon, one should replace . by its compactification in which any two lines indeed meet in a single point—so long as they are not the same line. ,

This brings us to one final issue of deformation-invariance that intersection theory is equipped to solve: can an intersection still be said to be deformation-invariant if the subvarieties are deformed so far that they meet along an entire curve? For instance, is there a sensible way in which to interpret the “number of intersection points” of two identical lines in as so that this number is truly insensitive to deformations of the lines? The answer to this question is “yes,” and it is precisely what the subject of excess intersection theory addresses. ,

Applying these ideas to the context of enumerative geometry led mathematicians, in the late twentieth century, to express curve counts as certain intersection numbers on a moduli space that are now called *Gromov–Witten invariants*. This development allowed the deformation-invariance of curve counts to finally be expressed in a robust and rigorous way, but the work was far from over. In particular, the project of actually computing Gromov–Witten invariants is difficult and ongoing, and moreover, there are other methods of formalizing curve counts (such as Donaldson–Thomas theory) whose relationship to Gromov–Witten theory is not obvious. One breakthrough in the subject came in the 1990s from an unexpected interaction between curve-counting and the theoretical physics of string theory, and in the decades since then, this interdisciplinary connection has continued to yield fruit.

## The Moduli Space of Stable Maps

We begin our journey toward defining Gromov–Witten invariants by fixing an ambient space in which we will count curves. For the reasons mentioned above, we will always assume that is compact and the ground field is for instance, if our goal is to count conics passing through five given points in the plane, “the plane” refers to ; We also fix the degree . of the curves being counted and the number of incidence conditions being imposed, so in the above example, and To put things a bit more precisely, . should be a smooth projective variety and an element of so setting , in our example really means where , is the homology class of a line in What we will count is maps . where , is a curve, and the image of , satisfies the requisite incidence conditions. In order for our count to be finite in general, we must fix one further piece of information: the genus of the source curve In our example of conics through five points, this choice is forced upon us if we want our count to include the embedded irreducible conics in . since the genus-degree formula implies that , for these.

Having fixed the data of , , and , we now define a moduli space in which we will interpret our curve count as an intersection theory problem. As we have seen, we should look for a compact moduli space if we want any hope that our count will be deformation-invariant. Unfortunately, this means that we cannot restrict ourselves to including only maps , for which is a smooth curve, nor for which is an embedding, even if these are the types of maps we really care about; the issue is that these “nice” maps may degenerate to less nice ones.

To produce a compact moduli space, one must allow some degeneracies. This can be done while keeping the singularities of the curves mild; specifically, we will consider nodal curves, which can roughly be viewed as the result of gluing together a collection of smooth curves at finitely many pairs of points, as illustrated in Figure 1. The trade-off for the mildness of these singularities is that the map may become quite degenerate, possibly collapsing entire components to a point. The result is the following key player in our story.

We will abbreviate for now. The last condition in the definition may appear technical—and relies on a definition of “isomorphism” that we have not specified—but, as we will see momentarily, it turns out to be crucial in ensuring that the moduli space is well-behaved from a geometric perspective.

But what is

Thus, in order to give *family* of stable maps over any base

This, in particular, relies on giving

Up to this point, we have been purposefully vague about what we mean by “geometry.” A topologist might hope that

There is a fix to this problem, which is to give *orbifold* (or, in more modern language, a *Deligne–Mumford stack*); very roughly, this is a space that looks locally like the quotient of a manifold by a finite group. In the setting of

## Gromov–Witten Invariants

Now that we have a moduli space, our goal is to use it to count genus-*evaluation maps*

for each

Then

provided this intersection is finite. Inverse images generally preserve codimension, and intersections generally add codimension, so if

A more refined version of 2, which captures its insensitivity to deformations of the subvarieties

We will see shortly that this definition has some serious deficiencies that will need to be repaired, but taking it as a working definition for the moment, one sees that using it requires first of all knowing the dimension of

If

This reasoning leads to the guess that

which equals

3(The interested reader with some background in algebraic geometry is encouraged to verify this computation; the key ingredients are the short exact sequence

and the Riemann–Roch theorem.)

We refer to the quantity 3 as the *virtual dimension* of

The idea, from a deformation-theoretic perspective, is that while

If it happens that

If, however,

A simplified perspective may give a flavor of these ideas: imagine that

With all of this in mind, we now see two problems with our preliminary definition of Gromov–Witten invariants. First, because

The solution to these problems lies in constructing a *virtual fundamental class*, an element

that agrees with the fundamental class in the special situations where the moduli space is smooth of dimension equal to the virtual dimension. It is a hard theorem, first due to Behrend–Fantechi in the algebro-geometric setting

To give a very rough intuition in the simplified vision of

Once the virtual fundamental class is constructed, we are at last ready to give the true definition of Gromov–Witten invariants.

At this point, we have gotten quite far afield of our initial goal of counting curves. Thus, at least two questions are in order. First, do Gromov–Witten invariants agree with a more naïve notion of curve counts, when the latter is possible? And second, how can Gromov–Witten invariants be computed?

The answer to the first question is sometimes—though admittedly rarely—yes. For instance, we have mentioned that the virtual fundamental class is the ordinary fundamental class on

where

This situation is rare, though. In fact, the possibility that

as counting the number of degree-

Instead, we will simply content ourselves with attempting to compute Gromov–Witten invariants, out of the philosophy that this is a worthwhile goal even when we are not in the rare situations when the invariants are enumerative. In particular, Gromov–Witten invariants have beautiful structure that is worth studying in its own right; we will see one example of this at the end of the article when we discuss mirror symmetry. Various other connections to theoretical physics as well as to more classical algebro-geometric subjects like the moduli space of curves have motivated mathematicians to study Gromov–Witten theory. Thus, as often happens in mathematics, our initial goal (curve-counting) has led us to an object (Gromov–Witten invariants) that is interesting regardless of the extent to which it actually achieves the goal.

How, then, to calculate Gromov–Witten invariants? This is a difficult question, in many cases prohibitively difficult, but there are important situations in which computation is possible. The first such situation we consider is when

## Kontsevich’s Formula

To explain Kontsevich’s computation, we first note that in the case where the target

whose codomain can be understood very concretely. First, when

Because any two points in

Pulling this relation back under

is equal to the corresponding sum where we instead insist that

This yields a host of relations among genus-zero Gromov–Witten invariants, collectively known as the *WDVV relations*. When

Interpreting the invariant

which recovers, via much more modern machinery, the ancient Greeks’ assertion that there is a unique conic passing through five general points in the (complex, projective) plane. In a celebrated theorem from the early days of Gromov–Witten theory

Although Kontsevich’s proof was entirely mathematical, there is a different interpretation of his result that passes through the unexpected world of theoretical physics—more specifically, string theory. Curves arise in that setting as the “worldsheet” traced out by a string as it travels through spacetime, and Gromov–Witten invariants appear in the definition of a structure known as the “quantum product” on