Paul Johnson

As a topological space, the surface of genus g is unique – that is, if X and Y are two genus g surfaces, there is a homeomorphism f : X → Y. However, if we give X and Y complex structures and think about them as Riemann Surfaces, they no longer have to be “the same” – we can't necessarily turn the homeomorphism f into a biholomorphic map.

Understanding the different ways we can view a genus g topological surface as a Riemann Surface is a fundamental (if vague) question in geometry. It turns out that for g = 0 there is a unique way to put a complex structure on a sphere – this is why it makes sense to talk about the Riemann Sphere. However, for g > 0, there are infinitely many distinct complex structures. Thus, counting the number of complex structures is not an interesting problem, and we need a more precise meaning of what it means to “understand” them. The answer lies in the concept of a moduli space.

Moduli Spaces

A moduli space M is, first of all, a topological space. What makes a moduli space more than just a topological space is that the underlying set of points is naturally in bijection with some interesting geometric objects. A bit more precisely, each point of M corresponds to an isomorphism class of some object we want to study, and the intuitive idea is that two points in M are “close” to each other if the corresponding isomorphism classes of geometric objects are “close” to one another. Let us now look at some examples to refine our understanding.

Example C.1.1 (Projective space). In Definition 2.3.1, was introduced as the set of lines in through the origin. Another way of saying this is that is “the moduli space of lines through the origin in”. Each point of the moduli space corresponds to a line, and moving continuously from a point to a nearby point amounts to continuously wiggling the corresponding lines.

Example C.1.2 (Moduli space of curvesMg). Similarly, we have the moduli space of genus g curves Mg. Each point in Mg corresponds to an isomorphism class of genus g Riemann Surfaces. That is, we can view any genus g Riemann Surface Xg as a point of Mg.