moduli space of Riemann surface

Question

For the definition of moduli space of Riemann surface, it seems that it is not based on the definiton in terms of functor(representable...) but rather put a topology on the set of isormophism classes. Is there any difference between these? Or if I miss something? Is there any reference ? Thank you!

Answer 1

From the mention of representability I assume you mean Riemann surfaces as algebraic curves, not the complex-analytic or conformal moduli problems.

Representing the moduli functor would mean you have object in a "reasonable" category (e.g., a variety, scheme, or algebraic space) that is a universal family of curves of whatever kind you are considering. But this doesn't exist for algebraic curves without further restrictions (eg., stability) and rigidification data (such as marked points) to get a "fine moduli space". However, if by topology you mean a Grothendieck topology then you can define such a thing, i.e., a suitable category of coverings associated to the problem, as a virtual moduli object that you can consider as a space in the sense that cohomology and other geometrical invariants can be associated to it.

The canonical and much admired exposition that explained this to the world is "Picard Groups of Moduli Problems" by David Mumford. Among other achievements it shows how to work with moduli stacks, without using the word "stack" (which may have been coined later, I don't know).

Answer 2

In the complex analytic setting one can also work in terms of representing functors and so on; unfortunately, just as in the algebraic setting, it will turn out that the natural functor is not representable without either imposing level structure or passing to a larger category of stack-like objects.

However, the complex analytic study of Riemann surfaces and their moduli has its own traditions, coming from Teichmuller theory, the theory of the mapping class group, the analytic theory of Ahlfors and Bers, and so on, and these traditions come with their own view-points, in which emphasizing the representability of the functor by a certain stack-like object may not be of paramount importance.

Just to see why this might be, consider that if $f: X \to S$ is a smooth proper map of complex analytic manifolds (e.g. a family of Riemann surfaces over a smooth base), then by Ehresmann's theorem, $f$ is a fibre-bundle in the $\mathcal C^{\infty}$-category. So one sees that the fibres of $f$ don't vary topologically, and one can try to consider directly how the complex structure varies. (This is the Kodaira--Spencer view-point on deformation theory.) The fact that one has these vivid geometric view-points available means that there is less pressure to use categorical language to describe the situation.