Editing Complex geometry (section)

=== Riemann surfaces ===

The term ''moduli'' was coined by [[Bernhard Riemann]] during his original work on Riemann surfaces. The classification theory is most well-known for compact Riemann surfaces. By the [[Surface_(topology)#Classification_of_closed_surfaces|classification of closed oriented surfaces]], compact Riemann surfaces come in a countable number of discrete types, measured by their [[genus (topology)|genus]] <math>g</math>, which is a non-negative integer counting the number of holes in the given compact Riemann surface.

The classification essentially follows from the [[uniformization theorem]], and is as follows:<ref>Forster, O. (2012). Lectures on Riemann surfaces (Vol. 81). Springer Science & Business Media.
</ref><ref>Miranda, R. (1995). Algebraic curves and Riemann surfaces (Vol. 5). American Mathematical Soc.</ref><ref>Donaldson, S. (2011). Riemann surfaces. Oxford University Press.</ref>

*''g = 0'': <math>\mathbb{CP}^1</math>
*''g = 1'': There is a one-dimensional complex manifold classifying possible compact Riemann surfaces of genus 1, so-called [[elliptic curves]], the [[modular curve]]. By the [[uniformization theorem]] any elliptic curve may be written as a quotient <math>\mathbb{C}/(\mathbb{Z} + \tau \mathbb{Z})</math> where <math>\tau</math> is a complex number with strictly positive imaginary part. The moduli space is given by the quotient of the group <math>\operatorname{PSL}(2,\mathbb{Z})</math> acting on the [[upper half plane]] by [[Möbius transformation]]s.
*''g > 1'': For each genus greater than one, there is a moduli space <math>\mathcal{M}_g</math> of genus g compact Riemann surfaces, of dimension <math>\dim_{\mathbb{C}} \mathcal{M}_g = 3g-3</math>. Similar to the case of elliptic curves, this space may be obtained by a suitable quotient of [[Siegel upper half-space]] by the action of the group <math>\operatorname{Sp}(2g, \mathbb{Z})</math>.