Editing Complex geometry (section)

== Classification in complex geometry ==

One major theme in complex geometry is [[Classification theorem|classification]]. Due to the rigid nature of complex manifolds and varieties, the problem of classifying these spaces is often tractable. Classification in complex and algebraic geometry often occurs through the study of [[moduli space]]s, which themselves are complex manifolds or varieties whose points classify other geometric objects arising in complex geometry.

=== 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>.

=== Holomorphic line bundles ===

Complex geometry is concerned not only with complex spaces, but other holomorphic objects attached to them. The classification of holomorphic line bundles on a complex variety <math>X</math> is given by the [[Picard variety]] <math>\operatorname{Pic}(X)</math> of <math>X</math>.

The picard variety can be easily described in the case where <math>X</math> is a compact Riemann surface of genus g. Namely, in this case the Picard variety is a disjoint union of complex [[Abelian varieties]], each of which is isomorphic to the [[Jacobian variety]] of the curve, classifying [[divisor (algebraic geometry)|divisors]] of degree zero up to linear equivalence. In differential-geometric terms, these Abelian varieties are complex tori, complex manifolds diffeomorphic to <math>(S^1)^{2g}</math>, possibly with one of many different complex structures.

By the [[Torelli theorem]], a compact Riemann surface is determined by its Jacobian variety, and this demonstrates one reason why the study of structures on complex spaces can be useful, in that it can allow one to solve classify the spaces themselves.
<!--- === Enriques-Kodaira classification ===

=== Minimal model program ===

=== Moduli spaces === --->