Editing Complex geometry (section)

=== 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 === --->