Editing Complex geometry (section)

=== Toric varieties ===
{{Main article|Toric variety}}

[[File:Moment polytope of first Hirzebruch surface.png|thumb|Moment polytope describing the first [[Hirzebruch surface]].]][[Toric varieties]] are complex algebraic varieties of dimension <math>n</math> containing an open [[dense subset]] biholomorphic to <math>(\mathbb{C}^*)^n</math>, equipped with an action of <math>(\mathbb{C}^*)^n</math> which extends the action on the open dense subset. A toric variety may be described combinatorially by its ''toric fan'', and at least when it is non-singular, by a ''[[moment map|moment]] polytope''. This is a polygon in <math>\mathbb{R}^n</math> with the property that any vertex may be put into the standard form of the vertex of the positive [[orthant]] by the action of <math>\operatorname{GL}(n,\mathbb{Z})</math>. The toric variety can be obtained as a suitable space which fibres over the polytope. 

Many constructions that are performed on toric varieties admit alternate descriptions in terms of the combinatorics and geometry of the moment polytope or its associated toric fan. This makes toric varieties a particularly attractive test case for many constructions in complex geometry. Examples of toric varieties include complex projective spaces, and bundles over them.