Editing Complex geometry (section)

== Types of complex spaces ==

=== Kähler manifolds ===
{{Main article|Kähler manifold}}

Complex manifolds may be studied from the perspective of differential geometry, whereby they are equipped with extra geometric structures such as a [[Riemannian metric]] or [[symplectic form]]. In order for this extra structure to be relevant to complex geometry, one should ask for it to be compatible with the complex structure in a suitable sense. A [[Kähler manifold]] is a complex manifold with a Riemannian metric and symplectic structure compatible with the complex structure. Every complex submanifold of a Kähler manifold is Kähler, and so in particular every non-singular affine or projective complex variety is Kähler, after restricting the standard Hermitian metric on <math>\mathbb{C}^n</math> or the [[Fubini-Study metric]] on <math>\mathbb{CP}^n</math> respectively.

Other important examples of Kähler manifolds include [[Riemann surface]]s, [[K3 surface]]s, and [[Calabi–Yau manifold]]s.

=== Stein manifolds ===
{{Main article|Stein manifold}}

Serre's GAGA theorem asserts that projective complex analytic varieties are actually algebraic. Whilst this is not strictly true for affine varieties, there is a class of complex manifolds that act very much like affine complex algebraic varieties, called [[Stein manifold]]s. A manifold <math>X</math> is Stein if it is holomorphically convex and holomorphically separable (see the article on Stein manifolds for the technical definitions). It can be shown however that this is equivalent to <math>X</math> being a complex submanifold of <math>\mathbb{C}^n</math> for some <math>n</math>. Another way in which Stein manifolds are similar to affine complex algebraic varieties is that [[Cartan's theorems A and B]] hold for Stein manifolds.

Examples of Stein manifolds include non-compact Riemann surfaces and non-singular affine complex algebraic varieties.

=== Hyper-Kähler manifolds ===
{{Main article|Hyperkähler manifold}}

A special class of complex manifolds is [[hyper-Kähler manifold]]s, which are [[Riemannian manifold]]s admitting three distinct compatible [[Almost complex manifold#Integrable almost complex structures|integrable almost complex structures]] <math>I,J,K</math> which satisfy the [[Quaternion|quaternionic relations]] <math>I^2 = J^2 = K^2 = IJK = -\operatorname{Id}</math>. Thus, hyper-Kähler manifolds are Kähler manifolds in three different ways, and subsequently have a rich geometric structure.

Examples of hyper-Kähler manifolds include [[Gravitational instanton|ALE spaces]], [[K3 surface]]s, [[Higgs bundle]] moduli spaces, [[Quiver_(mathematics)#Quiver_Variety|quiver varieties]], and many other [[moduli space]]s arising out of [[gauge theory]] and [[representation theory]].

=== Calabi–Yau manifolds ===
{{Main article|Calabi–Yau manifold}}

[[File:CalabiYau5.jpg|thumb|A real two-dimensional slice of a quintic [[Calabi–Yau manifold|Calabi–Yau]] threefold]]
As mentioned, a particular class of Kähler manifolds is given by Calabi–Yau manifolds. These are given by Kähler manifolds with trivial canonical bundle <math>K_X = \Lambda^n T_{1,0}^* X</math>. Typically the definition of a Calabi–Yau manifold also requires <math>X</math> to be compact. In this case [[Shing-Tung Yau|Yau's]] proof of the [[Calabi conjecture]] implies that <math>X</math> admits a Kähler metric with vanishing [[Ricci curvature]], and this may be taken as an equivalent definition of Calabi–Yau. 

Calabi–Yau manifolds have found use in [[string theory]] and [[Mirror symmetry (string theory)|mirror symmetry]], where they are used to model the extra 6 dimensions of spacetime in 10-dimensional models of string theory. Examples of Calabi–Yau manifolds are given by [[elliptic curve]]s, K3 surfaces, and complex [[Abelian varieties]].

=== Complex Fano varieties ===
{{Main article|Fano variety}}
A complex [[Fano variety]] is a complex algebraic variety with [[ample line bundle|ample]] anti-canonical line bundle (that is, <math>K_X^*</math> is ample). Fano varieties are of considerable interest in complex algebraic geometry, and in particular [[birational geometry]], where they often arise in the [[minimal model program]]. Fundamental examples of Fano varieties are given by projective space <math>\mathbb{CP}^n</math> where <math>K=\mathcal{O}(-n-1)</math>, and smooth hypersurfaces of <math>\mathbb{CP}^n</math> of degree less than <math>n+1</math>.

=== 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.