Editing Complex geometry (section)

==Idea==
[[File:RiemannKugel.svg|thumb|A typical example of a complex space is the [[complex projective line]]. It may be viewed either as the [[sphere]], a smooth manifold arising from [[differential geometry]], or the [[Riemann sphere]], an extension of the complex plane by adding a [[point at infinity]].]]

Broadly, complex geometry is concerned with [[Space_(mathematics)|spaces]] and [[Geometry|geometric objects]] which are modelled, in some sense, on the [[complex plane]]. Features of the complex plane and [[complex analysis]] of a single variable, such as an intrinsic notion of [[orientability]] (that is, being able to consistently rotate 90 degrees counterclockwise at every point in the complex plane), and the rigidity of [[Holomorphic_function|holomorphic functions]] (that is, the existence of a single complex derivative implies complex differentiability to all orders) are seen to manifest in all forms of the study of complex geometry. As an example, every complex manifold is canonically orientable, and a form of [[Liouville's theorem (complex analysis)|Liouville's theorem]] holds on [[Compact space|compact]] complex manifolds or [[Projective variety|projective]] complex algebraic varieties.

Complex geometry is different in flavour to what might be called ''real'' geometry, the study of spaces based around the geometric and analytical properties of the [[real number line]]. For example, whereas [[smooth manifold]]s admit [[Partition of unity|partitions of unity]], collections of smooth functions which can be identically equal to one on some [[open set]], and identically zero elsewhere, complex manifolds admit no such collections of holomorphic functions. Indeed, this is the manifestation of the [[identity theorem]], a typical result in complex analysis of a single variable. In some sense, the novelty of complex geometry may be traced back to this fundamental observation.

It is true that every complex manifold is in particular a real smooth manifold. This is because the complex plane <math>\mathbb{C}</math> is, after forgetting its complex structure, isomorphic to the real plane <math>\mathbb{R}^2</math>. However, complex geometry is not typically seen as a particular sub-field of [[differential geometry]], the study of smooth manifolds. In particular, [[Jean-Pierre Serre|Serre]]'s [[Algebraic geometry and analytic geometry|GAGA theorem]] says that every [[Projective variety|projective]] [[analytic variety]] is actually an [[algebraic variety]], and the study of holomorphic data on an analytic variety is equivalent to the study of algebraic data. 

This equivalence indicates that complex geometry is in some sense closer to [[algebraic geometry]] than to [[differential geometry]]. Another example of this which links back to the nature of the complex plane is that, in complex analysis of a single variable, singularities of [[meromorphic functions]] are readily describable. In contrast, the possible singular behaviour of a continuous real-valued function is much more difficult to characterise. As a result of this, one can readily study [[Singularity_(mathematics)|singular]] spaces in complex geometry, such as singular complex [[analytic varieties]] or singular complex algebraic varieties, whereas in differential geometry the study of singular spaces is often avoided.

In practice, complex geometry sits in the intersection of differential geometry, algebraic geometry, and [[analysis]] in [[several complex variables]], and a complex geometer uses tools from all three fields to study complex spaces. Typical directions of interest in complex geometry involve [[Classification theorem|classification]] of complex spaces, the study of holomorphic objects attached to them (such as [[holomorphic vector bundle]]s and [[coherent sheaves]]), and the intimate relationships between complex geometric objects and other areas of mathematics and physics.