Editing Complex geometry (section)

== Definitions ==
Complex geometry is concerned with the study of [[complex manifold]]s, and [[complex algebraic variety|complex algebraic]] and [[complex-analytic variety|complex analytic varieties]]. In this section, these types of spaces are defined and the relationships between them presented.

A '''complex manifold''' is a [[topological space]] <math>X</math> such that:
*<math>X</math> is [[Hausdorff_space|Hausdorff]] and [[second countable]].
*<math>X</math> is locally [[homeomorphic]] to an open subset of <math>\mathbb{C}^n</math> for some <math>n</math>. That is, for every point <math>p\in X</math>, there is an [[open neighbourhood]] <math>U</math> of <math>p</math> and a homeomorphism <math>\varphi: U \to V</math> to an open subset <math>V\subseteq \mathbb{C}^n</math>. Such open sets are called ''charts''.
*If <math>(U_1,\varphi)</math> and <math>(U_2,\psi)</math> are any two overlapping charts which map onto open sets <math>V_1, V_2</math> of <math>\mathbb{C}^n</math> respectively, then the ''transition function'' <math>\psi \circ \varphi^{-1}:\varphi(U_1\cap U_2) \to \psi(U_1\cap U_2)</math> is a [[biholomorphism]].
Notice that since every biholomorphism is a [[diffeomorphism]], and <math>\mathbb{C}^n</math> is isomorphism as a [[real vector space]] to <math>\mathbb{R}^{2n}</math>, every complex manifold of dimension <math>n</math> is in particular a smooth manifold of dimension <math>2n</math>, which is always an even number.

In contrast to complex manifolds which are always smooth, complex geometry is also concerned with possibly singular spaces. An '''affine complex analytic variety''' is a subset <math>X\subseteq \mathbb{C}^n</math> such that about each point <math>p\in X</math>, there is an open neighbourhood <math>U</math> of <math>p</math> and a collection of finitely many holomorphic functions <math>f_1, \dots, f_k: U \to \mathbb{C}</math> such that <math>X\cap U = \{z\in U \mid f_1(z) = \cdots = f_k(z) = 0\} = Z(f_1,\dots,f_k)</math>. By convention we also require the set <math>X</math> to be [[irreducible algebraic set|irreducible]]. A point <math>p\in X</math> is ''singular'' if the [[Jacobian matrix]] of the vector of holomorphic functions <math>(f_1,\dots,f_k)</math> does not have full rank at <math>p</math>, and ''non-singular'' otherwise. A '''projective complex analytic variety''' is a subset <math>X\subseteq \mathbb{CP}^n</math> of [[complex projective space]] that is, in the same way, locally given by the zeroes of a finite collection of holomorphic functions on open subsets of <math>\mathbb{CP}^n</math>.

One may similarly define an '''affine complex algebraic variety''' to be a subset <math>X\subseteq \mathbb{C}^n</math> which is locally given as the zero set of finitely many polynomials in <math>n</math> complex variables. To define a '''projective complex algebraic variety''', one requires the subset <math>X\subseteq \mathbb{CP}^n</math> to locally be given by the zero set of finitely many ''[[homogeneous polynomials]]''.

In order to define a general complex algebraic or complex analytic variety, one requires the notion of a [[locally ringed space]]. A '''complex algebraic/analytic variety''' is a locally ringed space <math>(X,\mathcal{O}_X)</math> which is locally isomorphic as a locally ringed space to an affine complex algebraic/analytic variety. In the analytic case, one typically allows <math>X</math> to have a topology that is locally equivalent to the subspace topology due to the identification with open subsets of <math>\mathbb{C}^n</math>, whereas in the algebraic case <math>X</math> is often equipped with a [[Zariski topology]]. Again we also by convention require this locally ringed space to be irreducible.

Since the definition of a singular point is local, the definition given for an affine analytic/algebraic variety applies to the points of any complex analytic or algebraic variety. The set of points of a variety <math>X</math> which are singular is called the ''singular locus'', denoted <math>X^{sing}</math>, and the complement is the ''non-singular'' or ''smooth locus'', denoted <math>X^{nonsing}</math>. We say a complex variety is ''smooth'' or ''non-singular'' if it's singular locus is empty. That is, if it is equal to its non-singular locus.

By the [[implicit function theorem]] for holomorphic functions, every complex manifold is in particular a non-singular complex analytic variety, but is not in general affine or projective. By Serre's GAGA theorem, every projective complex analytic variety is actually a projective complex algebraic variety. When a complex variety is non-singular, it is a complex manifold. More generally, the non-singular locus of ''any'' complex variety is a complex manifold.