In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an analytic variety is defined locally as the set of common zeros of finitely many analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

s. It is analogous to the included concept of complex algebraic variety

Algebraic variety

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

, and every complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

 is an analytic variety. Since analytic varieties may have singular points

Mathematical singularity

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...

, not all analytic varieties are complex manifolds. An analytic variety is also called a (real or complex) analytic set.

See also

• Algebraic variety
  Algebraic variety
  In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
  
  
• Analytic space
  Analytic space
  An analytic space in the theory of functions of several complex variables is a generalization of the concept of an analytic manifold: heuristically, an analytic space is a set which is locally isomorphic to an analytic variety, while an analytic manifold is locally isomorphic to a complex euclidean...
  
  
• Complex manifold
  Complex manifold
  In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
  
  
• Several complex variables
  Several complex variables
  The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...