---
tag: GB7M582M5Y
kind: concept
name: Complex analytic variety
msc: [32B15, 32C25, 32C15]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Summary

A complex analytic variety, or complex analytic set, in a [[complex manifold|JSGWAN2RJV]] $X$ is a closed subset with a local description.
Near each point of $X$ it is the common zero set of finitely many [[holomorphic functions|507C34W0ZT]].
It is the analytic counterpart of an algebraic variety, with polynomials replaced by holomorphic functions.
Analytic varieties may have singular points, and their local structure is studied through [[germs of analytic sets|Q9B74BYZ8C]] and the ring of convergent power series.
Abstract varieties of this kind, defined by gluing local models, are the [[complex analytic spaces|HXSZA44J5J]].

## Explanations

### L1 human

Take a few holomorphic functions on $\C^n$ and look at the points where all of them vanish.
Such a set may be smooth, like the line $\set{z_2 = 0}$ in $\C^2$, or have singular points, like the curve $\set{z_1 z_2 = 0}$, which is two lines crossing at the origin.
An analytic variety is a set that looks like this near every point, even if no single finite family of functions defines it everywhere.
For example, the zero set of $\sin z$ in $\C$, the set of integer multiples of $\pi$, is an analytic variety but not the zero set of a polynomial.