---
tag: Q9B74BYZ8C
kind: concept
name: Germ of an analytic set
msc: [32B10, 32B05]
specializes: [VZXF7RNFKS]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_analytic_variety
---

## Summary

The germ of an analytic set at a point $a$ records the shape of a [[complex analytic variety|GB7M582M5Y]] arbitrarily close to $a$.
It forgets the rest of the set.
Two analytic sets have the same germ at $a$ when they coincide on some neighbourhood of $a$.
Germs of analytic sets at $a$ correspond to radical ideals of the ring of [[germs of holomorphic functions|7PJ8KVZZCQ]] at $a$.
This makes local analytic geometry a branch of commutative algebra.

## Explanations

### L1 human

Zoom in on an analytic set at one point and keep only what is visible at every scale.
For instance, the curve $\set{z_2^2 = z_1^2(1 + z_1)}$ in $\C^2$ is connected and irreducible as a whole, but its germ at the origin looks like two smooth branches crossing.
Far away points, such as other components of the set, do not affect the germ.