# analytic set

Let $G\subset {\u2102}^{N}$ be an open set.

###### Definition.

A set $V\subset G$ is said to be locally analytic
if for every point $p\in V$ there exists a neighbourhood $U$ of $p$ in $G$
and holomorphic functions^{} ${f}_{1},\mathrm{\cdots},{f}_{m}$ defined in $U$ such that
$U\cap V=\{z:{f}_{k}(z)=0\text{for all}1\le k\le m\}.$

This basically says that around each point of $V,$ the set $V$ is analytic. A stronger definition is required.

###### Definition.

A set $V\subset G$ is said to be an analytic variety in $G$ (or analytic set in $G$) if for every point $p\in G$ there exists a neighbourhood $U$ of $p$ in $G$ and holomorphic functions ${f}_{1},\mathrm{\cdots},{f}_{m}$ defined in $U$ such that $U\cap V=\{z:{f}_{k}(z)=0\text{for all}1\le k\le m\}.$

Note the change, now $V$ is analytic around each point of $G.$ Since the zero sets of holomorphic functions are closed, this for example implies that $V$ is relatively closed in $G,$ while a local variety need not be closed. Sometimes an analytic variety is called an analytic set.

At most points an analytic variety $V$ will in fact be a complex analytic manifold. So

###### Definition.

A point $p\in V$ is called a regular point^{} if there is a neighbourhood
$U$ of $p$ such that $U\cap V$ is a complex analytic manifold. Any other
point is called a singular point^{}.

The set of regular points of $V$ is denoted by ${V}^{-}$ or sometimes ${V}^{*}.$

For any regular point $p\in V$ we can define the dimension as

$${\mathrm{dim}}_{p}(V)={\mathrm{dim}}_{\u2102}(U\cap V)$$ |

where $U$ is as above and thus $U\cap V$ is a manifold with a well defined dimension. Here we of course take the complex dimension of these manifolds.

###### Definition.

Let $V$ be an analytic variety, we define the dimension of $V$ by

$$\mathrm{dim}(V)=sup\{{\mathrm{dim}}_{p}(V):p\text{a regular point of}V\}.$$ |

###### Definition.

The regular point $p\in V$ such that ${dim}_{p}(V)=dim(V)$ is called a top point of $V$.

Similarly as for manifolds we can also talk about subvarieties. In this case we modify definition a little bit.

###### Definition.

A set $W\subset V$ where $V\subset G$ is a local variety is said to be a subvariety of $V$ if for every point $p\in V$ there exists a neighbourhood $U$ of $p$ in $G$ and holomorphic functions ${f}_{1},\mathrm{\cdots},{f}_{m}$ defined in $U$ such that $U\cap W=\{z:{f}_{k}(z)=0\text{for all}1\le k\le m\}$.

That is, a subset $W$ is a subvariety if it is definined by the vanishing of analytic functions near all points of $V$.

## References

- 1 E. M. Chirka. . Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
- 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.

Title | analytic set |

Canonical name | AnalyticSet |

Date of creation | 2013-03-22 14:59:28 |

Last modified on | 2013-03-22 14:59:28 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 10 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32C25 |

Classification | msc 32A60 |

Synonym | analytic variety |

Synonym | complex analytic variety |

Related topic | IrreducibleComponent2 |

Defines | regular point |

Defines | simple point |

Defines | top simple point |

Defines | singular point |

Defines | locally analytic |

Defines | dimension of a variety |

Defines | subvariety of a complex analytic variety |

Defines | complex analytic subvariety |