analytic set

Let $G\subset {ℂ}^N$ 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

$${\dim }_p(V)={\dim }_{ℂ}(U\cap V)$$

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

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

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 $V$.

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