analytic set

Let $G\subset {ℂ}^N$ be an open set.

Definition.

A set $V\subset G$ is said to be locally analyticif 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 thezero 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 complexanalytic 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 otherpoint 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 defineddimension. 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 bea 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 $V$.