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 $f_{1},\cdots,f_{m}$ defined in $U$ such that $U\cap V=\{z:f_{k}(z)=0\text{for all}1\leq k\leq m\}.$

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 $f_{1},\cdots,f_{m}$ defined in $U$ such that $U\cap V=\{z:f_{k}(z)=0\text{ for all }1\leq k\leq m\}.$

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.

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

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

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 $f_{1},\cdots,f_{m}$ defined in $U$ such that $U\cap W=\{z:f_{k}(z)=0\text{ for all }1\leq k\leq m\}$.