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Let $G \subset {\mathbb{C}}^N$ be an open set.
Definition 1 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 $U \cap V = \{ z : f_k(z) = 0 {for all} 1\leq k \leq m \}.$
This basically says that around each point of $V,$ the set $V$ is analytic. A stronger definition is required.
Definition 2 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 $U \cap V = \{ z : f_k(z) = 0 { for all } 1\leq k \leq m \}.$
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 3 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 \begin{equation*} \operatorname{dim}_p(V) = \operatorname{dim}_{\mathbb{C}}(U \cap V) \end{equation*}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 4 Let $V$ be an analytic variety, we define the dimension of $V$ by \begin{equation*} \operatorname{dim}(V) = \sup \{ \operatorname{dim}_p(V) : p \text{ a regular point of } V \} . \end{equation*}
Definition 5 The regular point $p \in V$ such that $\dim_p(V) = \dim(V)$ is called a top simple point of $V$ .
Similarly as for manifolds we can also talk about subvarieties. In this case we modify definition a little bit.
Definition 6 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 $U \cap W = \{ z : f_k(z) = 0 { for all } 1\leq k \leq m \}$ .
That is, a subset $W$ is a subvariety if it is definined by the vanishing of analytic functions near all points of $V$ .
- 1
- E. M. Chirka. Complex Analytic Sets. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
- 2
- Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.
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