| Version 2 |
Version 1 |
| \theoremstyle{definition} |
\theoremstyle{definition} |
| \newtheorem*{defn}{Definition} |
\newtheorem*{defn}{Definition} |
| \theoremstyle{theorem} |
\theoremstyle{theorem} |
| \newtheorem*{thm}{Theorem} |
\newtheorem*{thm}{Theorem} |
|
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| Let $G \subset {\mathbb{C}}^N$ be an open set. |
Let $G \subset {\mathbb{C}}^N$ be an open set. |
|
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| \begin{defn} |
\begin{defn} |
| A set $V \subset G$ is said to be {\em locally analytic} |
A set $V \subset G$ is said to be {\em locally analytic} |
| if for every point $p \in V$ there exists a neighbourhood $U$ of $p$ in $G$ |
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 |
and holomorphic functions $f_1,\cdots,f_m$ defined in $U$ such that |
| $U \cap V = \{ z : f_k(z) = 0 \text{for all} 1\leq k \leq m \}$. |
$U \cap V = \{ z : f_k(z) = 0 \text{for all} 1\leq k \leq m \}$. |
| \end{defn} |
\end{defn} |
|
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| This basically says that around each point of $V$, the set $V$ is analytic. |
This basically says that around each point of $V$, the set $V$ is analytic. |
| A stronger definition is required. |
A stronger definition is required. |
|
|
| \begin{defn} |
\begin{defn} |
| A set $V \subset G$ is said to be an {\em analytic variety} in $G$ |
A set $V \subset G$ is said to be an {\em analytic variety} in $G$ |
| (or {\em analytic set} in $G$) |
|
| if for every point $p \in G$ there exists a neighbourhood $U$ of $p$ 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 |
and holomorphic functions $f_1,\cdots,f_m$ defined in $U$ such that |
| $U \cap V = \{ z : f_k(z) = 0 \text{ for all } 1\leq k \leq m \}$. |
$U \cap V = \{ z : f_k(z) = 0 \text{ for all } 1\leq k \leq m \}$. |
| \end{defn} |
\end{defn} |
|
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| Note the change, now $V$ is analytic around each point of $G$. Since the |
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 |
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. |
$V$ is relatively closed in $G$, while a local variety need not be closed. |
| Sometimes an analytic variety is called an {\em analytic set}. |
Sometimes an analytic variety is called an {\em analytic set}. |
|
|
| At most points an analytic variety $V$ will in fact be a complex |
At most points an analytic variety $V$ will in fact be a complex |
| analytic manifold. So |
analytic manifold. So |
|
|
| \begin{defn} |
\begin{defn} |
| A point $p \in V$ is called a {\em regular point} if there is a neighbourhood |
A point $p \in V$ is called a {\em regular point} if there is a neighbourhood |
| $U$ of $p$ such that $U \cap V$ is a complex analytic manifold. Any other |
$U$ of $p$ such that $U \cap V$ is a complex analytic manifold. Any other |
| point is called a {\em singular point}. |
point is called a {\em singular point}. |
| \end{defn} |
\end{defn} |
|
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| The set of regular points of $V$ is denoted by $V^-$ or sometimes $V^*$. |
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| For any regular point $p \in V$ we can define the dimension as |
For any regular point $p \in V$ we can define the dimension as |
| \begin{equation*} |
\begin{equation*} |
| \operatorname{dim}_p(V) = |
\operatorname{dim}_p(V) = |
|
\operatorname{dim}_{\mathbb{C}}(U \cap V)
|
\operatorname{dim}}(U \cap V)
|
| \end{equation*} |
\end{equation*} |
| where $U$ is as above and thus $U \cap V$ is a manifold with a well defined |
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. |
dimension. Here we of course take the complex dimension of these manifolds. |
|
|
| \begin{defn} |
\begin{defn} |
| Let $V$ be an analytic variety, |
Let $V$ be an analytic variety, |
| we define the dimension of $V$ by |
we define the dimension of $V$ by |
| \begin{equation*} |
\begin{equation*} |
| \operatorname{dim}(V) |
\operatorname{dim}(V) |
| = |
= |
| \sup \{ \operatorname{dim}_p(V) : p \text{ a regular point of } V \} . |
\sup \{ \operatorname{dim}_p(V) : p \text{ a regular point of } V \} . |
| \end{equation*} |
\end{equation*} |
| \end{defn} |
\end{defn} |
|
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| \begin{defn} |
|
| The regular point $p \in V$ such that $\dim_p(V) = \dim(V)$ is called a {\em top |
|
| \PMlinkescapetext{simple} point} of $V$. |
|
| \end{defn} |
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| Similarly as for manifolds we can also talk about subvarieties. In this case we modify definition a little bit. |
|
|
|
| \begin{defn} |
|
| A set $W \subset V$ where $V \subset G$ is a local variety is said to be |
|
| a {\em 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 |
|
| $U \cap W = \{ z : f_k(z) = 0 \text{ for all } 1\leq k \leq m \}$. |
|
| \end{defn} |
|
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| That is, a subset $W$ is a subvariety if it is definined by the vanishing of analytic functions near all points of $V$. |
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|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Whitney:varieties} |
\bibitem{Whitney:varieties} |
| Hassler Whitney. |
Hassler Whitney. |
| {\em \PMlinkescapetext{Complex Analytic Varieties}}. |
{\em \PMlinkescapetext{Complex Analytic Varieties}}. |
| Addison-Wesley, Philippines, 1972. |
Addison-Wesley, Philippines, 1972. |
| \end{thebibliography} |
\end{thebibliography} |
