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5
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'analytic set'
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| Title of object: |
analytic set |
| Canonical Name: |
AnalyticSet |
| Type: |
Definition |
| Created on: |
2005-02-01 11:01:20 |
| Modified on: |
2007-12-12 16:48:07 |
| Classification: |
msc:32A60, msc:32C25 |
| Defines: |
regular point, simple point, top simple point, singular point, locally analytic, dimension of a variety, subvariety of a complex analytic variety |
| Synonyms: |
analytic set=analytic variety
analytic set=complex analytic variety |
Revision comment (for changes between this and next version):
| add reference to chirka (another good book) |
Preamble:
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\newtheorem*{defn}{Definition} |
Content:
Let $G \subset {\mathbb{C}}^N$ be an open set.
\begin{defn}
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 $G$
and holomorphic functions $f_1,\cdots,f_m$ defined in $U$ such that
$U \cap V = \{ z : f_k(z) = 0 \text{for all} 1\leq k \leq m \}.$
\end{defn}
This basically says that around each point of $V,$ the set $V$ is analytic.
A stronger definition is required.
\begin{defn}
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 $G$
and holomorphic functions $f_1,\cdots,f_m$ defined in $U$ such that
$U \cap V = \{ z : f_k(z) = 0 \text{ for all } 1\leq k \leq m \}.$
\end{defn}
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 {\em analytic set}.
At most points an analytic variety $V$ will in fact be a complex
analytic manifold. So
\begin{defn}
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
point is called a {\em singular point}.
\end{defn}
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.
\begin{defn}
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*}
\end{defn}
\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}
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}
That is, a subset $W$ is a subvariety if it is definined by the vanishing of analytic functions near all points of $V$.
\begin{thebibliography}{9}
\bibitem{Whitney:varieties}
Hassler Whitney.
{\em \PMlinkescapetext{Complex Analytic Varieties}}.
Addison-Wesley, Philippines, 1972.
\end{thebibliography} |
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