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analytic set (Definition)

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

$\displaystyle \operatorname{dim}_p(V) = \operatorname{dim}_{\mathbb{C}}(U \cap V)$    

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
$\displaystyle \operatorname{dim}(V) = \sup \{ \operatorname{dim}_p(V) : p$    a regular point of $\displaystyle V \} .$    

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$.

Bibliography

1
E. M. Chirka. Complex Analytic Sets. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
2
Hassler Whitney. Complex Analytic Varieties. Addison-Wesley, Philippines, 1972.



"analytic set" is owned by jirka.
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See Also: irreducible component

Other names:  analytic variety, complex analytic variety
Also defines:  regular point, simple point, top simple point, singular point, locally analytic, dimension of a variety, subvariety of a complex analytic variety, complex analytic subvariety

Attachments:
intersection of complex analytic varieties is a complex analytic variety (Theorem) by jirka
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Cross-references: near, subset, subvarieties, complex, well defined, manifold, dimension, complex analytic manifold, variety, implies, closed, zero sets, analytic, holomorphic functions, neighbourhood, point, open set
There are 23 references to this entry.

This is version 7 of analytic set, born on 2005-02-01, modified 2008-02-04.
Object id is 6696, canonical name is AnalyticSet.
Accessed 8212 times total.

Classification:
AMS MSC: 32A60 (Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Zero sets of holomorphic functions)
 32C25 (Several complex variables and analytic spaces :: Analytic spaces :: Analytic subsets and submanifolds)
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