# analytic set

Let $G\subset{\mathbb{C}}^{N}$ be an open set.

###### Definition.

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 $U\cap V=\{z:f_{k}(z)=0\text{for all}1\leq k\leq m\}.$

This basically says that around each point of $V,$ the set $V$ is analytic. A stronger definition is required.

###### Definition.

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 $U\cap V=\{z:f_{k}(z)=0\text{ for all }1\leq k\leq m\}.$

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.

A point $p\in V$ is called a if there is a neighbourhood $U$ of $p$ such that $U\cap V$ is a complex analytic manifold. Any other point is called a .

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

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

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.

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

 $\operatorname{dim}(V)=\sup\{\operatorname{dim}_{p}(V):p\text{ a regular point % of }V\}.$
###### Definition.

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

Similarly as for manifolds we can also talk about subvarieties. In this case we modify definition a little bit.

###### Definition.

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 $U\cap W=\{z:f_{k}(z)=0\text{ for all }1\leq k\leq m\}$.

That is, a subset $W$ is a subvariety if it is definined by the vanishing of analytic functions near all points of $V$.

## References

• 1 E. M. Chirka. . Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
• 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
 Title analytic set Canonical name AnalyticSet Date of creation 2013-03-22 14:59:28 Last modified on 2013-03-22 14:59:28 Owner jirka (4157) Last modified by jirka (4157) Numerical id 10 Author jirka (4157) Entry type Definition Classification msc 32C25 Classification msc 32A60 Synonym analytic variety Synonym complex analytic variety Related topic IrreducibleComponent2 Defines regular point Defines simple point Defines top simple point Defines singular point Defines locally analytic Defines dimension of a variety Defines subvariety of a complex analytic variety Defines complex analytic subvariety