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# 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 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

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

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

## Mathematics Subject Classification

32C25*no label found*32A60

*no label found*

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