analytic set (original) (raw)

Definition.

A set V⊂G is said to be locally analyticif for every point p∈V there exists a neighbourhood U of p in Gand holomorphic functions f1,⋯,fm defined in U such thatU∩V={z:fk⁢(z)=0⁢for all⁢1≤k≤m}.

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

Definition.

A set V⊂G is said to be an analytic variety in G(or analytic set in G) if for every point p∈G there exists a neighbourhood U of p in Gand holomorphic functions f1,⋯,fm defined in U such thatU∩V={z:fk⁢(z)=0⁢ for all ⁢1≤k≤m}.

Note the change, now V is analytic around each point of G. Since thezero sets of holomorphic functions are closed, this for example implies thatV is relatively closed in G, while a local variety need not be closed. Sometimes an analytic variety is called an analytic set.

The set of regular points of V is denoted by V- or sometimes V*.

For any regular point p∈V we can define the dimension as

where U is as above and thus U∩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

Definition.

The regular point p∈V such that dimp⁡(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⊂V where V⊂G is a local variety is said to be a subvariety of Vif for every point p∈V there exists a neighbourhood U of p in Gand holomorphic functions f1,⋯,fm defined in U such thatU∩W={z:fk⁢(z)=0⁢ for all ⁢1≤k≤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.