Let be an open set.
A set is said to be an analytic variety in (or analytic set in ) if for every point there exists a neighbourhood of in and holomorphic functions defined in such that
Note the change, now is analytic around each point of Since the zero sets of holomorphic functions are closed, this for example implies that is relatively closed in while a local variety need not be closed. Sometimes an analytic variety is called an analytic set.
At most points an analytic variety will in fact be a complex analytic manifold. So
The set of regular points of is denoted by or sometimes
For any regular point we can define the dimension as
where is as above and thus is a manifold with a well defined dimension. Here we of course take the complex dimension of these manifolds.
Let be an analytic variety, we define the dimension of by
The regular point such that is called a top point of .
Similarly as for manifolds we can also talk about subvarieties. In this case we modify definition a little bit.
A set where is a local variety is said to be a subvariety of if for every point there exists a neighbourhood of in and holomorphic functions defined in such that .
That is, a subset is a subvariety if it is definined by the vanishing of analytic functions near all points of .
- 1 E. M. Chirka. . Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
- 2 Hassler Whitney. . Addison-Wesley, Philippines, 1972.
|Date of creation||2013-03-22 14:59:28|
|Last modified on||2013-03-22 14:59:28|
|Last modified by||jirka (4157)|
|Synonym||complex analytic variety|
|Defines||top simple point|
|Defines||dimension of a variety|
|Defines||subvariety of a complex analytic variety|
|Defines||complex analytic subvariety|