1. Analytic Subsets and Complex submanifolds
Let X be an n-dimensional complex manifold. A subset Y of X is a complex submanifold of codimension k if there exists an open subset V ⊂ Cn−kand a holomorphic map f : V → Y such that
rank J (f )(z) = n − k, for all z ∈ V and Y = f (V ).
A subset Y of X is called an analytic subset1 if for each y ∈ Y, there exists an open neighborhood U ⊂ X of y and a finite numbers of holomorphic functions f1, · · · , fr on U such that
Y ∩ U = {p ∈ U : f1(p) = · · · = fr(p) = 0}.
The functions {fi : 1 ≤ i ≤ r} are called the locally defining functions of Y at y.
Definition 1.1. Let Y be an analytic subset of X. We say that y ∈ Y is a smooth point if there exists an open neighborhood U ⊂ X of y so that Y ∩ U is a complex submanifold of X.
Let Y be an analytic subset of X. We denote Y∗ the subset of Y consisting of smooth points of Y. The singular locus Ys of Y is the set Y \ Y∗ whose elements are called singular points. An analytic subset Y is called nonsingular (or smooth) if Y = Y∗, i.e. Y is a complex submanifold.
Definition 1.2. An irreducible analytic subset of X is called a complex analytic variety of X. We define dimCV = dimCV∗. An analytic subset of X is said to be purely d-dimensional if all of its irreducible components are d-dimensional.
An algebraic manifold is an algebraic subset of Pnsuch that it is also a complex subman- ifold of Pn.
1In some books, it is called an analytic variety.
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