On the Zariski tangent space to a complex analytic variety

Abstract

Let (X, (9) be a complex space and x a point of X. T(X, x), the Zariski tangent space to X at x, is defined as the space of (U-linear derivations from the local ring (9 x to (U. As such it is an algebraic object. T(X, x) is a vector space and it is the minimal dimensional vector space into which a sufficiently small neighborhood of x in X may be embedded. In the case when X is a complex analytic subvariety of an open subset of (I~", Whitney [6, 7] has defined certain cones associated to the germ of X at x. These cones are subsets of the Zariski tangent space T (X, x). The largest of the Whitney cones Cs(X, x) is, in general, a proper subset of the Zariski tangent space. In fact, the vector subspace of T(X, x) spanned by Cs(X, x) is, in general, only a proper subspace of T(X, x). The cone Cs(X, x) is defined as the limiting positions of certain sequences of vectors. The purpose of this article is to define, via a geometric procedure analogous to that used to define Cs(X, x), an increasing sequence of cones Dj, I(X, x) in T(X, x), with D 1 ,! (X, x) -C s ( X , x) and for some integer Jl, Dj,,1 (X, x) = T(X, x). The minimal values Jl for which Di,,1 (X, x) is equal to T(X, x) will depend on the singularity of X at x. There are hig;ler order tangent spaces Tk(X, x) (see Definition 1.2) associated to the germ of X at x and we will define a similar sequence of cones in each one of these.