The book I am using is Chirka's Complex Analytic Set.
Suppose A is a complex analytic set in $\mathbb{C}^n$ containing $0$ as a singular point. The tangent cone $ c(A,0)$ of $A$ at $0$ is the set of all vectors $v$ such that there exists $a_j\in A$ and $t_j>0$ such that $a_j\to 0$ and $t_ja_j\to v$.
$C(A,0)$ is a homogeneous algebraic variety. It is the common zeros of the leading terms of holomorphic functions vanishing on $A$.
My question is: As an analytic set, $C(A,0)$ itself has singularities. So is there any relation between $sgn C(A,0)$ and $C(sgn(A),0)$?(Assuming $0$ is a singularity of $A$) For example, is one included in the other? If neither inclusion holds in general, then is there any condition that ensures one side of inclusion?
I am in functional analysis but my research concerns analytic subsets. I will appreciate it if any one can explain this in a relatively analytic way.
Thank You!