Let $X$ be a complex algebraic variety (not necessarily irreducible, nor reduced). Then the local Euler obstruction is a group isomorphism $$\textrm{Eu}: Z_\ast X\to F_\ast X,$$ where $Z_\ast X$ is the group of cycles and $F_\ast X$ is the group of constructible functions on $X$. For a prime cycle $V\subset X$, $\textrm{Eu}(V):X\to\mathbb Z$ is the function defined by $$P\mapsto\int_{\nu^{-1}(P)}c(\tilde T)\cap s(\nu^{-1}(P),\tilde V),$$ where $\nu:\tilde V\to V$ is the Nash blow-up, $\tilde T$ is the Nash tangent bundle on $\tilde V$, and $s(\nu^{-1}(P),\tilde V)$ denotes the Segre class of the cone of the closed immersion $\nu^{-1}(P)\subset\tilde V$.
If $X$ is smooth, the fundamental cycle $[X]$ corresponds to the constant function $\textbf 1_X:P\mapsto 1$ under $\textrm{Eu}$.
Question. What is the image of the fundamental cycle $[X]=\sum m_V[V]\in Z_\ast X$ if $X$ is singular? What is the preimage of the constant function $\textbf 1_X$?
It seems to me that $[X]$ and $\textbf 1_X$ do not correspond to each other anymore in the singular case. I tried to do the explicit computation but I'm not sure I got it right. Is the following true? If $V\subset X$ is a prime cycle of multiplicity $m_V$ in $X$, then $\textrm{Eu}(V)(P)$ takes the value $$0,\color{red}{\qquad 1},\qquad \color{blue}{m_V\int_{\nu^{-1}(P)}c(\tilde T)\cap [\tilde V]}$$ according to whether $$P\in X\setminus V,\color{red}{\qquad P\in V_{smooth}},\qquad \color{blue}{P\in V_{sing}}.$$ A reasonable candidate for $\textrm{Eu}([X])$ seems to be the function $\sum m_V\textbf 1_V\in F_\ast X$, but that seems too easy. And I have no idea about the preimage of $\textbf 1_X$. Thank you for any help!
