Let $A_\ast$ and $F_\ast$ be the functors $\textrm{Var}_\mathbb C\to \textrm{Ab}$ of Chow groups and constructible functions, respectively, with respect to proper maps. Then the *Chern-Schwartz-MacPherson class* is the unique natural transformation $$c_{\textrm{SM}}:F_\ast\to A_\ast$$ taking the value $c_{\textrm{SM}}(\textbf{1}_X)=c(TX)\cap [X]\in A_\ast X$ on $\textbf{1}_X\in F_\ast X$ for nonsingular $X$, and commuting with proper pushforwards. It can be a tool for computing the Euler characteristic of singular varieties.

The Chern-Schwartz-MacPherson class of $X$ is by definition the class $c_{\textrm{SM}}(\textbf{1}_X)$. It respects the excision relation exactly as (generalized) Euler characteristics, meaning that if $Z\subset X$ is closed an $U=X\setminus Z$, then $$c_{\textrm{SM}}(\textbf{1}_X)=c_{\textrm{SM}}(\textbf{1}_Z)+c_{\textrm{SM}}(\textbf{1}_{U})$$

Question. When is $c_{\textrm{SM}}(\textbf{1}_X)$ computable in practice for singular $X$? Is it true that (at least the degree zero part of) $c_{\textrm{SM}}(\textbf{1}_X)$ vanishes if $\chi_{top}(X)=0$?