Let $X$ be a, possibly singular, algebraic variety embedded as a closed subvariety of a manifold $M$ with map $i : X \rightarrow M$, and $\pi : \tilde{M} \rightarrow M$ be a proper birational map with $\tilde{M}$ a manifold and $X^{'} = (\pi^{-1} (X))_{red}$ be a divisor possibly singular. Denote by $j : X^{'} \rightarrow \tilde{M}$ the inclusion map.
We consider $c_{\ast} : \mathbb{F}(X^{'}) \rightarrow H_{\ast}(X^{'})$ be the natural transformation conjectured by Deligne and Grothendieck in 1969. The Chern-Schwartz-MacPherson class of $X^{'}$ can be defined by $c_{SM}(X^{'}) = c_{\ast}(1_{X^{'}})$, where $1_{X^{'}}$ denote the characteristic function whose value is 1 over $X^{'}$ and $0$ elsewhere. We know that
$$c(T\tilde{M}) \cap [\tilde{M}] - j_{\ast}c_{SM}(X^{'}) = c_{\ast}(1_{\tilde{M}\setminus X^{'}}).$$
Moreover, by functoriality of Chern-Schwartz-MacPherson's classes
$$\pi_{\ast}c_{\ast}(1_{\tilde{M}\setminus X^{'}}) = c_{\ast}\pi_{\ast} (1_{\tilde{M}\setminus X^{'}}) = c_{\ast}(1_{M \setminus X}) = c(TM) \cap [M] - i_{\ast}c_{SM}(X).$$
$\textbf{Question 1:}$ If we let $c_{F}(X^{'})$ be the Chern-Fulton class of $X^{'}$ (defined in W. Fulton - Intersection theory-Springer (1998)), is it true that
$$c(T\tilde{M}) \cap [\tilde{M}] - j_{\ast}c_{F}(X^{'}) = c_{F}(\tilde{M} \setminus X^{'}) ?$$
$\textbf{Question 2:}$ Is it true the push-forwards commutes?
$$\pi_{\ast}j_{\ast} c_{F}(X^{'}) = i_{\ast}c_{F}(X)?$$
Thank you for advanced.
