Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. I have a morphism which is close to being a blow-up along $Z$, namely:

(*) Let $\pi : Y\rightarrow X$ be a morphism with the property that $\pi$ is an isomorphism over $X\setminus Z$, and moreover $\pi$ is a $\mathbb{P}^n$- bundle over $Z$ for some $n$.

(Note that $\pi$ may not be a blow-up if codim$Z\neq n+1$, for example.)

My question is: Can we compute the Chow groups of $Y$ in terms of the Chow groups of $X$ and $Z$?

In the case when $\pi$ is a blow-up, it has been proved in Claire Voisin's *Hodge Theory and Complex Algebraic Geometry, Vol II*, Theorem 9.27 that such formula indeed exists, namely

When $\pi$ is a blow-up, for each $l$, there exists an isomorphism

$\bigoplus_{0\leq k\leq n-1}CH_{l-n+k}(Z) \bigoplus CH_{l}(X)\xrightarrow{\cong} CH_l (Y)$

I believe a similar result should be true for my case (*) as well, but I can't prove it. Any help or reference is welcome.

**P.S.** Actually, even knowing $CH_1(Y)$ is good enough for me for now. If even that is possible in terms of $X$ and $Z$'s Chow groups then please let me know.

J. Number Theory128(2008), 926–944), §2 (specifically ¶2.6): I don't remember the details, which are tedious, but I remember that, while I couldn't find a general result that applied, it was possible (albeit tedious) to compute the maps explicitly on the explicit case at hand. $\endgroup$ – Gro-Tsen Jul 8 at 8:43