# Computing Chow group of a variety which is almost a blow-up of another variety

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.

• I studied a fairly similar situation (on an explicit case) in an arithmetic context in my paper “Équivalence rationnelle sur les hypersurfaces cubiques de mauvaise réduction” (J. Number Theory 128 (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. – Gro-Tsen Jul 8 '19 at 8:43
• The map $\pi_*$ in your situation is surjective. Combining this with the localisation sequence on $Y$ and the projective bundle formula (for the restriction of $\pi$ to the inverse image of $Z$) gives quite a lot of information about the Chow groups. There is probably no natural map from $CH(X)$ to $CH(Y)$ though, so it seems unlikely that there is a direct sum decomposition as in the displayed formula. – ulrich Jul 8 '19 at 12:28
• There's a lot known about Chow rings of projective bundles (which is one way to establish the result for blow-ups, using the fact that a blowup is a subvariety of a projective bundle). You could either see if a similar result is true for your situation, or perhaps try to exploit the fact that $\pi$ is a projective bundle over $Z$. – A Nonny Mouse Jul 11 '19 at 14:33