In the article "Hodge theory for combinatorial geometries" by Adiprasito, Huh and Katz, it it claimed in the proof of theorem 5.12 that there is a Chow equivalence between the de Concini-Processi wonderful model $Y$ of an arrangement, and a certain toric variety $X$.


However, the source they cite only proves that


$$H^\*(Y)\cong Ch(X)$$


It seems that to draw the conclusion that $Ch(Y)\cong Ch(X)$ it is used that $Ch(Y)\cong H^\*(Y)$, but I do not know why that is the case?




---


Here is what I know: $Y\subset X$ as a closed subset, and the relevant maps are (should be) induced by this inclusion. We have a diagram
$$
\require{AMScd}
\begin{CD}
Ch(X) @>{ch}>> H^\*(X)\\
@VVV @VVV \\
Ch(Y) @>{ch}>> H^\*(Y)
\end{CD}
$$
and we know that the diagonal arrow is an isomorphism. To establish that $Ch(X)\cong Ch(Y)$ it is enough to show that $Ch(Y)\cong H^\*(Y)$. Since the diagonal is an isomorphism, the map $Ch(Y)\to H^\*(Y)$ is surjective. I do not see how to prove injectiveness though.



Is the Chow ring of a wonderful model for a hyperplane arrangement isomorphic to the singular cohomology ring?
The isomorphism $H^\cdot(Y)\cong Ch^\cdot(X)$ is shown by 


Eva Feichtner and Sergey Yuzvinsky in Feichtner, E. & Yuzvinsky, S. Invent. Math. (2004) 155: 515. <https://doi.org/10.1007/s00222-003-0327-2>


It's worth emphasizing that the inclusion $Y\subset X$ is not a homotopy equivalence, and generally $H^\cdot(X)\not\cong H^\cdot(Y)$.


The property that $H^\cdot(Y)\cong Ch^\cdot(Y)$ for iterated blowups of projective spaces I think is due to Sean Keel, Trans. Amer. Math. Soc. 330 (1992), 545-574, 
<https://doi.org/10.1090/S0002-9947-1992-1034665-0>, in line with [Libli's](https://mathoverflow.net/users/37214/libli "Libli") comment above.


