I am reading the cycle class map for singular projective varieties as mentioned by Laterveer in this article (see Definition $1$). The article does not define the map but refers to an article of Totaro, which also does not define the map but refers to the article "Nonarchimedean Arakelov theory" by Bloch, Gillet and Soule, which is published in JAG in 1995 and is not available online. Can someone give a bit more details on this map (or an alternate reference)? For example, the authors say that this map is functorial. I do not quite understand what this means, as the cohomology group is functorial under pullback and the Chowgroup is functorial under proper pushforward and flat pullback? Moreover, does the image of the composition $$\mathrm{A}^p(X) \xrightarrow{cl} \mathrm{Gr}^{W}_{2p}H^{2p}(X,\mathbb{Q}) \xrightarrow{rest.} \mathrm{Gr}^{W}_{2p}H^{2p}(X_{\mathrm{smooth}},\mathbb{Q})$$ is the same the usual cycle class map $$A^p(X_{\mathrm{smooth}}) \xrightarrow{cl} H_{2n2p}^{\mathrm{BM}}(X_{\mathrm{smooth}}) \xrightarrow{P.D.} H^{2p}(X_{\mathrm{smooth}})?$$ What is the pullback of the image of the singular cycle class map to the cohomology on the resolution of singularities?

$\begingroup$ In the paper you refer to $A^i(X)$ is the operational Chow group. These groups are functorial for arbotrary pullbacks (and when $X$ is singular (usually) not equal to the usual Chow groups). $\endgroup$– nafCommented May 1, 2021 at 1:18

$\begingroup$ Perhaps what you need can be found in ias.ac.in/public/Volumes/pmsc/103/03/02090247.pdf $\endgroup$– KapilCommented May 1, 2021 at 3:39

$\begingroup$ @naf and Kapil I wanted to ask: is the relative Chow group (of rationally equivalent classes supported on the regular locus of the variety) as studied in the above mentioned article of Srinivas, the same as the operational Chow group? $\endgroup$– user45397Commented May 1, 2021 at 10:54

$\begingroup$ No, it is not the same thing. There is no (known) good cycle theoretic description of the operational Chow groups. $\endgroup$– nafCommented May 1, 2021 at 13:12
1 Answer
Suppose for simplicity $X$ is proper. Choose a smooth simplicial resolution $X_\bullet \to X$. We can assume that $X_0\to X$ is a resolution of singularities. By (a careful reading of) Deligne, Théorie de Hodge III, we have an exact sequence $$0\to Gr^W_{i} H^i(X)\to H^i(X_0)\to H^i(X_1)$$ where the second map is the difference of pullbacks along the structure maps $X_1\to X_0$. Now as naf says, in a comment, the operational Chow group is functorial. Therefore the we can compose to get $$A^p(X)\to A^p(X_0)\to H^{2p}(X_0)$$ The image maps to $0$ in $H^{2p}(X_1)$. So you have your cycle map.
(I don't have access to BlochGilletSoulé either, but I assume it's the same map as theirs.)