Cycle class map for singular varieties

Question

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 "Non-archimedean 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 pull-back and the Chow-group is functorial under proper pushforward and flat pull-back? 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 $$A^p(X_{\mathrm{smooth}}) \xrightarrow{cl} H_{2n-2p}^{\mathrm{BM}}(X_{\mathrm{smooth}}) \xrightarrow{P.D.} H^{2p}(X_{\mathrm{smooth}})?$$ What is the pull-back of the image of the singular cycle class map to the cohomology on the resolution of singularities?

Answer 1

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 Bloch-Gillet-Soulé either, but I assume it's the same map as theirs.)