# Cycle class map for singular varieties

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?

• 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).
– naf
May 1 '21 at 1:18
• Perhaps what you need can be found in ias.ac.in/public/Volumes/pmsc/103/03/0209-0247.pdf May 1 '21 at 3:39
• @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? May 1 '21 at 10:54
• No, it is not the same thing. There is no (known) good cycle theoretic description of the operational Chow groups.
– naf
May 1 '21 at 13:12

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.