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The basic question, still open as of, is whether there is a cycle map:

${\displaystyle A^{*}(X)\to \operatorname {H} ^{*}(X,\mathbb {Z} ).}$

If X is smooth, such a map exists since ${\displaystyle A^{*}(X)}$ is the usual Chow ring of X. (Totaro 2014) harv error: no target: CITEREFTotaro2014 (help) has shown that there might not be such a map even if X is a linear variety (a variety admitting a certain cell decomposition.)

References

• W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, ‘Intersection theory on spherical varieties’, J. Alg. Geom. 4 (1995), 181–193.

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