A generalized Lefschetz (1,1)-theorem holds for singular projective varieties after replacing the operational Chow group by an inverse limit over resolutions, the extended operational Chow group.
Chow groups, Chow cohomology, and linear varieties
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Extended operational Chow group and Lefschetz (1,1)-theorem
A generalized Lefschetz (1,1)-theorem holds for singular projective varieties after replacing the operational Chow group by an inverse limit over resolutions, the extended operational Chow group.