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.
Descent, Motives and K-theory.Journal f¨ ur die reine und angewandte Mathematik, 478:127–176, 1996
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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.