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.
Fractional intersection and bivariant theory.Communications in algebra, 20(1):285–302, 1992
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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.