Lefschetz (1,1)-theorem for singular varieties

Lefschetz (1,1)-theorem states that every (1,1) class  in a smooth projective variety is the first Chern class of a line bundle. Such a statement fails when the variety is singular. There have been various attempts at extending the Lefschetz (1,1) to singular varieties. The most universal statement so far is due to Arapura. However, it follows from the work of Totaro that the map studied by Arapura does not look at all the possible (1,1) classes, in the singular case. Totaro suggests looking at the Bloch-Gillet-Soule cycle class map from the operational Chow group to the space of Hodge classes. In a joint work with I. Kaur, we study this map and give a criterion under which this map is surjective, thereby giving a possible Lefschetz (1,1) theorem for singular varieties. In the talk, I plan to present these results and give various examples where the surjectivity holds.