A Lefschetz (1,1) theorem for singular varieties

The goal of this article is to try understand where Hodge cycles on a singular complex projective variety X come from. As a first step we consider Hodge cycles on the maximal pure quotient $H^{2p}(X)\mathrm{/}{W}_{2p-1}$, and introduce a class of algebraic cycles that we call homologically Cartier, that should conjecturally describe all such Hodge cycles. Secondly, given a singular complex projective variety $X$, we show that there is a cycle map from motivic cohomology group $H_M^{2p}(X,Q(p))$ to the space of weight 2p Hodge cycles in $H^{2p}(X,Q)$. We conjecture that this is surjective when X is defined over the algebraic closure of $\mathbb{Q}$. We show that this holds integrally when p=1, and we also give a concrete interpretation of motivic classes in this degree. Finally, we show that the general conjecture holds for a self fibre product of elliptic modular surfaces.