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The goal of this article is to try understand where Hodge cycles on asingular complex projective variety X come from. As a first step we considerHodge cycles on the maximal pure quotient $H^{2p}(X)/W_{2p-1}$, and introduce aclass of algebraic cycles that we call homologically Cartier, that shouldconjecturally describe all such Hodge cycles. Secondly, given a singularcomplex projective variety $X$, we show that there is a cycle map from motiviccohomology group $H^{2p}_M(X,Q(p))$ to the space of weight 2p Hodge cycles in$H^{2p}(X,Q)$. We conjecture that this is surjective when X is defined over thealgebraic 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 ofelliptic modular surfaces.

A Lefschetz (1,1) theorem for singular varieties