A Lefschetz (1,1) theorem for singular varieties

الأصل · EN

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 H²ᵖ(X)/W₂ₚ₋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 H²ᵖₘ(X,Q(p)) to the space of weight 2p Hodge cycles in H²ᵖ(X,Q). We conjecture that this is surjective when X is defined over the algebraic closure of 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.