A Lefschetz (1,1) theorem for singular varieties

TL;DR

This paper investigates the origins of Hodge cycles on singular complex projective varieties, proposing new classes of algebraic cycles and establishing connections with motivic cohomology, with results confirmed for specific cases like elliptic modular surfaces.

Contribution

It introduces homologically Cartier cycles to describe Hodge cycles on singular varieties and establishes a cycle map from motivic cohomology, conjecturing surjectivity over algebraic closures of rationals.

Findings

Cycle map from motivic cohomology to Hodge cycles is surjective for p=1.

Homologically Cartier cycles potentially characterize all Hodge cycles on singular varieties.

The conjecture holds for self fibre products of elliptic modular surfaces.

Abstract

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 $H^{2p}(X)/W_{2p-1}$, 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 $H^{2p}_M(X,Q(p))$ to the space of weight 2p Hodge cycles in $H^{2p}(X,Q)$. We conjecture that this is surjective when X is defined over 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.