Extended operational Chow group and Lefschetz (1,1)-theorem

Abstract

Let $X$ be a singular, projective variety. For every $p>0$, $H^{2p}(X,\mathbb{Q})$ is equipped with a mixed Hodge structure. The elements of $\mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{Q}) \cap H^{p,p} \mathrm{Gr}^W_{2p}H^{2p}(X,\mathbb{C})$ will be called Hodge (p,p)-classes. The purpose of this article, is to study the Bloch-Gillet-Soul\'{e} (BGS) cycle class map from the $p$-th operational Chow group $A^p(X)$ to the space of $(p,p)$-Hodge classes. We show that if $p=1$ and $X$ is a normal surface with at worst rational singularities, then the BGS cycle class map is surjective. This extends the Lefschetz $(1,1)$-theorem to the setup of rational surface singularities. However, the BGS map is not always surjective. For this reason we introduce extended operational Chow group $A^p_{\mathrm{ext}}(X)$ which contains the operational Chow group. We show that the BGS cycle class map extends to $A^p_{\mathrm{ext}}(X)$. Moreover, if $p=1$ and $X$ has at worst isolated singularity (not necessarily a surface), then the extended BGS map is surjective. This further extends the Lefschetz $(1,1)$-theorem to the case of isolated singularities.