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 ${\mathrm{Gr}}_{2p}^W{H}^{2p}(X,\mathbb{Q})\cap H^{p,p}{\mathrm{Gr}}_{2p}^W{H}^{2p}(X,\mathbb{C})$ will be called Hodge (p,p)-classes. The purpose of this article, is to study the Bloch-Gillet-Soulé (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 $A_{\mathrm{ext}}^p(X)$ which contains the operational Chow group. We show that the BGS cycle class map extends to $A_{\mathrm{ext}}^p(X)$. 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.

11 pages, Results generalised to the case of isolated singularities for all varieties, title has been changed to reflect this. Some inconsistencies have also been corrected