Chow cohomology and Lefschetz (1,1)-theorem

TL;DR

This paper investigates the surjectivity of a cycle class map for singular varieties, providing criteria under which the singular Hodge conjecture in codimension one holds for varieties with certain mild singularities.

Contribution

It offers a sufficient criterion for the Bloch-Gillet-Soulé cycle class map to be surjective on singular varieties, confirming the singular Hodge conjecture in specific cases.

Findings

Several singular varieties with isolated singularities satisfy the singular Hodge conjecture in codimension 1.

The paper establishes conditions under which the cycle class map is surjective for certain singularities.

It extends the classical Lefschetz (1,1)-theorem to a broader class of singular varieties.

Abstract

Any smooth, projective variety satisfies the Hodge conjecture in codimension one, known as the Lefschetz (1,1) theorem. Totaro formulated a version for singular varieties. He asked whether the natural Bloch-Gillet-Soul\'{e} cycle class map from the operational Chow group to the (1,1)-classes in the weight graded piece of the cohomology group is surjective? In this short article, we give a sufficient criterion for this to hold. In particular, we show that several singular varieties with at worst isolated singularities (log canonical, divisorial log terminal, ADE-singularities) satisfy the singular Hodge conjecture in codimension 1.