Correspondences and singular varieties

TL;DR

This paper extends the Bloch--Srinivas method to singular and quasi-projective varieties, providing new tools for understanding their cohomological properties and implications for the Hodge conjecture.

Contribution

It generalizes the decomposition of the diagonal technique to singular varieties and applies it to Mumford's theorem and the Hodge conjecture in this broader context.

Findings

Extended the Bloch--Srinivas method to singular varieties

Provided a version of Mumford's theorem for singular varieties

Addressed the Hodge conjecture for singular varieties

Abstract

What is generally known as the "Bloch--Srinivas method" consists of decomposing the diagonal of a smooth projective variety, and then considering the action of correspondences in cohomology. In this note, we observe that this same method can also be extended to singular and quasi--projective varieties. We give two applications of this observation: the first is a version of Mumford's theorem, the second is concerned with the Hodge conjecture for singular varieties.