Correspondences and singular varieties

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.