Hodge Conjecture via Singular Varieties
Abstract
In this article we study the cohomological and homological (due to Jannsen) Hodge conjecture for singular varieties. The motivation for studying singular varieties comes from the fact that any smooth projective variety X is birational to a (possibly singular) hypersurface Y in a projective space. We prove that odd dimensional hypersurfaces with singularities satisfy both versions of the conjecture and moreover their (smooth) resolutions satisfy the classical Hodge conjecture, thus producing new examples of smooth varieties satisfying the classical Hodge conjecture.
Cite
@article{arxiv.2509.25273,
title = {Hodge Conjecture via Singular Varieties},
author = {Ananyo Dan and Inder Kaur},
journal= {arXiv preprint arXiv:2509.25273},
year = {2025}
}
Comments
This article supersedes arXiv:2301.01005. The current article expands on the previous preprint substantially; we study the cohomological singular Hodge conjecture but following feedback, we also consider the Jannsen's homological version. Furthermore, we give new examples of smooth varieties satisfying the classical Hodge conjecture. Comments are welcome!