Hodge Conjecture via Singular Varieties

TL;DR

The paper develops a framework to study the Hodge conjecture on singular varieties by leveraging Jannsen's SHC and a cycle-class-theoretic approach via limit mixed Hodge structures. It introduces Mumford–Tate families to control how Hodge classes degenerate and shows MT is preserved under key geometric operations and correspondences, enabling the transfer of HC information from smooth fibers to singular central fibers. A main result proves that odd-dimensional hypersurfaces with $A_n$ singularities satisfy SHC and Jannsen's conjecture, and their smooth resolutions satisfy the classical Hodge conjecture, providing new HC examples arising from singular geometry. The work also develops practical tools (degeneration, semi-stable reduction, and MT-preserving correspondences) to generate further MT families and apply them to examples such as Fano varieties of lines and moduli spaces of sheaves, with concrete results for $A_n$-singularities.

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 $A_n$ 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.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: pet
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5