Singularities of admissible normal functions

TL;DR

This paper demonstrates that the existence of certain singular admissible normal functions is equivalent to the Hodge conjecture, using mixed Hodge modules and extending key theorems in the field.

Contribution

It introduces a new approach linking singularities of normal functions to the Hodge conjecture via mixed Hodge modules and proves a relative weak Lefschetz theorem for perverse sheaves.

Findings

Singularities of admissible normal functions are equivalent to the Hodge conjecture.

Established a relative weak Lefschetz theorem for perverse sheaves.

Extended the understanding of normal functions using mixed Hodge modules.

Abstract

In a recent paper, M. Green and P. Griffiths used R. Thomas' works on nodal hypersurfaces to establish the equivalence of the Hodge conjecture and the existence of certain singular admissible normal functions. Inspired by their work, we study normal functions using M. Saito's mixed Hodge modules and prove that the existence of singularities of the type considered by Griffiths and Green is equivalent to the Hodge conjecture. Several of the intermediate results, including a relative version of the weak Lefschetz theorem for perverse sheaves, are of independent interest.