A remark on singularities of primitive cohomology classes

TL;DR

This paper introduces a new approach to understanding singularities of primitive cohomology classes using elementary theorems, linking these singularities to the triviality of classes and relating the Hodge conjecture to primitive class non-vanishing.

Contribution

It defines global and local singularities of primitive cohomology classes via elementary methods and connects these to the triviality of classes and the Hodge conjecture.

Findings

Singularities detect triviality of primitive classes.

Elementary methods can analyze primitive cohomology singularities.

Relation between Hodge conjecture and primitive class non-vanishing.

Abstract

Green and Griffiths have introduced several notions of singularities associated with normal functions, especially in connection with middle dimensional primitive Hodge classes. In this note, by using the more elementary aspects of the Decomposition Theorem, we define global and local singularities associated with primitive middle dimensional cohomology classes and by using the Relative Hard Lefschetz Theorem, we show that these singularities detect the global and local triviality of the primitive class. In a final section, we write-up a classical inductive argument relating the Hodge conjecture to the local non-vanishing of primitive classes.