Generalized Thomas hyperplane sections and relations between vanishing cycles

TL;DR

This paper explores the relations between vanishing cycles and primitive Hodge classes, providing a criterion for generalized Thomas hyperplane sections related to the Hodge conjecture, extending Griffiths' construction.

Contribution

It establishes a new connection between vanishing cycle relations and primitive Hodge classes, generalizing Griffiths' approach to generalized Thomas hyperplane sections.

Findings

Relation between vanishing cycles and primitive Hodge classes identified

Criterion for generalized Thomas hyperplane sections based on pairings established

Extension of Griffiths' construction to broader context

Abstract

As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge conjecture is inductively equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between the vanishing cycles of type (0,0) with unipotent monodromy around a singular hyperplane section defines a primitive Hodge class such that this singular hyperplane section is a generalized Thomas hyperplane section if and only if the pairing between a given primitive Hodge class and some of the constructed primitive Hodge classes does not vanish. This is a generalization of a construction by P. Griffiths.