Griffiths heights and pencils of hypersurfaces

TL;DR

This paper explores the Griffiths height associated with variations of Hodge structures over curves, providing formulas that relate it to characteristic classes for hypersurface pencils.

Contribution

It introduces formulas linking Griffiths heights of hypersurface cohomology to characteristic classes, extending previous definitions to more general settings.

Findings

Formulas expressing Griffiths height via characteristic classes

Extension of Griffiths height definition to include bad reduction points

Connection between Griffiths height and Kato height for motives

Abstract

The Griffiths height of a variation of Hodge structures over a projective curve is defined as the degree of its canonical line bundle, as defined by Griffiths and generalized by Peters to allow bad reduction points. It may be seen as a geometric analog of the Kato height attached to pure motives over number fields. In this paper, we establish various formulas expressing the Griffiths height of the middle-dimensional cohomology of a pencil of projective complex hypersurfaces in terms of characteristic classes.