Pencils of projective hypersurfaces, Griffiths heights and geometric invariant theory. I

TL;DR

This paper investigates the relationship between Griffiths heights and GIT heights for pencils of projective hypersurfaces, aiming to establish bounds and properties that relate to conjectures in arithmetic geometry.

Contribution

It provides new estimates and comparisons between Griffiths and GIT heights for hypersurface pencils, extending previous computations to singular cases and analyzing stability properties.

Findings

Established bounds between Griffiths and GIT heights.

Extended Griffiths height computations to hypersurfaces with singularities.

Proved lower semicontinuity of stable Griffiths height.

Abstract

We study the Griffiths heights associated to the middle-dimensional cohomology of pencils of projective hypersurfaces, by comparing them to heights defined by means of geometric invariant theory (GIT). Kato and Koshikawa have conjectured a Northcott property for the Kato heights attached to motives over number fields, and investigated its consequences. Bounding these Griffiths heights in terms of the GIT heights would constitute a geometric counterpart, valid over function fields of characteristic zero, of Kato and Koshikawa's conjecture. Part of our results follows from our earlier works on the computation of these Griffiths heights in the case of pencils with generic singularities, and on semistability criteria for singular projective hypersurfaces, combined with a general formalism of GIT heights over function fields. We also establish estimates between the Griffiths and GIT heights associated to pencils of projective hypersurfaces, which are valid beyond the case of generic singularities. To achieve this, we establish diverse results of independent interest. Notably we extend our previous computations of Griffiths heights to pencils of projective hypersurfaces with semihomogeneous singularities, and we show the lower semicontinuity of the stable Griffiths height attached to polarized variations of Hodge structures over complex algebraic curves.