Genus three Ceresa cycles and limit of archimedean heights

TL;DR

This paper investigates the limit of archimedean heights for genus three Ceresa cycles, linking it to Deligne splittings of associated mixed Hodge structures in degenerating families.

Contribution

It establishes a connection between limit heights of Ceresa cycles and Deligne splittings in the context of degenerating mixed Hodge structures.

Findings

Limit height is given by Deligne splitting for genus three Ceresa cycles.

The limit height depends on the choice of a parameter in the variation.

The work extends understanding of heights in degenerating geometric families.

Abstract

For a one-parameter variation of biextension mixed Hodge structures, Brosnan and Pearlstein showed that the limit of the asymptotic height of the variation is given by a certain limit height of the nilpotent orbit. This limit height depends on the choice of a parameter. In the case of a variation of geometric origin related to Ceresa cycles associated with curves of genus three, after fixing a parameter, we show that this limit height is given by the Deligne splitting of a biextension mixed Hodge structure associated with cycles in the boundary.