Rank of normal functions and Betti strata

TL;DR

This paper generalizes geometric results on the rank of normal functions and Betti strata to broader families of cycles, providing formulas, closedness results, and exploring arithmetic implications.

Contribution

It extends previous work by establishing a general formula for Betti rank and proving Zariski closedness for any admissible normal function in variations of Hodge structures.

Findings

Derived a formula for Betti rank in general settings

Proved Zariski closedness of Betti strata for admissible normal functions

Explored degeneracy loci and arithmetic questions related to Betti strata

Abstract

In a recent work of the authors, we proved the generic positivity of the Beilinson-Bloch heights of the Gross-Schoen and Ceresa cycles. The geometric part of the proof was to prove the maximality of the rank of the associated normal function and the Zariski closedness of the Betti strata. In this paper, we generalize these geometric results to an arbitrary family of homologically trivial cycles. More generally, we prove a formula to compute the Betti rank and prove the Zariski closedness of the Betti strata, for any admissible normal function of a variation of Hodge structures of weight $-1$. We also define and prove results about degeneracy loci. In the end, we go back to the arithmetic setting and ask some questions about the rationality of the Betti strata and the torsion loci.