Normal functions for algebraically trivial cycles are algebraic for arithmetic reasons

TL;DR

This paper proves that normal functions from algebraically trivial cycles are algebraic and defined over the base field, confirming a conjecture and impacting the understanding of algebraic cycles in complex geometry.

Contribution

It establishes that normal functions for algebraically trivial cycles are algebraic and defined over the field of definition, confirming Charles's conjecture.

Findings

Normal functions from algebraically trivial cycles are algebraic.

Zero loci of these functions are algebraic and defined over the base field.

Confirms Charles's conjecture in complex algebraic geometry.

Abstract

For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.