Normal functions and spread of zero locus

TL;DR

This paper investigates the behavior of admissible normal functions in algebraic geometry, showing conditions under which their zero loci are constant or defined over certain fields, generalizing previous results.

Contribution

It extends previous work by demonstrating that zero loci of admissible normal functions are algebraically defined under broader conditions, using spreading out techniques.

Findings

Zero locus components are defined over the base field if they contain a rational point.

Normal functions are constant on fibers if they are constant on some fiber.

Generalizes Charles's result to broader classes of normal functions.

Abstract

If there is a topologically locally constant family of smooth algebraic varieties together with an admissible normal function on the total space, then the latter is constant on any fiber if this holds on some fiber. Combined with spreading out, it implies for instance that an irreducible component of the zero locus of an admissible normal function is defined over k if it has a k-rational point where k is an algebraically closed subfield of the complex number field with finite transcendence degree. This generalizes a result of F. Charles that was shown in case the normal function is associated with an algebraic cycle defined over k.