Complex vs etale Abel Jacobi map and algebraicity of the zero locus of etale normal functions

TL;DR

This paper establishes a deep connection between complex and etale Abel-Jacobi maps for algebraic varieties, proving that the vanishing of the etale map implies the vanishing of the complex map, and deduces algebraicity of zero loci of normal functions.

Contribution

It proves that the complex Abel-Jacobi map vanishes whenever the etale Abel-Jacobi map does, and shows the algebraicity of zero loci of etale normal functions over fields of finite type over Q.

Findings

Vanishing of etale Abel-Jacobi map implies vanishing of complex Abel-Jacobi map.

Zero locus of etale normal functions is contained in that of complex normal functions.

Zero locus of complex normal functions is defined over the algebraic closure of the base field.

Abstract

We prove, using $p$-adic Hodge theory for open algebraic varieties, that for a smooth projective variety over a subfield $k\subset\mathbb C$ which is of finite type over $\mathbb Q$, the complex abel jacobi map vanishes if the etale abel jacobi map vanishes. This implies that for a smooth projective morphism $f:X\to S$ of smooth complex algebraic varieties over $k\subset\mathbb C$ which is of finite type over $\mathbb Q$ and $Z\in\mathcal Z^d(X,n)^{f,\partial=0}$ an algebraic cycle flat over $S$ whose cohomology class vanishes on fibers, the zero locus of the etale normal function associated to $Z$ is contained in the zero locus of the complex normal function associated to $Z$. From the work of Saito or Charles, we deduce that the zero locus of the complex normal function associated to $Z$ is defined over the algebraic closure $\bar k$ of $k$ if the zero locus of the etale normal function associated to $Z$ is not empty. We also prove an algebraicity result for the zero locus of an etale normal function associated to an algebraic cycle over a field of finite type over $\mathbb Q$. By the way, for a smooth morphism $f:X\to S$ of smooth algebraic varieties over a field of finite type over $\mathbb Q$, we embed the locus of Hodge-Tate classes of $f$ inside the locus of Hodge classes of $f$.