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