Griffiths–Green algebraicity conjecture for zero loci of admissible normal functions

Let $S$ be a smooth complex algebraic variety, let $\mathcal H$ be a variation of pure Hodge structure on $S$, and let $\nu$ be an admissible normal function, with zero locus $\mathcal Z\subset S$. Griffiths–Green conjecture. The zero locus $\mathcal Z$ of $\nu$ is an algebraic subvariety of $S$. This conjecture concerns the algebraicity of loci that are initially known to be complex analytic. The paper proves the assertion when the admissible normal function has torsion singularities, while the unrestricted statement is not established here.