Green–Griffiths conjecture on singular normal functions

Let $X$ be a smooth complex projective variety, let $L\to X$ be a very ample line bundle, and let $\zeta$ be a non-torsion, primitive Hodge class of type $(n,n)$ on $X$. For the associated admissible normal function

$$\nu_{\zeta}:P-\hat X\to J(\mathcal H),$$

where $P=\mathbb P H^0(X,\mathcal O(L))$ and $\hat X$ is the dual variety, call $\nu_{\zeta}$ singular if there is a point $\hat x\in\hat X$ such that its singularity $\operatorname{sing}_{\hat x}(\nu_{\zeta})\in \mathit{IH}_{\hat{x}}^1(\mathcal H)$ is non-torsion. Green–Griffiths conjecture. There exists an integer $k>0$ such that, after replacing $L$ by $L^k$, the associated normal function $\nu_{\zeta}$ is singular. This conjecture is part of the Green–Griffiths program relating the algebraicity of infinitesimal invariants and the Hodge conjecture; the supplied text does not give a resolution status.