The Green–Griffiths–Brosnan–Fang–Néron–Pełř conjecture for singular normal functions

Let $X$ be a smooth projective variety of dimension $2m$, let $L$ be a very ample line bundle on $X$, and let $H_{\mathrm{prim}}^{m,m}(X,\mathbb{Z}(m))$ denote the primitive integral Hodge classes. Green–Griffiths–Brosnan–Fang–Néron–Pełř conjecture. For every non-torsion class $\zeta\in H_{\mathrm{prim}}^{m,m}(X,\mathbb{Z}(m))$, there exists an integer $d>0$ such that the associated normal function $AJ(\zeta)$ is singular on $\bar P=|L^{\otimes d}|$. This predicts that sufficiently high linear systems detect every non-torsion primitive Hodge class through a singularity of its normal function; the statement is presented as part of the program relating normal functions to the Hodge conjecture.