Rational Singularities of Admissible Normal Functions and Integral Torsion Orders along Curves

We consider admissible normal functions associated with polarizable variations of integral Hodge structure of weight $-1$ on a polydisk $\Delta^r$ with a normal-crossings boundary. When the coordinate monodromies are unipotent, vanishing of the multivariable rational singularity is equivalent to uniform boundedness of the orders of the integral singularities obtained by restriction to curves with arbitrary positive integral weights. Moreover, if the rational singularity is nonzero, then along at least two of the $r$ prescribed families $n(1,\ldots,1)+ε_i$ in fixed coordinates, a common positive linear lower bound, depending on the input, holds for all sufficiently large $n$. The proof uses a criterion in terms of weight-zero residuals in the limiting mixed Hodge structure and the discreteness of a fixed integral lattice. Neither square-zero monodromy logarithms nor constancy of the kernel of their positive linear combinations is assumed. We construct a geometric example of a curve with two nodes, an indecomposable rank-$8$ example coupled in two variables, and an example of finite torsion that is lost upon passage to weight zero. For the rank-$8$ example, we verify admissibility of the extension and obtain an exact formula whose values along a prescribed family have cubic order of growth and are not monotone. Finally, we discuss the relation to known singularity detection theorems, but we do not assert a result supplying nonzero singularities from arbitrary Hodge classes or a resolution of the usual rational Hodge conjecture.