On the Divisorial Geometry of Volume Asymptotics of Sublevel Sets

The real log canonical threshold (RLCT) is a central invariant in birational geometry and singularity theory, measuring the complexity of a singularity through discrepancy and valuation data on a log resolution. Beyond this algebro-geometric definition, it also admits a metric interpretation, reflecting how neighbourhoods of the singular locus degenerate at small scales.
In this work, we investigate these degenerations via sublevel sets associated with an analytic ideal. We show that the asymptotic behaviour of their volume determines the \emph{visible} intrinsic divisorial spectrum (i.e.\ the set of actual poles of the local zeta function), a finite set contained in the resolution-dependent set of multiplicity ratios of any log resolution. Conversely, this intrinsic spectrum, together with its multiplicities and coefficients, can be recovered from the volume function through a finite reconstruction procedure.
We also describe intrinsic interpretations in terms of arc spaces: the divisorial exponents appear both as ratios of vanishing orders along generic arcs and as asymptotic codimension growth rates of divisorial cylinders.
Taken together, these results show that certain divisorial invariants admit a metric realisation through the asymptotic behaviour of sublevel-set volumes, and that the birational structure of an analytic singularity can be reconstructed from the geometry of its infinitesimal neighbourhoods.