On the Divisorial Geometry of Volume Asymptotics of Sublevel Sets

Abstract

The log canonical threshold (LCT) is a fundamental invariant in birational geometry and singularity theory, measuring the complexity of an analytic singularity through discrepancy and valuation data on a log resolution. In this work we investigate the asymptotic behaviour of sublevel-set volumes associated with principal analytic ideals, equivalently with holomorphic function germs. Building on the classical theory of local zeta functions and Mellin asymptotics, we introduce the visible spectrum, the set of actual poles of the local zeta function, and show that it is determined by the asymptotic expansion of the volume. Conversely, we prove that this spectrum, together with its multiplicities and coefficients, can be recovered recursively from the volume asymptotics by an explicit reconstruction procedure. We also give complementary interpretations in terms of arc spaces, where the divisorial exponents appear both as ratios of vanishing orders along generic divisorial arcs and as normalized codimension growth rates of divisorial cylinders. Taken together, these results establish an explicit correspondence between the visible spectrum and the asymptotic expansion of sublevel-set volumes, providing an intrinsic metric characterization of the visible spectrum itself.