Divisorial Persistence and Asymptotic Homology of Analytic Pairs

We introduce \emph{Divisorial Asymptotic Homology} (DAH) for analytic pairs $(X,\mathcal I)$, a homological theory governed by the asymptotic concentration of Hausdorff measure inside shrinking sublevel sets of the intrinsic analytic energy $K_{\mathcal I}=\sum_j|f_j|^2$.

DAH defines a covariant functor to the category of persistent graded abelian groups and distinguishes three layers of asymptotic information: the real log canonical threshold, the intrinsic divisorial spectrum, and the homological spectrum, the latter governing persistence and recording the asymptotic concentration rates realized by admissible cycles.

The admissibility filtration is a tame persistence module whose critical values are precisely the finitely many elements of the homological spectrum.

We establish functoriality, relative long exact sequences, Mayer--Vietoris sequences, and birational invariance.

For normal analytic surface germs we prove a rigidity theorem identifying the critical DAH groups with the homology of critical weighted dual resolution graphs.

Examples show that DAH detects asymptotic topological information invisible to the real log canonical threshold and that divisorial criticality and homological persistence may diverge in higher dimension.