Divisorial persistent homology and volume asymptotics of analytic pairs

Abstract

A log-resolution of an analytic pair $(X,I)$ attaches to every prime divisor $E$ of its total transform a vanishing order $\nu_E$, a Jacobian order $a_E$, and hence a divisorial exponent $\gamma_E=(a_E+1)/(2\nu_E)$. The smallest exponent is one half of the real log canonical threshold and governs the volume of the sublevel sets $\{\sum |f_j|^2 \le \varepsilon\}$; the multiplicity of the logarithmic correction is a second intrinsic invariant. We organize all the exponents into two persistence modules and study what survives when the resolution is changed. The first module lives on $X$: each compact subanalytic chain receives a valuative threshold, the infimum of the normalized log discrepancy $A(v)/(2v(I))$ over the divisorial valuations reaching it; its superlevel sets filter the chain complex. We prove that the resulting module is the persistent homology of the complements of the sublevel sets of the local threshold; tameness, Mayer-Vietoris, independence of the resolution, and invariance under bi-Lipschitz subanalytic homeomorphisms preserving the energy up to equivalence follow. The second lives on the resolution: the dual complex of the total transform, filtered by the exponents. An admissible blow-up changes each sublevel complex by a stellar subdivision or by attaching a cone over a contractible subcomplex, the only input being a mediant inequality for the exponent of the new divisor. Under an explicit factorization hypothesis (unconditional for surface germs, expected from relative weak factorization in the complex algebraic case) the persistent homology is independent of the resolution. For normal surface singularities it is the persistent homology of the weighted dual graph, computed for the $A_n$ points. The two modules share the first critical value and the list of exponents, have opposite persistence directions, and are different shadows of the same data.