Divisorial persistent homology and volume asymptotics of complex pairs

Abstract

To a log-resolution of a complex pair $(X,\I)$ we associate the weights ${\lambda }_E=(a_E+1)/{\nu }_E$ of the prime divisors $E$ of the total transform of $\I$, where ${\nu }_E$ is the vanishing order of $\I$ and $a_E$ the Jacobian (Mather discrepancy) order along $E$. On Jacobian-adapted resolutions, ${\min }_E{\lambda }_E$ is the volume-growth exponent of $K_\I=\sum_j|f_j|^2$ for the $2n$-dimensional Hausdorff measure: the log canonical threshold for smooth $X$, Ishii's Mather one for singular $X$, with a logarithmic correction given by a resonance multiplicity. The main object is the divisorial persistent homology $\DPH$: the persistent homology of the dual complex filtered by the subcomplexes on divisors of weight at most $\alpha$. It records incidences of divisors, which the threshold ignores. We prove that $\DPH$ is invariant under blow-ups along smooth centres in normal crossing with the boundary, for adapted resolutions; $A_2$ shows adaptedness is necessary. Independence of the resolution holds for complex algebraic pairs (weak factorization) and normal surface singularities, and is a hypothesis for non-algebraic analytic pairs of dimension $\ge 3$. When $V(\I)$ is complete, $\DPH$ equals the weight-zero part of the cohomology of the union of divisors of weight at most $\alpha$ (Deligne). For $A_n$ the minimal resolution is adapted iff $n$ is odd, and $\DPH_0$ has one finite bar of length $(n-1)/2$, so within this family it detects $n$ (detection, not classification). We compare with Nicaise--Xu and de Fernex--Kollár--Xu.