{
  "abstract": "To a log-resolution of a complex pair $(X,\\I)$ we associate the weights $λ_E=(a_E+1)/ν_E$ of the prime divisors $E$ of the total transform of $\\I$, where $ν_E$ is the vanishing order of $\\I$ and $a_E$ the Jacobian (Mather discrepancy) order along $E$. On Jacobian-adapted resolutions, $\\min_Eλ_E$ 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 $α$. 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 $\\ge3$. When $V(\\I)$ is complete, $\\DPH$ equals the weight-zero part of the cohomology of the union of divisors of weight at most $α$ (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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.06717",
  "author_total": 1,
  "authors": [
    {
      "name": "Nivaldo Grulha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4bcez9fsqrpn6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.06717",
  "fulltext_url": null,
  "id": "4bcez9fsqrpn6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.06717",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7167828303",
  "pdf_url": "https://arxiv.org/pdf/2607.06717",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4bcez9fsqrpn6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2607.06717",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.06717"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Divisorial persistent homology and volume asymptotics of complex pairs",
  "title_text": "Divisorial persistent homology and volume asymptotics of complex pairs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/4bcez9fsqrpn6",
  "venue": "arXiv",
  "versions": []
}