{
  "abstract": "Let $\\overline{Z}$ be a smooth projective variety with an simple normal crossing divisor $D$ such that $Ω_{\\overline{Z}}^1(\\log D)$ is nef. We prove that if $Y\\subset Z:=\\overline Z\\setminus D$ is a smooth closed subvariety with nonzero Euler characteristic, and $P$ is a perverse sheaf on $Y$ with full support, then $χ(Y,P)>0$. This is a strict version of an inequality obtained in arXiv:2408.15788. Applying this to the trace-zero part of a finite direct image yields a logarithmic Riemann--Hurwitz inequality: if $Y$ has dimension $n$, any finite surjective morphism $f\\colon X\\to Y$ of degree $d$ with $X$ smooth satisfies $(-1)^nχ(X)\\ge d\\,(-1)^nχ(Y)$, the difference being an explicit sum of nonnegative intersection numbers. When $(-1)^nχ(Y)>0$ this forces any such $f$ with $χ(X)=χ(Y)$ to be an isomorphism. We verify the nef hypothesis for subvarieties of semiabelian varieties, and for $\\overline{\\mathscr M}_{g,n}$---the moduli of curves, obtaining in particular that every finite surjective self-morphism of a moduli space of curves with level structure is an isomorphism.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13081",
  "author_total": 3,
  "authors": [
    {
      "name": "Donu Arapura"
    },
    {
      "name": "Chikako Mese"
    },
    {
      "name": "Deepam Patel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a8n9zh9xsbab0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13081",
  "fulltext_url": null,
  "id": "a8n9zh9xsbab0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13081",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213091177",
  "pdf_url": "https://arxiv.org/pdf/2609.13081",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a8n9zh9xsbab0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.13081",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13081"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers",
  "title_text": "A Higher dimensional log Riemann--Hurwitz inequality and rigidity of covers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/a8n9zh9xsbab0",
  "venue": "arXiv",
  "versions": []
}