{
  "id": "2505.01716",
  "version": "v1",
  "published": "2025-05-03T06:59:21.000Z",
  "updated": "2025-05-03T06:59:21.000Z",
  "title": "Variation of Tannaka groups of perverse sheaves in family",
  "authors": [
    "Anna Cadoret",
    "Haohao Liu"
  ],
  "comment": "Comments are welcome!",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $k$ be a field of characteristic $0$, let $S$ be a smooth, geometrically connected variety over $k$, with generic point $\\eta$, and $f:\\mathbb{X}\\rightarrow S$ a morphism separated and of finite type. Fix a prime $\\ell$. Let $\\mathbb{P}$ be an $f$-universally locally acyclic relative perverse $\\overline{\\mathbb{Q}}_\\ell$-sheaf on $\\mathbb{X}/S$. We prove that if for some (equivalently, every) geometric point $\\bar \\eta$ over $\\eta$ the restriction $\\mathbb{P}|_{\\mathbb{X}_{\\bar \\eta}}$ is simple as a perverse $\\overline{\\mathbb{Q}}_\\ell$-sheaf on $\\mathbb{X}_{\\bar \\eta}$, then there is a non-empty open subscheme $U\\subset S$ such that, for every geometric point $\\bar s$ on $U$, the restriction $\\mathbb{P}|_{\\mathbb{X}_{\\bar s}}$ is simple as a perverse $\\overline{\\mathbb{Q}}_\\ell$-sheaf on $\\mathbb{X}_{\\bar s}$. When $f:\\mathbb{X}\\rightarrow S$ is an abelian scheme, we give applications of this result to the variation with $s\\in S$ of the Tannaka group of $\\mathbb{P}|_{\\mathbb{X}_{\\bar s}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-03T06:59:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tannaka group",
      "perverse sheaves",
      "geometric point",
      "non-empty open subscheme",
      "restriction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}